spingeometry

Spin Geometry H. BLAINE LAWSON, JR. and MARIE-LOUISE MICHELSOHN PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY 1989 Copyright © 1989 by Princeton University Press Published by Princeton University Press, 41 WilIiam Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Chichester, West Sussex Library of Congress Cataloging-in-Publication Data Lawson, H. Blaine. Spin geometry/H. B. Lawson and M.-L. Michelsohn. p. cm.---{Princeton mathematical series: 38) Includes index. ISBN 0-691-08542-0 (alk. paper): 1. Nuclear spin-Mathematics. 2. Geometry. 3. Topology. 4. Clifford algebras. 5. Mathematical physics. I. Michelsohn, M.-L. (Marie-Louise), 1941-11. Title. Ill. Series. QC793.3.S6L39 1989 539.7'25-<lc20 89-32544 Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America For Christie, Didi, Michelle, and Heather PREFACE ACKNOWLEDGMENTS INTRODUCTION Contents CHAPTER I Clifford Algebras, Spin Groups and Their Representations §1. Clifford algebras §2. The groups Pin and Spin §3. The algebras Crn and cr"s §4. The classification §5. Representations §6. Lie algebra structures §7. Some direct applications to geometry §8. Some further applications to the theory of Lie groups §9. K-theory and the Atiyah-Bott-Shapiro construction §10. KR-theory and the (l,l)-Periodicity Theorem CHAPTER 11 Spin Geometry and the Dirae Operators §l. Spin structures on vector bundles §2. Spin manifolds and spin cobordism §3. Clifford and spinor bundles §4. Connections on spinor bundles §5. The Dirac operators §6. The fundamental elliptic operators §7. Crk-linear Dirac operators §8. Vanishing theorems and some applications CHAPTER III Index Theorems §l. Differential operators §2. Sobolev spaces and Sobolev theorems §3. Pseudodifferential operators §4. Elliptic operators and parametrices ix XII 3 7 7 12 20 25 30 40 44 49 58 70 77 78 85 93 101 112 135 139 153 166 167 170 177 188 viii CONTENTS §5. Fundamental results for elliptic operators 192 §6. The heat kernel and the index 198 §7. The topological invariance of the index 201 §8. The index of a family of elliptic operators 205 §9. The G-index 211 §1O. The Clifford index 214 §11. Multiplicative sequences and the Chern character 225 §12. Thorn isomorphisms and the Chern character defect 238 §13. The Atiyah-Singer Index Theorem 243 §14. Fixed-point formulas for elliptic operators 259 §15. The Index Theorem for Families 268 §16. Families of real operators and the Ctk-index Theorem 270 §17. Remarks on heat and supersymmetry 277 CHAPTER IV Applications in Geometry and Topology 278 §l. Integrality theorems 280 §2. Immersions of manifolds and the vector field problem 281 §3. Group actions on manifolds 291 §4. Compact manifolds of positive scalar curvature 297 §5. Positive scalar curvature and the fundamental group 302 §6. Complete manifolds of positive scalar curvature 313 §7. The topology of the space of positive scalar curvature metrics 326 §8. Clifford multiplication and Klihler manifolds 330 §9. Pure spinors, complex structures, and twistors 335 §1O. Reduced holonomy and calibrations 345 §11. Spinor cohomology and complex manifolds with vanishing first Chern class 357 §12. The Positive Mass Conjecture in general relativity 368 ApPENDIX A Principal G-bundles 370 ApPENDIX B Classifying Spaces and Characteristic Classes 376 ApPENDIX C Orientation Classes and Thorn Isomorphisms in K -theory 384 ApPENDIX D Spine-manifolds 390 BmLIOGRAPHY 402 INDEX 417 NOTATION INDEX 425 viii CONTENTS §5. Fundamental results for elliptic operators 192 §6. The heat kernel and the index 198 §7. The topological invariance of the index 201 §8. The index of a family of elliptic operators 205 §9. The G-index 211 §10. The Clifford index 214 §11. Multiplicative sequences and the Chern character 225 §12. Thorn isomorphisms and the Chern character defect 238 §13. The Atiyah-Singer Index Theorem 243 §14. Fixed-point formulas for elliptic operators 259 §15. The Index Theorem for Families 268 §16. Families of real operators and the Cf,k-index Theorem 270 §17. Remarks on heat and supersymmetry 277 CHAPTER IV Applications in Geometry and Topology 278 §1. Integrality theorems 280 §2. Immersions of manifolds and the vector field problem 281 §3. Group actions on manifolds 291 §4. Compact manifolds of positive scalar curvature 297 §5. Positive scalar curvature and the fundamental group 302 §6. Complete manifolds of positive scalar curvature 313 §7. The topology of the space of positive scalar curvature metrics 326 §8. Clifford multiplication and Kiihler manifolds 330 §9. Pure spinors, complex structures, and twistors 335 §W. Reduced hoionomy and caiibrations 345 §1l. Spinor cohomology and complex manifolds with vanishing first Chern class 357 §12. The Positive Mass Conjecture in general relativity 368 ApPENDIX A Principal G-bundles 370 ApPENDIX B Classifying Spaces and Characteristic Classes 376 ApPENDIX C Orientation Classes and Thorn Isomorphisms in K-theory 384 ApPENDIX D Spine-manifolds 390 BIBLIOGRAPHY 402 INDEX 417 NOTATION INDEX 425 viii CONTENTS §5. Fundamental results for elliptic operators 192 §6. The heat kernel and the index 198 §7. The topological invariance of the index 201 §8. The index of a family of elliptic operators 205 §9. The G-index 211 §10. The Clifford index 214 §11. Multiplicative sequences and the Chern character 225 §12. Thorn isomorphisms and the Chern character defect 238 §13. The Atiyah-Singer Index Theorem 243 §14. Fixed-point formulas for elliptic operators 259 §15. The Index Theorem for Families 268 §16. Families of real operators and the Cf,k-index Theorem 270 §17. Remarks on heat and supersymmetry 277 CHAPTER IV Applications in Geometry and Topology 278 §1. Integrality theorems 280 §2. Immersions of manifolds and the vector field problem 281 §3. Group actions on manifolds 291 §4. Compact manifolds of positive scalar curvature 297 §5. Positive scalar curvature and the fundamental group 302 §6. Compiete maniiolds of positive scalar curvature 3D §7. The topology of the space of positive scalar curvature metrics 326 §8. Clifford multiplication and Kiihler manifolds 330 §9. Pure spinors, complex structures, and twistors 335 §10. Reduced holonomy and calibrations 345 §11. Spin or cohomology and complex manifolds with vanishing first Chern class 357 §12. The Positive Mass Conjecture in general relativity 368 ApPENDIX A Principal G-bundles 370 ApPENDIX B Classifying Spaces and Characteristic Classes 376 ApPENDIX C Orientation Classes and Thorn Isomorphisms in K-theory 384 ApPENDIX D Spine-manifolds 390 BIBLIOGRAPHY 402 417 NOTATION INDEX 425 Priface In the late 1920's the relentless march of ideas and discoveries had carried physics to a generally accepted relativistic theory of the electron. The physicist P.A.M. Dirac, however, was dissatisfied with the prevailing ideas and, somewhat in isolation, sought for a better formulation. By 1928 he succeeded in finding a theory which accorded with his own ideas and also fit most of the established principles of the time. Ultimately this theory proved to be one of the great intellectual achievements of the period. It was particularly remarkable for the internal beauty of its mathematical structure which not only clarified much previously mysterious phenomena but also predicted in a compelling way the existence of an electron-like particle of negative energy. Indeed such particles were subsequently found to exist and our understanding of nature was transformed. Because of its compelling beauty and physical significance it is perhaps not surprising that the ideas at the heart of Dirac's theory have also been discovered to play a role of great importance in modern mathematics, particularly in the interrelations between topology, geometry and analysis. A great part of this new understanding comes from the work of M. Atiyah and I. Singer. It is their work and its implications which form the focus of this book. It seems appropriate to sketch some of the fundamental ideas here: In searching for his theory, Dirac was faced, roughly speaking, with the problem of finding a Lorentz-invariant wave equation Dt/! = At/! compat-ible with the Klein-Gordon equation Dt/! = At/! where 0 = (a/axo)2 -(%x1? -(O/OX2)2 -(O/OX3)2. Causality required that D be first order in the "time" coordinate Xo. Of course by Lorentz invariance there could be no preferred time coordinate, and so D was required to be first-order in all variables. Thus, in essence Dirac was looking for a first-order differential operator whose square was the laplacian. His solution was to replace the complex-valued wave function t/! with an n-tuple 'P = (t/!1, ... ,t/!n) of such functions. The operator D then became a first-order system of the form 3 a D= L YIl-1l=0 aXil x PREFACE where Yo, ... ,Y3 were n x n-matrices. The requirement that led to the equations YVY/L + Y/LYv = ±2c5vw These were easily and explicitly solved for small values of n, and the analysis was underway. This construction of Dirac has a curious and fundamental property. Lorentz transformations of the space-time variables (xo, ... ,X3) induce linear transformations of the n-tuples '¥ which are determined only up to a sign. Making a consistent choice of sign amounts to passing to a non-trivial2-fold covering L ofthe Lorentz group L. That is, in transforming the 'P's one falls upon a representation of L which does not descend to L. The theory of Dirac had another interesting feature. In the presence of an electromagnetic field the Dirac Hamiltonian contained an additional term added on to what one might expect from the classical case. There were strong formal analogies with the additional term one obtains by introducing internal spin into the mechanical equations of an orbiting particle. This "spin" or internal magnetic moment had observable quan-tum effects. The n-tuples 'P were thereby called spinors and this family of transformations was called the spin representation. This physical theory touches upon an important and general fact con-cerning the orthogonal groups. (We shall restrict ourselves for the moment to the positive definite case.) In the theory of Cartan and Weyl the repre-sentations of the Lie algebra of SOn are essentially generated by two basic ones. The first is the standard n-dimensional representation (and its ex-terior powers). The second is constructed from the representations of the algebra generated by the Y /L's as above (the elt.fford algebra associated to the quadratic form defining the orthogonal group). This second represen-tation is called the spin representation. It does not come from a represen-tation of the orthogonal group, but only of its universal covering group, called Spinn. It plays a key role in an astounding variety of questions in geometry and topology: questions involving vector fields on spheres, immersions of manifolds, the integrality of certain characteristic numbers, triality in dimension eight, the existence of complex structures, the exis-tence of metrics of positive scalar curvature, and perhaps most basically, the index of elliptic operators. In the early 1960s general developments had led mathematicians to con-sider the problem of finding a topological formula for the index of any elliptic operator defined on a compact manifold. This formula was to gen- PREFACE xi eralize the important Hirzebruch-Riemann-Roch Theorem already estab-lished in the complex algebraic case. In considering the problem, Atiyah and Singer noted that among all manifolds, those whose SOn-structure could be simplified to a Spinn-structure had particularly suggestive prop-erties. Realizing that over such spaces one could carry out the Dirac con-struction, they produced a globally defined elliptic operator canonically associated to the underlying riemannian metric. The index of this operator was a basic topological invariant called the A-genus, which was known always to be an integer in this special class of spin manifolds. (It is not an integer in general.) Twisting the Dirac-type operator with arbitrary coefficient bundles led, with some sophistication, to a general formula for the index of any elliptic operator. Atiyah and Singer went on to understand the index in the more proper setting of K-theory. This led in particular to the formulation of certain KO-invariants which have profound applications in geometry and topology. These invariants touch questions unapproachable by other means. Their study and elucidation was a principal motivation for the writing of this tract. It is interesting to note in more recent years there has been another pro-found and beautiful physical theory whose ideas have come to the core of topology, geometry and analysis. This is the non-abelian gauge field theory of C. N. Yang and R. L. Mills which through the work of S. Donaldson and M. Freedman has led to astonishing results in dimension four. Yang-Mills theory can be plausibly considered a highly non-trivial generalization of Dirac's theory which encompasses three fundamental forces: the weak, strong, and electromagnetic interactions. This theory involves modern dif-ferential geometry in an essential way. The theory of connections, Dirac-type operators, and index theory all play an important role. We hope this book can serve as a modest introduction to some of these concepts. H. B. LAWSON AND M.-L. MICHELSOHN Stony Brook Acknowledgments This book owes much to the fundamental work of Michael Atiyah and Iz Singer. Part of our initial motivation in writing the book was to give a leisurely and rounded presentation of their results. The authors would like to express particular gratitude to Peter Land-weber, Jean-Pierre Bourguignon, Misha Katz, Haiwan Chen and Peter Woit, each of whom read large parts of the original manuscript and made a number of important suggestions. We are also grateful to the National Science Foundation and the Brazilian C. N. Pq. for their support during the writing of this book. H. B. LAWSON AND M.-L. MICHELSOHN This author would like to take this opportunity to express her deep appreciation to Mark Mahowald who held out a hand when one was so dearly needed. M.-L. MICHELSOHN Spin Geometry Introduction Over the past two decades the geometry of spin manifolds and Dirac operators, and the various associated index theorems have come to play an increasingly important role both in mathematics and in mathematical physics. In the area of differential geometry and topology they have be-come fundamental. Topics like spin cobordism, previously considered exotic even by topologists, are now known to play an essential role in such ciassicai questions as the existence or non-existence of metrics of po-sitive curvature. Indeed, the profound methods introduced into geometry by Atiyah, Bott, Singer and others are now indispensible to mathemati-cians working in the field. It is the intent of this book to set out the fun-damental concepts and to present these methods and results in a unified way. A principal theme of the exposition here is the consistent use of Clifford aigebras and their representations. This reflects the observed fact that these algebras emerge repeatedly at the very core of an astonishing variety of problems in geometry and topology. Even in discussing riemannian geometry, the formalism of Clifford mul-tiplication will be used in place of the more conventional exterior tensor calculus. There is a philosophical justification for this bias. Recall that to any vector space V there is naturally associated the exterior algebra A * V, and this association carries over directiy to vector bundies. Appiied to the tangent bundle of a smooth manifold, it gives the de Rham bundle of ex-terior differential forms. In a similar way, to any vector space V equipped with a quadratic form q, there is associated the Clifford algebra C.e(V, q), and this association carries over directly to vector bundles equipped with fibre metrics. In particular, applied to the tangent bundle of a smooth riemannian manifold, it gives a canonically associated bundle of algebras, caned the Ciifford bundie. As a vector bundie it is isomorphic to the bundle of exterior forms. However, the Clifford multiplication is strictly richer than exterior multiplication; it reflects the inner symmetries and basic identities of the riemannian structure. In fact fundamental curvature identities will be derived here in the formalism of Clifford multiplication and applied to some basic problems. 4 INTRODUCTION Another justification for our approach is that the Clifford formalism gives a transparent unification of all the fundamental elliptic complexes in differential geometry. It also renders many of the technical arguments involved in applying the Index Theorem quite natural and simple. This point of view concerning Clifford bundles and Clifford multiplica-tion is an implicit, but rarely an explicit theme in the writing of Atiyah and Singer. The authors feel that for anyone working in topology or geometry it is worthwhile to develop a friendly, if not intimate relationship with spin groups and Clifford modules. For this reason \ve have used them ex= plicitly and systematically in our exposition. The book is organized into four chapters whose successive themes are algebra, geometry, analysis, and applications. The first chapter offers a detailed introduction to Clifford algebras, spin groups and their represen-tations. The concepts are illuminated by giving some direct applications to the elementary geometry of spheres, projective spaces, and low-dimensional Lie groups. K-theory and KR=theory are then introduced, and the fundamental relationship between Clifford algebras and Bott pe-riodicity is established. In the second chapter of this book, the algebraic concepts are carried over to define structures on differentiable manifolds. Here one enters prop-erly into the subject of spin geometry. Spin manifolds, spin cobordism, and spinor bundles with their canonical connections are all discussed in detail, and a general formalism of Dirac bundles and Dirac operators is developed. Hodge-de Rham Theory is reviewed in this formalism, and each of the fundamental elliptic operators of riemannian geometry is derived and examined in detail. Special emphasis is given here to introducing the notion of a Ce,,-linear elliptic operator and discussing its index. This index lives in a certain quotient of the Grothendieck group of Clifford modules. For the fun-damental operators (\vhich are discussed in detail here) it is one of the deepest and most subtle invariants of global riemannian geometry. The systematic discussion of Ctk-linear differential operators is one of the important features of this book. In the last section of Chapter II a universal identity of Bochner type is established for any Dirac bundle, and the classical vanishing theorems of Bochner and Lichnerowicz are derived from it. This seems an appropriate time to make some general observations about spin geometry. To begin it should be emphasized that spin geom-etry is really a special topic in riemannian geometry. The central concept of a spin manifold is often considered to be a topological one. It is just a manifold with a simply-connected structure group. This is understood systematically as follows. On a general differentiable n-manifold (n 3), the tangent bundle has structure group GLn• The manifold is said to be INTRODUCTION 5 oriented if the structure group is reduced to GL';-(the connected com-ponent of the identity). The manifold is said to be spin if the structure group GL';-can be "lifted" to the universal covering group GC --+ GL';-. This approach is perfectly correct, but there is a hidden obstruction to the viability of the concept: namely, the group atn (for n 3) has no finite dimensional representations that do not come from GL';. This means that in terms of standard tensor calculus, nothing has been gained by this re-finement of the structure. However, if one passes from GLn to the maxima! compact subgroup On' that is, if one introduces a riemannian metric on the manifold, the story is quite different. An orientation corresponds to reducing the struc-ture group to SO", and a spin structure corresponds to then lifting the structure group to the universal covering group Spinn --+ SOn. Maximal compact subgroups are homotopy equivalent to the Lie groups which contain them, and there is essentially no topological difference in viewing spin structures this \vay .. HO'Never, there do exist Jinite dimensional repre-sentations of Spinn which are not lifts of representations of SOn. Over a spin manifold one can thereby construct certain new vector bundles, called bundles of spinors, which do not exist over general manifolds. Their exis-tence allows the introduction of certain important analytic tools which are not generally available, and these tools play a central role in the study of the global geometry of the space. It is, by the way, an important fact that this construction is metric-dependent; the bundle of spinars itself depends in an essential way on the choice of riemannian structure on the manifold. These observations lead one to suspect that there must exist a local spinor calculus, like the tensor calculus, which should be an important component of local riemannian geometry. A satisfactory formalism of this type has not yet been developed. However, the spinors bundles have yielded profound relations between local riemannian geometry and global topology. The main tools by which we access the global structure of spin mani-folds are the various index theorems of Atiyah and Singer. These are pre-sented and proved in Chapter III of the book. They include not just the standard G-Index Theorem but also the Index Theorem for Families and the Theorem (for elliptic operators). There are in existence today many elegant proofs of index theorems which use the methods of the heat equation. These do not apply to the C.ek-Index Theorem however, because of the non-local nature of this index. For this reason our exposition follows the "softer," or more topological, arguments given in the original proofs. Chapter IV of the book is concerned with applications of the theory. There is no attempt to be exhaustive; such an attempt would be pointless 6 INTRODUCTION and nearly impossible. We have tried however to demonstrate the broad range of problems in which the considerations of spin geometry can be effectively implemented. It is of some historical interest to note that while Dirac did essentially use Clifford modules in the construction of his wave operator, he was not really responsible for what is commonly called the "Dirac operator" in riemannian geometry. The construction of this operator is due to Atiyah and Singer and is, in our estimation, one of their great achievements. It required for its discovery an understanding of the subtle geometry of spin manifolds and a recognition of the central role it would play in the general theory of elliptic operators. Even the formidable Elie Cartan, who sensed the importance of the question and, of course, authored the general theory of spinors and who was not unaware of the fundamentals of global anal-ysis, never reached the point of defining this operator in the proper con-text of spin manifolds. In keeping with historical developments we shall call the general construction of operators from modules over the Clifford bundle, the Dirac construction, and we shall call the specific operator so defined on the spin or bundle, the Atiyah-Singer operator. It is this operator which in a very specific sense generates all elliptic operators over a spin manifold. It introduces a direct relationship between curvature and topology which exists only under the spin hypothesis. The Ctk-linear version of this operator carries an index in KO-theory. In fact its index gives a basic ring homomorphism KO-*(pt) which generalizes to KO-theory the classical A-genus. The applications of this to geometry include the fact that half the exotic spheres in dimensions one and two (mod 8) do not carry metrics of positive scalar curvature. The presentation in this book is aimed at readers with a knowledge of elementary geometry and topology. Important things, such as the concept of spin manifolds and the theory of connections, are developed from basic definitions. The Atiyah-Singer index theorems are formulated and proved assuming little more than a knowledge of the Fourier inversion formula. There are several appendices in which principal bundles, classifying spaces, Thorn isomorphisms, and spin manifolds are discussed in detail. The references to theorems and equations within each chapter are made without reference to the chapter itself (e.g., 2.7 or (5.9». Refer-ences to other chapters are prefaced by the chapter number (e.g., III.2.7 or (IY.5.9». CHAPTER I Cl!fford AJnebras, Spin Groups and Their Representations The object of this chapter is to present the algebraic ideas which lie at the heart of spin geometry. The central concept is that of a Clifford algebra. This is an algebra naturally associated to a vector space which is equipped with a quadratic form. Within the group of units of the algebra there is a distinguished subgroup, called the spin group, which, in the case of the positive definite form on !Rn (n > 2), is the universal covering group of SOn. It is a striking (and not commonplace) fact that Clifford algebras and their representations play an important role in many fundamental aspects of differential geometry. These include such diverse topics as Hodge-de Rham Theory, Bott periodicity, immersions of manifolds into spheres, families of vector fields on spheres, curvature identities in riemannian geometry, and Thorn isomorphisms in K-theory. The effort invested in becoming comfortable with this algebraic formalism is well worthwhile. Our discussion begins in a very general algebraic context but soon moves to the real case in order to keep matters simple and in the domain of most interest. In §§7 and 8 we present some applications of the purely algebraic theory to topology and to the appearence of exceptional phe-nomena in the theory of Lie groups. The last part of the chapter is devoted to K-theory. Basic definitions are given and fundamental results are reviewed. The discussion culminates with the Atiyah-Bott-Shapiro isomorphisms which directly relate the periodicity phenomena in Clifford algebras to the classical Bott Period-icity Theorems. In particular, explicit isomorphisms are given between K-*(pt) = EBnK(sn) (and KO-*(pt) = EBn Ko(sn)) and a certain quotient of the ring of Clifford modules. Section 10 is concerned with KR-theory which later plays a role in the index theorem for families of real elliptic operators. This is a bigraded theory and the corresponding Atiyah-Bott-Shapiro isomorphism entails representations of Clifford algebras Ctr•s for quadratic forms of indefinite signature. §l. Clifford Algebras Let Vbe a vector space over the commutative field k and suppose q is a quadratic form on V. The Ciifford aigebra Ct(V,q) associated to V and 8 I. CLlFFORD ALGEBRAS AND SPIN GROUPS q is an associative algebra with unit defined as follows. Let CX) ff(V) = L @rv denote the tensor algebra of V, and define .?"q(V) to be the ideal in ff(V) generated by all elements of the form v ® v + q(V)l for v E V. Then the Clifford algebra is defined to be the quotient Ct(V,q) == ff(V)/.?"q(V). There is a natural embedding Vc......-. Ct(V,q) (1.1) which is the image of V = @1 V under the canonical projection 1tq: ff(V) Ct(V,q). (1.2) We prove that 1tqlv is injective as follows. We say that an element qJ E ff(V) is of pure degree s if qJ E @V. (Every element of ff(V) is a finite sum of elements of pure degree.) We want to show that any element qJ E .?"g(V) n V is zero. Any such element can be written as a finite sum qJ = L ai ® (Vi ® Vi + q(Vi» ® bi where we may assume that the a;'s and b;'s are of pure degree. Since qJ E V = @1 V, we conclude that L at ® (Vi' ® vd ® bi' = 0, where this sum is taken over those indices with deg ai + deg bi maximal. This equation implies, by contraction with q, that L ai,q(vr) . bi' = 0. Proceeding inductively, we prove that qJ = 0. The algebra Ct(V,q) is generated by the vector space Vc Ct(V,q) (and the identity 1) subject to the relations: v· V = -q(v)l (1.3) for v E V. If the characteristic of k is not 2, then for all v,w E V, V· W + W· v = -2q(v,w) (1.4) where 2q(v,w) == q(v + w) -q(v) -q(w) is the polarization of q. The re-lations (1.3) can be used to give the following universal characterization of the algebra. Proposition 1.1. Let f: V -> d be a linear map into an associative k-algebra with unit, such that f(v)· f(v) = -q(v)l (1.5) for all v E V. Then f extends uniquely to a k-algebra homomorphism f: Ct(V,q) -> d. Furthermore, Ct(V,q) is the unique associative k-algebra with this property. §l. CLIFFORD ALGEBRA::> ':I Proof. Any linear map f: V d extends to· a unique algebra homo-morphism l: ff(V) d. Property (1.5) implies that 1 = 0 on .?"iV), and so I descends to C£(V,q). Suppose now that CC is an associative k-algebra with unit and that i: V 4 CC is an embedding with the property that any linear map f: V d (d as above) with property (1.5) extends uniquely to an algebra homomorphism 1: C(j .01. Then the isomorphism from Vc Ct(V,q) to i(V) c CC clearly induces an algebra isomorphism Ct(V,q) CC. • This characterization of Clifford algebras is extremely useful. It shows, for example, that they are functorial in the following sense. Given a morphism f: (V,q) (V',q'), i.e., a k-linear map f: V V' between vector spaces which preserves the quadratic forms (f*q' = q), there is, by Propo-sition 1.1, an induced homomorphism j: ct(V,q) ct(V',q'). Given another morphism g:(V',q') (V",q"), we see from the uniqueness in Proposition 1.1, that 0 = 9 0 I A particular consequence of this is that the orthogonal group O(V,q) == {f E GL(V) :f*q = q} extends canonically to a group of automorphisms of Ct(V,q). We shall see later that this embedding O(V,q) c Aut(Ct(V,q)) actually lies in the subgroup of inner automorphisms. An element here of particular importance is the automorphism (X: ct(V,q) --+ Ct(V,q) (1.6) (1.7) which extends the map (X(v) = -v on V. Since (X2 = Id, there is a decom-position Ct(V,q) = CtO(V,q) EB ct l(V,q) (1.8) where ct i(V,q) = {<p E ct(V,q) : (X( <p) = ( -1 )i<p} are the eigenspaces of (x. Clearly, since (X(<P1<P2) = (X(<P1)' (X(<P2), we have that (1.9) where the indices are taken modulo 2. An algebra with a decomposition (1.8) satisfying (1.9) is called a Z2-graded algebra. Note that Ct°(V,q) is a subalgebra of Ct(V,q). It is called the even part of Ct(V,q). The subspace ct1(V,q) is called the odd part. It is an observation of Atiyah, Bott and Shapiro that this Z2-grading plays an important role in the analysis and application of Clifford algebras. There exist some elementary and important relationships between the ClitTord algebra Cf,(V,q) of a space and its exterior algebra A * V (whose definition is, of course, independent of the quadratic form q). There is a natural filtration #0 c #1 C #2 C ... c ff(V) of the tensor algebra, 10 1. CLIFFORD ALGEBRAS AND SPIN GROUPS which is defined by ff;r == L @V, and has the property that ff;r ® ff;r' ff;r+r'. If we set ffi = nq(ff;i) we obtain a filtration ;y;o c /¥1 C fF2 C ... c Cf(J.T,q) of the Clifford algebra, which also has the property that (1.10) for all r,r'. This makes Ct(V,q) into a filtered algebra. It follows from (1.10) that the multiplication map descends to a map x (JFs/JFS-1) (JF,+s/JFr+s-l) for all r,s. Setting rg* == rgT where fliT == !F' /!Fr -1, we obtain the associated graded algebra. Proposition 1.2. For any quadratic form q, the associated graded algebra of ct(V,q) is naturally isomorphic to the exterior algebra A*V. Proof. The map @V JFr 1, which is given by vit ® ... ® vir 1-+ [Vii' .. Vd clearly descends to a map NV by property (1.4). (When the characteristic of k is 2, we llse the fact that v' w + w . V = 0.) This map is evidently surjective and is easily seen to give a homomorphism of graded algebras A*V rg*. To see that this map is injective we proceed as follows. The kernel of @V rgT consists of the r-homogeneous pieces of eiements <P E Jq(V) of degree r. Any such <p can be written as a finite sum <p = L ai ® (Vi ® Vi + q(Vi)) ® bi where Vi E Vand where we may assume that the ai and bi are of pure degree with deg 0i + deg bi r -2. The r-homogeneous part of <p is then of the form <Pr = L ai ® Vi ® Vi ® bi (where deg ai + deg bi = r -2 for each i). Since Vi /\ Vi = 0 for all i, we see that the image of <p in the exterior algebra is zero. Hence the map NV rgT is injective. • Proposition 1.2 says that Clifford multiplication is an enhancement of exterior multiplication which is determined by the form q. Note that ct(V,O) A*V. Proposition 1.3. There is a canonical vector space isomorphism A*V Ct(V,q) compatible with the jiltrations. (1.11) REMARK 1.4. The map (1.11) is, of course, not an isomorphism of al-gebras unless q = O. The point here is that the map is canonical. Thus we may speak of the embeddings NV c Ct(V,q) for all r O. (1.12) §l. CLlFFORD ALGEBRAS 11 Proof. We define a map of the r-fold direct product f: V x ... x V--+ ct(V,q) by setting f(vl, •.. , vr) = L sign(a)vO"(I) ... vO"(r) r. 0" (1.13) where the sum is taken over the symmetric group on r elements. (If the characteristic of k is not zero, one must drop the factor l/r!.) Clearly f determines a linear map j: A'V --+ ct(V,q) whose image lies in :Fr. The composition off with the projection :Fr --+ :Frj:Fr-1 is easily seen to be the map discussed in the proof of Proposition 1.2. Hence {is injective, and the direct sum of these maps (1.11) is an isomorphism. We now take up the question of tensor products. Recall that if d and PA are algebras with unit over k, then the tensor product of the algebras d ® PA is the algebra whose underlying vector space is the tensor product of d and PA and whose multiplication is given (on simple elements) by the rule (a ® b) . (a' ® b') = (aa') ® (bb'). If, however, and are ;E2-graded algebras, then we can introduce a second ";E2-graded" multiplication, determined by the rule (a ® b) . (a' ® b') = (_I)deg(b)deg(a')(aa') ® (bb') (1.14) whenever b and a' are of pure degree (even or odd). The resulting algebra is called the ;E2-graded tensor product and is denoted d ® PA. The ;E2-graded tensor product is again ;E2-graded with (d ® PA)1 = dl ® PAD + dD ® PAl. It also carries a filtration ;goD c ;gol C ;go2 C ... c d ® 81, where ;gor = L ;goi(d) ® ;goj(PA). i+j=r The importance of the ;E2-graded tensor product for Clifford algebras is evident from the following proposition. Proposition 1.5. Let V = VI $ V2 be a q-orthogonal decomposition Qf the vector space V (i.e., q(VI + v2) = q(vl) + q(v2) for all VI E VI and V2 E V2). Then there is a natural isomorphism of Clifford algebras ct(V,q) -Ct(VI,qd ® ct(V2,q2) where qi denotes the restriction of q to V; and where ® denotes the ;E2-graded tensor product. 12 I. CLlFFORD ALGEBRAS AND SPIN GROUPS Proof. Consider the map f: V -+ Ct(V1,q1) ® Ct(V2,q2) given by f(v) = V1 ® 1 + 1 ® V2 where v = V1 + V2 is the decomposition of v with re spec to the splitting V = V1 $ V2. From (1.14) and the q-orthogonality of thi splitting we see that f(v)' f(v) = (V1 ® 1 + 1 ® V2)2 = ® 1 + 1 = -(q1(V1) + q2(v2))1 ® 1 = -q(v)l ® 1. Hence, by Proposition 1.1 f extends to an algebra homomorphism j:Ct(V,q) -+ Ct(V1,q1) Ct(V2,q2)' The image of j is a subalgebra which contains ct(V1,q1) ® and 1 ® CqV2,q2)' Therefore, j is surjective. Injectivity follows easily b considering a basis for Ce(V,q) generated by a basis of Vwhich is com patible with the splitting. _ We finish this section by introducing a second fundamental involutio: on the algebra. The tensor algebra 2/"(V) has an involution, given 0 simple elements by the reversal of order, i.e., v1 ® ... ® Vr f--+ Vr ... ® v1• This map clearly preserves the ideal ..?"(V,q) and so descends t, a map ( Y: cqv,q) -C£(V,q) called the transpose. Note that ( r is an antiautomorphism, i.e., (qJt/lY = t/ltqJt. §2. The Groups Pin and Spin We now consider the multiplicative group of units in the Clifford algebn which is defined to be the subset C£ x (V,q) = {qJ E cqv,q) : 3qJ -1 with qJ -lqJ = qJqJ -1 = I} (2.: This group contains all elements v E V with q(v) =I-O. When dim V = n < 00, and k is either IR or 1[, this is a Lie group of dimension 2n. I general, there is an associated Lie algebra cl x (V,q) = ct(V,q) with Li bracket given by [X,y] = xy -yX. (2.: The group of units always acts naturally as automorphisms of tll algebra. That is, there is a homomorphism Ad: cc (V,q) -Aut(Ct(V,q)) called the adjoint representation, which is given by Adq>(x) = qJxqJ -1. Taking the "derivative" of this gives a homomorphism ad: cl x (V,q) -Der(C£(V,q)) (2.: (2.' (2.: 12 I. CLIFFORD ALGEBRAS AND SPIN GROUPS Proof. Consider the map f: V -+ C£(V1,Ql) ® C£(Vz,qz) given by f(v) = VI ® 1 + 1 ® V2 where v = VI + V2 is the decomposition of V with respect to the splitting V = VI $ V2. From (1.14) and the q-orthogonality of this splitting we see that f(v)' f(v) = (VI ® 1 + 1 ® V2)2 = vi ® 1 + 1 ® ? , J' 'I.. J' \ 'I." .-'" J' \ '" .-of 'rT ........ • .,., vi = + Q9 1 = Q9 1. Hence, oy rroposlUon 1.1, f extends to an algebra homomorphism j:C£(V,q) -+ C£(Vl,ql) ® C£(V2,q2)' The image of j is a subalgebra which contains C£(V1,ql) ® 1 and t ® C£(Vz,qz)' Therefore, j is surjective. Injectivity follows easily by considering a basis for C£(V,q) generated by a basis of V which is com-patible with the splitting. _ We finish this section by introducing a second fundamental involution on the algebra. The tensor algebra 5"(V) has an involution, given on simple elements by the reversal of order, i.e., VI ® ... ® Vr 1---+ Vr ® ... ® V1. This map clearly preserves the ideal .9"(V,q) and so descends to a map ( y : C£( V,q) --+ C£(V,q) (1.15) called the transpose. Note that ( )' is an antiautomorphism, i.e., (<pI/J)' = I/Jt<pt. §2. The Groups Pin and Spin We now consider the multiplicative group of units in the Clifford algebra, which is defined to be the subset CC(V,q) == {<p E C£(V,q): 3<p-l with <p-l<p = <p<p-l = I} (2.1) This group contains all elements V E V with q(v) =1= O. When dim V = n < 00, and k is either IR or C, this is a Lie group of dimension 2". In general, there is an associated Lie algebra cl x (V,q) = C£( V,q) with Lie bracket given by [X,y] = xy -yx. (2.2) The group of units always acts naturally as automorphisms of the algebra. That is, there is a homomorphism Ad: CC (V,q) --+ Aut(C£(V,q» called the adjoint representation, which is given by Adq>(x) == <px<p -1. Taking the "derivative" of this gives a homomorphism ad: cl x (V,q) --+ Der(C£( V,q» (2.3) (2.4) (2.5) §2. PIN AND SPIN 13 into the derivations of the algebra, defined by setting ady(x) == [y,x]. REMARK 2.1. Suppose Vis finite dimensional, and defined over IR or IC. Then there is a natural exponential mapping exp: cl x (V,q) -+ Ct x (V,q), defined by setting Cl) 1 exp(y) = L --, ym. m. (2.6) Note that this series converges since for any choice of positive definite inner product on Ct(V,q), we have IlxY11 cllxllllyll for some c > 0. It is easy to see that (2.7) From this point on \ve shall assume that the characteristic of the Jield k is different from 2. Under this assumption, we have the following important facts concerning the adjoint representation: Proposition 2.2. Let v EVe ct(V,q) be an element with q(v) =1= 0. Adv(V) = V. In fact,for all WE V, the following equation holds: A A (, .. \ _ .. , ., q(v,w) " -,,"uv\'" -rv -.. q(V) v. Proof. Since v-1 = -v/q(v), we have from (1.4) that -q(v)Adv(w) = -q(v)vwv-1 = vwv = -v2w -2q(v,w)v = q(v)w -2q(v,w)v. • Then (2.8) This leads us naturally to consider the subgroup of elements ({J E Cf!(V,q) such that Adq>(V) = V. By Proposition 2.2, this group contains all elements v E Vwith q(v) =1= 0. Furthermore, we see from equation (2.8) that whenever q(v) =f. 0, the transformation Adv preserves the quadratic form q. That is, == q(Adv{w)) = q(w) for all w E V. Therefore, we define P(V,q) to be the subgroup of Ct x (V,q) generated by the elements v E V with q( v) =1= 0, and observe that there is a representation Ad P(V,q) ----+ O(V,q) (2.9) where O(V,q) = P. E GL(V): A*q = q} (2.10) is the orthogonal group of the form q. The group P(V,q) has certain im-portant subgroups. 14 1. CLIFFORD ALGEBRAS AND SPIN GROUPS DEFINITION 2.3 The Pin group of (V,q) is the subgroup Pin(V,q) of P(V,q) generated by the elements v E V with q(v) = ± L The associated Spin group of (V,q) is defined by Spin(V,q) = Pin(V,q) n O;O(V,q). We observe now that the right-hand side of equation (2.8) is just the map Pv: V -+ V given by reflection across the hyperplane v.l = {w E V: q(w,v) = O}. That is, the map Pv fixes this hyperplane and maps v to -v. Unfortunately, there is a minus sign on the left in equation (2.8). This means, for example, that if dim V is odd, then Adv is always orientation preserving. This defect can be removed by considering the twisted adjoint representation Ad:Cf x (V,q) __ GL(Cf(V,q» defined· by setting """"'d/\ /\ -1 """"\ f\. = -. (.L.ll) Clearly, Ad'l'l'l'2 = Adrpl 0 Ad'l'2 and Adrp = Ad'l' for even elements ({J (i.e., for ({J E CfO(V,q». Furthermore, from (2.8) we have '""-' q(v,w) Adiw) = w -2--) v. q(v We then define the subgroup P(V,q) == {({J E Cf x (V,q): Adrp(V) = V}. (2.12) (2.13) It is clear that P(V,q) c P(V,q). Furthermore, we have the following. Proposition 2.4. Suppose that V is finite dimensional and that q is nonde-generate. Then the kernel of the homomorphism P(V,q) -.EL GL(V) is exactly the group k x of non-zero multiples of 1. Proof. Choose a basis {v1; ..• ;v.} for V such that q(v;) =f. 0 for and q(vj,vj) = 0 for all i =f.j. Suppose ({J E CC (V,q) is in the kernel of Ad, that is, suppose ({J has the property that a.«({J)v = V({J for all v E V. Write ({J = ({Jo + ({Jl' where ({Jo is even and ({Jl is odd, and observe that V({Jo = ({JoV (2.14) -V({Jl = ({J1V for all v E V. The terms ({Jo and ({Jl can be written as polynomial expres-sions in Vl"" 'Vn' Successive use of the fact (1.4) that ViVj = -vjvi-2q(vi,vj) shows that ({Jo can be expressed as ({Jo = ao + vlal where ao and a1 are polynomial expressions in vz, ... Applying r:L shows that ao is §2. PIN AND SPIN 15 even and al is odd. Setting v = Vl in (2.14), we see that vlaO + vial = aOvl + vl al Vi = vlao -vIal' Hence, vial = -q(vl)al = 0, and so ai = O. This implies that <Po does not involve Vi' Proceeding inductively, we see that <Po does not involve any of the terms Vl, ... 'Vn and so <Po = t . 1 for t E k. The analogous argument can now be applied to <Pl' Write <Pl = al + VIaO, where ao and aI do not involve VI' Note that aI is odd and ao is even; and therefore, from (2.14), -Vial -viao = aivi + VlaOVi = -vlal + viao. Hence, ao = 0 and so <Pl is independent of Vl. By induc-tion, <Pl is independent of Vb ... 'Vn and so <Pl = O. Now we have <P = <Po + <Pi = t· 1 E k. But <P =1= 0, so <P E e. • Note that this proposition requires the twisted adjoint representation and not the adjoint representation. The minus sign in (2.14) is crucial to the proof. Proposition 2.4 is false if we do not assume that q is non-degenerate. To see this, consider the extreme case Ct(V,O) = A*V. For all Vl,V2 E V, we have 1 + VIV2 E CC (V,O). In fact, (1 + t1IV2)-1 = 1 -VIV2' However, for any V E V, we see that 0!(1 + V1V2)V(1 + V1V2)-1 = (1 + V1V2)· v(1 -Vl v2) = v. Hence, the kernel of the homomorphism includes many non-scalar terms. We now introduce the norm mapping N: ct(V,q) --+ ct(V,q) defined by setting (2.15) Here <pt denotes the transpose of <P introduced in (1.15). It is easy to see that O!(<pl) = (O!(<p))t. Note that N(v) = q(v) for v E V. (2.16) The importance of the norm is evident from the following proposition. Proposition 2.5. Suppose that V is finite dimensional and that q is non-degenerate. Then the restriction of N to the group P(V,q) gives a homomorphism kX (2.17) into the multiplicative group of non-zero multiples of the identity in ct(V,q). Proof. To begin we observe that N(P(V,q)) c k x. Choose <P E P(V,q) and recall that by definition, O!(<p)V<p-l E V for all v E V. Applying the trans-pose antiautomorphism, which is the identity on V, we see that (<pI) -1 VO!( <pI) = O!( <P )v<p -1. 16 I. CLIFFORD ALGEBRAS AND SPIN GROUPS Hence, = M<x(q>t)q>(V) = v for all v E V. Hence, !X(q/)qJ is in the kernel of M. It is easy to check that !X(qJl) belongs to P(V,q), and therefore so does !X(qJl)qJ. Hence, by Proposi-tion 2.4 we have !X( qJl)qJ E k x. Applying !X shows that qJl!X( qJ) = N( qJl) E k x . Since the transpose antiautomorphism preserves P(V,q), we conclude that N(qJ) E k x for all qJ E P(V,q). We now observe that if qJ,!jJ E P(V,q), then N(qJ!jJ) = qJ!jJ!X(!jJI)!X(qJt) = qJN(!jJ)!X(qJt) = qJ!X(qJt)N(!jJ) = N(qJ)N(!jJ). Thus, N is a homomorphism on P(V,q). • Continuing to assume that dim V < 00 and q is non-degenerate, we have the following. Corollary 2.6. The transformations Mq>: V -+ V for qJ E P(V,q) preserve the quadratic form q. Hence, there is a homomorphism M:P(V,q) --O(V,q) (2.18) Proof. To begin we note that N(!XqJ) = N(qJ) for qJ E P(V,q) since N(!XqJ) = !X(qJ)qJt = !XN(qJ) = N(qJ). Consequently, if we set VX = {v E V: q(v) =f. O}, (2.19) then for each v E VX(cP(V,q», we have N(Mq>(v» = N(!X(qJ)VqJ-i) = N(!XqJ)N(v)N(qJ)-l = N(qJ)N(qJ)-lN(v) = N(v). Since N(qJ) = q(v) for v E V (cf. (2.16», we See that Adq> preserves all non-zero q-Iengths. ApplyingMq>-l now shows that Adq>(V X) = and so Adq> leaves invariant the set of vec-tors of zero q-Iength. Thus, Adq> is q-orthogonal. • We now return to the group P(V,q) P(V,q) and observe that by definition P(V,q) = {Vi' .. Vr E C£(V,q) : Vi ... ,Vr is a finite sequence from VX}. (2.20) Recall that the twisted adjoint representation gives a homomorphism Ad: P(V,q) -+ O(V,q) such that (2.21) where q(w,V) p (w) = w -2--v v q(v) (2.22) §2. PIN AND SPIN 17 is reflection across v.L. Thus the image of P(V,q) under Ad is exactly the group generated by reflections. It is an important and classical result that this is always the entire orthogonal group. Theorem 2.7 (Cartan-Dieudonne). Let q be a non-degenerate quadratic form on a finite dimensional vector space V. Then every element 9 E O( V,q) can be written as a product of r reflections 9 = PV! 0 ••• 0 PVr where r dim( V). We refer the reader to Artin's book [1] for the general proof. In the special case where V = !Rn and q(x) = IIxl12 is the standard norm, this theorem is easily proved by putting the orthogonal matrix 9 in "diagonal" form: ±1 ±1 where each R9i is a 2 x 2 rotation matrix (which can be expressed as a product of two reflections). Theorem 2.7 says that the homomorphism Ad: P(V,q) -+ O(V,q) is sur-jective. Furthermore, we could consider the group SP(V,q) = P(V,q) n CfO(V,q) and, since dim V is finite, the special orthogonal group SO(V,q) = {A E O(V,q) : det(A) = I}. Theorem 2.7 also says that the homomorphism Ad: SP(V,q) -+ SO(V,q) is surjective. To see this, we first show that det(pv) = -1 for any v E V. To prove this, choose a basis V1, .•. ,Vn such that Vi = v and q(v,v) = 0 for j 2. It follows from the definition that Pv(v1) = -Vl and Pv(vj) = Vj for j 2, and so det (pv) = -1 as claimed. Thus from Theorem 2.7 we conclude that SO(V,q) = {PVl o· .• 0 Pvr: q(v) =1= 0 and r is even}. (2.23) From the definition (cf. (2.20)) we see that SP(V,q) = {Vi' .. Vr E P(V,q): r is even}. The surjectivity of Ad: SP(V,q) -+ SO(V,q) follows immediately (see (2.21)). 18 I. CLlFFORD ALGEBRAS AND SPIN GROUPS We now return to the groups Pin and Spin. Recall that these are the groups generated by the generalized unit sphere S = {v E V: q(v) = ± 1} in V. That is, Pin(V,q) = {VI· .. Vr E P( V,q) : q(V) = ± 1 for all j} (2.24) and Spin(V,q) = {VI·· . Vr E Pin(V,q): r is even}. (2.25) In light of the above it is natural to ask whether the homomorphism Ad restricted to Pin(V,q) and Spin(V,q) maps onto O(V,q) and SO(V,q) respec-tively. This seems quite likely since at a glance one can see that Ptv = Pv (2.26) for any non-zero scalar t E k, and so one should be able to renormalize any V E VX to have q-Iength ± 1. Of course since q is quadratic, q(tv) = t2q(v), and the equation t2q(v) = ± 1, i.e., the equation t2 = ±a for a given a, mayor may not be solvable in a general field k. If k = IR or C, of course, it is always solvable. If k = Q (the rational numbers) it is very often not solvable. (The group Q x /(Q X)2 is infinitely generated.) If k is a finite field of characteristic l' 2, then k x /(k X)2 7L2 and -1 mayor may not lie in (k X)2. In the cases where Ad is not surjective, we still have the following general fact, which is interesting because the group SO(V,q) is often almost a simple group (see Artin [1]). (The reader interested only in the real and complex cases can skip this proposition.) Proposition 2.8. Each of the images M(Pin(V,q)) and M(Spin(V,q)) is a normal subgroup ofO(V,q). Proof. Recall (cf. (1.6)) that from the universal property of Cf(V,q), the action of O(V,q) on Vextends to automorphisms of Cf(V,q). It is easy to see that these automorphisms commute with IX. Suppose then that we have V,W E V with q(v) l' 0 and choose 9 E O(V,q). Then Actg(V)(w) = lX(gv)w(gv) -1 = g(lXv)wg(v -1) = g(lX(v)g -I(W)V -1) = gAilv(g -lW). Conse-quently, we have that 1 Adgv = goAd 0 g-(2.27) for all v E V with q(v) l' 0 and for all 9 E O(V,q). The proposition now follows immediately from (2.24), (2.25) and (2.27). • We now come to the main result of this section. We are primarily in-terested in the real and complex cases, so we shall focus on fields k that have the property discussed above. We shall say that a field k of charac-teristic l' 2 is spin if at least one of the two equations t2 = a and t2 = -a can be solved in k for each non-zero element a E k x. That is, k is spin if k x = (k X)2 u ( -(k x )2). The fields IR, C and 7Lp for p a prime with p == 3(mod 4), are spin. §2. PIN AND SPIN 1" Theorem 2.9. Let V be a finite-dimensional vector space over a spin field k, and suppose q is a non-degenerate quadratic form on V. Then there are short exact sequences 0---F ---Spin(V,q) SO(V,q) ---1 (2.28) 0---F ---Pin(V,q) A O(V,q) -1 (2.29) where f71.2 = {1,-1} F--l71.4 = {± 1,±.J=i} otherwise. These sequences hold for general fields provided that SO(V,q) and O(V,q) are replaced by appropriate normal subgroups of O(V,q). Proof. Suppose <p = V1 ••• Vr E Pin(V,q) is in the kernel of Ad. Then <p E k x by Proposition 2.4, an<lJo <p2 = N(<p) = N(vd ... N(vr) = ± 1. This es-tablishes the kernel of Ad in both caseS. The surjectivity of the homo-morphisms follows from Theorem 2.7, the fact that Pv = Ptv, and the fact that since k is spin, any v E VX can be renormalized to have q-Iength 1. • It is interesting to observe that if k is a spin field, then either P(V,q) = P(V,q) or P(V,q)/P(V,q) 7L2o The proof (which the reader may skip) is as follows. Since P(V,q) is generated by V x, we know that t2q(v) E P(V,q) for all t E k x and v E V x. Since k is spin, this implies that P(V,q) contains (kX)2 or _(kX)2 (and possibly more). In fact, if we set k; = {t E kX: t·l E P(V,q)}, then from the above and from the definition of a spin field, we see that kX = k; u (-k;). Thus, kX /k; = 0 or 71.2. Now we have the sequence -k; P(V,q) P(V,q) O(V,q) where k x = ker(Ad) and where M(P(V,q» = O(V,q). It follows that O(V,q) P (V,q)/k x P(V,q)/k;. It then follows without difficulty that p(V,q)/P(V,q) k x /k; 0 or 71.2 as claimed. We now examine the real caSe in some detail. Let V be an n-dimensional vector space over and suppose q is a non-degenerate quadratic form on V. Then we may choose a basis for V' [Rn so that q(x) = xi + ." + x; -X;+l -... -x;+s (2.30) where r + s = nand 0 :$; r :$; n. It is standard notation to write: qr,s == q, Or.s == O(V,q) and SOr,s == SO(V,q). In accordance we write Pinr,s == Pin(V,q) and Spinr,s == Spin(V,q). (2.31) Similarly, it is conventional to write On == On,O Oo,n and SOn == SOn,o SOo,n' Thus, we set (2.32) We also write P"s == P(V,q) and i\s == P(V,q), and note from the para-graph above that P"s = P"s' (2.33) It is a classical fact (cf. Helgason [1]) that SOn is connected and that SO"s' for r,s 1, has exactly two connected components. It is also a classical fact that 1tl(SOn) 'Z2 for n 3 and 1tl(SO?,s) 1tl(SO,) x 1tl(SOs) for all r,s. Hence, 1tl(SO?) = 1tl(SO?'l) = 'Z2 and 1tl(SO?,s) = 'Z2 x 'Z2 for all r,s 3. (Here SO?,s denotes the connected component of the identity.) The main result of this section is the following. Theorem 2.10. There are short exact sequences o ----+ 'Z2 ----+ Spin"s ----+ SO"s ----+ 1 o ----+ "'£2 ----+ Pin"s ----+ Or,s --1 for all (r,s). Furthermore, if(r,s) #-(1,1), these two-sheeted coverings are non-trivial over each component of Or,s' In particular, in the special case (2.34) the map eo == Ad represents the universal covering homomorphism of SOn for all n 3. Proof. The exact sequences are a direct consequence of Theorem 2.9. The kernel in each case is explicitly given by 'Z2 = {1,-t}. To prove that the coverings are non-trivial, it suffices to join - 1 to 1 by a path in Spinr,s' Choose orthogonal vectors el,e2 E IRn with q(el) = q(e2) = ± 1. (This is possible since (r,s) #-(1,1).) Then y(t) = ±c o s ( 2 t ) + ele2sin(2t) = (elcos t + e2sin t)(e2sin t -elcos t) does the job. • The above argument also shows that restricting Ad to the identity component Spin?'l of Spin',l gives the universal covering homomorphism (2.35) for all r 3. §3. The Algebras CC" and Ctr,s We shall now study the Clifford algebras Ct"s == Ct(V,q) where V = 1R'+s and qlx) = xi + ... + x; -x;+ 1 -..• -x;+s' Of particular interest are the cases and (3.1) (3.2) §3. THE ALGEBRAS ct. AND ctr" 21 reason for studying these algebras is the following. As seen in §2, the ra ctr,s contains the groups Spinr,s and Pinr,s, and so any representa-)f the algebra ctr,s restricts to a representation of these groups which tl-trivial on the element -1. (Such representations are therefore not :ed from representations of Or.s or SOr.s.) ese algebras have a simple classical presentation: [)sition 3.1. Let el, ... ,er+s be any q-orthonormal basis of [Rr+s C Then ctr•s is generated (as an algebra) by el' ... ,er+s subject to the ons {-21Jij eiej + ejei = 2"" if' + Vij I I> r. (3.3) (, This follows easily from the discussion in §1. • e also have a pretty decomposition in terms of the Zz-graded tensor uct. osition 3.2. There is an isomorphism ctr•s ctl ® ... ® Ctl ® ct! ® .. , ® cq (3.4) e ctl appears r times and cq appears s times on the right in (3.4), {. Decompose [Rr+s into one-dimensional q-orthogonal subspaces apply Proposition 1.5 inductively, • is not difficult to see that as algebras over [R, (3,5) ,llows immediately that dimuiCtr.s) = 2r+s• Proposition 3,2 is, how-I not so useful if we wish to represent Ctr•s as a matrix algebra, For it is more useful to find decompositions in terms of ungraded tensor lucts. We shall do this in the next section. Dr the remainder of this section we shall examine some of the general >erties of the algebras ctr•s' We begin with a discussion of the volume lent. Choose an orientation for [Rr+s and let el' . , , ,er+s be any posi-y-oriented, q-orthonormal basis. Then the associated (oriented) vol-element is defined to be (J) = e I . , , er + s (3.6) [, .. , ,e;+s is any other such basis, then e; = :L gijej for g = ((gij)) E .s' From (3.3) we easily see that e'l'" e;+s = det(g)el ' " er+s = .. er+s' Hence the definition (3.6) is independent of the choice of the s, §3. THE ALGEBRAS ct. AND ct .. , 21 )ne reason for studying these algebras is the following. As seen in §2, the tlgebra ct"s contains the groups Spin"s and Pin,.s' and so any representa-:ion of the algebra Ct,.s restricts to a representation of these groups which s non-trivial on the element -1. (Such representations are therefore not nduced from representations of O,.s or SO, .•. ) These algebras have a simple classical presentation: Proposition 3.1. Let el,'" ,e,+. be any q-orthonormal basis of IR'+' c et,.s. Then Ct, .• is generated (as an algebra) by el' ... ,e,+s subject to the relations {-21Jij eiej + ejei = 2"" if' + Vij I I> r. (3.3) Proof. This follows easily from the discussion in §1. • We also have a pretty decomposition in terms of the Zz-graded tensor product. Proposition 3.2. There is an isomorphism ct,.s ctl ® ... ® ctl ® ct! ® ... ® cq (3.4) where etl appears r times and cq appears s times on the right in (3.4). Proof. Decompose 1R'+s into one-dimensional q-orthogonal subs paces and apply Proposition 1.5 inductively. • It is not difficult to see that as algebras over IR, (3.5) It follows immediately that dimuiCt"s) = 2'+s. Proposition 3.2 is, how-ever, not so useful if we wish to represent Ct,.s as a matrix algebra. For this it is more useful to find decompositions in terms of ungraded tensor products. We shall do this in the next section. For the remainder of this section we shall examine some of the general properties of the algebras ct,.s. We begin with a discussion of the volume element. Choose an orientation for 1R'+s and let el' ... ,e,+. be any posi-tively-oriented, q-orthonormal basis. Then the associated (oriented) vol-ume element is defined to be (3.6) If efl, •.. is any other such basis, then et = Lj gijej for g = ((gi)) E SO, .•. From (3.3) we easily see that efl··• = det(g)el ... e,+. = el ••• e,+s. Hence the definition (3.6) is independent of the choice of the basis. 22 I. CLIFFORD ALGEBRAS AND SPIN GROUPS Proposition 3.3. The volume element (3.6) in etr,. has the following basic properties. Let n = r + s. Then n(n+1)+. vi = (-1) 2 for all v E [Rn, (3.7) (3.8) In particular, if n is odd, then the element (jJ is central in etr ••. If n is even, then (3.9) for all ({J E etr ••. Proof. Choose a q-orthonormal basis and apply the relations (3.3). • We note that property (3.7) can be rewritten as 2 _ {( -1)' (jJ -(_1),+1 if n == 3 or 4 (mod 4) if n == 1 or 2 (mod 4) (3.7)' We now make the following elementary but important observation. Lemma 3.4. Suppose the volume element (jJ in etr •• satisfies (jJ2 = 1, and set and Then 11: + and 11: -satiify the relations 11:+ + 11:-= 1 Proof. This is a trivial consequence of the fact that (jJ2 = 1. • This leads to two basic but important facts: (3.10) (3.11) (3.12) (3.13) Proposition 3.5. Suppose that the volume element (jJ in etr •• satisfies (jJ2 = 1, and that r + s is odd. Then etr,. can be decomposed as a direct sum (3.14) of isomorphic subalgebras, where = 11: ± . etr .• = etr, •. 11: ± and where = •. Proof. Since r + s is odd, we know from Proposition 3.3 that (jJ is central. Hence 11:+ and 11:-are central and the decomposition (3.14) into ideals §3. THE ALGEBRAS ct. AND ctr•s 23 'ollows directly from (3.11), (3.12) and (3.13). Since w is an odd element, l(n± ) = n+ and so = Since 0( is an automorphism, we con-that these two ideals are isomorphic. • Proposition 3.6. Suppose that the volume element w in ct,.s satisfies w2 = 1 2nd that r + s is even. Let Y be any ct"s-module (i.e., Y is a real vector ,pace with an algebra homomorphism ct"s --+ Hom(Y, V)). Then there is :1 decomposition Y = Y+ EEl Y-(3.15) into the + 1 and -1 eigenspaces for multiplication by w. In fact, y+ = n+ . Y and Y-= n-. Y, and for any e E 1R'+s with q(e) #-0, module multiplication by e gives isomorphisms e:Y+ --Y-and e:Y---Y+. (3.16) Proof. The decomposition (3.15) is a direct consequence of (3.11), (3.12) and (3.13), together with the observation that w'n± = ±n ±. The isomorphisms (3.16) follow directly from the facts that by (3.8), {en+ = e(1 + w) = (1 -w)e = n-e en-= n+e and e . e = -q(e) . 1. • REMARK. The above construction will prove useful when we are dealing with vector bundles in the next chapter. We now come to an important and basic fact. Recall the even-odd decomposition Ct, s = ct? s EEl Ct: s given in (1.8), where the sub algebra et? s is the set the 0(. Theorem 3.7. There is an algebra isomorphism Ct,.s ct?+ 1,s for all r,s. In particular, for all n. (3.17) (3.18) 24 I. CLIFFORD ALGEBRAS AND SPIN GROUPS Proof. Choose a q-orthonormal basis el, ... , er+s+l of [Rr+s+l so that q(ej) = 1 for 1 i r + 1 and q(ej) = -1 for i > r + 1. Let [Rr+s = span{eili #-r + 1} and define a map f: [Rr+s -+ by setting f(ej) = er+lei for i #-r + 1, and extending linearly. For x = Li*r+ 1 xjej, we have that: f(x)2 = L xixjer+leier+lej i,j = L xixjeiej i.j = x . x = -q(x) . 1 since er+ 1 . er+ 1 = -1 and er+ lei = -eier+ 1 for i #-r + 1. It follows from the universal property (Proposition 1.1) that f extends to an algebra homomorphism Checking j on a linear basis shows that j is an isomorphism. • We now specialize to the case of ct". Proposition 3.8. Let L : ct" -+ ct" be the linear map defined by setting L(cp) = -L ejcpej j (3.19) where el' ... ,en is any orthonormal basis of [R". Set L = Il( 0 L. Then the eigenspaces of L are the canonical images of AP == AP[R" in Ct". In fact LIAP = (n -2p)ld (3.20) for p = 0, ... ,no Proof. It suffices to consider cp = e1 ... ep. Then P " L(cp) = -L ejel ... epej -L el!l'" epej j=l j=p+1 P " = -L (-1)P-leJe1 ... ep -L (-!)peJel'" ep j=1 j=p+l = (_1)r1pel'" ep + (-1)P(n -p)el'" ep = (-1)P(n -2p)el ... ep = (n -2p)ll((cp) • Under the canonical isomorphism ct" A*[R", Clifford multiplication has a nice interpretation. Using the inner product on [R" we can identify [R" with its dual. We can thereby talk about the interior product or con-traction in A* [R". For v E [R", this is a linear map (v L): AP[R" -+ AP-l[R" §4. THE CLASSIFICATION 25 on simple vectors by p ( _ '\' i+ 1< > A V LVI /\ ... /\ v p) = L... (-1) V i, V VI /\ ... /\ Vi /\ ... /\ V P (3.21) i= 1 I,Ihere () indicates deletion. This gives a skew-derivation of the algebra, .e., v L(ep /\ t/J) = (v L ep) /\ t/J + (-!)Pep /\ (v L t/J) for any ep E N!Rn. It is not lifficult to see that v L(V L) = 0 for any v E !Rn. Hence, by universality the nterior product extends to all elements of A *!Rn, i.e., to a bilinear map \*!Rn x A*!Rn --+ A*!Rn. )roposition 3.9. With respect to the canonical isomorphism Ctn A * !Rn, multiplication between v E !Rn and any ep E ctn can be written as (3.22) Choose an orthonormal basis el, ... , en for !Rn with v = tel for ome t E !R. Let ep = ei1 .•. eip for i1 < ... < ip• Then if il = 1 {-tei ••• ei (v /\ -V L)ep V'ep= 2 P teleit " . eip (v /\ -V L)ep if il > 1. ;ince (3.22) holds on an additive basis of ctn, it holds in general. _ §4. The Classification n this section we shall give an explicit description of the algebras ctr.s as natrix algebras over !R, C, or IHl ( = quaternions). With little difficulty the eader can check the first few cases: ctl•O = C ctz,o = IHl CtO,l = !R EB !R cto,z = !R(2) ctl,l = !R(2) vhere !R(2) denotes the algebra of 2 x 2 real matrices. The key facts to the classification are the following: ['heorem 4.1. There are isomorphisms or all n,r,s O. ctn,o ® Cto,2 Cto,n+z cto,n ® ctz,o ctn+z,o ctr,s ® Ctl,1 ctr+ 1,s+ 1 \,fote that here we are using the ungraded tensor product. (4.0) (4.1) (4.2) (4.3) 26 I. CLIFFORD ALGEBRAS AND SPIN GROUPS Proof. Let el' ... ,en + z be an orthonormal basis for !Rn + Z in the standard inner product, and let q(x) = -llxIl2• Let e'l' ... denote standard genera-tors for ctn.o and let denote standard generators for cto.z (in the sense of Proposition 3.1). Define a map f: !Rn + 2 --+ ctn.O ® Cto.z by setting _ {e; ® e'{ for 1 i n f(ei) -1 ® e;'-n for i = n + 1, n + 2 and extending linearly. Note that for 1 i,j n, we have f(e;)f(e) + f(ej)f(ei) = (e;ej + eje;) ® (-1) = 2c5ij1 ® 1; and for n + 1 1Y.,/3 n + 2 we have f(el1.)f(ep) + f(ep)f(el1.) = 1 ® + = 2c5l1.p1 ® 1. Also we see that f(e;)f(el1.) + f(el1.)f(ei) = O. It follows that f(x)f(x) = IIxW 1 ® 1 for all x E !Rn + z. Hence, by the universal property (Proposition 1.1), f extends to an algebra homomorphism j: cto n + Z --+ ctn 0 ® cto z· Since j maps onto a set of generators for ® cto.z, it must be Then, since dim cto.n+z = dim ctn.o ® cto.z, we conclude that f must be an isomorphism. This proves (4.1). The proof of (4.2) is entirely analogous. For (4.3) we proceed in a similar manner. We choose a q-orthogonal basis el" .. ,e.+ l,el' ... ,es+ 1 for !R'+s+z such that q(ei) = 1 and q(ej) = -1 for all i,j. We then let e'l, ... , ... and e'{,e'{ be corresponding bases for !R'+s and !Rz, and we define a map f: !R'+s+z --+ Ct •. s ® Ctl•l by setting and {e; ® e'{e'{ f(ei) = 1 ® e'l f(e.) = {ej ® e'; e'{ J 1®e'{ for 1 i r for i = r + 1, for j = s + 1, and then extending linearly. We now apply Proposition 1.1 and complete the argument as in the previous cases. • To apply this basic proposition we shall need the following elementary facts concerning the tensor products of algebras over !R. For K = !R, C or /HI, we denote by K(n) the algebra of n x n-matrices with entries in K. Proposition 4.2. !R(n) ® !R(m) !R(nm) for all n,m. (4.4) !R(n) ®ffil K K(n) for K = C or /HI and for all n. (4.5) (4.6) §4. THE CLASSIFICATION C ®Iffi IHl C(2) IHl ®Iffi IHl 1R(4). (4.7) (4.8) Proof. The isomorphisms (4.4) and (4.5) are obvious. The isomorphism IC EEl IC --+ IC ® IC is determined by sending (1,0) --! (1 ® 1 + i ® i), (0,1) --! (1 ® 1 -i ® i). For the isomorphism (4.7) we consider IHl as a C-module under left scalar multiplication, and we define an IR-bilinear map <1>: C x IHl --+ HomdlHl,lHl) by setting <l>z,q(x) == zxq. This extends (by the universal property of ®) to an IR-linear map <1>: IC_ ®Iffi IHl --+ HomdlHl,lHl) C(2). Since <l>z,q 0 = Ibzz',qq" we have that <I> is an algebra homomorphism. Checking <I> on a natural basis shows that it is injective. Hence, since dimlffi(C ®Iffi 1Hl) = dimlffi(C(2», iD is an isomorphism. The isomorphism (4.8) is proved similarly. Consider the IR-bilinear map \}I:1Hl x IHl --+ Homlffi(IHl,IHl) 1R(4) given by setting 'Pql,qz(x) == qlxqz. The resulting IR-linear map 'P: IHl ®Iffi IHl --+ Homlffi(IHl,IHl) is an algebra homo-morphism between algebras of the same dimension. The injectivity of iD can be checked on a natural basis for IHl ® 1Hl. • We now come to the first main result of the section. Before stating the result, we make the observation that for any (r,s), the complexification of the algebra ct"s is just the Clifford algebra (over C) corresponding to the complexified quadratic form, i.e., Ct"s ®Iffi C ct(C+s, q ® C). (This fol-lows easily from Proposition 1.1.) However, all non-degenerate quadratic forms on en are equivalent over ct.(C). Hence, setting • qdz) = L z; j= I and defining (4.9) we have that Ct. ct.,o ®Iffi C 3; Ct.-l,l ®Iffi C ... ®Iffi C (4.10) Theorem 4.3. For all n 0, there are "periodicity" isomorphisms ct.+s,o ct.,o ® cts,o Cto,.+s cto,. ® cto,s ct.+ z Ct. ®c ctz (4.11) (4.12) (4.13) 28 where 1. CLIFFORD ALGEBRAS AND SPIN GROUPS cts,o = Cto,8 = 1R(16) Ct2 = q2), (4.14) (4.15) Therefore, by using the identities (4.4) and (4.5), all the algebras ctn,o, cto,n and ctn can be easily deduced from the following table. Table I 1 2 3 4 5 6 7 8 ct •. o C IHJ IHJ(BIHJ 1HJ(2) C(4) 1R(8) 1R(8) (B IR(S) 1R(16) Cio,. IR(BIR 1R(2) C(2) 1HJ(2) 1HJ(2) (B 1HJ(2) 1HJ(4) C(8) 1R(16) ct. C(BC C(2) C(2) (B C(2) C(4) C(4) (B C(4) C(8) C(8) (B C(S) C(16) Proof. From (4.1) and (4.2) we see that for any n, we have ctn+s,o ctn,o ® cto,z ® ctz,o ® cto,z ® ctz,o' Using (4,0) and Proposition 4,2 we see that ctn + S,o ctn,o ® IHl ® IHl ® 1R(2) ® 1R(2) ctn,o ® 1R(4) ® 1R(4) ctn,o ® 1R(16), This establishes (4,11). The periodicity (4.12) is proved similarly, To prove (4,13), note from (4,10) that Ctn+Z Ctn+Z,o ® C ctn,o ® cto,z ® C ctn ®c ctz, Using the isomorphisms (4.1) and (4.2), and the facts (4.4) to (4.8), one can work out the first two rows of the table in "cri ss-cross" fashion (start-ing with the initial data (4.0)). The third row of the table now follows by taking the tensor product of corresponding terms in either of the first two rows with C. • Combining Table I with the fundamental periodicity isomorphism (4.3) and the fact that Ctl,l 1R(2), we achieve the complete classification in Table n. By now the reader has probably noticed some of the intrinsic beauty of this constellation of algebras and its interrelationships. There are some observations one can make from the table that are interesting exercises to prove. For example, Ct"s O",-4,s+4 (4.16) Ct"S+l Cts,'+l (symmetry about the axis y = x + 1). (4.17) REMARK. The above classification reduces the Clifford algebras to fa-miliar matrix algebras over K = IR, C or 1Hl. Of course it is also useful to think of this result as introducing hidden and unexpected structure in the algebras K(2m). This information can be quite interesting as we shall see when we discuss vector fields on spheres in §8. Table H. et,., in the box (r,s) 8 1hl(16) (;(16) G-Il(16) G-Il(16) EEl G-Il(16) 7 (;(8) G-Il(8) G-Il(8) EEl G-Il(8) G-Il(16) 6 G-Il(4) G-Il(4) EEl G-Il(4) G-Il(8) (;(16) 5 G-Il(2) EEl G-Il(2) G-Il(4) (;(8) 1hl(16) 4 G-Il(2) (;(4) 1hl(8) 1hl(8) EEl 1hl(8) 3 (;(2) 1hl(4) 1hl(4) EEl 1hl(4) 1hl(8) 2 1hl(2) 1hl(2) EEl 1hl(2) 1hl(4) (;(4) 1 IhlEElIhl 1hl(2) (;(2) G-Il(2) 0 Ihl IC G-Il G-IlEElG-ll 0 1 2 3 G-Il(32) (;(64) 1hl(128) (;(32) 1hl(64) 1hl(64) EElIhl(64) 1hl(32) 1hl(32) EEl 1hl(32) 1hl(64) 1hl(16) EElIhl(16) 1hl(32) (;(32) 1hl(16) (;(16) G-Il(16) (;(8) G-Il(8) G-Il(8) EEl G-Il(8) G-Il(4) G-Il(4) EEl G-Il(4) G-Il(8) G-Il(2) EEl G-Il(2) G-Il(4) (;(8) G-Il(2) (;(4) 1hl(8) 4 5 6 1hl(128) EEl 1hl(128) 1hl(128) (;(64) G-Il(32) G-Il(16) EEl G-Il(16) G-Il(16) (;(16) 1hl(16) 1hl(8) EEl 1hl(8) 7 1hl(256) (;(128) G-Il(64) G-Il(32) EEl G-Il(32) G-Il(32) (;(32) 1hl(32) 1hl(16) EEl 1hl(16) 1hl(16) 8 I I I ! ! --l ::t: tT1 (j t""' :> u:> u:> :;; fi o z IV \Cl 30 I. CLIFFORD ALGEBRAS AND SPIN GROUPS §5. Representations Most of the important applications of Clifford algebras come through a detailed understanding of their representations. This understanding fol-lows rather easily from the classification given in §4. We begin with a general definition. Let V be a vector space over a field k and let q be a quadratic form on V. DEFINITION 5.1. Let K k be a field containing k. Then a K-representa-tion of the Clifford algebra Ct(V,q) is a k-algebra homomorphism p: ct(V,q) ----HomK(W, W) into the algebra of linear transformations of a finite dimensional vector space W over K. The space W is called a ct(V,q)-module over K. We shall often simplify notation by writing p(<p)(w) == <P' w (5.1) for <p E Ct(V,q) and WE W, when no confusion is likely to occur. The product <p . W in (5.1) is often referred to as Clilford multiplication. Note. By a k-algebra homomorphism we mean a k-linear map p which satisfies the property p(<pl/!) = p(<p) 0 p(l/!) for all <p,1/! E Ct(V,q). We shall be interested in K-representations of Ct"s where K = C or lHl. Note that a complex vector space is just a real vector space W to-gether with a real linear map J: W -+ W such that j2 = -Id. A complex representation of ct"s is just a real representation p: Ct"s -+ HomlHl(W, W) such that p(<p) 0 J = J 0 p(<p) (5.2) for all <p E Ct"s' Thus the image of p commutes with the subalgebra span{Id,J} C. (This algebra is called a "commuting subalgebra" for p.) Strictly analogous remarks apply to quaternionic representations of Ct"s' Here the real vector space W carries three real linear transfor-mations I, J and K such that 12 = J2 = K2 = -Id IJ = -JI = K, JK = -KJ = I, KI = -IK = J. This makes W into an lHl-module. A representation p: ct"s -+ is quaternionic if p( <p) 0 1 = I 0 p( <p ), p(<p) 0 J = J 0 p(<p), p(<p) 0 K = Ko p(<p) (5.3) for all <p E Ct"s' That is, p has a commuting subalgebra spanlHl{Id,I,J,K} isomorphic to lHl. §5. REPRESENT A TIONS 31 REMARK 5.2. Any complex representation of ct.,s automatically ex-tends to a representation of Cir,s C ;-:; C£r+so quaternionic repre= sentation of Ct.,s is automatically complex (by restricting to IC c lHl). Of course the complex dimension of any lHl-module is even. The above remarks will prove useful when we carry these constructions over to vector bundles in Chapter 11. We now come to the notion of irreducibility. DEFINITION 5.3. Let V,q,k K, be as in definition 5.1. A K-represen-tation p: ct( V,q) HomK( W, W) will be said to be reducible if the vector space W can be written as a non-trivial direct sum (over K). W=W1EBW2 such that p(<p)(»j) »j for j = 1,2 and for all <P E Ct(V,q). Note that in this case we can write where pi<p) '= p(<p)!Wj for j = 1,2. A representation is called irreducible if it is not reducible. It is more conventional to call a representation "irreducible" if it has the property that there are no proper invariant subspaces. However, since et. is the algebra of a finite group (see the discussion following Proposi-tion 5.15), the two concepts are easily seen to agree in this case. Proposition 5.4. Every K-representation P of a Clifford algebra Ct(V,q) can be decomposed into a direct sum P = PlEB ... EB Pm of irreducible rep-resentations. Proof. If P is reducible, it can be decomposed as a direct sum P = PI EB P2' If either PI or P2 is reducible, P can be further decomposed. This process must stop because of the finite dimensionality of the module. • We shall be interested here, of course, only in equivalence classes of representations. DEFINITION 5.5. Two representations Pj: Ct(V,q) HomK(»j, »j) for j = 1,2 are said to be equivalent if there exists a K-linear isomorphism F: W1 W2 such that F 0 PI(<P) 0 F-1 = P2(<P) for all <P E Ct(V,q). From §4 we know that every algebra ct.,s is of the form K(2m) or K(2m) EB K(2m) for K = IC or lHl. The representation theory of such algebras is particularly simple. Theorem 5.6. Let K = IC or lHl, and consider the ring K(n) of nx n K-matrices as an algebra over Then the natural representation P of K(n) on the vector space K" is, up to equivalence, the only irreducible real rep re-sentation of K(n). 32 I. CLIFFORD ALGEBRAS AND SPIN GROUPS The algebra K(n) Et> K(n) has exactly two equivalence classes of irre-ducible real representations. They are given by and acting on Kn. Proof. This follows from the classical fact that the algebras K(n) are sim-ple and that simple algebras have only one irreducible representation up to equivalence. See Lang [1]. • From the classification of §4 (see Table 11) we immediately conclude the following: Theorem 5.7. Let vr•s denote the number of inequivalent irreducible real rep-resentations of C£r ... and let denote the number of inequivalent irre-ducible complex representatons of C£n-Then and _ {2 if r + 1 -s == 0 (mod 4) Vr s -. . 1 otherwIse VC = {2 n 1 if n is odd if n is even. This is a good time to recall (cf. Theorem 3.7) that there are isomorphisms (5.4) for all r,s, and consequently (5.5) for all n. Since (5.6) we see that it is the irreducible representations of C£r-1,s and C£r+s-1 that are relevant to constructing irreducible real and complex representa-tions of Spinr•s. From this point on we shall restrict our attention to the algebras C£n = C£n,o (and C£n = C£n ®1Hl q in order to simplify the exposition. Corresponding facts for the general case C£r,s are easy to deduce if the reader is interested. We shall begin with a summary of information easily deduced from the classification theorem 4.3. We begin with some definitions. For each n, let dn = dimlHl(W) where W is an irreducible for C£n' Similarly, let = dimdW') where W' is an irreducible complex module for C£n (and therefore for C£n = §5. REPRESENTATIONS 33 etn q. Let Kn = IR, C or IHI denote the maximal commuting subal-gebra for an irreducible real representation of etn• Thus if Kn = C, this representation is automatically complex. If Kn = IHI, it is automatically quaternionic. (Note that in the cases where etn has two distinct irreduc-ible representations, dn, and Kn are the same for both.) An object which will be of interest later on is the following. Let Wln (or denote the Grothendieck group of equivalence classes of irreducible real (respectively, complex) representations of etn• This is merely the free abelian group generated by the distinct irreducible representations over IR (or q. Since any representation can be decomposed into irreducibles, it naturally corresponds to an element in this group (with positive coefficien ts). Theorem 5.8. For 1 S n S 8, the elements Vn == vn,o, dn, Kn, Wln and (defined above) are as given in Table Ill. Table III n ct. v. d. K. 9Jl. u. 1 C I 2 C 7L CEBC 2 1 7LEB7L 2 IHl 1 4 IHl 7L (:(2) 1 2 7L 3 IHlEBIHl 2 4 IHl 7LEB7L (:(2) EB (:(2) 2 2 7LEB7L 4 1Hl(2) 1 8 IHl 7L (:(4) 1 4 7L 5 (:(4) 1 8 C 7L (:(4) EB (:(4) 2 4 7LEB7L 6 1 8 7L (:(8) 1 8 7L 7 EB 2 8 7LEB7L (:(8) EB (:(8) 2 8 7LEB7L 8 1 16 7L (:(16) 1 16 7L For n > 8 these elements can be computed from the following facts, which hold for all m,k 1. = 2kdm + 2k Proof. This is a direct consequence of Theorem 4.3. • (5.7) (5.8) (5.9) (5.10) We shall now consider the key role played by the volume element in determining irreducible representations. Recall from §3 that the volume element in etn is defined as (5.11) 34 I. CLlFFORD ALGEBRAS AND SPIN GROUPS where el' ... ,en is an orthonormal basis of It is well defined up to sign and is fixed after a choice of orientation on In the complex case we have a corresponding element WIC E ICtn given by [n+ 1] WI[ = i -2-W, (5.12) called the complex volume element. Note that when n = 2m, we have (Note also that WIC = w only in dimensions seven and eight modulo 8. Other conventions for a complex volume element are possible. This one is particularly useful in studying elliptic operators.) Recall from Proposition 3.3 that if n is odd, then wand WIC are central. Furthermore, by (3.7)' we have that w2 = 1 if n == 3 or 4 (mod 4), (5.13) (wd2 = 1 for all n. (5.14) Thus there are algebra decompositions Ctn = Ct: EB Ct; for n == 3 (mod 4) (5.15) ICtn = ICtn+ EB ICt; for n odd (5.16) where Ctn± = (1 ± w)Ctn and ICtn± = (1 ± wdlCtn-(5.17) (see Proposition 3.5). These decompositions correspond to the ones given in Table Ill. Proposition 5.9. Let p : Ctn -+ HomlHl(W, W) be any irreducible real repre-sentation where n = 4m + 3. Then either p(w) = Id or p(w) = -Id. Both possibilities can occur, and the corresponding representations are in-equivalent. (They represent the two generators of 9Jln.) The analogous statements are true in the complex case for ICtn, n odd. Proof. Since p(m)2 = p(m2) = Id, \ve can decompose l"l into l'l' = W+ EB W-where W+ and W-are the + 1 and -1 eigenspaces for p(w) respectively. Since w is central, the spaces W+ and W-are Ctn-invariant. By irreducibility either W+ = W or W-= W. This proves the first statement. The inequivalence of representations p+ and p_ with p±(w) = ±Id is evident, since if F: W -+ W' is an isomorphism and if p(w): W -+ W is a scalar multiple of Id, then F 0 p(w) 0 F-1 is the same scalar multiple of Id. §5. REPRESENTATIONS 35 To see that both possibilities exist we take irreducible factors of Ctn acting on Ct: and on Ct;. by multiplication from the left. The complex case is proved in the analogous manner by using mc. • Proposition 5.10. Let p: ctn --+ HomlliW, W) be an irreducible real repre-sentation where n = 4m, and consider the splitting W= W+ E9 W-lvhere W± = (1 ± p(w))' W (as in Proposition 3.6). Then each of the sub-spaces W+ and W-is invariant under the even subalgebra Under the isomorphism (3.18) Ctn -I, these spaces correspond to the two distinct irreducible real representations of Ctn -1 . The analogous statements are true in the complex case for Ctn, n even. Proof. The invariance of W+ and W-under is evident from the fact that (J) commutes with everything in (see (3.9)). Under the isomor-phism Ctn _ 1 given in (3.18), we see that the volume element m' = el ... en-I of Ctn-I goes to the volume element m E (To see this 1 nntp th"t { ( .0) = ... 0 )n-I = ... 0 ..... "" .. -......... _ ... ,el .... n! \en-1--nJ --\ .... ....1 .... n-l' ..... nl el .... n since n = 4m.) It follows that m' Id on W+ and m' -Id on W-. Hence, by Proposition 5.9 these representations of Ctn_1 are inequivalent. The complex case is proved in the same manner using the volume form Wc for Ct"" • The representations of the algebras Ctn give rise to important represen-tations of certain groups. Consider the spin group ( 5.18) DEFINITION 5.11. The real spinor representation of Spinn is the homo-morphism .1n: Spinn ----GL(S) given by restricting an irreducible real representation Ctn --+ HomlHl(S,S) to Spinn C c Ct"" Proposition 5.12. When n == 3 (mod 4) this definition of .1n is independent of which irreducible representation of Ctn is used. For n =1= 0 (mod 4) the representation .1n is either irreducible or a direct sum of two equivalent irre-ducibie representations. (The second possibiiity occurs exactiy when n == j or 2 (mod 8).) In the other cases there is a decomposition (5.19) where .1:rn and .14m are inequivalent irreducible representations of Spin4m' 36 I. CLIFFORD ALGEBRAS AND SPIN GROUPS !'roof. Recall that if n == 3 (mod 4), then the automorphism IX: Ctn -+ Ctn mterchanges the factors et: and et; (since oc(01) = -01). Consequently, sits diagonally in the decomposition Ctn = ct: EB ct; , i.e., = {(<p, 1X(<p)) E Ct: EB Ct; : <p E Ct:} (5.20) The two irreducible representations of Ctn differ by the automorphism IX, and are clearly equivalent when restricted to This proves the first statement of the proposition. It is evident from Table III that the restriction of an irreducible real representation of Ctn to Ctn-1 is still irreducible if n == 3, 5, 6, or 7 (mod 8), and must be two copies of an irreducible representation when n == 1 or 2 (mod 8). When n == 0 (mod 4), we know from Proposition 5.10 that the restriction to splits into two inequivalent irreducible repre-sentations. To complete the proof we observe that any irreducible repre-sentation of restricts to an irreducible representation of Spinn because Spinn contains an additive basis for • REMARK 5.13. Note that the spin representations are complex for n == 2 or 6 (mod 8), and are quaternionic for n == 3, 4 or 5 (mod 8). (The maximal commuting algebra is determined by Ctn-1 The analysis above carries over to the complex case. DEFINITION 5.14. The complex spin representation of Spinn is the homo-morphism Spinn ---+ GLdS) given by restnctmg an irreducible complex representation ICtn-+ HomdS,S) to Spinn C c ICtn. Proposition 5.15. When n is odd, this definition of is independent of which irreducible representation of ICtn is used. Furthermore, when n is odd, the representation is irreducible. When n is even, there is a de-composition (5.21) into a direct sum of two inequivalent irreducible complex representations of Spinn· Proof. The proof is entirely analogous to that of Proposition 5.12. • It should be pointed out that the spin representations defined above do not descend to the group SOn = Spinn/Z2 since -1) = -Id. It is worthwhile noting that representations of Ctn also give rise to re-presentations of the Ciiiiord group. This is the finite group F n c ct; §5. REPRESENTATIONS 37 generated by an orthonormal basis el, ••. ,en of IRn. It can be presented by the abstract eiements ell' .. ,en, -i subject to the rdation that -i is central and that (_1)2 = 1, (ei)2 -1 and eiej = for all i ¥ j. The Clifford algebra is nearly the group algebra !RFn of Fn. More explicitly et. iRFn/iR· {( -1) + l} It is clear that representations of ctn correspond exactly to linear repre-sentations of F n such that ( -1) acts by -Id. This group enabies us to draw an important conclusion: Proposition 5.16. Let Ctn --+ HomlHl(W,W) be a real representation ofCtn• Then there exists an inner product <-,. > on W such that Clffford multipli-cation by unit vectors e E IRn is orthogonal, i.e., such that <e' w,e' w') = <w, w') (5.22) for aU w,w' E Wand for aU e E iR" with Ilell = 1. If K. (= !R, iC or iHi) is a commuting subalgebra for the representation, then the inner product can be chosen to be Kn-invariant, that is, so that J is orthogonal when Kn = IC and so that I, J and K are each orthogonai if K" = IHI. In particular, the spin representations are unitary if n == 2 or 6 (mod 8) and symplectic if n == 3, 4 or 5 (mod 8). Proof. Choose a Kn-invariant inner product and average it over the finite group F •. Note that if e = L ah where L aJ = 1, then <ew,ew) = L aJ<ejw,ejw) + L aiaj<eiW,ejw) = <w, w) ii=j since <eiw,eiw) = <w,w) and for i #-j, <eiw,ejw) = <ejeiw, -w) = <eiejw, w) = -<ejw, eiw) = O. For the last statement, recall that comes from a representation of -l' • Corollary 5.17. Let <.,.) be the metric discussed in Proposition 5.16. Then for any v E !Rn, (v'w,w')= -<w,v'w') (5.23) for all w,w' E W. That is, Clffford multiplication by any vector v E !Rn is a skew-symmetric transformation of W. Proof. Assume v#-O. Then <v· w, w') = «v/llvll)' V· w,(v/llvll)' w') = (1/llvI12)<V2 . W, V· w') = -<w, V· w'). • It is worth noting that the irreducible representions of 1C£2" have a par-ticularly nice description. Introduce on en the standard hermitian metric n (z,O == L Zlj' j=l (5.24) 38 I. CLIFFORD ALGEBRAS AND SPIN GROUPS and use this inner product to define a complex linear contraction map (v L) : AflC" ----Af-11C" for v E Cn by formula (3.21). We then define Iv: -+ by setting (5.25) Then since v 1\ v = 0, (v L)(V L) = 0, and v L (v 1\ ((1) = IIv1l2((1 -V 1\ (v L ((1), we see that (5.26) Note that since the inner product is C-antilinear in the second variable, the map v -+ Iv is only Nevertheless, writing cn, we see from universality (Proposition 1.1) that property (5.26) determines a unique extension of I to a representation j : Ct2n ----HomdA *Cn, A *IC"). (5.27) Since the complex dimension of this representation is 2n, we see that it must be the irreducible one. We now make some remarks concerning tensor products. Suppose W is a K-module for Ctn (where K = or IC) and let V be any vector space over K. Then W ® K V is also a K-module for Ctn where by definition ((1 . (w ® v) = «((1w) ® v. (5.28) Therefore, if W1 and W2 are K -modules for Ctn, then W1 ® K W2 is a K-module for Ctn in two distinct ways. We set A<p(W1 ® w2) = «((1wd ® W2, PiW1 ® w2) = W1 ® «((1w2)· Then A and p are commuting representations of Ctn. Furthermore, the product Cl><P • .<P2(W1 ® W2) = «((11wd ® «((12W2) is a representation of the (ungraded) tensor product ctn ® Ctn. Proposition 5.18. Let ct2n -+ HomdS,S) be an irreducible complex repre-sentation ofct2n• Then the tensor product representation oIct2n ®c ct2n on S ®c S is equivalent to the representation Cl> on Ct2n itself given by setting Cl><P"<P2«((1) = ((11' ((1' Proof. Since Ct2n ®c Ct2n Cf4n (see Theorem 4.3), we see that S ®c S must be an irreducible module for reasons of dimension. Since dimdS ® S) = 22n = dimdCt2n), the representations must be equivalent. • §5. REPRESENTATIONS 39 Corollary 5.19. Let Pn : Spin. -+ SO(IR·) denote the standard n-dimensional representation of Spin •. Then in the complex representation ring of Spin2m (cf. Adams [1]) we have the equation + + = 2(1 + + + ... + + where denotes the complexification of P2m' Proof. The tensor product ® is obtained by embedding Spin2m into ICt2m ® ICt2m diagonally (g f-+ g ® g) and restricting the tensor product representation. By Proposition 5.18 this is equivalent to the adjoint representation <1>g(q» = gq>l = gq>g-l = Adg(q». Under the correspondence ICt2n AllC2m, this representation is equivalent to (1 + P2m + A2P2m + ... + A2mP2m) ® IC. However, by Hodge duality APP2m A2m-PP2m' • Results analogous to Proposition 5.18 and Corollary 5.19 hold for the algebras ctsm and for the real spin representation = + Note that the tensor product of irreducible real representations of ct. and cts gives an irreducible real representation of ctn+8 ctn ® Cts. Similarly the complex tensor product of irreducible complex representa-tions of ICtn and ICt2 gives an irreducible complex representation of ICtn+2 ICtn ® ICt2. In general, however, ctn ® ctm is not a Clifford algebra. Thus, to find a multiplicative structure in the representations of Clifford algebras it is natural to consider the category of 1'2-graded mod-ules. A 1'l-graded module for ctn is a module W with a decomposition W = WO EB W1 such that Wi W(i+i)(mod2) for 0 i, j 1. Proposition 5.20. There is an equivalence between the category of 1'2-graded modules over ctn and the category of ungraded modules over ctn -l' It is defined by passing from the graded module WO EB W1 over Ctn to the module WO over Cf. -l' Proof. The inverse procedure is given by assigning to a WO, the E2-graded module W == Ctn ® W ° (Left multiplication by Ctn on Cfn makes W into a E2-graded module.) The remainder of the proof is straightforward. • There is a natural definition of the 1'2-graded tensor product of E2-graded modules W = WO EB Wl and V = VO EB Vi over etn and etm 40 I. CLlFFORD ALGEBRAS AND SPIN GROUPS respectively. We set (W ® V)O = WO ® VO + W1 ® Vi (W ® V)i = WO ® Vi + W1 ® Vu. The action of ctn ® ctm on W ® V is given by (cp ® 1jJ) . (w ® v) == (-I)pQ(cpw) ® (IjJv) where deg(ljJ) = p and deg(w) = q. Under the isomorphisrn ctn ® ctm, induced from mapping !Rn EB !Rm --+ Ctn+m fe 1----+ e ® 1 le' 1----+ 1 ® e' if e E !Rn c ctn if e' E !Rm c ctm, r-.. 0 ....... , \'-{,n+m == V ® W becomes a Z2-graded module over ctn+m• This construction holds fer either real er complex modules. In analogy with the above we define 9Rn to be the Grothendieck group of real (complex) Z2-graded modules over ct .. Note that by Prop-osition 5.20 there are natural isomorphisms and The arguments just given have established the following: Proposition 5.21. There are natural pairings 9Jln ® 9Jlm -9Jln + m ® -+ m (5.29) (5.30) (5.31) induced by the Z2-graded tensor product. These pairings are associative and give 9)1* == EB..?:o Win and == EB..?:o the structure of graded rings. These pairings are important in the relation of Clifford algebras to real and complex K-theory (see §9). §6. Lie Algebis Stiuctures This section shall be concerned with the Lie algebra of Spin .. Recall that the group of units ctnX is a Lie group with Lie algebra clnx == (Ctn,[·,·]) where [<p, i/iJ == <p . i/i -i/i . <po There is an exponential mapping exp: er: --+ ctnX given by the standard series (see Remark 2.1). The group Spinn is an explicitly defined, compact subgroup of ctnx, We shall now investigate its associated Lie sub algebra spinn in ctn• §6. LIE ALGEBRA STRUCTURES 41 Recall that there are canonical embeddings AP!Rn c ctn for all p. Proposition 6.1. The Lie subalgebra of (ctn,[·,· J) corresponding to the subgroup Spin. c ct: is spinn = A 2 !Rn. (6.1) I n particular, A 2 !Rn is closed under the bracket operation. Proof. The Lie subalgebra spinn is the vector subspace of ctn spanned by the tangent vectors to the submanifold Spinn at 1. Fix an orthonormal basis el' ... ,en of !Rn and consider for each pair i < j, the curve y(t) == (ej cos t + ej sin t) . ( -ej cos t + ej sin t) = (cos2t -sin2t) + 2ejej sin t cos t = cos(2t) + sin(2t)ejej. This curve lies in Spinn by definition of Spinn, and its tangent vector at y(O) = 1. is 2ejej• Hence, spinn contains the vector sub-space = A2!Rn. Since = n(n -1)/2, we conclude they are eq ual. _ We now·recall that the Lie algebra of the orthogonal group SOn is ex-actly the space SOn = {)\.: IRn ----+ )\. is linear and ske\v-symmetric} (6.2) There is a natural isomorphism A 2 !Rn SOn induced by associating to a pair of vectors V,W E !Rn, the skew-symmetric endomorphism "v 1\ w" de-fined by (v 1\ w)(x) == <v,x)w -<w,x)v, (6.3) and then extending by universality. Note that ej 1\ ej, for i < j, corresponds to the elementary skew-symmetric (i,j) matrix: ·10 ] I I \ I I! This is a standard basis of SOn. Recall now that the adjoint representation gives a surjective homomor-phism Spinn SOn. (Since Spinn c we have M!sPjnn = Ad!sPinn.) This induces an asso-ciated Lie algebra isomorphism (6.4) 42 I. CLIFFORD ALGEBRAS AND SPIN GROUPS Proposition 6.2. The Lie algebra isomorphism (6.4) induced by the adjoint representation is given explicitly on basis elements {eiej};<j by Consequently for v,w E IRn, 1 301(v 1\ w) = 4: [v,w] (6.5) (6.6) Note. This factor of t plays a delicate role in geometric applications. Proof. Consider the curve y(t) = cos(t) + sin(t)eiej in Spin" with y(O) = 1 and y'(O) = eiej' Then and to compute this we apply it to a vector x E [R". Since )(x) = y(t)xy(t) -1, and since (y -1 )'(0) = -y'(O) = -eiej, we have that 30(eiej)(x) = eiejx -xeiej = eiejx + (eix + 2<ei,x»)ej = eiejx -eiejx -2<ej,x)ei + 2<ei,x)ej = 2(ei 1\ ej)(x) To prove (6.6), note that on basis elements, Hei,ej] = t(eiej -ejei) = teiej' • This Proposition has the following immediate corollary: Corollary 6.3. Let A: Spinn -+ SO(W) be a representation obtained by re-striction of a representation ctn -+ Hom(W, W) of the Clifford algebra C£n Spinn. Let ,1* : SOn -+ so(W) be the associated representation of the Lie algebra (obtained by first pulling back SOn to the double covering via 301). Then on the elementary transformations v 1\ WE SOn' A*(v 1\ W) = Hv,w] . where the dot indicates Clifford module multiplication on W. In terms of the standard basis {ei 1\ ej};<j A*(ei 1\ ej) = teiej (6.7) (6.8) Suppose now that ctn -+ Hom(W, W) is a complex representation of C£n, and fix an element WE W. The subgroup Gw == {g E Pinn: gw = w} (6.9) §6. LIE ALGEBRA STRUCTURES is called the isotropy group of w. Its Lie algebra is the su balgebra 9w == {cp E spinn : cP . w = O} 43 (6.10) Two elements w,w' E Ware considered to be different as spinors (or more precisely, to have distinct orbit types) if their isotropy groups Gw and Gw' are not conjugate in Pinn• One crude measure of this difference is the following: DEFINITION 6.4. The rank of the (generalized) spinor w is the rank of the Lie group GW' This is the dimension of a maximal torus in Gw or, equiv-alently, of a maximal abelian subalgebra of 9w' In a compact Lie group, every abelian subgroup is contained in a maxi-mal torus, and all maximal tori are conjugate (cf. Adams [1]). Hence any maximal torus T w of Gw is contained in a maximal torus T of Pinn which we can assume to be the following standard one associated to a fixed orthonormaI basis {e 1, ... ,en} of !Rn: T == {[rf (cos Ok + sin 0ke2k-l e2k) : 0 Ok < 2n for each k 1 k= 1 f --The Lie algebra of T is given by (In/2J ) t == y Ake2k-te2k: Ak E IR for each k 1 --J We now use our distinguished orthonormal basis to decompose the module W. For each k, 1 2k n, define and note that Wb' .• ,w[n/2J pairwise commute, wf = 1 for each k, Wkek = -ekwk for each k. (6.11) (6.12) (6.13) Suppose now that Vc W is a linear subspace which is e2k-cinvariant and e2k-invariant for some fixed k. Then by (6.12) we know that V = V+ 63 V_ where V± = (l ± wk)V are the ± 1 eigenspaces of wk on V. Fur-thermore, by (6.13) we see that ekV+ = V_ and ekV-= V+. In particular, dim V± = t dim V. We shall now use this process to decompose the module W. We begin with the decomposition W = W+ EB W_ by Wl' Since W1 commutes with e],e4' ... ,en, we see that each of the subspaces W+ and W_ is e3' ... ,en-invariant. Hence we can similarly decompose each subspace W+ and W_ by W2 to get W+ = W+ + w+ _ and W_ == W_ + $ W __ 0 Each sub-space W± ± is es, ... ,en-invariant. Continuing inductively, we produce a 44 I. CLlFFORD ALGEBRAS AND SPIN GROUPS decomposition W= EB W± ... ± = EB w;, (6.14) a where dim w;, = (dim W)/2[n/2] for each a and where a = (al, ... ,a[n/2]) ranges over the 2[n/2tpossibilities having ak = + or -for each k. (Note that if W is an irreducible module, then dime w;, = 1 for all a.) The maximal torus T preserves each subspace w;,. In fact, given g = TI (cos Ok + i sin 0kWk) E T, we see that glw« = en;ak9k == ei<a,9> Similarly for qJ = L Ake2k -1 e2k = i L AkWk E t we have qJlw. = i L akAk == i<a,A.) (6.15) The set of vectors !a E t* are called the weights of the representation. The! occurs because in general theory the weights are normalized by relating them to the weights of the adjoint representation. We now return to our given spinor W E W. With respect to the decom-position (6.14) we write W=LWa a From (6.10) and (6.15) we conclude the following: Proposition 6.5. The maximal abelian subalgebra of Gw is tw = {i L AkWk : <a,A.) = 0 for all a such that Wa =f. O} Corollary 6.6. rank W = [nI2] -dim a : Wa =f. O} If W = Wa for some a, i.e., if all but one component vanish, then W is clearly of maximal rank. Those elements that take the simple form W = Wa for some choice of orthonormal basis in \Rn, are called pure. Pure spinors are related to complex structures, twistors spaces and calibrations. They will be discussed in detail in Chapter IV. §7. Some Direct Applications to Geometry In this section we shall use the classification of Clifford modules given above to construct families of pointwise linearly independent vector fields on spheres, projective spaces and other elliptic space forms. We shall also apply the methods to study the "hyperplane" bundle over complex and quaternionic projective space. This allows us to estimate the geometric dimension of TlPn(lC). In almost all cases the families constructed in this manner are maximal. §7. APPLICATIONS TO GEOMETRY We begin with the following observation: 45 Proposition 7.1. Suppose !RN + 1 is a module for the algebra ctn. Then there exist n pointwise linearly independent tangent vector fields on the sphere SN and also on the projective space jp>N(!R) = SN/7L2• Proof. Choose an inner product in !RN + 1 so that Clifford multiplica-tion by unit vectors in !Rn is orthogonal (see Proposition 5.16). Let SN = {x E 1: IIxll2 = I}. Choose a basis v1, ••• ,Vn for and to each Vj as-sociate the vector field lJ on !RN + 1 defined by -lJ(x) == vj' x j = 1, ... ,n (where the dot denotes Clifford multiplication). Since the linear transfor-mation x 1-+ v . x is skew-symmetric (see Corollary 5.17), we have that (lJ(x), x) == (vr' x) == O. Hence, the vector fields lJ are tangent to SN. It remains to show that Vb' .. , v" are pointwise linearly independent. Fix x E SN and consider the linear map ix: !Rn -+ T xSN C + 1 given by ix(v) = V' x The image of ix is the linear span of V1 (x), ... ,v,,(x), so it suffices to prove that ix is injective. However if ixv = v . x = 0, then v . v . x = -llvl12x = 0 and so v = O. Since lj( -x) = -lj(x), these vector fields descend to (pointwise linearly independent) vector fields on • The question now is: given an integer N, what is the largest number of independent vector fields on SN that can be constructed in this manner? That is, what is the largest integer n such that !RN + 1 is a Ctn-module? We recall that the dimension of an irreducible Cfn-module is always a power of 2. Hence, we want to find the largest power of 2 which divides N + 1. That is, we write N + 1 = p2m where p is odd, and then we consult Table III to find the largest n such that dn = 2m. The result is the following clas-sical result of Radon and Hurwitz. Theorem 7.2. On the sphere SN (and on the projective space there exist n pointwise linearly independent vector fields where n is computed as follows. Write N + 1 = 24a+b(2t + 1),0 S b S 3. Then n = 8a + 2b -1. (7.1) Proof. One need only check this when a = 0, and then note that for each increase ofn by 8 the dimension of the vector space for an irreducible representation of etn increases by 24. Note that when N is even, the num-ber of such vector fields is zero as it must be since the Euler characteristic is non-zero in this case. Note also that this construction gives three vector fields on S3, seven on S7 and eight on S15. !!! 46 I. CLlFFORD ALGEBRAS AND SPIN GROUPS One of the deep results of algebraic topology is the following: Theorem 7.3 (1. F. Adams [2J). The number of vector fields constructed on SN above is the largest possible number of pointwise linearly independent vector fields that can exist on SN. It is worth noting that this construction also gives rise to vector fields on many elliptic space forms. If the representation of ctn on [RN + I is com-plex or quaternionic, then Clifford multiplication by v E [Rn commutes with complex scalar multiplication. Therefore, if f3 = e21!i/p is a pth root of unity, then the vector field V(x) == V· x on SN has the equivariance property V(f3x) = f3 . V(x) (7.2) for all x E SN. This means precisely that the vector field V(x) is invariant under the diffeomorphism SN -+ SN given by scalar multiplication by f3. The .lp-action on SN generated by f3 is free. From (7.2) we conclude that V descends to a vector field on the quotient SN /?Lp which is a lens space of simple type LN(p) = LN(p;l, ... ,1). Analogous remarks hold when the representation of ctn is quaternionic. Here we may replace ?Lp by any finite multiplicative subgroup of IHI. Such subgroups are constructed as follows. The unit sphere S3 c IHI is a Lie group isomorphic to Spin3 (see the paragraph below). Let S3 -+ S03 be the 2-fold covering homomorphism. Then for any finite subgroup roe S03, the inverse image r = l(r 0) is a finite subgroup of S3 c IHI. Of course, the symmetry groups of the regular polygons, the so-called di-hedral groups, and the symmetry groups of the Platonic solids give many examples of finite subgroups of S03' The I-images of dihedral groups are called binary dihedral groups. There are also the binary tetrahedral group, the binary octahedral group, and the binary icosahedral group, cor-responding to the I-images of the symmetry groups of the tetrahedron, octahedron and icosahedron, respectively. From our remarks above we conclude the following two theorems. . Theorem 7.4. On each simple lens space L N(p) = SN /.lp, for p ;:::: 1, there exist k point wise linearly independent vector fields where, if N + 1 = 2m(2t + 1), then k = 2m -1 Moreover, this is the maximal number of point wise linearly independent vec-tor fields possible on L N (p) if m == 1 or 2 mod ulo 4. Theorem 7.5. There exist q pointwise linearly independent vector fields on sN/r where r is any finite subgroup of S3 c IHI and where if N + 1 = §7. APPLICATIONS TO GEOMETRY 2m(2t + 1) and m = 4a + b, 2 ::; b ::; S, then _ {8a + b + 1 q -8a + b + 2 ifbi'S if b = S· 47 Moreover, this is the maximal number of point wise linearly independent vec-tor fields possible on the elliptic space form SN /r if m is congruent to 1 or 2 modulo 4. Similar constructions can be made for complex and quaternionic pro-jective space. Of course in these cases the Euler characteristic of the mani-fold is not zero, and so every tangent vector field must vanish somewhere. However, we can pass to the stabiiized bundle TX Ee ii{m where ii{m de-notes the trivial bundle of dimension m. Clearly there exists some bundle E over X so that TX Ee IRm E Ee IRm+n-k where n = dim X, k = dimlJ;l E and k is as small as possible (for any choice of m). The dimension of E is called the geometric dimension of TX. Let iP>n(K) denote the n-dimensional projective space over the field K. For K = IR, IC or IHI there is a tautological K-line bundle 1] -+ iP>n(K) whose fibre above a point [t] E iP>n(K) is the one-dimensional linear subspace t c Kn+ 1 corresponding to [t]. For K = IR or IC we have the following fact: T(iP>n(K)) Ee K 1]* Ee··· Ee 1]* (n + 1 )-times where ;i* is the dual bundle 1]* == HomK(1], K). To see this, we first show that (7.3) (7.4) Each one-dimensional K-linear subspace t c Kn+ 1 can be canonically identified with the (K-linear) orthogonal projection map ne:Kn+1 -+ t. Note that n: = ne, nenp = npne = 0 and ne + np = Id. Let tt, It I < e, be a smooth family of K-lines with to = t, and set n = o. Then by deriving the identities above we have that nene + nene = ne, nenp + neneL = n(Lne + npne = 0, and ne + np = O. It follows that nenp = 0 and nene = O. Hence ne E HomK(t, t.L). On the other hand it is easy to construct an ele-mentary basis of HOffiK(t, {1-) as tangent vectors nt fer curves t{t) \vith t(O) = t. From the exact sequence (\ R ----------lo. vn + 1 -----lr.. jJ.l ----loo. {\ V -., J."-' v ' v 48 I. CLlFFORD ALGEBRAS AND SPIN GROUPS we obtain the exact sequence o ------+ t) ------+ HomK(t, Kn+ 1) ------+ HomK(t, t-L) ------+ O. (7.5) From (7.4) this gives an exact sequence of bundles over IPn(K): o ------+ HomK('1, '1) ------+ '1* EB ... EB '1* ------+ TIP"(K) ------+ 0 (7.5) (n + 1) times which holds for K = IR, C or IHJ. When K = IR or C, we have '1) '1* ® '1 K trivial K-line bundle), and this establishes (7.3). Suppose now that L: Kn + 1 -+ Km is a K -linear map (for any m 1). Then L defines a section of '1* EB ... EB '1* (m times) as follows. Fix a K -line t c K" + 1. Then Lit: t -+ Km is exactly m K -linear functions on t. Thus, we have an embedding HomK(Kn+l, Km) c r('1* EB ... EB '1*), m times (7.7) into the space of sections of the bundle '1* EB ... EB '1*. A section corre-sponding to a K-linear map L is nowhere zero if and only if L(x) i:-0 for any x i:-O. Similarly, L1, ... ,Lp give pointwise linearly independent sec-tions of '1* EB ... EB '1* if and only if L1(x), ... ,Lp(x) are linearly inde-pendent at each non-zero x E Kn+ 1. The K-representations of Ctp give us precisely p K-linear maps L1, ... ,Lp: KN -+ KN which satisfy this condi-tion of pointwise independence. Thus, by consulting Table Ill, we have the following result analogous to that of Theorem 7.4 (cf. Lawson-Michelsohn [1]). Theorem 7.6. Let (n + 1)'1* = '1* EB ... EB '1* denote (n + 1)-copies of the "hyperplane" bundle over complex projective n-space IPn(lC). If n + 1 = 2m(2t + 1), then there exist k sections of (n + 1)'1 * where k=2m-1. Therefore ybn :'5: 2n -2m + 3 where ybn denotes the geometric dimension of the tangent bundle TlPn(lC). We also have the following result which is analogous to that of The-orem 7.5. Theorem 7.7. Let (n + = EB ... EB denote (n + I)-copies of the "hyperplane" bundle over quaternionic projective n-space IPn(IHJ). Ifn + 1 = §8. APPLICATIONS TO LIE GROUPS 2m(2t + 1) then there exist k sections of(n + where Therefore {2m -3 k = 2m - 2 2m -1 (4n -2m + 7 gdn :::;; 4n -2m + 6 L4n -2m + 5 ifm == 0 (mod 4) ifm == 3 (mod 4) otherwise. ifm == 0 (mod 4) ifm == 3 (mod 4) otherwise 49 vJhere gdll denotes the geometric dimension of the bundle Tpn(lHl) (f) Note that we are unable to make a conclusion about the geometric dimension of the tangent bundie itseif because is not triviaL §S. Some Further Applications to the Theory of Lie Groups It is interesting to note that the classification of Clifford modules gives an immediate proof of certain well-known isomorphisms between low-dimensional Lie groups. It also leads to the fact that S7 = Spin7/G2 and to the principal of triality for Spins . We would like to thank Reese Harvey for pointing this out to us. ""le begin \rvith some classical definitions. Let en and !HIn carry the stan .. dard "hermitian" inner products n (x,y) == I. xiYi (S.l) j=-1 -where for a quaternion x = Xo + LX1 + jX2 + !s.X3, the conjugate is defined by x = Xo -iX1 -jX2 -!s.X3' Then we-have the following definitions of the classical groupS: Un = {g E Hom,dCn,Cn) : (gx,gy) = (x,y) for all x,y E en}. SUn = {g E Un: det,dg) = I} Spn = {g E HomlHl(!H1n,!HI") : (gx,gy) = (x,y) for all x,y E !HI"} Under the natural isometries 1R4n e2n IHln (and [R2n en) one finds that (S.2) 50 I. CLIFFORD ALGEBRAS AND SPIN GROUPS Elementary linear algebra proves that {dim(So.) = tn(n -1) dim(SU.) = n2 -1 dim(Sp.) = n(2n + 1), and it is not difficult to see that each of these groups is connected. (8.3) When the integers p,q,r are sufficiently large (say;:::; 7), the groups Spinp, SU q' SPr are all distinct. However, in low dimensions there are certain exceptional isomorphisms between these groups. These isomorphisms are easily deduced from the Dynkin diagrams. However, the approach re-quires the non-trivial classification of compact, simply-connected Lie groups. The same isomorphisms can also be deduced from Table n, which was comparatively easy to establish. Theorem 8.1. There exist the following isomorphisms between low-dimen-sional Lie groups: Spin3 SU2 SP1 Spin4 Spin3 x Spin3 Spins SP2 Spin6 SU4 Furthermore, Spin? has a faitliful 8-dimensional real representation, and Spins has three inequivalent 8-dimensional real representations. Proof. Recall that Spin. c C£. _ 1. Furthermore any representa-tion of C£.-1 on qf or !HIN can be assumed to have the property that, when restricted to the group Spin., it preserves the hermitian inner prod-uct (8.1). Consequently, since C£2 !HI has a faithful one-dimensional quaternionic representation, we get an injection Spin3 4 SPl. The first isomorphism follows easily. Since C£3 !HI EB !HI, we get an injective map Spin44 SP1 x SP1. Since dim(Spin4) = dim(SP1 x Spd and Spin4 is con-nected, we get the second isomorphism. Since C£4 !HI(2), we get an in-jection Spins 4 SP2 and the third isomorphism follows. Since C£s C(4), we get an injection Spin6 4 U4. By the simplicity of (the Lie algebra of) Spin6' or by Lemma 8.5 below, we see that Spin6 must lie in the kernel of the homomorphism detc:U4 U1. Thus, we have Spin6 4 SU4 and the fourth isomorphism follows. The existence of a representation of Spin? on !R8 is obvious since C£6 !R(8). Furthermore, since C£? !R(8) EB !R(8), we see that the two spin representations At and As of Spins are on !Rs. There is also the §S. APPLICATIONS TO LIE GROUPS 51 adjoint representation Ad of Spins on [Rs. To see that these representations are all distinct, it suffices to consider the central elements. Set = {l,-l,w,-w} 71.2 EB 71.2 where w denotes the oriented volume element of Ct7 Ctg. This group lies in the center of Spins. (In fact it is the center.) From Propositions 5.9 and 5.10 we know that A;(w) = Id and As(w) = -Id. Since come from representations of Ct7 we have A;(w) = Id, As(w) = -Id, A;(-w) = -Id, As( -w) = Id, A;(-l)=-Id As(-l)=-Id. (S.4) (S.5) From the definition it is clear that if Ad denotes the adjoint representation of Cts on [R8, restricted to Spins c ctg ct7, then Ad(-l) = Id. Recall now that equivalent irreducible representations must agree on central elements. (Note, for example, that if there exists an isomorphism F:llls [Rs with the property that F 0 A;(g) 0 F-1 = As(g) for all 9 E Spins, then in particular, A;(g) = As(g) for all g E Consequently the three representations A;, As and Ad are inequivalent. • It is an interesting and often useful fact that two S-dimensional rep-resentations of Ct7 can be explicitly generated using the Cayley numbers. Recall that the Cayley numbers 0 can be defined as pairs of quaternions with multiplication given by (a,b) . (c,d) = (ac -db, da + be). (S.6) The multiplication so defined is neither commutative nor associative. However, every non-zero element has a multiplicative inverse. Further-· more, given a Cayley number x = (a,b), we write x = (a, -b) and define real and imaginary parts of x by setting Re(x) = t(x + x); Im(x) = t(x -x) An inner product on 0 is defined by <x,y) = Re(xY). It has the property that Ixyl = Ixllyl for all x,y EO (where Ixl2 = <x,x) as usual). An impor-tant fact concerning the Cayley numbers is that any subalgebra of 0 gen-erated by two elements is associative. We now consider /R7 = Im(O) and [Rs = 0 with the above inner prod-uct. For any v E Im(O) we define a linear endomorphism Av of [Rs by setting (S.7) 52 I. CLIFFORD ALGEBRAS AND SPIN GROUPS for x EO = [Rs. From the associativity of the algebra generated by x and v, or by a direct computation from (8.6), \ve see that (8.8) for all v E Im(O) = [R7. Consequently, from the universal property (Propo-sition 1.1), we know that A extends to a representation (8.9) of et7. For dimensional reasons this representation must be irreducible. The other irreducible 8-dimensional representation is generated by the mapping Pv(x) = -v . x. Observe that under the con-jugation map c(x) = x, Pv becomes equivalent to right multiplication by v, i.e., Pv = c 0 Pv 0 c is given by J\(x) = (-v· x) = -x' v = X· v (since for v E lm 0 we have v = -v). These two representations are equivalent when restricted to Spin7' but they are inequivalent on Spins c Ctg ct7. We now consider the action of Pin? on 1R8 given by the representation A above. From (8.7) it is clear that the orbit of 1 contains all elements e E Im(O) with lel = 1. That is, this orbit contains the "equator" S6 = S7 n Im(O). It also contains e' S6, which is a great sphere passing through the "north pole" 1, for each e E S6. Since the orbit Pin7 . 1 is a compact embedded submanifold of S7, we conclude that it is S7. It now follows that the orbit of Spin7 must be 7=dimensional and, hence, also equal to 87. \l.le have proved the result of A. Borel. Theorem 8.2. The 8-dimensional spin representation of Spin? is transitive on the unit sphere S7. ----. §8. APPLICATIONS TO LIE GROUPS 53 With a little more work it is possible to prove that the isotropy sub-group {g E Spin7: Ail) = 1} of 1 E S7 is exactly the group G2 == Aut(O) (see Harvey-Lawson [3] for example). Hence, we have the diffeomorphism S7 Spin7/G2• We pass on now to the group Spins. Recall from Theorem 8.1 that this group has three distinct homomorphisms Ad: Spins SOs with kernels isomorphic to £:2' Writing the center 1E = {l,-l,w,-w} as before, we have the following table (cf. (8.4) and (8.5)). 9 (1) -(1) -1 A:(g) Id -Id -Id (8.10) A;(g) -Id Id -Id Ad(g) -Id -Id Id We now consider the pair of homo morph isms and From general covering space theory there must be a lifting of over which takes the identity to itself, i.e., a map a: Spins Spins such that the diagram Spins commutes and a(l) = 1. The map a satisfies the condition a(g 1 g 2) = a(g da(g 2) (8.11 ) (8.12) for all gl,g2 in a small neighborhood of 1, since both and are group homomorphisms. In fact, the relation (8.12) holds for all gbg2 E Spins. To see this note that both sides of (8.12) are well defined on Spins x Spins and that the set where (8.12) holds is both open and closed. (Alternatively, one could use the fact that (8.12) is an equation between real analytic maps.) Thus, a is a group homomorphism. Since a is a covering map between simply-connected spaces, it must be injective, and so a is a group automorphism. It is now clear from (8.11) that a carries the kernel of onto the kernel of Since = {l,-w} and = {l,w}, and since 0'(1) = 1, we conclude that a( -w) = w Since wand -ware central, a must be an outer automorphism. 54 I. CLlFFORD ALGEBRAS AND SPIN GROUPS Lifting Ad over the homomorphisms At and As and applying the arguments above we construct outer automorphisms r + and r -of Spins with the property that (8.13) By definition we have that 0 -r± = Ad and 0 (J = Furthermore, the associated Lie algebra maps and Ad* are isomorphisms, and we have that (8.14) At this point the automorphisms (J and t ± are only defined modulo inner automorphisms. To get concrete representatives we must choose concrete representatives for the maps and Ad. We shall give such an explicit construction for (J. Choose an ortho-normal basis el' ... ,es of /Rs and recall that the Lie algebra spins = A2/Rs has an orthonormai basis {ei· ej}i<j. The map Ct7 Ctg is induced by the assignment j = 1, ... ,7. Consequently, the preimage of spins under this map is just spins /R7 EB A2/R7 C Ct7 (8.15) (8.16) where ei eies for 1 si s 7 and where eiej eieSejeS = eiejfor 1 i <j 7. We now consider the two representations A and p of Ct7 on /Rs = 0 that were constructed above. Since Av(X) = V· x and Pv(x) = -v . x for v E /R7, we see that for <p E /R7 for qJ E A 2/R7. (8.17) Now restricted to spins /R7 EB A2/R7 we have that A = (At)* and p = (As)*. Consequently, (J* = (At); 1 0 has the property that I r-qJ (J*tqJ) = 1 qJ if qJ E /R7 if qJ E A2/R7" (8.18) In particular, «(J*)2 = Id and so by exponentiation we have that (J2 = Id. (8.19) We now let Os and 1 s denote respectively the groups of outer and inner automorphisms of Spins. There is a natural homomorphism Os/1 s (8.20) It is easy to see that Aut(!r) Aut(iE2 EB iE2) S3. In fact, we have natu-ra!!y that Perm(-l,w,-w). Since a(-w) = w and = 1L, §8. APPLICATIONS TO LIE GROUPS we have that either a( -1) = -1 or a( -1) = -wo Since a2 conclude that a( -1) = -1 and a(w) = -wo This information, together with (8.13), easily proves the following: 55 Id, we Theorem 8.3. The homomorphism Os/Is Aut(!r) ;;:: S3 is surjective. In particular, since Os/I s is a finite group (cf. Helgason [lJ) there exists an element in Os/I s of order three. Those readers similar with the classification of Lie groups know that this map Os/Is S3 is reflected in the representation of Os/Is as the group of symmetries of the Dynkin diagram of Spins. This representation is faithful, and so Os/Is ;;:: S3' It can be shown that in fact there exists a non-trivial element A E Os such that This element is called the triality automorphism. Continuing with calcula-tions as above, this automorphism can be constructed explicitly. It has most likely occurred to the reader that the methods of Theorem 8.1 can be applied more generally to the groups Spin"s for any rand s. Indeed, this does produce further exceptional isomorphisms between low-dimensional Lie groups. We recall the following classical groups. SLn(lR) = {g E : = 1} SLn(C) = {g E Homc(IC",Cn) : detc(g) = 1} SLn(!H1) = {g E Hom01l(!H1n,!HI") : detc(g) = 1} For the last definition we have fixed a presentation IHIn = (C2n,J) where J: C2n C2n is C-antilinear and J2 = -Id. We observe that the complex determinant is in fact real-valued on Hom01l(lHIn,lHIn) = {g E Homc(C2n,C2n): go J = Jog}. To see this, let c: c2n C2n denote complex conjuga-tion and set c(g) = cog 0 c. Then since c2 = 1 and since g and J com-mute, we have detc(c(g) 0 c 0 J) = detc(c 0 g 0 J) = detc(c 0 Jog). Hence, detdc(g)) = detc(g) and since detc(c(g)) = detc(g), the determinant is real. 56 I. CLIFFORD ALGEBRAS AND SPIN GROUPS Now, each of the groups SLn(K) is connected, and elementary linear algebra shows that = n2 -1 = 2(n2 -1) = 4n2 -1. (8.21 ) Recall that for r,s > 0, the group Spin"s has two connected components. We let Spin?,s denote the component containing the identity. It is obvious that there is an isomorphism Spin"s Spins." and an elementary calculation shows that dim(Spin,,,) = (r + s)(r + s -1) (8.22) for all r,s. A careful look at Table II (p. 29) now proves the following. Let SLn(lR) ...... SLilR) denote the (2-sheeted) universal covering group for n 3. Theorem 8.4. There exist the following isomorphisms between low-dimen-sional Lie groups. SL2(1R) Sping,l SLiC) Sping,l SLz(IHl) SL2(1R) x SLilR) Furthermore Spin4,4 has three inequivalent 8-dimensional real represen-tations. Proof. Recall from Theorem 3.7 that for r 1 we have Spin"s c Cf,-l,s' Consequently, from Table II, we have the following embeddings: SpinZ,l C GLz(IR), Spin3,l C GLz(C), SpinS,l c GLilHl) x GLz(IHl), Spinz,z C GL2(1R) x GLz(lR) and Spin3,3 C GL4(1R) x GL4(1R), where GLiK) denotes the set of invertible elements in HomK(Kn,Kn). We now observe that in each of these embeddings the identity component is actually contained in the subgroup SLn(K) or in SLn(K) x SLn(K) for the latter cases. This is an immediate consequence of the following lemma. §8. APPLICATIONS TO LIE GROUPS 57 Lemma 8.5. Let p: C£r-i, ....... HomK(V, V) be any K-representation for K = lR or C. If r + s 3, then detK(p(g)) = ± 1 for all g E Spinr,. C C£r-i, •. Proof. If r + s 3, then every element g E Spinr" can be written as a product g = gi ... gm where gj E Spinr,. satisfies gJ = ± 1 (8.23) for allj. To see this, write g = Vi ... V2m where Vj E lRr+s and q(vj) = ± 1. Write ViV2= +ViVVV2 where vElRr+. satisfies q(V) = ±1 and where q(v,vd = q(v,v2) = O. Set gi = ViV and g2 = +VV2· Then gi = ViVViV = -viv2 = ± 1, and similarly, = ± 1. Of course ViV2 = glg2' Continuing in this manner proves our claim. Since dimK(V) is even, we see that for g = gi ... gm as above, we have [detK(pg)Y = [If detK(pgj) J = If detK(pgJ) = n detK(±Id) = 1. • j The first, second and fourth isomorphisms of Theorem 8.4 now follow from dimension considerations (cf. (8.21) and (8.22)). For the fifth isomorphism, consider the image of Sping,3 under the two projections SL4(lR) x SL4(lR) :4 SL4(lR). At least one of these must be non-trivial and, therefore, locally injective by the simplicity of the Lie algebra. Since dim(Spin3,3) = dim(SL4(lR)) = 15, this projection is a cov-ering map Sping.3 ...... SL4(lR). §!Ece Sping,3 is simply connected, it lifts to an isomorphism Sping,3 ..::. SL4(lR). The third isomorphism is proved similarly. Looking more closely we can see that the two projections Sping,3 :4 SL4(lR) are both non-trivial and inequivalent. To prove this we consider the volume element (J) = ei ... e6 where ei = d = = -ei = = = -1 (see Proposition 3.3). This element is central in Spin3,3 and satisfies (J)2 = 1. It clearly lies in Spin3 x Spin3 C Spin3,3 and is therefore connected to the identity. The module V lRs for C£3,3 decomposes as V V+ E9 V-where V± = (1 ± (J))V are invariant subspaces for Spin3,3' Since (J) = ± Id on V± , we see that the representations are in equivalent. They are each non-trivial, since otherwise we would have the identity w = 1 (or (J) = -1) in Spin3,3' which is clearly false. To prove the final statement we again consider the volume element w = ei ... es in Spin4,4' As above we see that (J) is central in Spin4,4 and is connected to the identity. Since (J)2 = 1, the module W lR16 for C£4,4 58 1. CLIFFORD ALGEBRAS AND SPIN GROUPS decomposes as W = W+ E9 W-where W± = (1 ± w)W are each invari-ant under Spin4.4. Let A ± denote these two 8-dimensional representations, and consider the central subgroup 1E = {l,-l,w,-w} 7Lz E9 7Lz in Spin4.4' Then A +, A -and Ad have precisely the values given in the table (8.10) derived above for the case of Spins. It follows that these three 8-dimensional representations of Spin4.4 are distinct. • The above arguments actually prove slightly more. Let SL:(K) = {g E HomK(Kn,Kn) : detdg) = ± I} where K' = K ifK = IR or C and K' = Cif K = !HI. There is a short exact sequence 1 --+ SLn(K) --+ SL:(K) 7Lz --+ O. From the proof of Theorem 8.4 we can actually conclude that SpinZ.l sq(lR) Spin3•1 sq(C) SpinS.l sq(!HI) Note. For further results of the type given in Theorem 8.4, the reader is referred to the beautiful book of Reese Harvey [1]. §9. K-Theory and the Atiyah-Bott-Shapiro Construction In this section we shall present the essentials of K-theory in a fashion which will be useful later in our discussion of the Atiyah-Singer Index Theorem (Chap. Ill). Following these basics we shall present the con-struction of Atiyah, Bott and Shapiro which relates the Grothendieck groups of real Clifford modules to the KO-theory of spheres and that of complex Clifford modules to the K-theory of spheres. Their fundamental isomorphisms explicitly identify Bott periodicity with the periodicity phenomena in the theory of Clifford algebras. For an elaboration of the details presented here, the reader is referred to Atiyah-Bott-Shapiro [1] and Karoubi [2]. Throughout this section all spaces will be assumed to be compact. If X is any such space, we denote by V(X) the set of all isomorphism classes of complex vector bundles over X. The set V(X) is an abelian semigroup if we define addition by direct sum. We iet F(X) be the free abeiian group generated by the elements of V(X) and let E(X) be the subgroup of F(X) §9. K-THEORY AND THE ABS CONSTRUCTION 59 generated by elements of the form [V] + [W] -([V] Et> [W]) where + denotes addition in F(X) and EB denotes addition in V(X). DEFINITION 9.1. The K-group of X is defined to be the quotient K(X) = F(X)JE(X). Note that K(X) is an abelian group. The elements of K(X) are called virtual bundles. If Vand Ware bundles over X and Y respectively then V ® W is a bundle over X x Y. When X = Y the diagonal map L\: X -+ X x X can be used to define an interior tensor product in K(X) by [u]· [v] == L\*[u ® vJ. (9.1) This gives us Proposition 9.2. The group K(X) has a ring structure with multiplication given by (9.1). Let a: V(X) -+ K(X) be the composition V(X) 4 F(X) -+ F(X)/ E(X). Proposition 9.3. IfG is any abelian group andf: V(X) -+ G any semi-group homo!!,orphism, then there is a unique homomorphism 1: K(X) -+ G such that fa = f. Proposition 9.4. K(X) is the universal group with respect to maps of the type described in Proposition 9.3 Suppose that f: X -+ Y is a continuous map and consider the map f*: V(Y) -+ V(X) given by the induced bundle construction. Since this is a semi-group homomorphism, it descends to a homomorphism f* : K(Y) -+ K(X). One easily checks that K thereby becomes a contravariant functor from the category of compact spaces to the category of abelian groups. Suppose now that S is any abelian semi-group with unit and let 11: S -+ S x S be the diagonal map. This is a semi-group homomorphism. If we let $'(S) be the set of cosets of L\(S) in S x S, then $'(S) is a quotient semi=group. Since the interchange of factors in S x S induces an inverse in $'(S), $'(S) is actually a group. DEFINITION 9.5. $'(X) is defined to be $'(V(X)). Proposition 9.6. $'(X) is isomorphic to K(X). Proof. For an abelian semi-group S with 0 we define f3s : S -+ $'(S) to be 5 f-+ (5,0) followed by the natural projection S x S -+ .J¥'(S). If g: S -+ T 60 I. CLlFFORD ALGEBRAS AND SPIN GROUPS is a semi-group homomorphism, then there is induced a map %(g): %(S)--+ %(T) so that %(g) 0 f3s = f3T 0 g. Now let S be V(X) and Tbe any abe-lian group. Then f3T is an isomorphism. This shows that %(X) is uni-versal with respect to semi-group homomorphisms from V(X) to abelian groups. • Corollary 9.7. Every element of K(X) can be represented in the form [V] -[W] where [V],[W] E V(X). Lemma 9.S. Let n: V --+ X be a vector bundle over a compact Hausdorff space X. Then for some N there is a continuous map f: V --+ ([N which is injective and linear on each fibre. Proof. Cover X with a finite number of open sets U 1, .•. ,Ur over which there exist trivializations!Xj: n-1(Vj) V j X ([k and set aj = prj 0 !Xj where prj: Vj x ([k --+ ([k is projection. Let {I/tj}j= 1 be a partition of unity sub-ordinate to {VJj=1. Thenf:;;:: (1/t1al) EB··· EB (I/trar): V --+ ([k EB··· EB ([k is the desired map. • Corollary 9.9. To each bundle V over X there exists a "complementary" bundle V.L such that V EB V.L is trivial. Hence, each element of K(X) can be represented in the form [V] -[rN], where rN is the trivial bundle over X of dimension N. Proof. Let f: V --+ ([N be the map in Lemma 9.8 and define V.L = UXEX f(Vx).L· Clearly V EB V.L X X ([N. Hence, [V] = ErN] -[V.L] in K(X) and an arbitrary element [W] -[V] in K(X) can be rewritten as [WEB V.L] -ErN]. • DEFINITION 9.10. We define the real K-ring for X, KO(X), just as we defined K(X) by replacing V(X) by V[J;!(X), the set of isomorphism classes of real vector bundles. The same construction and considerations· apply. Analogues of the following definitions can also be made for KO-theory. We will assume them without specifically stating them. We would now like to use the K-groups to define a generalized coho-mology theory. For this reason we now consider X to be a space with a distinguished base point, pt E X. DEFINITION 9.11. The reduced K-ring, K(X), is defined to be the kernel of the natural projection K(X) --+ K(pt) Z, so that K(X) is an ideal of K(X). In fact the exact sequence o ---+ K(X) ---+ K(X) ---+ K(pt) ---+ 0 splits in an obvious fashion. (9.2) §9. K-THEORY AND THE ABS CONSTRUCTION 61 DEFINITION 9.12. Suppose Y is a non-empty closed subset of X. Then we define the relative K-groups as follows: K(X,y) == K(X/Y) where X/Yis taken to have Y as its basepoint. If Y is empty, X /Y is defined to be the space X+ == X u {Pt} (9.3) where pt is a disjoint point which will play the role of basepoint. OBSERVATION 9.13. On a non-basepointed space X we have the identi-fication K(X) K(X+) = K(X, 0). DEFINITION 9.14. We define the wedge X v Y and the smash product X " Y of two spaces X, Y with basepoints ptx and pty by X v Y == (X x pty) u (ptx x Y) c X x Y X " Y == X x Y / X v Y. We also define the (reduced) suspension of X by == S1 " X. Iterating this i times gives us the i-fold suspension For all i there is a homeomorphism Si " X. DEFINITION 9.15. When X is a compact base pointed space, or when (X, Y) is a compact pair, we define K-i(X) == v-i{v v\ -V-i(v/V\ _ Vf .... ifV/V\\ ''" \-"',') = ''" -'" / ' ) = ''"\-<-< \-'" ,)). For spaces X which are not necessarily basepointed we define, in the spirit of 9.13, K-i(X) == K-i(X, 0) == K(2:i(X+)). Since the functor K is representable (see Chap. Ill, Theorem 8.6) there is an exact sequence for basepointed pairs (X, Y) K(X,Y) --K(X) --K(Y) which we may now extend to a Barratt-Puppe sequence (Barratt [1]): ... __ R-i-l(y) K-i(X,y) __ R-i(X) __ (9.4) K-i(y) __ ... __ KO(X) --KO(y). We write R-* for the graded functor K-i, i O. 62 I. CLIFFORD ALGEBRAS AND SPIN GROUPS REMARK 9.16. If Y is a retract of X, then for all i 0--K-i(X,Y) --K-i(X) __ K-i(Y) --0 is a split short exact sequence and K-i(X) K-i(X,y) EB K-i(y). This follows from the Barratt-Puppe sequence (9.4). Now if X and Y are spaces with basepoints then K-i(X x Y) K-i(X /\ Y) EB K-i(X) EB K-i(Y) since X is a retract of X x Y and Y is a retract of X x Y / x. We would like now to use the ring structure on K(X) to enable us to define a ring structure on K -*(X). Proposition 9.17. Given X and Y and i,j 0 there is a pairing K -i(X) ® K-i(Y) -+ K-i-i(X /\ Y) which is given by tensor product. Proof. For a bundle E on Si /\ X and a bundle F on Si /\ Y we have the tensor product bundle E ® F on (Si /\ X) X (Si /\ Y). This induces a pairing K(Si /\ X) ® K(Si /\ Y) __ K((Si /\ X) X (Si /\ Y)). But K((Si /\ X) /\ (Si /\ Y)) is the kernel of K((Si /\ X) X (Si /\ Y)) -+ K(Si /\ X) EB KW /\ Y). So we have a pairing K(Si /\ X) ® K(Si /\ Y) -+ K((Si /\ X) /\ (Si /\ Y)) = K((Si+i) /\ X /\ Y) as desired .• Replacing X by X+ and Y by y+ in Proposition 9.17 gives us a pairing K-i(X) ® K-i(y) -+ K-i-i(X x Y). We easily conclude the following. Corollary9.18. The pairing of Proposition 9.17 makes K-*(pt) a graded ring. Furthermore, for any basepointed space (X,pt), this pairing makes K -*(X) a graded module over K -*(pt). Thus far everything we have said for K-theory holds equally true for KO-theory. We will now describe periodicity and at this point the descrip-tions must diverge. We will discuss the basic complex case first and then proceed to the real case. We will simply state the results. A proof based on the theory of Fredholm operators is given in §1O of Chapter Ill. For alternative proofs the reader is referred to Bott [1], [4]. Bott Periodicity Theorem 9.19 (the complex case). The ring K -*(pt) is a polynomial algebra generated by an element E K-2(pt) K(S2), i.e., there is a ring isomorphism (9.5) §9. K-THEORY AND THE ABS CONSTRUCTION 63 Note. The element is in fact represented by the virtual bundle [H] -,1 E K(S2) where H denotes the "tautologically defined" complex line bun-dle over S2 = iP'1(C) and ,1 denotes the trivial line bundle. The isomorphism (9.5) says in particular that the map K-i-2(pt) induced by multiplication by is an isomorphism for all i. In this form, the theorem extends to arbitrary compact Hausdorff spaces. Recall from 9.18 that for any pointed space (X,pt) the ring K-*(X) is a module over K-*(pt). General Bott Periodicity Theorem 9.20 (the complex case). Let X be any compact Hausdorff space. Then the map K-i-2(X) given by module multiplication by is an isomorphism for all i 0. Note. Replacing X by X/Y, we get corresponding isomorphisms K-i(X,Y) --K-i-2(X,y) for any pair (X, Y) of compact Hausdorff spaces. The situation in KO-theory is slightly more complicated. Bott Periodicity Theorem 9.21 (the real case). The ring KO-*(pt) is gene-rated by elements yE KO-4(pt), subject only to the relations 2'1 = 0, '13 = 0, '1Y = 0, i.e., there is a ring isomorphism (9.6) As before we have that for any space X, KO -*(X) is a KO -*(pt)-module. Theorem 9.22. Let X be a compact Hausdorff space. Then the map Jlx:KO-i(X) __ KO-i-8(X) given by module multiplication by x E KO -8(pt), is an isomorphism for all As before there are also isomorphisms KO-i(X,y) for any pair (X, Y) of compact Hausdorff spaces. 64 I. CLIFFORD ALGEBRAS AND SPIN GROUPS The best explicit representatives for the elements '1, x and y in KO -*(pt) are given via the Atiyah-Bott-Shapiro isomorphism. To present this iso-morphism, and also to adapt K-theory easily to the theory of elliptic operators, we present now an alternative definition of the groups K(X, Y) and KO(X,Y). The discussion is completely parallel in the real and com-plex cases. We shall just use the generic term "vector bundle." We begin with the following definition. We assume throughout that Y is a closed subspace of X. DEFINITION 9.23. For each integer n 1, consider the set !l'iX, y) of elements V = (Vo, V1, ... ,v,,; a 1, ... ,an) where VD' ... ,Vn are vector bundles on X and where 0--... v"ly--O is an exact sequence of bundle maps for the restriction of these bundles to Y. Two such elements V = (Vo, ... ,Vn;a1,· •• ,an) and V' = (VO,··· a'l' ... are said to be isomorphic if there are bundle isomorphisms <Pi: Vi -+ v; over X so that the diagram Vi-11y Vily 1 V;-lly V;ly commutes for each i. An element V = (Vo, ... ,v,,; a 1, ... ,an) is said to be elementary if there is an i such that (a) Vi = Vi-1 and ai = Id (b) = {O} for j t= i or i -1. There is an operation of direct sum EB defined on the set !l'iX, Y) in the obvious way. Two elements V,V' E !l'iX, Y) are defined to be equiv-alent if there exist elementary elements El' ... ,Ek,F 1, ... ,Ft E !l'n(X, Y) and an isomorphism V EB El EB ... EB Ek V' EB F1 EB··· EB Ft· The equivalence class of an element (Vo, ... ,v,,; a 1, ... ,an) will be denoted by [Vo, ... ,v,,; a1,··· ,an]. The set of all equivalence classes in !l'n(X,Y) will be denoted by LiX, Y). The set LiX,y) is an abelian group under the operation EB. Our first main proposition is the following, whose proof is left to the reader. Consider the natural map !l'n(X, Y) -+ !l'n + l(X, Y) which asso-ciates to each element (Vo, ... , v,,; a l' ... ,an) the trivially extended element (Vo, ... ,v",0; a l' ... ,a n'O). §9. K-THEORY AND THE ABS CONSTRUCTION 65 Proposition 9.24. For each n 1, the induced map LiX, Y) -+ Ln+ l(X, Y) is an isomorphism. We set L(X,y) = lim LiX,Y). Each inclusion LiX,y) -+ L(X,y) is an isomorphism, and it would have sufficed for many purposes to consider only the case n = 1. Our second main proposition is the following. Proposition 9.25. There exists a unique equivalence offunctors x: L(X, Y) -+ K(X,y) with the property that n X([Vo, ... ,v,,]) = L (-1 t[v,,] when Y = 0. (9.7) k;Q Proof. This equivalence will be essentially determined by defining it on L1(X,Y). Given an element V = [VO,v1;O"] E L1(X,Y) we associate to it an element x(V) E K(X, Y) by the following "difference bundle construction". Set Xk = X x {k} for k = 0,1 and consider the space Z = Xo Uy Xl ob-tained from the disjoint union X ° II Xl by identifying y x {O} with y x {1} for all yE Y. The natural sequence 0--K(Z,X1) K(Z) K(X1) --0 is split exact since there is an obvious retraction P:Z--X1· Furthermore, there is an isomorphism q>: K(Z,X 1) K(X, Y) induced by the map of pairs (X, Y) -+ (Z,X 1) which identifies X with Xo. From our element V = [VO,v1;O"] we define a vector bundle W over Z (wen defined up to isomorphism) by setting l"/lxk Vk and identifying over Y via the isomorphism (J. Setting W1 == p*(Vd we have [W]-[W1] E ker(i*). Hence, there is a unique element x(V) E K(X, Y) with j*cp-1X(V) = LW] -[W1]. This defines the homomorphism X:L1(X,Y)-+ K(X,Y). It clearly has property (9.7). It is now straightforward to verify that any homomorphism L1 (X, Y) -+ K(X,y) with property (9.7) is an isomorphism, and furthermore, any two such homomorphisms agree. The reader is referred to Atiyah-Bott-Shapiro [1] for details. • As a result of this proposition we shall henceforth drop the notation L(X,y). We win however discuss eiements [VO,v1; (J] E K(X,y) whose meaning is now obvious. For our later discussion of elliptic operators it will be useful to note that the multiplication in K(X,y) can be realized explicitly in It'l(X,Y) 66 I. CLIFFORD ALGEBRAS AND SPIN GROUPS as follows. Choose V = (VO,vl; 0"), W = (Wo, W1; r) E ,!Z\(X, Y) and for con-venience introduce metrics in each of the bundles. We then define the tensor product U = V ® WE !i\(X,y) to be the element U = (UO,U1;p) where and P = (0" ® 1 1 ® r -1 ® r*). 0"* ® 1 Under the construction above, this tensor product on ,!Z\(X, Y) carries over to the standard product on K(X, Y). This is a convenient time to introduce K-theory for locally compact spaces. K-theory in this setting will be important for certain geometric constructions we shall make in Chapter Ill. DEFINITION 9.26. For any locally compact space X we define Kcpt(X) = K(X+) where X+ = X u {pt} denotes the one point compactification of X. The higher groups are defined by setting = Kcpt(X x for i O. The groups K.;;,:(X) are functors on the category of locally compact spaces and proper maps. Collectively they comprise the K-theory of X with compact supports. They enjoy the multiplicativity properties that we pre-sented above in the compact case. Note incidentally that if X is compact, then K';;':(X) = K -i(X). Any element in Kcpt(X) can be represented as the formal difference of two bundles E and F on X, each of which is trivialized at infinity (i.e., trivialized outside some compact set in X). In fact, if (9 c X is an open subset of the locally compact space X, then there is a natural extension homomorphism Kcpt((9) -+ Kcpt(X) induced by the map X+ -+ X+ / (X+ -(9) = (9+. Taking products with gives extension homomorphisms K;p:((9) --K';;'\(X) for all i O. Of course for any closed subset Ye X we have the functorial restriction homomorphism K;p:(X) --K;p\(Y). The kernel of this map is defined to be the relative group K;p\(X,y). For any pair (X, Y) where X is a locally compact space and Y is a closed sub- §9. K-THEORY AND THE ABS CONSTRUCTION 67 space, there is a long exact sequence for the K';-p:-groups, analogous to (9.4) above. We leave as an exercise for the reader the verification of the following isomorphism: K';-p:(X, Y) Kcpt( (X -Y) X There exist definitions of "L-type" for the groups Kcpt(X). In analogy with the above we can define Ln(X)cpt to be the equivalence classes [Vo, ... ,v,,; 0" 1, ... ,0" n] where Vo, ... , v" are vecctor bundles on X, and where 0 ----+ Vo v;. ... v" ----+ 0 is an exact sequence of bundle maps defined on the complement of some compact subset in X. As above there are natural isomorphisms L1(X)cpt"; L2(X)cpt ..; ... Kcpt(X). Thus, in particular, any element of Kcpt(X) can be represented by a triple [Vo, V1; 0"] where 0": Vo V1 is a bundle isomorphism defined in a neighborhood of infinity. All of the discussion above applies equally well to real bundles and yields groups KO.;-p:(X) for any locally compact space X. The Bott Periodicity theorems carry over to locally compact spaces in the following elegant form: Kcpt(X) Kcpt(X x C) KOcpt(X) KOcpt(X x (9.8) These isomorphisms are induced by multiplication by fundamental ele-ments E Kcpt(C) and x E respectively. In the last part of this chapter we shall produce explicit representatives for these elements using Clifford modules. We are now in a position to discuss the isomorphism of Atiyah, Bott and Shapiro. Let W = WO EB W1 be a Z2-graded module over the Clifford algebra et,," == Let D" == {x E Ilxll ::; 1} be the unit disk and set S"-l = aD". We now associate to the graded module W, the element (9.9) where Ek == Dn x Wk is the trivial product bundle, and where /1: Eo ..; El is the isomorphism over S" -1 given by Clifford multiplication: /1(x,w) == (x, x . w). One easily checks that the element q>(W) depends only on the isomor-phism class of the graded module Wand, furthermore, that the map W 1--+ q>(W) is an additive homomorphism. Hence, (9.9) gives us a homo-morphism (9.10) where is the Grothendieck group of complex graded Cfn-modules defined in §5. 68 I. CLlFFORD ALGEBRAS AND SPIN GROUPS Consider for a moment the natural inclusion i: 4 1 given by setting i(X1' ... ,xn) = (Xl' ... ,xn,O). This induces an algebra homomor-phism i* : ctn ctn + 1. Restris.ting the from ctn + 1 to ctn thereby ind uces a homomorphism i*: + 1 Suppose now that W is a graded Ctn-module which can be obtained from a ctn + 1-module in the above fashion. This means that the Clifford multiplication of on W extends to all of + 1. Hence we may extend the isomorphism J.1., defined on sn-1 = aDn, to all of Dn by setting J.1.(x,w) = (x, (x + Jl -Ilxl12en+ d . w) where en+ 1 E + 1 is a unit vector orthogonal to Since this extended map is an isomorphism over all of Dn, the associated element q>(W) must be zero. It now follows that the homomorphism (9.10) descends to a homo-morphism K(Dn,sn-1) (9.11) where i*: 1 ---+ is the restriction map defined above. Exactly the same construction applied to the real case, gives us a homo-morphism (9.12) The isomorphisms (5.29), defined by taking the even part, determine isomorphisms anc/·*anc onC /·*onc and an /.*an on /·*on ;V'n I ;V'n + 1 = -1 I ;V'n I ;V'n + 1 JJ'n _ I I Set = + 1 and Qn = IDln/i*IDln + 1. The periodicity pheno-mena (4.11), (4.13) and (5.9) determine periodicity isomorphisms and Elementary algebraic arguments show that c {Z Qn = 0 if n is even if n is odd if n == 0 or 4 (mod 8) if n == 1 or 2 (mod 8) otherwise. As an example consider Q2 = IDldi*IDl2 considered as the quotient of the groups of ungraded modules. Since ct I = C, the ct 1-modules are just complex vector spaces, and the isomorphism 9Jl1 .; Z is generated by taking the complex dimension. Similarly, since ce2 = !HI, the Ct2-modules are quaternionic vector spaces and 9Jl2 .; Z is generated by taking the quaternionic dimension. The map i* : 9Jl2 ---+ 9Jl1 is generated by consider-ing a quaternionic vector space to be a complex vector space under restriction of scalars. Clearly this is just the map Z ---+ Z given by mul- §9. K-THEORY AND THE ABS CONSTRUCTION 69 tiplication by 2, since the complex dimension is twice the quaternionic dimension. Thus we have Q2 7L2• The reader may find the verification of these isomorphisms in general to be an interesting exercise. Full details are contained in Atiyah-Bott-Shapiro [1]. We recall now from Proposition 5.21 that the graded tensor product of modules gives a multiplication (8) -+ for all m,n. This makes + 1) == EBn 0 + 1) into a graded ring. On the other hand, by definition we know that K(Dn,sn-1) = K(sn) = K-n(pt) and therefore the direct sum = K-*(pt) is also naturally a graded ring (cf. Corollary 9.18). (The analogous comments apply in the real case.) One of the main results of Atiyah-Bott-Shapiro [1] is the following. Theorem 9.27 (the Atiyah-Bott-Shapiro Isomorphisms). The maps (9.11) and (9.12) defined above induce graded ring isomorphisms: «J*: + 1) K -*(pt) «J*: (9Jl*/i*9Jl*+ 1) KO-*(pt). Since the periodicity of the quotients 9Jl*/i*9Jl* + 1 is an elementary al-gebraic fact, this theorem appears to give a new algebraic proof of the Bott Periodicity theorems. However, the argument given in Atiyah-Bott-Shapiro [1] to establish the isomorphisms of 9.27 actually invokes the Bott result. Nevertheless the existence ofthese isomorphisms is a profound and important fact. It goes a long way towards explaining the fundamental role played by Clifford algebras in the index theory for elliptic operators. REMARK 9.28. Theorem 9.27 gives us explicit generators for K-*(pt) and KO-*(pt) defined via representations of Clifford algebras. For ex-ample, let Se = EB Se be the fundamental 7Lrgraded representation space for Ct2n where Sf = (1 ± wdSe. (The sign of the complex volume element We depends on a choice of orientation in [R2n.) There is an iso-morphism 7L EB 7L with distinguished generators given by Se and its "flip" Se, the same graded module with the factors interchanged. (This corresponds to a reversal of orientation in [R2n.) The generator of + 1 is [Se] + [Se]. Hence, the group K-2n(pt) Kcpt([R2n) 7L is generated by the element Gc.n == where I1x: -+ Se denotes Clifford multiplication by x E [R2n. 70 1. CLlFFORD ALGEBRAS AND SPIN GROUPS The real case is entirely analogous. Let S = S+ Et> S-be the fundamen-tal graded module for ct4n where S± = (1 ± w)S. Then 0"4n == [S+,S-;I1] is a generator of the group KO-4n(pt) KOcpt(lR4n) 7L Using the structure of Clifford modules we can easily compute that O"rc.n = (O"rc.l)", O"Sn == (O"s)n and 40"s = (0"4)2. §IO. KR-Theory and the (I,I)-Periodicity Theorem In this section we present a theory which, in a sense, contains both K-theory and KO-theory. It was invented by Atiyah in the middle 1960s and was motivated in part by the study of indices for families of real operators. (There is a detailed discussion of this at the beginning of §16 in Chapter III.) The higher groups in this theory carry a natural double-indexing KW's, r s O. Interestingly, there is an isomorphism of the Atiyah-Bott-Shapiro type given in §9, which relates Kr.s(pt) with real modules over the Clifford algebra ctr•s. In this sense, KR-theory is the analogue of K-theory and KO-theory suggested by passing from Ctn and ctn to Ctr•s• Basic references for this material are Atiyah [2] and Karoubi [2]. We consider here the category of Real spaces, i.e., spaces with involu-tion. This is the category of pairs (X,cx) where X is compact and Cx: X -+ X is a map with cl = Idx. The map Cx may itself be the identity. Natural examples of such spaces are provided by complexifications of real algebraic varieties where c is given by complex conjugation. Such an example is (IPn(C),c) where in homogeneous coordinates c([z]) = [z]. The fixed point set here is IPn(lR) c IPn(c). DEFINITION 10.1. By a Real vector bundle over a Real space (X,cx) we mean a pair (V,cv) where n: V -+ X is a complex vector bundle over X and where cv: V -+ V is an involution such that the diagram commutes and Cv is C-antilinear on the fibres of V. We denote by VR(X,cx) the abelian semigroup of isomorphism classes of Real bundles over (X,cx). Proceeding as in §9 we then define the associated Grothendieck group KR(X,cx). It is called the Real-K-group of X. We shall generally drop the explicit mention of the involution Cx and simply write KR(X). §10. KR-THEORY AND (1, I)-PERIODICITY 71 REMARK 10.2. Note that if Cx = Idx then we have a natural identifica-tion KO(X) KR(X). This identification associates to any real bundle on X, the pair (8) C, c) where c denotes complex conjugation in the fibres. is recovered as the fixed-point set of c. The groups KR(X) == ker(KR(X) -+ KR(pt)) and KR(X,Y) == KR(X/Y) are defined exactly as in §9 and give functors on the obvious categories of spaces. The higher groups: J(R-i(X) == KR(LiX) and KR-i(X,Y) == KR(I:i(X/Y)), are also defined as in §9. We have the exact sequence for compact pairs (X, Y): ... --KR-i(X,Y) __ KR-i(X) __ KR-i(Y) (10.1) __ KR-i+1(X,Y) --... An interesting facet of KR-theory is that it carries naturally a doubly-indexed family of higher groups. Let fRr,s == fRr EB fR' = "fRr EB ifRs" be the Real linear space with involution c(x,y) = (x, -y). A basic case is \Rl,l C with complex conjugation. Consider now the compact Real subspaces Dr,s == {(x,y) E fRr,s : IIxl12 + IIyl12 1} sr,s == {(x,y) E fRr,s : IIxl12 + IIyl12 = 1}, and for any compact Real pair (X, Y) define Kr,S(X, Y) == KR(X x Dr,s, X x sr,s U Y x Dr,S). (10.2) (The order of (r,s) here is the opposite of that in Atiyah [2].) Note that it is immediate from the definitions that for all i O. The sequence (10.1) generalizes to give exact sequences ... __ KRr,S(X,Y) __ KRr,S(X) __ KR"S(Y) __ KR'-l,s(X,Y) __ ... (10,3) (10.4) for all s O. To prove this one establishes the exact sequence for compact triples in KR-theory above, and then applies it to the triple (X x DO,s, X x So" u Y x DO,s, X x SO,S). One of the basic facts about KR*'* is that the exterior tensor product induces a bigraded multiplication KRr,S(X,Y) (8) KR"'S'(X',Y') --KRr+r';+s'(X",Y") (10.5) where X" = X X X' and Y" = X X Y' u X' X Y. In particular, as in §9, the "coefficients" KR*'*(pt) are a bigraded ring, and for any compact space X, KR*'*(X) is a graded KR*'*(pt)-module. 72 I. CLlFFORD ALGEBRAS AND SPIN GROUPS Consider for a moment the group KR1,I(pt) == KR(D1,I,SI,I) = KR([P>I(C)). One can show that this group is isomorphic to 7L with genera-tor == [H] -Tt where H -+ [P>1(C) is the tautological (or "Hopf") com-plex line bundle over [P>1(C) with its natural Real structure. (The fibre Ht for t E [P>1(C) is the line t itself.) One of the fundamental results in this theory is the following (see Atiyah [2]): The (l,l)-Periodicity Theorem 10.3. Let X be any compact Hausdorffspace. Then the map (10.6) given by module multiplication by is an isomorphism for all r 0, s O. Consequently, for any compact Real pair (X, Y) we have the isomorphism KR"S(X, Y) KR'+ l,s+ I(X, Y) (10.7) Corollary 10.4. There exist natural isomorphisms (10.8) for all r s 0 generalizing (10.3). Using (10.8) we can extend the definition of KR-i to all integers i. The exact sequence (10.1) then extends to infinity in both directions. REMARK 10.5. There are many further internal symmetries in this the-ory. Using the multiplication in the fields IR, C and IHI, Atiyah [2] shows that for any compact space X there are isomorphisms (10.9) for p = 1, 2, and 4. (Recall that dim So,p = p -1.) This isomorphism for p = 1 gives the complex Bott Periodicity Theorem. From the case p = 4 one can deduce the real periodicity theorem. REMARK 10.6. All of the discussion in §9 concerning the functors Ln and their equivalence with K carries over to the Real situation. Therefore, in particular, elements in KR(X, Y) can be represented by classes of the form [VO,vl; 0] where Vo and VI are Real bundles on X and where (J: Vo -+ Viis a Real bundle isomorphism defined over Y. REMARK 10.7. If X is a locally compact space, then the groups KRcpt(X) are defined exactly as in 9.26. This group is generated by Real bundles on X which are trivialized at 00. Using the L-construction, it can also be generated by triples [Vo, VI; (J] where Vo and VI are Real bundles on X and where (J: Vo -+ VI is a Real bundle isomorphism defined outside of some compact subset of X. §10. KR-THEORY AND (1, I)-PERIODICITY 73 In this context the higher KR-groups can be written in a particularly nice form: (10.10) Of course if X is compact, then KR"S(X). If we identify C = \RI,I as before, then the (l,l)-Periodicity Theorem has the particularly nice form: KRcpt(X) KRcpt(X x q for any locally compact space X. (10.11) In light of the Remark 10.6 above it is natural to examine the Atiyah-Bott-Shapiro construction in this theory. To do this we define the Real Clifford algebra ct(lRr.s) to be the Clifford algebra generated by IRr.s and the positive definite quadratic form q(x,y) = IIxl12 + Ily112, taken together with the algebra involution c: ct(lRr.s) -+ ct(lRr.s) generated by the involu-tion (x,y) (x, -y) of IRr.s. DEFINITION 10.8. By a Real module over the algebra ct(lRr.s) we mean a finite-dimensional complex representation space W for ctr.s together with a C-antilinear involution c: W -+ W so that c(cp' w) = c(<p) . c(w) for all cp E ct(lRr., and all WE W. (10.12) If in addition W = WO EB WI is a 1:2-graded module with the property that C(Wi) = Wi for i = 0,1, then W is called a Real1:2-graded module for ct(lRr.s). We denote by 9JlRr.s and rom •. s the Grothendieck groups of equivalence classes of Real modules (and Real 1:2-graded respectively) for ct(lRr.S). There is an important basic relationship between these Real algebras and the algebras Ct •. s defined in §3. To begin, consider the complexifi-cation ct(lRr.s) of ct(IR"S) and note that the involution c has a unique extension to a C-antilinear involution on Ct(IR'·S). Any Real ct(IR"S)-module is naturally a Ct(lRr.S)-module by extension, and the condition (10.12) continues to hold. Considering these algebras and their modules is equivalent to considering those above. Now the main claim is that Ct(lRr.s) is the appropriate complexification of Ct •. s. Under the ordinary complexification, all the algebras ct •. s with the same value of r + s become isomorphic. However, if we also introduce the involution, this is not so. Recall that ctr s is the Clifford algebra generated by IRr EB IRS with the quadratic form qr:s(x,y) = IIxl12 -Ily112. Let 9Jl •. s and IDlr.s denote the Gro-thendieck groups of IR-modules (and 1:2-graded IR-modules respectively) for the Clifford algebra Ct •. s. 74 1. CLlFFORD ALGEBRAS AND SPIN GROUPS Proposition 10.9. There is a natural equivalence Wl"s WlR"s (10.13) for each (r,s) defined by assigning to a real ct"s-module W, the complex module W C with involution given by complex conjugation and with the ct(IR",S)-multiplication engendered by setting (x,y) . w == xw + iyw for (x,y) E IR,' EB IR,s. (10.14) Furthermore, under the natural inclusion i: IR,' EB IR,s 4 IR,'+ 1 EB IR,s given by i(x1, ... ,X"Y1'" . ,Ys) = (Xl" .. ,X"O'Yb' .. ,Ys) the diagram commutes. Hence, there are natural graded isomorphisms: Wl*,*/i*Wl* + 1 ,* WlR*,*/i*WlR* + 1,*' All the analogous statements hold in the ;E2-graded case. (10.15) Proof. The multiplication given in (10.14) has the property: (x,y)· (x,Y)· w = -(llxI12 + Ily112)W and therefore extends to Ct(IR,"'). Furthermore, we see that c[(x,y) . w] = (x + iy)w = (x -iy)w = c(x,y) . c(w) where both (-) and c denote complex conjugation. Hence, the map (10.13) is well defined. The inverse is given by taking the real module to be the fixed-point set of c and replacing multiplication by (x,y) E IR,' EB IR,S with multiplication by x -iy. The remaining details are left as an exercise for the reader. _ REMARK 10.10. As in previous cases the graded tensor product of mo-dules makes 9RR*,* and IDl*,* into graded rings. The multiplication is preserved by the equivalence (10.13) in the graded case. In particular, the map IDl*,*/i*IDl*+ 1,* IDlR*,*/i*ru*+ 1,* is a ring isomorphism. (10.16) The advantage of the groups IDlR"s is that they are natural for extend-ing the Atiyah-Bott-Shapiro construction given in §9. On the other hand, the groups 9Jl"s and the quotients 9Jl"s/i*9Jl,+ l,s are particularly easy to compute by using the results of §§4 and 5. Suppose that W = WO EB W1 is a Real ;E2-graded module for ct(IR,"S), and consider the associated element <p(W) E KRcpt(IR,"S) KR',S(pt) given by §10. KR-THEORY AND (1, i)-PERIODICITY 75 where Ek = IR"S X Wk for k = 0,1, are the trivial product bundles and where J.1: Eo -+ El is defined by J.1(z,w) = (z, z . w) for z = (x,y) E IR"S and for wE Wo. Since z' z' w = -llzI12W, the map J.1 is a bundle isomorphism outside the origin and so [Eo,E 1; J.1] is an ele-ment of KRcpt(IR"S) (cf. Remark 10.7). This gives a graded ring homomor-phism (10.17) Arguing as in §9 one can show that qJ = 0 on i*ru*+ 1.* and so qJ des-cends to the quotient. The arguments of Atiyah-Bott-Shapiro [I] then carry through to prove the following (see Atiyah [2]). Theorem 10.11. The map defined above, qJ:1mR*.*/i*IDlR*+l.* KR*'*(pt), is a graded ring isomorphism. Via the isomorphism (10.16) this relates the groups KR'·S(pt) to the alge-bras ct,.s. In particular, the (I,I)-Periodicity Theorem is reflected in the (I,I)-periodicity for these algebras (cf. (4.3) of Theorem 4.1). Furthermore, since KRs.o(pt) = KR -'(pt) = KO -S(pt) and cts ° cts we recapture here the real ABS-isomorphism of Theorem 9.27. From (l,I)-periodicity we see that KRO,O(pt) KR1,1(pt) KR2,2(pt) ... 71.. For our later discussion ofthe Ct.-linear Atiyah-Singer operator (in 111.16) it will be useful to have certain explicit generators for these groups, which we shall now present. Recall that as a Real space IRn,. is just cn with involution given by complex conjugation, and therefore we can write KRn'"(pt) KRcpt(C"). We will give an explicit generator for KRcpt(C") by using the Clifford algebra CCn = CC(Cn) and its natural 7L2-grading CC. = (£) For qJ E Ctn, let R",: Ctn -+ Ctn and L",: Ctn -+ Ct. denote right and left multiplication by qJ respectively. For x + iy E en, we consider the map (Rx + (10.18) which, since (Rx + iLy)(Rx -iLy) = -(llxI12 + IlyI12)Id, is invertible when x + iy of. O. This map (10.18) is clearly a Real endomorphism, i.e., (Rx + iLy)qJ = (Rx + iL_y)q5. Proposition 10.12. The element en = Rx + iLy] E KRcpt(C") is a generator of the group KRcpt(Cn) = KR"'"(pt) 71. for all n 1. 76 1. CLlFFORD ALGEBRAS AND SPIN GROUPS Proof. Because of the isomorphism of graded algebras I[tn = I[t! ® ... ® I[t1 (which follows from Proposition 3.2) it suffices to con-sider the case n = 1. The element e1 can be shown to coincide with the generator of the image of ({J in Theorem 10.11 as follows. To begin note that since ct? = I[ . 1, right and left multiplication coincide on ct? Consequently, e1 [1[,1[; m] where m(x.y): I[ --+ I[ is given by scalar multi-plication: m(X,y)(w) = (x + iy)w. _ ' On the other hand we have WlR1,1 7L EB 7L with generators given as follows. For (x,y) E [Rl,l, define a I[-linear map I[ EB I[ --+ I[ EB I[ by the matrix (10.19) At (x,y), the square is -(llxI12 + IlyI12)Id. Hence, this action extends to make I[ EB I[ (with involution given by complex conjugation) into a com-plex Z2-graded Ct([Rl,1 )-module. This is the first generator. The second generator is obtained by interchanging factors. That these are indepen-dent generators is easily verified by working through the natural iso-morphisms WlR1•1 Wl1,1 WlO,I (cf. Proposition 10.9 and Theorem 3.7). These isomorphisms commute with i*, and working out the details for the simple case Wlo,di*Wl1,1 shows that either of our distinguished generators becomes a generator of the quotient: WlR1,di*WlR2,1 KR1,I(pt). Clearly under the isomorphism ({J of Theorem 10.11 our module above with multiplication 10.18 becomes the element [1[,1[; m] as claimed. CHAPTER II Spin Geometry and the Dirae Operators In this chapter one finds the soul of the book. It is here that we examine the structures and concepts that are central to differential manifolds, vector bundles, connections, curvature, etc. If one also intro-duces metrics (in the sense of Riemann), then it is unavoidable that Clifford algebras and spin groups will enter the discussion. This is for the following reason. There is a principle by which the natural operations on vector spaces such as direct sum, tensor product, exterior power, etc., carry over canoni-cally to vector bundles. In the same fashion, the natural operations for vector spaces with quadratic forms carry over to vector bundles with metrics. In particular, suppose that n: E -+ X is a riemannian vector bundle. Then in each fibre, Ex = n-1(x), the quadratic form IIvl12 = (v, v) can be used to construct a Clifford algebra Ct(Ex). The result is a bundle Ct(E) -+ X of algebras over X called the Clifford bundle of E. It carries all the natural properties of Clifford algebras such as the Z2-grading, the transpose endomorphism and the L-operator. This rich structure is basic for the study of E itself. In the light of Chapter I it is natural to ask whether one can also find a vector bundle $(E) -+ X with the property that each fibre $(Ex) is an irreducible module over Ct(Ex). The answer is in general no. We shall discuss the obstruction involved. By taking E = TX this will lead to the notion of a spin manifold and spin cobordism. The bundle $(TX), when it exists, plays a central role in the study of the global geometry and to-pology of X. In general, given any bundle of modules S over Ct(TX), furnished with a suitable metric and connection, one can associate to S a self-adjoint, first-order elliptic operator D: r(S) -+ r(S) called the Dirac operator for s. (This is done in §5.) From the algebraic operations of Chapter lone can often construct natural splittings S = S+ Et> S-with respect to which the Dirac operator has the form D = (:+ DO-) where D+ : r(S+) -+ qs-) and D-: r(S-) -+ r(S+) are formal adjoints of one another. This gives a unified procedure for constructing all the "clas- 78 H. SPIN GEOMETRY AND DIRAC OPERATORS sical" elliptic operators on a manifold, namely: the Euler characteristic operator, the signature operator, and the Atiyah-Singer operator. Details of this are given in §6. In §7 we introduce the notion of a Ctk-linear operator and discuss in detail some of the basic examples. Over a compact manifold any ctk-linear Dirac operator D has an analytic index indk(D) E KO-k(pt) defined by applying the Atiyah-Bott-Shapiro isomorphism to the residue class of the Clifford module [ker DJ (see 1.9). Roughly speaking, every Dirac operator represents the square root of a Laplace operator. In euclidean space this statement is unambiguously true. Over general manifolds, however, the difference D2 -V*V between the square of the Dirac operator and the standard "connection laplacian" is a certain universal expression involving curvature and Clifford multi-plication. Deriving such formulas and using them to draw global con-clusions about curvature and topology is referred to as "Bochner's method." We shall show how our universal formula specializes to give the classical Bochner formulas on exterior differential forms, as well as the Lichnerowicz formula for spinors. Using curvature identities derived in §5, we shall then give a succinct proof of the theorem of Gallot and Meyer concerning curvature and homology spheres. We shall also derive formulas for the Atiyah-Singer operator with coefficients in a bundle. This will be quite useful later in studying manifolds of positive scalar curvature. §1. Spin Structures on Vector Bundles Let n: E --+ X be a real n-dimensional vector bundle over a manifold X. We assume this bundle is equipped with a riemannian structure, that is, a positive definite inner product continuously defined in the fibres. Such a structure always exists. We assume also that the bundle is oriented, i.e., that there is an orienta-tion continuously defined on the fibres. This structure does not always exist. To analyze the situation, we consider the bundle P o(E) of ortho-normal frames in E. This is the principal On-bundle whose fibre at a point x E X is the set of orthonormal bases of Ex == n-1(x). The bundle of orien-tations in E is then just the quotient Or(E) = P o(E)/SOn' where two bases of Ex are identified if the orthogonal matrix transforming one to the other has determinant + 1. Note that Or(E) is a 2-sheeted covering space of X and that E is orientable if and only if this covering space is the trivial one. We recall now the following elementary fact: Lemma 1.1. Let Cov 2(X) be the set of equivalence classes of 2-sheeted covering spaces of X. Then there is a natural isomorphism (1.1) §1. SPIN STRUCTURES ON VECTOR BUNDLES 79 Note. This is a special case of the isomorphism: Hl(X;G) {equivalence classes of principal G-bundles on X}, which is proved in Appendix A. Proof. Assuming X is connected, we can decompose the isomorphism (1.1) as follows: =. COV2(X), The second isomorphism follows from the fact that H l(X) is the abelianization of 1t1(X). The third isomorphism is a restatement of the elementary fact that 2-sheeted coverings of X are in one-to-one correspondence with subgroups of index 2 in 1t1(X). The case where X is not connected now follows immediately. • From (1.1) we now see that, for each vector bundle E over X, the 2-sheeted covering space Or(E) determines an element w1(E) E Hl(X; called the first Stiefel-Whitney class of E. Directly from the definitions we have the following fact. Theorem 1.2. A vector bundle E over X is orientable ifand only ifw1(E) = O. Furthermore if w1(E) = 0, then the distinct orientations on E are in one-to-one correspondence with elements of HO(X; The second statement simply says that there are two possible orienta-tions of E over each connected component of X. The definition of wl(E) given above is in accord with the one given via classifying spaces (see App. B). To prove this it suffices to show the following: (i) This definition of Wl is natural, i.e., wl(f*E) = f*w1(E) for any bundle E over X and any continuous map f : X' -+ X. (ii) This definition of Wl(lE) gives the non-zero element in H1(BOn; when lE is the universal n-plane bundle over the classifying space BOn• Fact (i) is obvious since Po(f*E) = f*Po(E) and, therefore, Or(f*E) = f*Or(E). Fact (ii) is true since otherwise every n-plane bundle would be orientable. By establishing properties (i) and (ii) it is possible to prove the equiv-alence of a number of quite different definitions of wl(E). For example, suppose X is connected. Then from the fibration On -+ P o(E) -+ X, there is an exact sequence. o ------(1.2) We can define wl(E) = WE(gl) where gl is the generator of HO(On; This definition of W1 (E) has property (i) since the sequence (1.2) is natural and property (ii) since the sequence (1.2) is exact and for the universal 80 n. SPIN GEOMETRY AND DIRAC OPERATORS bundle lE, P o(lE) = EO. is contractible. From the exactness of (1.2) it is clear that wl(E) = 0 iff Po(E) is disconnected, i.e., iff Eis orientable. We shall examine other equivalent definitions of W1 later in this section. At the moment we pass on to the next possible simplification of the structure group of a bundle E. Note that if Eis orientable, then choosing an orientation is equivalent to choosing a principal SO.-bundle Pso(E) c Po(E). This embedding is, of course, compatible with the action of SO. c a •. Having thereby made the structure group of EO-connected, one might ask whether it is possible to make the structure group I-connected. This leads us to the concept of a spin structure. Let E be an oriented n-dimensional riemannian vector bundle over a manifold X, and let Pso(E) be its bundle of oriented orthonormal frames. Recall that for n 3 we have the universal covering homomorphism Spin. -+ SO. with kernel {l, -I} 71.2, DEFINITION 1.3. Suppose n 3. Then a spin structure on E is a principal Spin.-bundle PSPin(E) together with a 2-sheeted covering : PsPin(E) --Pso(E) such that = for all p E PSpin(E) and all g E Spin •. When n = 2, a spin structure on E is defined analogously, with Spin. replaced by S02 and S02 -+ S02 the connected 2-fold covering. When n = 1, Pso(E) X and a spin structure is simply defined to be a 2-fold covering of X. Note that the diagram (1.3) where nand n' are the bundle projections, is commutative. Note also that restricted to the fibres corresponds to the covering The diagram of fibrations is On the other hand, suppose PsPin(E) -+ Pso(E) is a 2-sheeted covering §1. SPIN STRUCTURES ON VECTOR BUNDLES which is non-trivial on the fibres of X, i.e., so that the diagram 81 commutes. Then setting n' = n 0 makes PSpin(E) a fibre bundle over X. To make this a principal Spinn-bundle we must lift the action of SOn on Pso(E) to a compatible action of Spinn on Pspin(E). The proof that this lifting exists is a straightforward application of elementary covering space theory. We conclude the following (even for n = 1 and 2). Theorem 1.4. The spin structures on E are in natural one-to-one corre-spondence with 2-sheeted coverings of Pso(E) which are non-trivial on the fibres of n. This can be reinterpreted via Lemma 1.1 as follows: Corollary 1.5. Suppose X is connected. Then the spin structures on E are in natural one-to-one correspondence with elements of Hl(Pso(E); £:''1) whose restriction to the fibre of Pso(E) is non-zero. We are now in a position to discuss the question of existence and uniqueness of spin structures. Associated to the fibration SOn Pso(E) 4 X there is an exact sequence 0--Hl(X; £:'2) H1(Pso(E); £:'2) Hl(SOn; £:'2) H2(X; £:'2) (1.4) which can be deduced from the Serre spectral sequence. In analogy with the definition ofwl via the sequence (1.2) we make the following definition: DEFINITION 1.6. The image w2(E) = WE(g2) E H2(X; £:'2) of the generator g2 of Hl(SOn; £:'2) £:'2 is the second Stierel-Whitney class of the oriented bundle E. To prove that this (or any other) definition agrees with the one given via classifying spaces, it again suffices to show (cf. (i) and (ii) above): (i') This definition of W2 is natural. (ii') This definition of wilE) gives the non-zero element in H2(BSOn; £:'2) £:'2 when lE is the universal oriented n-plane bundle over the classifying space BSOn. Property (i') follows from the naturality of the sequence (1.4). Property (ii') follows from the exactness of (1.4) and the fact that Pso(lE) = ESOn is contractible. 82 H. SPIN GEOMETRY AND DIRAC OPERATORS From Corollary 1.5 and the exactness of the sequence (1.4) we im-mediately conclude the following. Theorem 1.7. Let E be an oriented vector bundle over a manifold X. Then there exists a spin structure on E if and only if the second Stiefel-Whitney class of E is zero. Furthermore, if w2(E) = 0, then the distinct spin structures on E are in one-to-one correspondence with the elements of Hl(X; 1'2)' Note that this result holds for all n. When n = 2, the class W2 is just the mod 2 reduction of the Euler class. More generally, the following is true. REMARK 1.8 (cf. Milnor-Stasheff [1 J). If E is the real 2n-dimensional bundle underlying a complex n-dimensional vector bundle E, then w2(E) == cl(E)(mod 2) where Cl (E) is the first Chern class of E. (To see this, it suffices to observe that the map H2(BS02n; 1'2) --+ H2(BUn; 1'2), induced by the inclusion Un C S02n, is an isomorphism.) REMARK 1.9. Note that the spin structure on a bundle E is independent of the bundle metric on E in the following sense. A spin structure on E uniquely determines a spin structure for any other metric. This follows from Theorem 1.4 and the observation that the inclusion Pso(E) c PGL+(E), where PGL+(E) is the bundle of all oriented bases in E, is a homotopy equivalence. REMARK 1.10. To choose an orientation and a spin structure for E is in particular to find structure groups for E which are respectively 0 and I-connected. Conversely, suppose E is equivalent to a vector bundle with a Lie structure group G. If G is connected, then Eis orientable; and if G is simply-connected, then E is spin. It should be noted that the process of finding successively more highly connected structure groups for E terminates at the spin level. This is because for any simply-connected Lie group G, 1t2(G) = 0, and if 1t3(G) = 0, then G is contractible. The conditions Wl = 0 and W2 = 0 can be interpreted geometrically as follows: Proposition 1.11. Let E be a vector bundle over a manifold X. Then E is orientable if and only if the restriction of E to any circle embedded in X is trivial. The proof is obvious. There is an analogue for W2• §1. SPIN STRUCTURES ON VECTOR BUNDLES 83 Proposition 1.12. Let E be an oriented vector bundle oJ dimension 3 over X. Then E is spin if and only if Jor any compact surJace L and any continuous map J : L X, the bundle J* E is trivial. Suppose, Jurthermore, that X is simply-connected and oJ dimension> 4. Then E is spin if and only if the restriction oJ E to any 2-sphere embedded in X is trivial. ProoJ. The group H iX;£:'2) is generated by maps J : L X of compact surfaces. Hence, w2(E) = 0 -J*w2(E) = W2(J* E) = 0 for all such J -1* E is trivial for all such J (since an oriented bundle of dimension 3 over a surface is trivial if and only if W2 = 0). This proves the first state-ment. For the second statement it suffices to note that when nIX = 0 and dim(X) > 4, the group H2(X;£:'2) is generated by embedded 2-spheres. • REMARK 1.13. This second statement can be refined somewhat. If nIX = 0, then H 2(X;£:'2) is generated by immersions of S2, and if dim X > 4, the immersions can be deformed to embeddings. We now consider some alternative definitions of the classes WI and W2. Recall that (cf. App. A) for any topological group G, we can view the equivalence classes of principal G-bundles on X as elements in a "Cech cohomology space" HI(X;G). As noted in Milnor [7J the short exact sequence of topological groups 1 -----+ SOn On £:'2 -----+ 0 gives an exact seq uence HI(X; SOn) HI(X; On) HI(X; £:'2). (1.5) (1.6) Given an n-dimensional bundle E on X, we define the first Stiefel-Whitney class by setting wl(E) = p*([Po(E)]). This definition is easily seen to have properties (i) and (ii) and it therefore agrees with our previous definitions. In a similar way the short exact sequence 0-----+ £:'2 -----+ Spinn SOn -----+ 1 (1.7) gives an exact sequence HO(X; SOn) HI(X; £:'2) -----+ HI(X; Spinn) (1.8) HI(X; SOn) H2(X; £:'2) (cf. Hirzebruch [lJ). Thus, for an oriented bundle E we define the second Stiefel-Whitney class by setting w2(E) = b([Pso(E)J). This definition has properties (i') and (ii') and therefore agrees with our previous ones. With this definition it is transparent that w2(E) = 0 if and only if Pso(E) is equivalent to the £:'rquotient of a principal Spinn-bundle on X. 84 H. SPIN GEOMETRY AND DIRAC OPERATORS REMARK 1.14. This is a convenient time to inject a minor word of cau-tion. It was pointed out by Milnor that the Spinn-bundles associated to distinct spin structures on E may be equivalent as abstract principal bundles. Recall that spin-structures on E are in one-to-one correspondence with elements of H1(X;1'2)' However, by (1.8) we see that the equivalence classes of principal Spinn-bundles with 1'2-quotient equal to [Pso(E)] are in one-to-one correspondence with elements of H1(X;1'2)/(jO(HO(X;SOn))' Now by definition we have that HO(X;SOn) = C(X,SOn)' the space of continuous maps from X to SOn. The map (j0: C(X,SOn) H1(X;1'2) (1.9) is given as follows. For a map f: X --+ san, set (jo(f) = f*(l) where 1 is the generator of H1(SOn;1'2)' The map (1.9) is often surjective. For example, any class which is the mod 2 reduction of an integral class lies in the image. To see this let fo: X --+ S1 represent the integral class and define f = i 0 fo: X --+ SOn where i: S1 '-+ SOn is not homotopic to zero. Thus, if H1(X;1'2) H1(X;1') ® 1'2' then (j0 is surjective. Similarly if dim X < n, then (j0 is also surjective. To see this note that every class in H1(X;Z2) can be induced by a map to IPn(lR) C SOn. Thus, in either of these cases, all the principal Spinn-bundles associated to distinct spin structures on E are abstractly equivalent. As an example of this phenomenon consider an oriented bundle E over the circle SI. Since H1(SI;Z2) Z2, there are two spin structures on E. However, any principal Spinn-bundle over S1 is equivalent to the trivial one since it admits a cross-section (as does any fibre bundle over SI with connected fibre). Finally we mention a direct definition of Wl and W2 via homotopy theory. From the sequence (1.4) there is a fibration BSOn BOn B1'2 = K(1'2,l)· A map fE: X --+ BOn (classifying a bundle E) has a lifting to BSOn iff w 0 fE is homotopic to zero. Recall that [X,K(Z2,1)] Hl(X;Z2). The class wo fE is the first Stiefel-Whitney class w[(E). From the sequence (1.6) we have a fibration K(1'2,1) --+ BSpinn BSOn. It follows that the cofiber of is K(Z 2,2), i.e., there is a fibration BSpinn BSOn K(Z2,2). We can now define w2(E) as we defined wl(E) above. We complete this section with an observation concerning Whitney sums. Proposition 1.15. Given three vector bundles E', E" and E E' EB E" over a manifold X, a choice of orientation on any two of them uniquely determines §2. SPIN MANIFOLDS AND SPIN COBORDISM 85 an orientation on the third. Similarly, a choice of spin-structure on any two of them uniquely determines a spin structure on the third. Proof. The statement concerning orientations is obvious since for finite dimensional vector spaces V = V' EB V", an orientation on any two canon-ically determines an orientation on the third. Suppose now that E, E' and E" are orientable. Then w2(E) = w2(E') + w2(E"). Hence, if anv two are spin, so is the third. Suppose under the correspondence of Corollary 1.5 a' E H1(Pso(E'); £:'2) and a" E H1(Pso(E"); £:'2) represent spin structures on E' and E" respec-tively. Now consider the cartesian product bundle E' x E" --+ X x X. We let A: Pso(E' EB E") --Pso(E' x E") denote the diagonal map. The class in H1(Pso(E' EB E"); £:'2) which will represent the spin structure determined by a' and a" will be A*b where bE H1(Pso(E' x E"); £:'2) will be the unique class which extends a' x 1 + 1 x a" E H1(Pso(E') x Pso(E"); £:'2) under the inclusion Pso(E') x P so(E") c Pso(E' x E"). That there is a unique b can be seen from the following diagram: That any two of the classes a', a" and A*b determines the third is now easy to see. Let TCv: Pso(V) --+ X be the projection map where V = E, E' or E". Then any spin structure 1" (under the correspondence of Corollary 1.5) on E' may be written a' + TCt,U' and any spin structure 1''' on E" may be written a" + TCt"U" for u',u" E Hl(X;£:'2)' Then following the above pre-scription 1" and 1''' determine A*b + nlG1E"(u' + u"). Clearly any two of these determine the third. • §2. Spin Manifolds and Spin Cobordism We are now in a position to discuss the notion of spin structures on mani-folds. For convenience all manifolds are assumed to be of class COO. DEFINITION. A spin manifold is an oriented riemannian manifold with a spin structure on its tangent bundle. The Stiefel-Whitney classes wi(X) of a manifold X are defined to be the Stiefel-Whitney classes of its tangent bundle TX. Hence, by Theorem 1.7 we have the following. 86 n. SPIN GEOMETRY AND DIRAC OPERATORS Theorem 2.1. An oriented riemannian manifold X admits a spin structure if and only ifits second Stiefel-Whitney class is zero. Furthermore, ifw2(X) = 0, then the spin structures on X are in one-to-one correspondence with elements of Recall (cf. Remark 1.9) that the choice of spin structure for one rieman-nian metric on X canonically determines a spin structure for any other riemannian metric on X. Here we are using the fact that, for any metric, the inclusion Pso(X) c PGu(X) of the bundle of oriented orthonormal tangent frames into the bundle of all oriented tangent frames, is a homo-topy equivalence. Consider now a diffeomorphism f: X .; X of a spin manifold X. If f is orientation preserving, then there is an induced diffeomorphism df: P GU(X) P GU(X) of the bundle of oriented tangent frames. This map carries fibres to fibres and therefore induces a permutation of the possible spin structures on X (considered as 2-fold coverings of PGL+(X),) If the given spin structure remains fixed, then f is called a spin structure-preserving dilfeomorphism. It is a classical result of W.-T. Wu [1] that the Stiefel-Whitney classes of a compact manifold X depend only of the homotopy type of X. There-fore, the property of having or not having a spin structure is a homotopy invariant, and in particular, it remains the same under changes of the dif-ferentiable structure on X. Wu proves his result by giving a homotopy-theoretic formula for the total Stiefel-Whitney class w = 1 + Wl + W2 + ... of X. Define v = 1 + VI + V2 + ... E by the requirement that (v u a)[X] = Sq(a)[X] for all a E where Sq = 1 + Sq 1 + Sq2 + .. , is the Steenrod square automorphism, and where [X] E n = dim(X), denotes the fundamental class. Wu's formula states that w = Sq(v) (see Milnor-Stasheff [1] for details). Since Sqi(a) = 0 if i > deg(a), we see that Vk = 0 for k > [dim(X)/2]. Thus, for example, if dim(X) = 3, we see that v = 1 + VI = 1 + w1, and it follows that W2 = wi. In particular, if X is orientable, then it is automatically spin. Suppose now that dim(X) = 4 and W1 = O. Then v = 1 + V2 = 1 + W2' and we see that W2 is characterized by the fact that (W2 u a)[X] = (a u a)[X] for all a E We will now examine some examples of spin manifolds. For convenience we shall use the expression X is spin to mean that w1(X) = W2(X) = 0, (i.e., that X carries at least one spin structure for any riemannian metric on X). Recall that any complex manifold is canonically oriented. Further-more, by Remark 1.8 we know the following: §2. SPIN MANIFOLDS AND SPIN COBORDISM 87 REMARK 2.2. A complex manifold X is spin if and only if its first Chern satisfies c1(X) == 0 (mod 2). EXAMPLE 2.3 (the trivial examples). It follows immediately from Theo-rem 2.1 that any 2-connected manifold carries a unique spin structure. The obvious examples of this type include homotopy spheres, Stiefel mani-folds, and simply-connected Lie groups. Of course, any manifold whose tangent bundle is stably parallelizable is spin. This includes, for example, the inverse image of any regular value of a smooth map f: IRn+p -+ IRP. It also includes any Lie group and any orientable manifold of dimension 3. EXAMPLE 2.4. Let IJ1>n(k) denote the n-dimensional projective space over the (skew) field k. Then IJ1>n(lR) is spin iff n == 3 (mod 4) IJ1>n(C) is spin iff n is odd IJ1>n(!HI) is spin for all n. To see this recall that the total Stiefel-Whitney class of IJ1>n(K) is w = 1 + W1 + W2 + ... = (1 + gr+ 1 where g, the generator of the cohomology ring, has dimension 1, 2 and 4 for K = IR, C and !HI respectively. When K = IR, the conditions W1 = 0, W2 = 0 are equivalent to (n + 1) == n(n + 1)/ 2 == 0 (mod 2). The cases K = C and K = !HI are obvious. EXAMPLE 2.5. The manifold san has two distinct spin structures given as follows: P(SOn) = san x SON where N = n(n -1)/2. The two coverings are: i\(SOn) = san x SpinN p 2(SOn) = (Spinn x SpinN)/£:'2 where £:'2 acts on Spinn x SpinN by the map (g, h) f-+ ( -g, -h). EXAMPLE 2.6. Let be a compact Riemann surface of genus g. As observed in 2.4 above, this surface is spin. (The Euler characteristic is even.) There are exactly 22g distinct spin structures on X which can be constructed as follows. Let denote the equivalence classes of holomorphic complex line bundles on This is an abelian group under the operation of tensor product. There is a short exact sequence o __ J __ --0 where J T2g. Let '0 E denote the tan-gent bundle of and note that there exist exactly 22g elements ,E such that ,2 = '0' Recall that for any complex line bundle, the natural map, -+ ,2 is of the form z -+ Z2 in the fibres. Hence, each 88 n. SPIN GEOMETRY AND DIRAC OPERATORS bundle t with t2 = to determines a 2-fold covering of Pso(Lg) C to which is non-trivial on the fibres. These coverings realize all the distinct spin structures on Lg• This construction is a special case of the general fact (cf. Hitchin [1] and App. D) that for any compact Kiihler manifold X with c1(X) == 0 (mod 2), the spin structures on X are in one-to-one correspondence with holomorphic square roots of the canonical bundle of X. EXAMPLE 2.7. Let vn(d) denote the non-singular complex hypersurface of degree d in IPn+ 1(C). That is, Vn(d) is given in homogeneous coordinates [Zo, ... ,Zn+ 1] for IPn+ 1(C) as the zeros of a homogeneous polynomial p(Zo, ... ,Zn+ 1) of degree d which satisfies the condition (Vp)(Z) i= 0 when Z i= 0 and p(Z) = O. The diffeomorphism class of vn(d) is uniquely deter-mined by the integers nand d. The first Chern class of Vn(d) for n > 1 is Cl = (n + 2 -d) . g where g is the canonical generator of H2(vn(d); £:') (the Kiihler form induced from IPn+1(C)). It follows that Vn(d) is spin n + d is even. EXAMPLE 2.8. Let Vn(db ... ,dk) be the transverse intersection of hyper-surfaces Vn+k=1(d1), ... ,Vn+k+1(dk) in IPn+k(C). Then for n > k Cl = (n + k + 1 -d1 -••• -dk)· g where g is the canonical generator of H2(vn(d1, ... ,dk); Z) £:'. Hence, k n + k + 1 + I di is even. i= 1 EXAMPLE 2.9 (N. Hitchin [1]). Let f: M -+ 1P2(C) be a p-fold ramified cover, branched over a non-singular curve of degree pq. Then M is spin iff p is even and q is odd. As we observed above, every oriented manifold of dimension 3 is spin. In higher dimensions the spin condition has a nice geometric interpreta-tion. We shall restrict our attention to manifolds X with the property that the homomorphism H2(X;£:') -+ H2(X;£:'2)' given by reduction mod 2, is surjective. This holds whenever X is simply-connected. Theorem 2.10. Let X be an oriented n-dimensional manifold as above. If n 5, then X is spin if and only if every compact orientable surface embedded in X has trivial normal bundle. If n = 4, then X is spin if and only if the normal bundle to every compact orientable surface embedded in X has even Euler class. §2. SPIN MANIFOLDS AND SPIN COBORDISM 89 Proof. Since H 2(X;£,2) = H iX;£') ® £'2' the group H 2(X;£'2) is generated by maps of compact orientable surfaces. By the classical theorems of Whitney, every map of a surface into X is homotopic to an embedding if n 5, and to a self-transversal immersion if n = 4. In this latter case, we can remove small disks at each self-intesection point and attach an embedded handle, thereby producing an embedded surface in the same homology class. Consequently, H 2(X;£'2) is generated by smooth embedd-ings of compact orient able surfaces. Let i: 4 X be such an embedding. Then i*w2(X) = i*w2(TX) = w2(i*TX) = EB = + = Evaluating on the fundamental class, we have (w2(X), = (i*w2(X), = Consequently, if w2(X) = 0, then = O. On the other hand, H 2(X;£,2) is generated by such surfaces. We conclude that w2(X) = 0 if and only if = 0 for every compact orientable surface embedded in X. The normal bundle to is orientable. Therefore, when 3 (i.e., when n 5), we have = 0 if and only if is trivial. When = 2 (Le., when n = 4), we have == (mod 2). This completes the proof. • Corollary 2.11. Let X be a simply-connected manifold of dimension 5. Then X is spin if and only if every 2-sphere embedded in X has trivial normal bundle. Proof. Since H 2(X;£') 1!2(X) by the Hurewicz Theorem, we have that HiX;£'2) is generated by embedded 2-spheres. • Corollary 2.12. Let X be a compact, simply-connected 4-manifold. Then X is spin if and only if (y u y)[X] == 0 (mod 2) for each yE H2(X;£') (i.e., the intersection form is "even"). Proof. Let yE H2(X;£,) [X,IPOO(IC)] be given by a smooth map f: X -+ jpl3(C). Make f transversal to a hyperplane 1P2(1C) c 1P3(1C) and let = r 1(1fl>2(IC)). Then represents the Poincare dual of y, and (y u y)[ x] is the self-intersection number of Le., the Euler number of • From the classification of quadratic forms (e.g., Husemoller-Milnor [1]) we conclude that the signature of X must be a multiple of 8. This result holds, in fact, for any topological 4-manifold with Wl = W2 = O. For smooth manifolds there is the following deeper result (see Chap. IV). Theorem 2.13 (Rochlin). The signature of a smooth compact spin 4-manifold is a multiple of 16. 90 11. SPIN GEOMETRY AND DIRAC OPERATORS EXAMPLE 2.14. The complex hypersurface V2(4) = {(ZO,Zl,Z2,Z3):Zri + zt + + Zj = O} C [p>3(C) is a spin manifold with signature 16 (see Example 2.7 above). This is the so-called Kummer (or K3) surface. REMARK. The statement of Corollary 2.12 fails when 1!l(X) =f. O. How-ever, the following is true. For a compact orientable 4-manifold X, the intersection form is even if and only if w2(X) is the mod 2 reduction of a torsion class in H2(X;Z). This was proved by N. Habegger [1] who also showed the following. Let X = V2(4)jZ2 be the quotient of the Kummer surface by the involution (Zo, Zl,Z2,Z3) --+ (Zl,-ZO,Z3,-Z2). Then X is orient able and has even intersection form; however, sig(X) = 8. Of course, X cannot be a spin manifold. In fact, w2(X) is the mod 2 reduction of the unique torsion class in H2(X;Z). The remainder of this section will be devoted to a discussion of spin cobordism. We begin with two observations which follow immediately from Proposition 1.15. Proposition 2.15. The cartesian product of two spin manifolds is canonically a spin manifold. Any submanifold of a spin manifold with a spin structure on its normal bundle is canonically a spin manifold. In particular, if Y is a compact manifold with boundary a Y, then any spin structure on Y induces a spin structure on ay as follows. Let v be the field of interior unit normal vectors along ay. Using v, one obtains an embedding pso(ay) c Pso(Y), by completing each tangent frame to a Y with the given normal vector. The spin structure, considered for example as a 2-sheeted covering, can now be restricted to pso(ay). DEFINITION 2.16. Two spin manifolds are said to be dilferentiably equiv-alent if there is a diffeomorphism between them preserving orientations and spin structures. A compact (not necessarily connected) spin manifold is said to be spin cobordant to zero if it is differentiably equivalent to the boundary of a compact spin manifold Y with (orientation and) spin struc-ture induced from Y as above. Let denote the free abelian group generated by equivalence classes of compact connected n-dimensional spin manifolds, modulo the subgroup generated by elements [X 1] + ... + [X k] where XlII ... II X k is spin co-bordant to zero. is called the n-dimensional spin cobordism group. From Proposition 2.15 we know that the product of two spin manifolds has a uniquely determined spin structure. This multiplication makes nstn = EB,.""=0 into a graded ring, called the spin cobordism ring. Note that the equivalence class (and therefore also the cobordism class) of a spin manifold X is independent of the choice of riemannian metric onX. §2. SPIN MANIFOLDS AND SPIN COBORDISM 91 REMARK 2.17. Given two spin n-manifolds Xl and X2, we can form eir connected sum Xl # X 2 and equip it with a spin structure so that 1 # X 2 and XlII X 2 are spin cobordant. Thus every spin cobordism ass is represented by a connected manifold (see Milnor [7]). The connected sum operation is a special case of the general procedure doing surgery on spin manifolds (see Milnor [5] and Kervaire-Milnor ]). In particular one can show that for n 3 every spin cobordism class represented by a simply-connected manifold. For n 5, every spin bordism class is represented by a 2-connected manifold (see Corollary 11). Thus by Poincare duality and the h-cobordism theorem = o. For the special case n = 1, a spin structure is defined to be a 2-fold vering of PSO(Sl) = Sl. Consider Sl as the boundary of the 2-disk with s unique spin structure. Then the spin structure induced on Sl is the nnected 2-fold covering. Interestingly, the disconnected 2-fold covering not cobordant to zero, although two copies of it clearly is (a good xercise). Thus, &'.2. It is also true that the square of the circle with the bad spin structure not zero in &'.2 (another good exercise). We now briefly review the Thorn construction. Let xn be a compact -dimensional spin manifold and choose a smooth embedding xn -4 sn+k, > n. The spin structures on xn and sn+k determine a unique spin struc-ure on the normal bundle N(xn) of xn (see Proposition 1.15). Hence, there s a bundle map N(xn) -L IEk 1 1 xn BSpink lassifying N(xn), where IEk is the universal k-plane bundle. The map j escends to a map of Thorn spaces j: -r(N(xn» -4 -r(lEk) == MSpink. If we identify N(xn) with a tubular neighborhood of xn in sn+k, then -r(N(xn» is the space obtained by collapsing the complement of this tubular neigh-borhood to a point. Thus we get a map 1!: sn+k -4 r(N(xn», and the com-position j . re: sn+k -4 MSpink determines an element (xn) E == lim 1!n+k(MSpink). k-+ 00 The classic result of Thorn states that this map induces an isomorphism We observe now that there is a natural ring homomorphism p* : O!"in __ O!O (2.1) (2.2) 92 11. SPIN GEOMETRY AND DIRAC OPERATORS where O!O is the oriented cobordism ring. In fact, in low dimensions we have the following (cf. Milnor-Stasheff [1; §17] and Milnor [7]): n n:pin nso . 0 71. 71. 1 71.2 0 2 71.2 0 3 0 0 4 71. (generated by the Kummer surface 71. (generated by [P>2(C)) surface V2(4) in Example 2.14) 5 0 71.2 6 0 0 7 0 0 8 71. Efl 71. (generated by [P>2(1HJ) and a 71. Efl 71. (generated by [P>4(C) manifold L8 such that 4L8 is spin and [P>2(C) x [P>2(C)) cobordant to V2(4) x V2(4)) Of course the non-zero element x E given by the circle with the "bad" spin structure, goes to zero in Hence, we have x . nstn c ker p*. In fact, it can be proved that X· O!Pin = ker p* (see Anderson-Brown-Peters on [1]). The co kernel of p* is not so simple to describe. For example, the signa-ture gives an isomorphism 7L, and therefore using Theorem 2.13 and Example 2.14 we get a short exact sequence o ----. ----. 7L16 ----. O. Similarly, is exact. The map p* tensored with the rational numbers is an isomorphism. In fact, so is p* tensored with 7L[t] (cf. Milnor [5]). It is known that the oriented cobordism class of a manifold is determined by its Pontryagin and Stiefel-Whitney numbers. Similarly, it has been proved that the spin cobordism class of a manifold is completely deter-mined by its Stiefel-Whitney and KO-characteristic numbers (Anderson-Brown-Peterson [1]). A fundamental such KO-invariant is the ring homomorphism d*: O!"in ----. KO-*(pt) which we describe in §3 of this chapter. Recall that n (mod 8) 0 1 2 3 4 Ko-n(pt) 71. 71.2 71.2 0 71. 5 0 The homomorphism .si" is an isomorphism for n 7. (2.3) 6 7 0 0 §3. CLIFFORD AND SPINOR BUNDLES 93 One of the remarkable and very useful consequences of the Atiyah-Singer Index Theory is that this invariant can be computed as the topolo-gical index of an elliptic operator naturally defined on any spin manifold in the cobordism class. This will be discussed in Chapter Ill. One of the interesting uses of the invariant .si is the following. Let :En be an n-dimensional homotopy sphere, i.e., a compact differentiable mani-fold which is homotopy equivalent to the n-sphere sn. Then:En is cobordant to zero (cr. Kervaire-Milnor [1 ]), but it is not necessarily spin cobordant to zero. In fact the homotopy n-spheres form a finite abelian group, en, under the operation of connected sum, and.sin: en -4 KO -n(pt) is a homo-morphism. The following is a consequence of deep results of Adams [3] and Milnor [7]. Theorem 2.1S. For n == 1 or 2 (mod 8) and n > 8, the homomorphism .sin: en ----+ &'.2 is surjective. §3. Clilford and Spinor Bundles We begin this section by briefly describing the associated bundle construc-tion. Let re: P -4 X be a principal G-bundle over a space X, and let Homeo(F) denote the group of homomorphisms of another space F. Give Homeo(F) the compact-open topology. Then to each continuous homo-morphism p: G -4 Homeo(F), we construct a fibre bundle over X with fibre F as follows. Consider the free left action of G on the product P x F given by q; g(p,f) = (pg -l,p(g) f) for 9 E G and (p,f) E P x F. Define P xp F to be the quotient space (the space of orbits) of this action. One easily sees that the projection P x F -4 P X descends to a mapping rep: P xp F ----+ X which is the fibre bundle over X with fibre F. It is called the bundle asso-ciated to P by p. If X and F are manifolds, G is a Lie group, and P is differentiable, one can consider continuous homomorphisms p: G -4 Diff(F), where Diff(F) is the group of diffeomorphisms of F with the usual COO topology. In this case, the associated bundle rep: P x p F -4 X is differentiable. Note that if P is given by transition functions n Up ----+ G for E 1111, where 1111 is some open cover of X, then P xp F is given 94 11. SPIN GEOMETRY AND DlRAC OPERATORS by the transition functions: P 0 gap: U a n Up ----+ Difll:F). Note also that if p: G --+ GL(V) is a linear representation on a vector space V, then P x P V is a vector bundle over X. EXAMPLE 3.0. Let X be a manifold and let p: X --+ X be its universal covering space. Then X is a principal 1!l(X)-bundle. Choose PE Hom(1!l X'£'2) Hl(X;£'2)' and consider £'2 as the group of permutations of the set {O,l}. Then X x P {O,l} is a 2-sheeted covering of X (see Lemma 1.1). EXAMPLE 3.1. Let X be a manifold and let P = P GL(X) be the principal GLn(IR)-bundle of tangent frames. Let Pn: GLn(lR) --+ GL(lRn) denote the standard representation, and let P: denote the dual representation (p:(g) = Pn(g -1 n Then and where TX and T* X are the tangent and cotangent bundles of X respec-tively. Similarly, A kTX = P GL(X) X AkPn A klRn NT*X = PGdX) (NlRn)* @STX = P GL(X) x (@SlRn) where AkPm Akp: and @SPn are the induced exterior power and tensor product representations. EXAMPLE 3.2. Let X be an oriented riemannian manifold and let P = Pso(X) be the SOn-bundle of positively oriented orthonormal frames. If P. : SOn --+ SO(lRn) is the standard representation, then again TX = Pso(X) xPn IRn NTX = Pso(X) X AkPn (NlRn) @TX = Pso(X) X®rpn (@lRn) Note that in this case P: = Pn. This corresponds to the canonical iso-morphism TX T* X given by the riemannian metric. EXAMPLE 3.3. More generally, if E is any oriented riemannian vector bundle over a space X, then E = Pso(E) x Pn IRn N(E) = Pso(E) X Arpn (NlRn) @(E) = Pso(E) x ®rPn (@lRn) Again, since Pn = P:, E and E* are canonically isomorphic. §3. CLIFFORD AND SPINOR BUNDLES 95 These examples suggest the following. Recall that each orthogonal transformation of [Rn induces an orthogonal transformation of ct([Rn) = ctn• (It maps the tensor algebra to itself and preserves the ideal.) This induced map on ctn clearly preserves the multiplication. Hence we get a representation: (3.1) DEFINITION 3.4. The Clilford bundle of the oriented riemannian vector bundle E is the bundle Ct(E) = Pso(E) X cf(Pnl ct([Rn) associated to the representation (3.1). E is a bundle of vector spaces with inner products, and Ct(E) is just the associated bundle of Clifford algebras. In fact Ct(E) could be defined as the quotient bundle: Ct(E) = &fE) II(E) where I(E) is the bundle of ideals, i.e., the bundle whose fibre at x E X is the two-sided I(Ex) in @Ex, generated by elements v ® v + IIvl12 for v E Ex. It is evident that Ct(E) is in fact a bundle of (Clifford) algebras over x. The fibrewise mUltiplication in Ct(E) gives an algebra structure to the space of sections of Ct(E). It is also evident that each of the notions intrinsic to Clifford algebras carries over to Clifford bundles. For example, there is a decomposition (3.2) corresponding to the even-odd decomposition of the algebras. These are the + 1 and -1 eigenbundles of the bundle automorphism a:: Ct(E) --Ct(E) (3.3) which extends the map E --+ E sending v to -v. There is also an intrin-sically defined bundle map L:Ct(E) --Ct(E) which in any fibre Ct(Ex) is given by n L(cp) = -l: ekcpek k=l where {eh ... ,en} is an orthonormal basis of Ex (see Chap. I). The following is an elementary but important fact: Proposition 3.5. There is a canonical vector bundle isometry A. : A *(E) Ct(E) (3.4) (3.5) (3.6) 96 n. SPIN GEOMETRY AND DIRAC OPERATORS under which A(A even E} = CfO(E}; (3.7) and A(NE} = {q; E Ct(E}: r:J. 0 L(q;} = (n -2p)q;} (3.8) for p = 0, ... ,no Proof. The isomorphism A follows directly from the canonical isomor-phism A: A * [R" .; ct([R") and the fact that A 0 A * p" = cf(p"} 0 A. Equa-tions (3.7) and (3.8) follow from Proposition 3.8 in Chapter 1. • As mentioned in the introduction, it is now natural to look for bundles of irreducible modules over the bundle of Clifford algebras Cf(E}. Such bundles can be constructed if w2(E) = O. DEFINITION 3.6. Let E be an oriented riemannian vector bundle with a spin structure PsPin(E) --+ Pso(E). A real spinor bundle of E is a bundle of the form S(E) = PsPin(E) XII M, where M is a left module for Ct([R") and where J1: Spin" --+ SO(M) is the representation given by left multiplication by elements of Spin" c CfO([R"). Similarly, a complex spinor bundle of E is a bundle of the form Scr;(E) = PsPin(E) XII Mc where Mc is a complex left module for Cf([R"} ® C. lf the module M (or Md is Z2-graded, the corresponding bundle is said to be Zz-graded. EXAMPLE 3.7. Consider Ct([R") as a module over itself by left multiplica-tion t. The corresponding real spinor bundle CtsPin(E) = PSPin(E} Xl Ct([R") is a "principal Cf([R")-bundle", i.e., it admits a free action of Cf([R"} on the right. There is a natural embedding PsPin(E) C Ctspin(E} which comes from the embedding Spin" c Ct([R"). Hence, every real spinor bundle for E can be captured from this one. A similar remark holds for the complex case. Of course, the bundle ctsPin(E) differs from the Clifford bundle Cf(E). They can be compared as follows. Consider the representation Ad: Spin" ---+ Aut(Ct([R"» (3.9) given by Adg(q;) = gq;g-l for 9 E Spin" c Ct([R"}. Clearly Ad_1 = identity, §3. CLIFFORD AND SPINOR BUNDLES 97 and so this representation descends to a representation Ad' of SOn. One easily checks that Ad' is just the representation ct(Pn) given in (3.1). It follows that Ct(E) = Pspin(E) XAd Ct([Rn). This leads to the following. Proposition 3.8. Let S(E) be a real spinor bundle of E. Then S(E) is a bundle of modules over the bundle of algebras Ct(E). In particular the sections of the spinor bundle are a module over the sections of the Clifford bundle. The corresponding fact holds in the complex and Z2-graded cases. Proof. The diagram PSPin(E) X Ct([Rn) x M PSPin(E) X M given by (p,cp,m) ----..... , (p,cpm) 1 1 clearly commutes. Therefore, J1 descends to a mapping J1: Ct(E) Et) S(E) --+ S(E), (3.10) which is easily seen to have the desired properties. The corresponding argument goes through in the complex and Z2-graded cases .• We say that two spin or bundles of E are equivalent iffthey are equivalent as bundles of Ct(E)-modules. A bundle of (real or complex, graded or ungraded) Ct(E)-modules is called irreducible if at each x the fibre is irreducible as a (real or complex, graded or ungraded) module over Ct(Ex). Recall that every module for Ct([Rn) can be written as a direct sum of irreducible ones, and there are at most two equivalence classes of irreduc-ible modules. Consulting §5 of Chapter I we obtain the following: Proposition 3.9. Every spinor bundle of E (real or complex, graded or un-graded) can be decomposed into a direct sum of irreducible ones. With the assumption that X is connected, the number N of equivalence classes of ir-reducible ones depends on the dimension n of E as follows. 98 H. SPIN GEOMETRY AND DIRAC OPERATORS Real Complex Real Complex n (mod 8) Ungraded Ungraded Graded Graded 1 1 2 1 1 2 1 1 1 2 3 2 2 1 1 4 1 1 2 2 5 1 2 1 1 6 1 1 1 2 7 2 2 1 1 8 1 1 2 2 Thus, for n even, there is only one irreducible, ungraded spinor bundle of E (over [R or over IC). If n == 6 or 8 (mod 8), the complex one is just the complexification of the real one. If n == 2 or 4 (mod 8), the complexification of the real one splits into two copies of the complex one. This, and the corresponding information for n odd, can be easily deduced from the tables in Chapter I. We now observe that in certain dimensions any real spinor bundle auto-matically carries a natural complex or quaternion structure. This is a con-sequence of the fact that for certain n, each irreducible real Cf([Rn)-module carries a compatible IC or IHI structure, i.e., the module multiplication is IC or IHllinear (and hence, the complex or quaternion scalar multiplication in M descends to the quotient P Spin X /l M). The appearance of such struc-ture is periodic in n and can be deduced from Table III in Chapter I. We conclude the following. Proposition 3.10. Let E be a real n-dimensional bundle equipped with a spin structure, and let S(E) be any real ungraded spinor bundle of E. If n == 1 or 5 (mod 8), then S(E) carries a complex structure such that Clifford multipli-cation is complex linear in each fibre. If n == 2, 3 or 4 (mod 8), then S(E) carries a quaternion structure so that Clifford multiplication is quaternion linear in each fibre. Let us now say a word about the £'2-graded case. There is a natural one-to-one correspondence between classes of bundles of irreducible £'2-graded modules over Cf(E) = CfO(E) EB Cf l(E) and classes of bundles of irreducible modules over CfO(E). Given a bundle S(E) = SO(E) EB Sl(E) of the first kind, SO(E) is of the second. Given an SO(E) of the second kind, the bundle is of the first. Suppose now that n = 2m and Scr;(E) is the irreducible complex spin or bundle of E. We shall show explicitly how to split Scr;(E) into a direct sum SdE) = St(E) EB Sc(E) (3.11) §3. CLIFFORD AND SPINOR BUNDLES 99 of Cf,O(E)-modules. Interpreting St(E) as and Si(E) as Sb(E), or the other way around, gives a Z2-graded module structure to SdE). The two possibilities are the two inequivalent graded modules appearing in the table. The construction is as follows. Consider the global section Wc of Cf,(E) ® C which at x E X is given by (3.12) for any positively oriented orthonormal basis {e1, ••. ,e2m} of Ex. Then we have (3.13 ) (3.13') for any e E ef, l(E) ® c. We then define St(E) and Si(E) to be the + 1 and -1 eigenbundles for Clifford mUltiplication by Wc' One easily sees from (3.13') that these bundles have the same dimension and that they form the Z2-graded modules as stated above. These bundles can be written as associated bundles in the following way. Let and denote the two fundamental complex representa-tions of Spin2m' Then (3.14) For n == 0 (mod 4) there is an analogous construction in the real case. Let S(E) be the irreducible real spinor bundle of E and define a global section w of Cf,(E) by setting (3.15) at x E X where {el' ... ,en} is any positively oriented orthonormal basis of Ex. Again, we have that we = -ew for all e E Cf,O(E). (3.16) (3.17) For equations (3.16) and (3.17) it is necessary that n be a multiple of 4. This again determines a decomposition (3.18) into the + 1 and -1 eigenbundles of the operator given by Clifford multi-plication by w. They make S(E) a Z2-graded module in two distinct ways, thereby accounting for the 2 in Table 3.1 above. If n == 0 (mod 8), then S±(E) ® C Sf(E). This corresponds to the fact that in these dimensions, are the complexifications of real representations. 100 H. SPIN GEOMETRY AND DIRAC OPERATORS If n == 4 (mod 8), then S+(E) ® C S,E=(E) EB S,E=(E). In these dimen-sions Ll; are quaternionic. We now make a fundamental observation concerning these Z2-graded bundles. Let IO(E) = {e E E: lIell 1} be the unit disk bundle of E with boundary IO(E) = {e E E: Ilell = 1}, the unit sphere bundle. Let n: IO(E) -+ X be the bundle projection. Assume n is even, so that S,E=(E) are defined. Then the pull-backs of these bundles over IO(E) are canonically isomorphic on IO(E) by the map (3.19) given at e E IO(E) by /le((J) = e . (J that is, Clifford multiplication by e itself. Since e . e = -lleW = -1, each map /le is an isomorphism. The pair of bundles n*S,E= over IO(E), together with the isomorphism W n*St n*Si over IO(E), given by (3.19), deter-mine a "difference" element (3.20) where r(E) == IO(E)/IO(E) is the Thorn space of E. Here K denotes reduced complex K-theory (cf. I.9). If n == 0 (mod 4), the analogous construction clearly goes through in the real case. Here we obtain an element (3.21) We are now in a position to define the map (2.3) discussed in the last section. Let [Sk be the universal 8k-plane bundle over BSpinsk with its unique spin structure. (IESk is the pull-back of the universal8k-plane bundle over BSOSk by the map BSpinsk -+ BSOsd Let I](IESk) E Ko(MSpinsk) be the class (3.21) defined above. Now fix n and choose k sufficiently large that we have the isomorphism nn+sk(MSpinsk). Then a cobordism class [X] E determines a map Ix: sn + Sk -+ MSpinsk. We define (3.22) by §4. CONNECTIONS ON SPINOR BUNDLES 101 §4. Connections on Spinor Bundles Suppose E is a smooth riemannian vector bundle over a manifold X and that PSpin(E) -+ Pso(E) is a spin structure on E. Then, of course, any connection on Pso(E) can be lifted via to a connection on Pspin(E), and this, in turn, defines a connection on the associated spinor bundles. In this section we shall give an explicit computation of this spinor connection and its associated curvature tensor in terms of the connection on E. We begin by briefly recalling some facts from the theory of connections. Let n: P -+ X be a smooth principal bundle over a manifold X with group G. Then G is a Lie group whose Lie algebra will be denoted by g. G acts freely from the right on P. Each element V E 9 determines a vector field V on P by setting Vp = d/dt(p . eXp(tV))lt=o' The map V -+ Vp gives an isomorphism (4.1) where fp is the tangent space to the orbit through p. The orbits are the fibres of n, and the plane fp can be thought of as the "vertical" space through p. A connection is then a choice of an invariant field of comple-mentary "horizontal" spaces. DEFINITION 4.1. A connection on P is a G-invariant field of tangent n-planes 1: on P (n = dim(X)) such that the linear map n*: 1:p -+ T"p(X) is an isomorphism for all pEP. At each PEP, 1:p determines a linear projection Tp(P) -+ fp. The ca-nonical isomorphism (4.1) then gives a linear map (4.2) This defines a g-valued I-form w on P, called the connection I-form. It has the following properties. for all VE g. (4.3) for all g E G acting on the manifold P. (4.4) Note that given the connection I-form w, one can recapture the connec-tion by the relation 1:p = ker(wp). The curvature of the connection is the g-valued 2-form Q given by the equation Q = dw + [w,w]. (4.5) 102 n. SPIN GEOMETRY AND DIRAC OPERATORS (Note that by the skew-symmetry of the Lie bracket, [w,w](v,w) [w(v),w(w)] is a g-valued exterior 2-form.) This form has the following properties. QCV,) == 0 for all V E 9 g*Q = Adg-,(Q) for all g E G (4.6) (4.7) EXAMPLE 4.2 (orthogonal connections). Let P = Pso(E) where E is a smooth, oriented riemannian vector bundle. The Lie algebra of SOn is the space SOn of real, skew-symmetric n x n-matrices. Hence, a connection 1-form W can be considered as an n x n-matrix of I-forms W = «wij)) where wij = -Wji' The corresponding curvature is a matrix of2-forms Q = «Qij)) where n Qij = dWij + L Wik 1\ wkj· k=l For an orthogonal matrix g, Ady(w) = gwg-1. (4.8) Let ei 1\ ej denote the elementary skew-symmetric (i,j)-matrix. If {e 1, ... ,en} denotes the canonical basis of !Rn, this corresponds to the transformation (ei 1\ e)(v) = (ei,v)ej -(ej,v)ei. The connection and curvature forms can then be written as W = -L wijei 1\ ej i<j Q = -L QiA 1\ ej i<j (4.9) (4.10) (4.11) Given a connection on the bundle Pso(E) as above, we can define a rule for taking derivatives of sections of E. For any smooth vector bundle E' over X, let r(E') denote the space of smooth cross-sections of E'. DEFINITION 4.3. A covariant derivative on E is a linear map V:f'(E)---r(T*X ® E) such that V(fe) = dl ® e + IVe (4.12) for all I E CCXl(X) and all e E r(E). Thus, given a smooth vector field V on X, we obtain a map V y : r(E) -+ r(E) called the covariant derivative with respect to V. At a given point x E X, (Vye)x depends only on Vx and on the values of e in a neighborhood of x. §4. CONNECTIONS ON SPINOR BUNDLES 103 Proposition 4.4. Let w be a connection i10rm on Pso(E) as above. Then w determines a unique covariant derivative on E by the rule n Ve, = L Wj, ® ej j= 1 (4.13) where is = (el, ... ,en) is a local family of point wise orthonormal sections of E, i.e., a local section of Pso(E), and where W = is*w. This covariant derivative satisfies the rule V(e,e') = (Vye,e') + (e, Vye') (4.14) for all V E T(X) and e,e' E qE), where (., .) denotes the inner product in E. Conversely, any covariant derivative on E satisfying (4.14) determines a unique connection i-form by equations (4.13). Note. A covariant derivative with property (4.14) will be called riemannian. Proof. Let is = (el' ... ,en) be a frame defined on an open set U X, and let «wu)) be any skew-symmetric n x n-matrix of I-forms on U. Then equation (4.13) together with property (4.12) defines a unique covariant derivative on Elu. (To see this, note that any section e of E over U can be written uniquely as e = L fjej for fl' ... ,In E COO(U). Then by (4.12) and (4.13) we have Ve = L dfj ® ej + L fjWkj ® ek = L {dfk + L Wkjfj} ® ek· This definition of V has properties (4.12) and (4.14) and is therefore a riemannian covariant derivative on U.) Conversely, given is and a rieman-nian covariant derivative V on U, we have a skew-symmetric matrix of I-forms «wu)) on U uniquely defined by (4.13). Consequently, to define a global co variant derivative on E it suffices to assign to each local frame field is a matrix of local I-forms «w,)) sat-isfying the following compatibility condition. Suppose is = (el' ... ,en) and 8' = (e'l, ... are two orthonormal frame fields over an open set U, and let W = «wu)) and W' = «w;j)) be the associated matrices of I-forms. Then for each x E U, there is a unique orthogonal n x n-matrix g(x) = ((giJ{X))) such that is(x) = is'(x)g(x), i.e., n ei(x) = L e;{x)gj,(x). j= 1 Applying V and using (4.13) we find easily that w(x) = g-l(X)W'(x)g(x) + g-l(x)dg(x). This transformation rule is the required compatibility condition. ( 4.15) Suppose now that Pso(E) is provided with a connection I-form w. Then given a local section is = (el' ... ,en) of Pso(E) over an open set U, we get 104 n. SPIN GEOMETRY AND DIRAC OPERATORS a skew-symmetric matrix of 1-forms W = «wij)) on U by setting W = 8*w. Note that 8 determines a local trivialization qJ: U X SOn ---+ n-l(U) of n: Pso(E) -+ X by setting qJ(x,g) = 8(x)g. Conversely, qJ determines 8, since 8(x) = qJ(x,e). Note that qJ is SOn-equivariant. We now observe that in this local product determined by qJ, the connec-tion 1-form can be written as (4.16) To see this we first write qJ*w = Wo + W1 where Wo = L ai(x,g)dxi for local coordinates Xi on X and where W1 = L biix,g)dgij' Properties (4.3) and (4.4) imply that W1=g-ldg. By definition we have wo=8*w along U x {e} c: U x G. Property (4.4) then implies that Wo = Adg-,(w) = g-l 0 wog at a general point (x,g). If we choose a different cross section 8' = (e'!> ... over U, we get a new trivialization qJ': U X SOn -+ n-l(U) by the formula qJ'(x,g) = 8'(x)g. The change of trivializations <I> = (qJ') -1 0 qJ: U X SOn -+ U X SOn is given by <I>(x,g) = (x, g(x)g) (4.17) where g: U -+ SOn is the change of frames as above, i.e., 8(x) = 8'(x)g(x). Clearly we have that qJ*W = <1>*( qJ'*w). Using (4.16) and (4.17) we can re-express this as Adg-,(w) + g -ldg = Ad(g(x)g)-,(w') + (g(x)g) -ld(g(x)g) (4.18) = Adg-,{Adg-'(xiw') + g-l(x)dg(x)} + g-ldg. This equation immediately reduces to the compatibility condition (4.15). Consequently, a connection 1-form on Pso(E) determines a riemannian covariant derivative on E, as claimed. Conversely, given a riemannian co variant derivative, we obtain local 1-forms w transforming according to (4.15). This implies that the compatibility condition (4.18) for the ex-istence of a global connection 1-form on Pso(E) is satisfied. This completes the proof. • Given a covariant derivative V on E, it is natural to ask whether the second covariant derivatives commute in an appropriate sense. For this we consider the composition r(E) r(T* ® E) r(A2T* ® E) 104 H. SPIN GEOMETRY AND DIRAC OPERATORS a skew-symmetric matrix of I-forms W = ((w;)) on U by setting W = 8*w. Note that 8 determines a local trivialization qJ: U X SOn ----> n-l(U) of n: Pso(E) -+ X by setting qJ(x,g) = 8(x)g. Conversely, qJ determines 8, since 8(x) = qJ(x,e). Note that qJ is SOn-equivariant. We now observe that in this local product determined by qJ, the connec-tion I-form can be written as (4.16) To see this we first write qJ*w = Wo + Wl where Wo = L a;(x,g)dx; for local coordinates x; on X and where Wl = L bjx,g)dgij. Properties (4.3) and (4.4) imply that Wl = g-ldg. By definition we have Wo = 8*w along U x {e} c: U x G. Property (4.4) then implies that Wo = Adg-l(w) = g-l 0 wog at a general point (x,g). If we choose a different cross section 8' = (e'l, ... over U, we get a new trivialization qJ': U X SOn -+ n-l(U) by the formula qJ'(x,g) = 8'(x)g. The change of trivializations <I> = (qJ') -1 0 qJ : U X SOn -+ U X SOn is given by <I>(x,g) = (x, g(x)g) (4.17) where g: U -+ SOn is the change of frames as above, i.e., 8(x) = 8'(x)g(x). Clearly we have that qJ*W = <1>*( qJ'*w). Using (4.16) and (4.17) we can re-express this as Adg-l(w) + g-ldg = Ad(g(x)g)-l(W') + (g(x)g)-ld(g(x)g) (4.18) = Adg-l{Adg-l(x)(w') + g-l(x)dg(x)} + g-ldg. This equation immediately reduces to the compatibility condition (4.15). Consequently, a connection I-form on Pso(E) determines a riemannian covariant derivative on E, as claimed. Conversely, given a riemannian covariant derivative, we obtain local I-forms W transforming according to (4.15). This implies that the compatibility condition (4.18) for the ex-istence of a global connection I-form on P so(E) is satisfied. This completes the proof. • Given a co variant derivative V on E, it is natural to ask whether the second co variant derivatives commute in an appropriate sense. For this we consider the composition r(E) 2..... r(T* ® E) 2..... r(A2T* ® E) §4. CONNECTIONS ON SPINOR BUNDLES 105 where V is the natural prolongation of V defined on sections of the form IX ® e by V(IX ® e) = dlX ® e -IX 1\ Ve, and we set R = V 0 V Proposition 4.5. Let w, is and V be as in Proposition 4.4, and let Q be the curvature 2-form of the connection. Then where Q = is*Q. n Rei = L Qji ® ej j=l Proof. From equation (4.8) we have V(Vei) = v(t wji ® ej) J= 1 n n = L dWji ® ej + L Wkj 1\ Wji ® ek j=l j.k=l n =:= L Qji ® ej. • j= 1 (4.19) Proposition 4.6. Let Q, is and V be as in Proposition 4.5. Then for local tangent vector fields V and W on X, we have (4.20) Proof. Note that VVVWei = Vv( ejwji(w)) = L ekwk/V)Wji(W) + L ejV' wji(W) j,k j and therefore (VvVw -VwVv -VIV,Wj)ei = L ej{V' wji(W) -W· w)V) -wji([V,W])} j + L ej{wjk(V)Wki(W) -wjiW)wk;(V)} j,k = L ej{dwj;(V,W) + L wjk 1\ Wki(V,W)} j k = L ejQji(V,W). • (4.21) j Notice that from (4.14) we have the relation (Rv, we, e'> + (e, Rv,we'> = O. (4.22) Notice also that from the above it follows that the expression (Rv,we, e'> is a tensor, that is, at any point x E X, it depends only on the quantities Vx, w",e and not on the local fields V, W,e,e' extending them. Hence, 106 n. SPIN GEOMETRY AND DIRAC OPERATORS given two tangent vectors V,Wat x E X, the curvature gives a well-defined, skew-symmetric endomorphism (4.23) called the curvature transformation associated to V and W. Suppose now that n: P --+ X is a smooth principal G-bundle over X, that p: G --+ SOn is a representation of G, and that Ep = P xp !Rn is the associated riemannian vector bundle (cr. 11.3). Then given a connec-tion on P there is a canonical connection induced on P(Ep) as follows. Note that P(Ep) = P Xp SOn where an element g E G acts on SOn via left mUltiplication by p(g). Given a connection L on P, extend it trivially to P x SOn and then push it forward to P x p SOn. One can easily see that this gives a connection Lp on P(Ep). There is a canonical, G-equivariant mapping (4.24) given by p -[(p,e)]. (where [(p,h)] denotes the class of (p,h) E P X SOn in the quotient P(Ep).) If the representation p is faithful, the mapping (4.24) is an embedding. For simplicity we assume p to be faithful. Proposition 4.7. Suppose wand Q are the connection and curvature forms on P respectively, and let wp and Qp denote the corresponding forms for the induced connection on P(Ep). Then, considering P c: P(Ep) as above, we have that and Qp/p = p*Q where p*: 9 --+ SOn is the Lie algebra homomorphism associated to p: G --+ SOn· Proof. This follows straightforwardly from the fact that the embedding (4.24) is G-equivariant, i.e., that i(p . g) = i(p) . p(g). • We are now in a position to discuss the connections on Clifford and spinor bundles. Let E be an oriented riemannian vector bundle of dimen- §4. CONNECTIONS ON SPINOR BUNDLES 107 sion n, and suppose E is furnished with a riemannian connection, i.e., a connection r on Pso(E). The Clifford bundle Cf,(E) is associated to E by the representation (3.1): cf,(Pn): SOn Aut(Cf,(/Rn)). Therefore, by the construction above, r induces a unique connection, r', on Cf,(E). Note that since cf,(Pn) maps into the automorphisms of Cf,(/Rn), we have that the associated Lie algebra homomorphism is a map cf,(Pn)* : SOn Der(Cf,(/Rn)) where Der(·) is the Lie algebra of derivations; i.e., cf,(Pn)* has the prop-erty that for each element A E sOn {cf,(Pn)*A}(qJ· t/!) = ({cf,(Pn)*A}qJ)· t/! + qJ. ({cf,(Pn)*A}t/!) (4.25) for all qJ,t/! E Cf,(/Rn). Recall, furthermore, that under the canonical iden-tification Cf,(/Rn) A */Rn, the representation cf,(Pn) becomes A *Pn. To-gether with Propositions 4.4 and 4.7, this gives the following. Proposition 4.8. The covariant derivative V on Cf,(E) acts as a derivation on the algebra of sections, i.e., V(qJ . t/!) = (VqJ)· t/! + qJ . (Vt/!) (4.26) for any two sections qJ and t/! of Cf,(E). Furthermore, under the canonical identification Cf,(E) A *(E), the co-variant derivative V preserves the sub bundles AP(E) and agrees there with the covariant derivative induced by the representation APPn (i.e., the usual covariant derivative). Corollary 4.9. The sub bundles Cf,°(E) and Cf, l(E) are preserved by V. Furthermore, the "volume form" w = e1 •.. en is globally parallel; that is, vw=o. Therefore when n == 3 or 4 (mod 4), the eigenbundles Cf, ±( E ) = {qJ E Cf,(E): Q)qJ = ± qJ} are also preserved by V. Proof. The first statement follows from the fact that Cf,°(E) A even(E) and Cf, l(E) A odd(E). The second statement follows from the fact that w corresponds to the unit section of A n(E). • In the analogous way, Propositions 4.5 and 4.7 give the following. Proposition 4.10. For any pair of tangent vectors V and W at x E X, the curvature transformation 108 n. SPIN GEOMETRY AND DIRAC OPERATORS is a derivation, i.e., (4.27) for all cp,!/! E Ct(Ex). Furthermore, Ry,w preserves the subspaces CtO(Ex), Ct1(Ex) and Ct± (EJ as above. Suppose now that E carries a spin structure PSPin(E) --+ Pso(E) and let S(E) = P sPin(E) X fJ M be an associated real spinor bundle. Here M is a left module over Ct([Rn) and p.: Spinn --+ SO(M) is the resulting repre-sentation (cf. §3). The connection r on Pso(E) lifts via the covering map to a connection i on PSpin(E). This in turn induces a connection and therefore a covariant derivative on S(E). Recall (Proposition 3.8), that the sections of S(E) form a module over the sections of Ct(E). Proposition 4.11. The covariant derivative V on S(E) acts as a derivative with respect to the module structure over Ct(E), i.e., V(cp . a) = (Vcp)· a + CP' (Va) for any section cp of Ct(E) and any section a of S(E). (4.28) Corollary 4.12. If n = 3 or 4 (mod 4), the eigenbundles S ±( E ) = {cp E S(E): wcp = ± cp} are preserved by V. Proof of Proposition 4.11. The representations ct(Pn) (= Ad) and p. pre-serve the module multiplication, that is, p.(g)(cp . a) = {ct(Pn)(g)cp} . {p.(g)a} for all g E Spinm cp E Ct([Rn) and a E M (see the discussion surrounding Proposition 3.8.) Differentiating at the identity, we get that for each element A E sOn = spinn {p.*A}(cp· a) = ({ct(Pn)*A}cp)' a + CP' ({p.*A}a). (4.29) The argument is now completed using Propositions 4.4 and 4.7 as before . • We also have the analogous statement for curvature: Proposition 4.13. For any pair of tangent vectors V,W at x E X, the curva-ture transformation is a module derivation, i.e., (4.30) for all cp E Ct(Ex) and all a E S(Ex). Furthermore, Ry,w preserves the sub-spaces S± (Ex) when they are defined. §4. CONNECTIONS ON SPINOR BUNDLES 109 We shall now proceed to explicitly compute the connection and cur-vature forms on S(E). To do this we need to examine the representation iJ.. Recall that sOn is 'generated by the elementary transformations x /\ y; X,Y E [Rn, given by the formula (x /\ y)(v) == <x, v)y -<y, v)x. (4.31) From Chapter I, §6, we have iJ.*(x /\ y) = H x,y J. (4.32) That is, iJ.*(x /\ y)(a) = H x,y] . a for all a E M. Recall that on Spinn, we have c£(Pn) = Ad. Hence, (4.33) That is, Ad*(x /\ y)(cp) = H[x,y],cp] for cp E C£([Rn). Note that equation (4.29) follows immediately from (4.32) and (4.33). Suppose now that = (el' ... ,en) is an n-tuple of pointwise orthonor-mal sections of E defined over a contractible open set U x. is just a section of Pso(E) over U, and it can be lifted to a section i of PSpin(E) over U. There are two possible such liftings. They satisfy the relation: 0 8 The connection 1-form on PsPin(E) is just the lift of the connection 1-form (}J on Pso(E). To obtain a formula of the type given in Proposition 4.4 we want to pull down to U by the local section 8. This "pull-down" is just W = = 0 8)*(w) = Consequently, the scalar 1-forms wij are just the 1-forms we obtained earlier by pulling down the connection (}J by the local frame field Now for any spinor bundle S(E) we have a canonical embedding PSpin(E) C Pso(S(E)). Thus, 8 can be considered as a section of Pso(S(E)). Let (}JS denote the connection 1-form on Pso(S(E)). Then to apply Propo-sition 4.4 we want to compute WS = 8*(wS). However, by Proposition 4.7, we know that (}JS restricted to PSpin(E) C Pso(S(E)) is just Con-sequently, we have Writing W = -I wijei /\ ej (cf. (4.10)) and using (4.32), we can rewrite this as i<j 110 11. SPIN GEOMETRY AND DIRAC OPERATORS Finally, we note that the embedding PSPin(E) C Pso(S(E)) can be inter-preted to mean that every point of PSPin(E) <!,etermines an orthonormal frame in S(E). In particular, the local section $ determines a local section 9' = (a 1, ... ,aN) of Pso(S(E)). (The other choice of lifting j just deter-mines the negative frame -9' = ( -a l' ... , -aN)'} Combining the remarks above, we have the following. Theorem 4.14. Let w be the connection i-form on Pso(E) and let S(E) be any spinor bundle associated to E. Then the covariant derivative Vs on S(E) is given locally by the formula VSaa = t I Wji ® eiej . aa i<j (4.35) where $ = (el' ... ,en) is a local section of Pso(E), W = $*(w), and where 9' = (al, .. , ,aN) is a local section of Pso(S(E)) determined by $. Note. The frame field 9' = (a l' ... ,aN) is, in fact, only determined up to a constant orthogonal change of framing, that is, up to a choice of ortho-normal basis at some point. (This corresponds to a choice of basis in the module M, i.e., on the matrix realization of the representation p..) This family of framings is characterized by the following property. For any 1= (il, ... ,ip), we have that eil ... eipaj = L qPk where the coefficients are constants. An analysis similar to the one above can be carried out for the curva-ture 2-form. Since curvature is a tensor, we do not need to be concerned in this case with the distinguished frame field 9' on S(E). Hence, the cur-vature of S(E) can be expressed in the following very pretty way: Theorem 4.15. Let Q be the curvature 2-form on Pso(E) and let S(E) be any spinor bundle associated to E. Then the curvature RS of S(E) is given locally by the formula RSa = t I nji ® eiej' a i<j (4.36) where $ = (el, ... ,en) is a local section of Pso(E1 n = $*(Q) and where a is any section of S(E). In particular, for any two tangent vectors Vand Wat x E X, the curvature transformation S(Ex) S(Ex) is given by the formula (4.37) where Rv.w is the curvature transformation of Ex. §4. CONNECTIONS ON SPINOR BUNDLES Note that the expression 111 (4.38) is independent of the choice of an orthonormal basis (el' ... ,en) for Ex and is therefore invariantly defined. It is skew symmetric in V and W. Consequently, can be thought of as a 2-form on X with values in Ct(E). The formula (4.37) can now be succinctly expressed as = . 0'. (4.39) It is interesting to note that the arguments above can be carried through also for the Clifford bundle Ct(E) by using the equation (4.33). For tan-gent vectors V and W at x E X we define the invariant operator (4.40) where, as before, (el, ... ,en) is any orthonormal basis of Ex. We then ob-tain the following expression for the curvature tensor on Ct(E) A *(E) in terms of Clifford multiplication. Theorem 4.16. For any two tangent vectors V and W at x E X, the curva-ture transformation Ct(EJ -+ Ct(Ex) is given by the formula = (4.41) where w is the operator defined above in terms of the curvature trans-formation Rv.w of Ex. c)t is an easy exercise to check that the restriction of the operator .'£" c: Ct(Ex) agrees with Rv.w. One need only verify that Hejej,e] = (ej /\ ej)(e) for e E Ex. Note that Theorem 4.16 does not depend on the existence of a spin structure on E. In fact, it does not even depend on the existence of an orientation on E. We conclude this section with some remarks concerning the situation where E = T(X), the tangent bundle of X. In this basic case we abbre-viate notation by setting Pso(X) == Pso(T(X)), the (orthonormal) tangent frame bundle of X, and ct(X) == Ct(T(X)), the Clilford bundle of X. Suppose now that Pso(X) is furnished with a connection, and let V denote the corresponding covariant derivative. Then there is an invari-antly defined tensor field associated to V as follows. Let V and W be tangent vectors at a point x E X. Extend them to local vector fields (i.e., local sections of T(X)) and consider the expression (4.42) 112 11. SPIN GEOMETRY AND DIRAC OPERATORS where [V,W] is the Lie bracket. The value of Ty,w at x can be easily shown to be independent of the choice of vector fields extending Vx and Wx (see Helgason [1, Chap. I]). T V,w is clearly bilinear and skew-sym-metric. It therefore defines a global 2-form on X with values in T(X) called the torsion tensor of the connection. The following result can be found in any basic text in differential geometry. Theorem 4.17. (The Fundamental Theorem of Riemannian Geometry). Let Pso(X) denote the tangent frame bundle of a riemannian manifold X. Then there exists a unique connection on Pso(X) with the property that its torsion tensor vanishes identically. This connection will be called the canonical riemannian connection on X. It induces a canonical connection on Ct(X) A *(X). Note that if X admits a spin structure PSpin(X) -+ Pso(X), then by lifting, we obtain a canonical riemannian connection on PSPin(X). This, in turn, induces a connection on any spinor bundle associated to PSpin(X), This situation falls precisely into the general framework developed above. Thus any spinor bundle for X is a bundle of left modules over Ct(X), and the canonical covariant derivative is a derivation of the mod-ule multiplication (see Proposition 4.11). For most of the basic facts ofriemannian geometry, the reader is referred to the general literature which is extensive and quite good. However, there is one fact we use sufficiently often that it is worthwhile to cite it here. Proposition 4.18. Let R denote the curvature tensor of a riemannian mani-fold X, i.e., the curvature tensor of the canonical riemannian connection on the tangent bundle TX. Then R satisfies the following identities: RU,yW + Ry,wU + Rw,u V = 0, <RU,yW, Y) = <Rw,yU, V) for all tangent vectors U, V, W, YET xX, at all points x E X. §5. The Dirac Operators (4.43) (4.44) Let X be a riemannian manifold with Clifford bundle ct(X), and let S be any bundle of left modules over ct(X) (i.e., a vector bundle over X such that at each point x E X, the fibre Sx is a left module over the algebra Ct(X)x') Assume S is riemannian and is furnished with a remannian con-nection. Then under these general hypotheses, we can define a canonical first-order differential operator D: r(S) -+ r(S) called the Dirac operator §5. DlRAC OPERATORS 113 of S, by setting (S.O) at x E X, where et. ... ,en is an orthonormal basis of Tx(X), where V denotes the co variant derivative on S determined by the connection, and where "." denotes the Clifford module multiplication. The operator D2 is called the Dirac laplacian. Recall that the principal symbol of a differential operator D: r(E) --+ r(E) is a map which associates to each point x E X and each cotangent vector E a linear map Ex --+ Ex defined as follows. If in local coordinates we have and where m is the order of D, then = im I l"l=m The operator D is elliptic if is an isomorphism for all -# 0 (see 111.1 for further details). Recall also that the riemannian metric induces a canonical isomor-phism between T(X) and T*(X). Throughout this section we shall con-sider them as so identified. Lemma 5.1. Let D be the Dirac operator of the bundle S defined above. Then for any E T*(X) T(X) we have that = (S.1) (S.2) where the symbol on the right denotes Clijford multiplication by the vector in (S.1) and the scalar in (S.2). In particular, both D and D2 are elliptic operators. Proof. Fix x E X and an orthonormal basis el' ... ,en of Tx(X). Choose local coordinates (Xl> ... ,xn) on X at x such that x corresponds to 0 and ej corresponds to (ojOXj)o for each j. Under the identification TAX) we have that ej also corresponds to (dxj)o for each j. For any local trivialization of S near x, we have that Vej = (ojoxj)o + zero-order terms. Hence, at 0 we have that D = I ej(ojoXj)O + zero-order terms. Consequently, for any cotangent vector = I at 0, we have by definition of the symbol that = i I = This gives (S.1). Then = 0 = . = and the proof is complete . • 114 11. SPIN GEOMETRY AND DIRAC OPERATORS We now observe that in light of the basic examples it is natural to require that the bundle S have certain additional properties. The first property is that CIifford multiplication by unit vectors in T(X) be or-thogonal, i.e., that at each x E X, <eO'l,e0'2) = <0'1,0'2) (5.3) for all 0'1,0'2 E Sx and all unit vectors e E T x(X). Since e2 = -1, this is equivalent to the requirement that ( 5.3)' for all such 0' 1'0' 2 and e. Recall (from the end of §4) that the bundle C£(X) carries a canonical riemannian connection, whose associated covariant derivative will be denoted by V. Our second requirement is that the covariant derivative on S (which we also denote by V) be a module derivation, i.e., that V(cp' 0') = (Vcp)' 0' + cp' (VO') (5.4) for all cp E qc£(X» and all 0' E qS). There is a surprisingly large and important collection of bundles with the properties described above. Although these bundles can be quite varied in nature, a substantial part of the theory concerning them can be treated in a uniform way. For this reason we introduce the following concept: DEFINITION 5.2. A Dirac bundle over a riemannian manifold X is a bun-dle S of left modules over C£(X) together with a riemannian metric and connection on S having properties (5.3) and (5.4) above. Before presenting examples of these bundles, we shall investigate some of their elementary properties. Note that any Dirac bundle S has a canon-ically associated Dirac operator. Furthermore there is an inner product on qS) induced from the pointwise inner product <.,.) by setting (0'1,0'2) == Ix <0'1,0'2)' (5.5) Proposition 5.3. The Dirac operator of any Dirac bundle over a riemannian manifold is formally self-ad joint, i.e., (D0'1'0'2) = (0't>D0'2) for all compactly supported sections 0'1 and 0'2' Proof. Fix x E X and choose an orthonormal tangent frame field (e1, ••• ,en) in a neighborhood of x so that (Ve,e)x = 0 for all i,j. This can be done for example, by extending a frame at x by parallel translation §5. DIRAC OPERATORS 115 along geodesic rays emanating from x. Using properties (5.3)', (5.4) and (4.14), we have that at x, <Da1,(2)x = L <ejVepl>(2)x = -L <VeP1,ep2)x j j = -:E {ej<a1,ep2) -<a1,(Vejej)a2) -<a1,eFep2)}x J = -L (ej<a l' ep2»x + <ab D(2)x j = div(V)x + <abD(2)x where V is the vector field defined by the condition that <V, W) = -<a1, w· (2) for all tangent vectors W. The last line in (5.6) is established as follows div(V)x <VeX,ej)x J = L {ej<V, ej) -<V, Ve.ej)}x j J = L {e/V,ej)}x j = -L {ej<a1,ej' (2)}x' j The first and last expressions in (5.6) are independent of the frame field (el' ... ,en)' Hence, we have established the equation <Da1,(2) = div(V) + <a1,D(2) on X. The proposition follows immediately. • Note that if X is permitted to have a boundary oX, then the above argument proves that (Da1,a2) -(a1,Da2) = f <v· a1,(2) (5.7) J"x for compactly supported sections a 1 and a 2, where v denotes the outer unit normal field to oX in X. It is a general consequence of the ellipticity of D that any weak solution to the equation Dl/I = 0 is of class COO (cf. Theorem 5.2(i) of Chap. Ill). Because of the formal self-adjointness this may be expressed as follows. Let fcpt(S) denote the Frechet space of COO sections of S with compact support. Then any continuous linear functional F on fcpt(S) such that F(Dcp) = 0 for all cp E fcpt(S), can be represented as F(cp) = (cp,l/I) where '" E f(S) and Dl/I = O. In particular, any locally integrable section l/I of S which is orthogonal to Dfcpt(S) (i.e., which satisfies the equation Dl/I = o weakly), is of class COO and satisfies the equation Dl/I = 0 in the usual sense. 116 11. SPIN GEOMETRY AND DIRAC OPERATORS Similar comments apply to D2. With this in mind, we define ker D = {q> E r(S): Dq> = O} and ker D2 = {q> E r(S): D2q> = O}, and observe the following: Theorem 5.4. Let D be the Dirac operator of any Dirac bundle over a com-pact riemannian manifold X. Then ker D = ker D2 and this space has finite dimension. Proof. Finite dimensionality is a direct consequence of elementary elliptic theory (cf. Theorem 5.2(ii) of Chap. III). For the rest, observe that D2q> = 0 = IIDq>W = (Dq>,Dq» = (q>,D2q» = 0 = Dq> = O. • The results above have an important extension to L2-sections of a Dirac bundle over any complete manifold. We begin with the following pre-paratory lemma. Lemma 5.5. Let D, S and X be as above. Thenfor any f E COO(X) and any q> E qS), we have that D(f cp) = (grad f) . q> + fDq>. (5.8) Proof· D(fq» = I ej' Vej(fq» = I ej{(ejf)q> + fVejq>} = (I (ejf)ej)' cp + fDcp) = (grad f) . q> + fDcp. • REMARK 5.6. The relation (5.8) immediately extends to distributional sections <I> of S. To see this, consider q> E rcpt(S), and note that from (5.8) and (5.3)', (fD<I>,q» == (<I>,D(fcp)) = (<I>,(gradf)' q> + fDq» = (D(f<l»-(grad f) . <1>, q». We shall now consider the space L2(S) of L2-sections of S. This is the natural completion of rcpt(S) in the Hilbert space norm introduced above. We now consider D as a symmetric operator on rcpt(S) and take its closure (also denoted D), in e(S). This gives us an unbounded operator in L2(S). Its domain dom(D) consists of all q> E L2(S) for which there is a sequence <q>n>:= 1 c rcpt(S) such that q>n -+ q> and Dq>n -+ !/J = "Dq>" in L2(S). There is another extension of this operator to e(S) which we shall denote D*. The domain of D* consists of all q> E L2(S) such that the dis-tributional image Dq> is also in L2(S). Of course, dom(D) s;; dom(D*). We now observe that D* is simply the adjoint of D. Recall that !/J E e(S) is in the domain of the adjoint Dt if the function q> <Dq>,!/J> is continuous on dom(D). Since dom(D) is dense, there exists a unique ele-ment Dt!/J E L 2(S) such that < Dq>, !/J> = < q>, Dt!/J> for all q> E dom(D). Clearly D'!/J is just the distributional image of!/J under D, and so Dt = D*. §s. DIRAC OPERATORS 117 We now prove that on complete manifolds, any Dirac operator is essentially self-adjoint (cf. Wolf [1]). Theorem 5.7. Let X be a complete riemannian manifold and let D be the Dirac operator of any Dirac bundle S over X. Then the closure of D in L2(S) is a self-ad joint operator. Furthermore, ker(D) = ker(D2) on L2(S). Proof. Fix Xo E X and let d(x) be a regularization of the distance func-tion from Xo. Choose X E COO(IR) so that 0 X 1, X(t) = 0 for t 2, X(t) = 1 for t 1, and Ix'l 2. Then set Xn(x) = d(X)). We want to show that dom D = dom(D*). It suffices to prove that dom(D*) dom(D) since the reverse inclusion is obvious. Choose cP E dom(D*) and define CPn = XnCP for each positive integer n. Then by Re-mark 5.6 we know that DCPn = (grad Xn) . cP + XnDcp for each n. Clearly, X.Dq> -+ Dq> in L2(S). Furthermore, if we let Bp denote the metric ball of radius p centered at Xo in X, then II(grad Xn) . cpl12 r _ IIcpl12 o. JB2n Bn n Consequently, DCPn -+ Dcp in U(S). Thus, we are reduced to the case where cP E dom(D*) has compact support. By a partition of unity over supp cP we can assume that cP has compact support in a local coordinate system. Here, using F ourier trans-form methods, we can construct a parametrix for D, i.e. a pseudo-dif-ferential operator P such that DP = 1 -9' and PD = 1-9" where 9' and 9" are infinitely smoothing operators, and where P, 9' and fI" all have Schwartzian kernels supported near the diagonal. Observe now that since Dcp E U(S), there exists a sequence <!/In>:'= 1 C rcpt(S), with support uniformly bounded in this coordinate neighborhood, such that !/In -+ Dcp in U(S). We then set CPn = P!/In + 9"cP and observe that since the Schwartzian kernels of P and 9" are supported near the diagonal, each (smooth) CPn has compact support. That is, we have <CPn>:'=l c fcpt(S). From the relations above, we see that CPn -+ PDcp + 9"cP = cP, and that Dq>n = DP!/In + D9"cp = !/In -9'!/In + D9"cp -+ Dq> -9'Dcp + DfI"q> = Dcp (since, clearly, we have D9" = 9'D). This completes the proof of the essential self-adjointness of D. 118 H. SPIN GEOMETRY AND DIRAC OPERATORS We now prove that ker(D2) s.; ker(D). That is, Wf; shall show that any (necessarily smooth) U-section cp of S which satisfies the differential equation D2cp = 0, also satisfies the equation Dcp = O. To see this, let Xn be the sequence of functions above and note that o = = = (Dcp,2Xngrad(Xn) . cp + = IIXnDCPl12 + 2(XnDcp,grad(Xn)' cp). Consequently, by the Schwartz inequality we have Therefore, and we conclude that IIDcpl1 = lim IIXnDcpl1 = O. This completes the proof. n • Having discussed Dirac bundles in general terms, it is now time to look hard at some important examples. We begin with the basic ones. EXAMPLE (an historical case). Let X = !Rn, euclidean n-space, and let S = !Rn X V where V is some finite dimensional module for Ctn. In this case the Dirac operator is a constant coefficient operator (on V-valued functions) of the form n a D= L Yk-k; 1 aXk where each Yk is a linear map Yk: V -+ V and where YjYk + YkYj = -2Jjk for all j,k. If we choose a basis for V, these Yk'S will be represented by matrices. The relations above imply that where = -L is the positive laplacian in !Rn. This particular operator has historical roots in physics. In the 1920s, the physicist P.A.M. Dirac was searching for a Lorentz-invariant first-order differential operator whose square would be the Klein-Gordon operator. Thus he was essentially led to search for a first order operator D of the §5. DIRAC OPERATORS 119 form above which satisfied the equation DZ = Realizing that the "lk'S must be matrices, he was led immediately by this equation to the above relations, which we recognize now as the generating relations of a repre-sentation of Ctn• See the beautiful book of Dirac [2] for a discussion of the physics. It is illuminating to consider this euclidean operator in low dimensions. Let n = 1, so that Ct1 = V = C. Then we have o D=i-oX1 the generator of a basic semigroup of unitary operators on L z. Let n = 2, so that Ctz = V = C EB C. The decomposition of into C EB C is natural and corresponds to the Zz-grading Ctz = Ctg EB Cti. With respect to this Zz-grading, the Dirac operator inter-changes even and odd parts. In particular if we identify Ctg and Cti with C by setting u + veZe1 u + iv ue1 + vez, then D = e1(0/oxd + ez(%xz) has the form where %z == 0/ox1 + iO/oxz. Thus, the Dirac operator on IRz = C, con-sidered as mapping even to odd spinors, is exactly the Cauchy-Riemann operator. Let n = 3, so that Ct3 = EB and V = Ct3 has two representa-tions on given as follows. Identify 1R3 with and let {i,j,k} be the standard basis of imaginary quaternions. Then the two representations are generated by letting i,j,k act on either the right or the left in Choosing multiplication on the left, we get the following expression for the Dirac operator on functions o 0 0 uX1 uxz uX3 If we re-express left quaternion multiplication, with respect to the basis {1,i,j,k}, as 4 x 4 real matrices, then D becomes -01 -oz o -03 03 0 120 H. SPIN GEOMETRY AND DIRAC OPERATORS Let n = 4, so that C£4 = 1HI(2) and V = IHIz = IHI EB IHI. Here again the splitting corresponds to a Zz-grading of the module V, and D interchanges parts. To describe the full Dirac operator we consider first the following quaternion analogue of the Cauchy-Riemann operator. Identify 1R4 with IHI under the standard basis {1,i,j,k}, and define the following operators on functions from IHI to IHI: o 0 0 0 0 -=-+i-+j-+k-3 00.0.0 0 -=--l--j--k-oq oXo oX1 oXz OX3 oij oXo oX1 oXz ox Then the Dirac operator can be expressed with respect to the splitting IHI EB IHI as Note the analogy with dimension two. Note that left quaternion multiplication is always complex linear with respect to the complex structure given by multiplication by i on the right. Thus, left multiplication by i,j and k on IHI = CZ could be represented by complex 2 x 2-matrices 0"1' 0" Z and 0"3 respectively. Under this conven-tion, the operator %ij becomes o 0 0 0 0 :l-= + 0"1 + O"z + uq uXo uX1 uxz uX3 The matrices O"k can be chosen to be the classical Pauli matrices: -I (0 -1) O"z = 1 0' (0 i) 0"3 = i ° Note that these matrices generate the fundamental representation of C£3 in complex form. We could continue this analysis. For general n, one can calculate an enormous N x N-matrix whose entries are linear combinations of a/ox!> ... ,%xn. Here N is on the order of 2n. This matrix will have the property that its square is . I where I is the N x N identity matrix. How-ever, this explicit form of D is seldom, if ever, useful. It is always simpler to use the structure of the Clifford module. It is interesting to note that the concept of D as a generalized Cauchy-Riemann operator is a useful one. Let us fix a dimension n, and let D denote the euclidean Dirac operator acting on functions f: IRn -+ V where V is some fixed C£n-module. Recall that the fundamental solution for the §5. DIRAC OPERATORS 121 Laplace operator on for n > 2, is cp(x) = cn/llxlln-Z for an appropriate constant Cn' That is, we have f1cp = <50 where <50 is the Dirac <5-function at the origin. Since DZ = f1 . I, where I is the identity on V, we have that <I> == D(cpI) is the fundamental solution for D. That is, D<l> = <501. From this one can derive very pretty analogues of the Cauchy Integral Formula, and the analysis of D becomes quite accessible. We now examine Dirac bundles S over more general spaces. Let X be an arbitrary riemannian manifold. Then there are two basic cases. EXAMPLE 5.8. (the Clifford bundle). Let S = Ct(X) with its canonical riemannian connection, and view Ct(X) as a bundle of left modules over itself by left Clifford multiplication. (Note that Property (5.4) was estab-lished in Proposition 4.8.) The Dirac operator in this case is a square root of the classical Hodge laplacian. Most of the remainder of this section is devoted to a detailed analysis of this basic case. EXAMPLE 5.9. (the spin or bundles). Suppose X is a spin manifold with a spin structure on its tangent bundle. Let S be any spinor bundle asso-ciated to T(X). Then S is a bundle of modules over Ct(X), and as shown in §4, S carries a canonical riemannian connection which has property (5.4) (see Proposition 4.11 and the discussion following Theorem 4.17). The Dirac operator in this case was first written down by Atiyah and Singer in their work on the Index Theorem. Finding this operator was a major accomplishment, and for this reason we shall call it the Atiyah-Singer operator. Notation. For spin manifolds X of even dimension we shall denote the (unique) irreducible complex spinor bundle by $e; and when dim(X) ¥= 3 (mod 4), we denote the irreducible real spinor bundle by $. In both cases the Atiyah-Singer operator will be written I/J. These basic examples each generate large families of new examples by the following construction. Let S be a given Dirac bundle with connection Vs over a riemannian manifold X, and let E be an arbitrary riemannian vector bundle with connection "lE over X. Then the tensor product S ® E is again a bundle of left modules over Ct(X), where for cp E ct(X), a E S and e E E, the module multiplication is given by setting CP' (a ® e) == (cp' a) ® e. 122 H. SPIN GEOMETRY AND DIRAC OPERATORS It is clear that in the tensor product metric on S ® E we have that this Clifford multiplication by unit tangent vectors on X is orthogonal (i.e., Property (5.3) is satisfied). Furthermore, we can equip S ® E with the ca-nonical tensor product connection, V == VS ® VE, which is defined on sec-tions of the form (J ® e by the formula V«(J ® e) = (Vs(J) ® e + (J ® (VEe). It is straightforward to verify that with this riemannian connection, the bundle S ® E has the derivation property (5.4). Consequently, we have proved the following. Proposition 5.10. Let S be any Dirac bundle over a riemannian manifold X, and suppose E is any riemannian bundle with connection. Then the tensor product S ® E is again a Dirac bundle over X. An interesting example of this construction is the following. Let X be a spin manifold of dimension Sk, and let $ be the canonical (real) spin or bundle of X. Then under the above construction the bundle $ ® $ is a Dirac bundle over X in two canonical ways (by multiplication in the left or the right factor). It follows from the representation theory (see 1.5.18 and IY.10.16-17) that where the two module structures correspond to mUltiplication on the left and on the right (by the transpose) in Ct(X). Similarly, in all even dimensions we have ct(X) ® c $e ® $t REMARK 5.11. The construction above does indeed give rise to a large number of examples. As we shall see later (Remark 111.13.11), basically every elliptic operator on a spin manifold can be constructed, up to homo-topy and degree shift, as a Dirac operator of the form D: qSo ® E) -+ qS1 ® E) where S = SO EB Sl is the complex Z2-graded spinor bundle. Note that in each of the basic cases above (Examples 5.S and 5.9), the Dirac bundles and their associated Dirac operators are canonically defined in terms of the riemannian metric on X. Hence, any mathematical objects constructed using these operators are invariants of the riemannian struc-ture on X. . The remainder of this section will be devoted to an analysis of these operators. We begin with the Clifford bundle Ct(X). Our first observation is that it is also possible to view ct(X) as a bundle of right modules over Ct(X) (by right Clifford multiplication). Property (5.4) also holds for right multiplication. Hence, we can also define a "right- §5. DIRAC OPERATORS anded" Dirac operator D on Ct(M) by setting n Dep =.L (Vejep)·ej. J= 1 123 (5.9) his operator is also elliptic and formally self-adjoint. The principal sym-01 is just right multiplication by ie. Recall now that there is a canonical isomorphism Ct(X) A *(T*(X)) == *(X). The bundle A *(X) also has two canonical first order operators, namely the exterior derivative d: A *(X) -+ A *(X) and its formal adjoint d*: A *(X) -+ A *(X). This adjoint is given by the formula (5.10) on AP(X), where *: AP(X) -+ N-P(X) is the linear map defined by the condition that ep /\ *t/I = (ep,t/I)*l where *1 is the volume form. The Dirac operators and the exterior derivative are directly related as follows: Theorem 5.12. Under the canonical isomorphism Ct(X) A *(X), the Dirac oeerators of Ct(X) satisfy the following equations: D d + d* D (-1)P(d -d*) on AP(X). Consequently, since d2 = (d*)2 = 0, they also satisfy D2 = D2 = dd* + d*d == DD = DD. The operator defined in (5.13) is called the Hodge laplacian. (5.11 ) (5.12) (5.13) (5.14) Proof. Fix x E X and choose an orthonormal frame field (et. ... ,en) in a neighborhood U of x with the property that (Ve/j)x = O. We first observe the following: Lemma 5.13. The operators d and d* are given in U by the formulas n d=Lej/\Vej j=l n d* = -L ej L Ve j=l J where "L" denotes contraction in A *(X). Proof. Both expressions are invariantly defined; that is, they are indepen-dent of the choice of frame field (el' ... ,en)' To establish the first identity it suffices to show that the operator on the right satisfies the following 124 H. SPIN GEOMETRY AND DlRAC OPERATORS axioms for d: (i) d2ep = 0 (ii) d( ep 1\ 1jJ) = dep 1\ IjJ + ( -!)Pep 1\ dljJ (iii) df = grad(f) for all smooth functions f and all smooth p-forms ep and q-forms 1jJ. Property (iii) is obvious. Property (ii) is seen as follows: n d(ep 1\ 1jJ) = .L ej 1\ [(Vejep) 1\ IjJ + ep 1\ (Vei)] )=1 = dep 1\ IjJ + (-l)Pep 1\ dljJ. (Here we use the fact that V acts as a derivation on A *(X).) To prove prop-erty (i) we first note that from the independence of the choice of frame field, it suffices to verify this at the point x where we have (VeieA = O. Furthermore, by linearity it suffices to consider a ep of the form q> = ae1 1\ ... 1\ ep where a is some smooth function in U. Then one easily sees that at x, d2ep = L ([ej,ek]a)ej 1\ ek 1\ e1 1\ ., . 1\ ep. p<j<k Since [ej,ek]x = (Vejek -Veke)x = 0, property (i) is proved, and the first equation is established. For the second equation, we again consider ep = ae1 1\ ... 1\ ep. Hence, at x we have that Consequently, at x *(d * ep) = f (eja) * (ej 1\ ep+ 1 1\ ... 1\ en) j= 1 = f (_l)n(p+ 1)+ j-l(eja)e1 1\ ... 1\ ej 1\ ... 1\ ep j= 1 n = (-lr(p+ 1) L (eja)ej L (e1 1\ •.. 1\ ep) j= 1 n = (_l)n(p+1) ejL(V';jep). )-1 This completes the proof of Lemma 5.8. • We now recall that under the canonical isomorphism Ct(X) A*(X) we have that ep . e (-!)p(e 1\ ep + e L ep) §5. DlRAC OPERATORS 125 for all e E J\.l(X) and all qJ E J\.P(X) (see the discussion at the end of §3 in Chap. I). Equations (5.11) and (5.12) now follow directly from Lemma 5.13. This completes the proof of the theorem. _ This theorem has the following important consequences. On X the space of harmonic p-forms is defined to be n H == EEl HP = p=o Corollary 5.14. If X is compact and without boundary, then ker(D) = ker(D) = In particular, under the isomorphism ct(X) J\. *(X), the kernels of D and D correspond to the space of harmonic forms on X. Note that by applying Theorem 5.7 to a non-compact complete mani-fold we conclude that a differential form which is in L 2 is harmonic if and only if it is both closed and co-closed. In the compact case the harmonic forms are related to the topology of X as follows. Consider the so-called de Rham complex: 0--+ r(J\.°X) r(J\.lX) r(J\.2X) Since d2 = 0 we can form the quotient: 3{'*(X) == [ker(d)jimage(d)]*. The fundamental theorem of de Rham asserts that for each p = 0, ... , n, 3{'P(X) is isomorphic to the pth singular co homology group of X with real coefficients. Since we are given a riemannian metric, we can also consider t& adjoint sequence: o r(J\.°X) r(J\.lX) r(J\.2X) .... The fundamental result of harmonic theory is the following Hodge De-composition Theorem (Corollary 111.5.6): Theorem 5.15. Let X be compact, and without boundary. Then there is an orthogonal decomposition r(J\. * X) = H EB Im(d) EB Im(d*) (where Im(·) denotes the image of the operator on r(J\.*X)). In particular there is an isomorphism for each p = 0, ... ,n. A Note on Orientability. It is clear that orientability is not required for the definition of D, and so the spaces HP can be defined for a non-orientable 126 n. SPIN GEOMETRY AND DIRAC OPERATORS manifold X. Moreover, if X is compact the isomorphism HP also holds. This is proved as follows. Let n : X -+ X be the two-fold, orientable covering manifold, and let r = {1,g} 71./271. denote the group of covering transformations of X. Then r acts naturally on J\. *(X). The subspace of r-fixed elements is preserved by d, and its cohomology is just the cohomology of X. (Orientability is not required for the de Rham Theo-rem.) We claim that the inclusion of this co homology into the cohomology of X is injective. Suppose cP E J\.f(X) and cp = dljl for some IjI E J\.P-leX). Set 1jI' = 1(1jI + g*IjI). Then 1jI' E J\.f-l(X) and since d commutes with g*, cp = dljl'. This proves the injectivity. Given a riemannian metric on X, we lift the metric to X. The harmonic forms on X lift to r-invariant harmonic forms on X, and every r-invariant harmonic form is such a lift. That is, H*(X) We now observe that corresponds to the subspace c To see this we note that if cp is a harmonic form, so is g*cp. If cp is cohomologous to a closed r-invariant form, then cp and g*cp represent the same cohomology class. Hence cp = g*cp. We conclude that H*(X) as claimed. The fundamental identities (5.13) and (5.14), established above for the operators D and fj, imply certain important identities for the curvature tensor. In particular we have the following. Theorem 5.16. Suppose X is a riemannian manifold, and let R denote the Riemann curvature tensor acting by derivations on the bundle ct(X) J\. *(X). Then for any element cp E ct(X), L eiRe"eicp)ej = 0, i,j (5.15) (5.16) where (eh' .. ,en) is any orthonormal tangent frame at the point in question. In particular, from (5.15) we conclude that L eiejRe"e/cp) = -L Re"e/cp)eiej i<j i<j (5.17) 1 = 2" [eiej,Re"eicp)] 1 = -2 .L. ade,e/Re;,ej(Cp». '<j Note. Since {ei 1\ ej}i<j represents an orthonormal basis of J\.2(X), the formulas above could be easily re-expressed without the use of (el' ... ,en)' Proof. Let us fix a point x E X and choose a local orthonormal tangent frame field (el' ... ,en) such that (Ve)x = ° for eachj. Then for any section §5. DIRAC OPERATORS E qCt(X)), we have at x that D2cp = e;Ve,(e/vejCP) I,} = eiejVe,VejCP t,} = -L Ve,ve,CP + L eieiVe,ve. -Ve.Ve.)CP i i<j J J = -L Ve,ve,CP + L eiejRe"e/CP)· i i<j imilarly, we conclude that at the point x, D2cp = -L Ve,ve,CP + L Re"e/cp)ejei. i i<j 127 ince D2 = D2 (see (5.13)), we conclude that (5.15) holds at x, and therefore verywhere. The second equation is proved similarly. We observe that at x, DDcp = L elVe,Vejcp)ej i,j ubtracting these equations and recalling that DD = DD completes the roof .• Since equations (5.15) and (5.16) are for Clifford elements, each con-titutes 2n scalar equations. They include the Bianchi identities (consider he A1-component.) They also include a large number of new identities or the curvature transformation of the bundle A *(X). These identities ill prove useful in the Bochner-type vanishing arguments presented in 8. We now examine some of the basic operators on Ct(X) and analyse their relationships with D and D. Recall that ct(X) carries a canonical bundle mapping rJ. : ct(X) --+ Ct(X) (5.18) which is, on each fibre Ct(X), the algebra automorphism extending the map -Ion Tx(X). Since rJ.2 = 1, we obtain a decomposition (5.19) where CtO(X) and Ct 1(X) are the 1 and -1 eigenbundles of rJ. respectively. Under the isomorphism Ct(X) A*(X), we have CtO(X) Nven(x) and et 1(X) Ndd(X). For any non-zero vector e E Tx(X) left (or right) Clif-ford multiplication gives an isomorphism .; Ct!(X). Hence, if X admits a nowhere vanishing vector field, i.e., if the Euler characteris-tic of X is zero, then the bundles ctO(X) and Ct1(X) are isomorphic. 128 11. SPIN GEOMETRY AND DIRAC OPERATORS There is a second canonical bundle mapping L: ct(X) ct(X) (5.20) which on ctx(X) is defined by the formula L(<p) = -"LJ= 1 ej<pej for any orthonormal basis el' ... ,en of Tx(X). This map is globally diagonalizable and yields the canonical bundle decomposition n ct(X) = EB AP(X). (5.21) p=o In particular, L = ( -1)P(n -2p) on AP(X) (5.22) for each p = 0, ... ,n (see Chap. I). Clearly IX and L commute, and their composition satisfies IX 0 L = L 0 IX = (n -2p) on AP(X) (5.23) for each p. Finally, we consider the section w of N(X) c Cl',(X) given at each point x by setting (5.24) where el' ... ,en is any positively oriented orthonormal basis of Tx(X). Since w is independent of the choice of basis we may for any x E X choose local fields el' ... ,en such that (Vei)x = 0 for each i. This shows that Vw:=O. The section w satisfies the relations n(n+ 1) w2 = (-1)-2-we = ( -1)n -1 ew for any section e of T(X) c Cl',(X). We now define a canonical bundle map Aw : Cl',(X) Cl',(X) by setting Aw(<p) = W· <po H n := 3 or 0 (mod 4), then = 1 and we have a decomposition Cl',(X) = Cl', + (X) EB Cl', -(X) (5.25) (5.26) (5.27) (5.28) (5.29) where Cl', ±(X) are the ± 1 eigenbundles of Aw. From (5.27) we see that for any non-zero vector e E Tx(X) at any point x, left Clifford mUltiplication §5. DIRAC OPERATORS gives isomorphisms: e:Cl',;(X) --Cl',:(X) e: Cl',;(X) --Cl',;(X) ifn == 0 (mod 4) ifn == 3 (mod 4). The bundles Cl', ±( X ) can be explicitly written as Cl', ±( X ) = (1 ± w)Cl',(X). 129 (S.30) (S.31) (S.32) Each such bundle is evidently a submodule under right Clifford multiplication. The above construction is particularly important since it generalizes immediately to any Dirac bundle S over X. Again we have a bundle isomorphism Aw:S --S and a corresponding decomposition S = S+ EB S-(S.33) (S.34) where S± = (1 ± w)' S. Moreover, for any non-zero e E T;x(X), the for-mulas analogous to (S.30) and (S.31) hold. Note that for n == 1 or 2 (mod 4), A! = -1. Hence if we complexify 8 and consider the operator iAw we obtain a splitting S ® C = (8 ® et EB (S ® q-. In these dimensions, Aw defines a complex struc-ture in S, and this splitting is the usual (1,0), (0,1 )-decomposition for the complex structure. Let us turn our attention back to the Clifford bundle Cl',(X) and ex-amine some of the elementary properties of the operators defined above. Lemma 5.17. The operators a, Land Aw satisfy the following relations (i) aL -La = 0 (ii) aAw + (-It-1 Awa = 0 (iii) LAw + ( -1)n AwL = O. Proof. (i) was observed above. For (ii) we note that a(w<p) = a(w)a(<p) = (-1)"wa(<p). For (iii) we see that by (S.27), L(w<p) = -L eJ.(jJ<pej = -( _1)"-1 L wej<pej = (_1)n-1wL(<p). • It follows from (iii) and (S.22) above that Aw(N(X)) = N-P(X). In fact .1. .. is related to the *-operator (cf. (S.10)) as follows: 1 p(n-p)+2"P(p+1) w<p=(-1) *<p 1 2" p(p+ 1) <pw = ( -1) *<p for <p E N(X). (S.3S) 130 11. SPIN GEOMETRY AND DIRAC OPERATORS To see this it suffices to consider <p = e1 ••. ep• Then *<p = ep+ 1 .•• en and the computation is easy. Lemma 5.18. The operators iX, Land Aw considered as sections of Hom(Cl',(X),Cl',(X)), are globally parallel. That is, [V,iX] = [V,L] = [V,Aw] = O. Proof. That iX is parallel follows from the fact that -1 is parallel in Hom(T(X),T(X)) and V is a derivation. That Aw is parallel follows from (5.25). For L we fix a point x E X and choose a local orthonormal frame field el' ... ,en such that (VeJ. = 0 for eachj. Then at the point x, V(L<p) = -V L efPej= -L {(Vej)qJej+e,;(V<p)ej+ej<p(Vej)} = -L This completes the proof. • Corollary 5.19. The subbundles AP(X) and Cl', ±(X) (when defined) are preserved by covariant differentiation. We now examine the relationship of these operators to the Dirac operators. Proposition 5.20. Let D and D be the Dirac operators on C£(X) defined above. Then the following relations hold (i) Da. + a.D = Da. + a.D = 0 (ii) DAw + (-I)nAwD = DAw -AP = 0 (iii) DL + LD = 2D; DL + LD = 2D. Corollary 5.21. The operator Ll = D2 = D2 satisfies the relations [a., Ll] = [Aw,Ll] = [L,Ll] = O. In particular we have that Aw: RP Rn-p. This is the Poincare Duality Isomorphism. Proof of Proposition 5.20. We shall consider only D. The arguments for D are similar. Let <p be a section of C£(X). Then using the lemmas above, we have (i) D(iX<p) = L ejVeiiX<p) = L ep,(Vej<p) j j = -iX( ejvej<p) = -iX(D<p) (ii) D(w<p) = 4: ejVeiw<p) = L ejwVej<p J = (_I)"-lW L ejVeJ<p = (_I)"-lWD<p §5. DIRAC OPERATORS (iii) D(Lcp) = I e/Ve.(Lcp) = I ejL(Ve'cp) . J • J ] ] = -I eh(Ve·cp)ek j,k J = -(-ekej -2bk)(V ejCP )ek ],k = -L(Dcp) + 2D(cp). This completes the proof. • The above arguments carry directly over to the following case. 131 Proposition 5.22. Let S be any Dirac bundle over X. Then the Dirac opera-tor on S satisfies the relation We shall conclude this section with some remarks on the variation of the Atiyah-Singer operator under changes of metric. Notice that all of the standard bundles of riemannian geometry-the tangent bundle, the cotangent bundle, and their tensor products-have structure group GLn(lR). They exist independently of any metric con-siderations. In fact, the introduction of a riemannian metric amounts to a (simultaneous) reduction of the structure group of these bundles to SOn. When X is a spin manifold, the situation for the canonical spinor bundle of X is completely different. The spinor bundle itself depends on the choice of riemannian metric. This last statement can be made precise as follows. Let PGv(X) be the oriented frame bundle of X and suppose that dim (X) = n > 2. Denote by Gt:(IR) -+ GL:(IR) the 2-fold, universal covering group of GL:(IR). Since X is spin, there exists a principaIGLn+(IR)-bundle Pa+(X) with a GLn+(IR)-equivariant bundle map PG'v(X) -+ PGL+(X), Introducing a riemannian metric gives a reduction of the structure groups and a commutative diagram 1 1 Now one could hope for the existence of a finite dimensional represen-tation a:GL:(IR) -+ GL(V) whose restriction to Spinn is an irreducible 132 11. SPIN GEOMETRY AND DIRAC OPERATORS spinor representation. The associated vector bundle S = PGi..+(X) xa V would then be the canonical spinor bundle, and choosing a metric on X would induce a metric on S (as in the case of the tensor bundles.). This would give us a fixed vector bundle on which we could consider a family of Atiyah-Singer operators associated to each variation of the riemannian structure on the base. Unfortunately, this hope cannot be fulfilled. Lemma 5.23. The Lie group Gln+(IR) (n> 2) has no finite dimensional representations other than those which descend to GL: (IR). Proof. Consider the subgroup SLn(lR) c GLn+(IR) and its 2-fold, universal covering group SLn(lR) c at: (IR). Since SLn(lR) contains the kernel of the homomorphism Gl:(IR) -+ GL: (lRli,t will clearly suffice to prove the assertion for this subgroup. Let qJ: SLn(lR) -+ GLN(IR) be any Lie group homomorphism. Let qJ*: sI.(IR) -+ gIN(IR) denote the associated Lie algebra homomorphism, and consider its complexification qJ* ® IC: sIn(1C) -+ gIN(IC). Since SLn(1C) is simply-connected, the elementary theory of Lie groups tells us that qJ* ® IC is induced by a homomorphism of Lie groups <1>: SLn(1C) -+ GLN(IC). It follows that qJ = <I> iSLn(lkl) in a neighborhood of the identity. By the uniqueness of analytic continua-tion this identity holds everywhere, and so the representation descends to SLilR) as claimed. • Note that the argument just given does not apply to the conformal group Cn == {g E GLn(IR):g = Ago for A E IR+ and go E SOn}. Indeed, con-formal changes in the metric on the base manifold can be lifted to a fixed spinor bundle, and one can study there the associated change in the Atiyah-Singer operator. A basic and important fact is that the Atiyah-Singer Dirac operator remains essentially invariant under all conformal changes of the metric. We now make this last statetnent precise. Fix a riemannian spin mani-fold X with metric <.,. ), and consider the conform ally related metric <.,.)' = e2u<.,.) where u is some smooth function on X. Let X' denote this riemannian manifold with metric <-,. )'. To each orthonormal tangent frame 8 = {el, ... ,en} on X we can associate the orthonormal frame "'(8) = {e'l, ... on X', where ej = exp( -u)ej for each j. This gives us an SOn-equivariant map"': Pso(X) -+ Pso(X') which lifts to a Spinn-equivariant §5. DIRAC OPERATORS 133 map (5.36) between principal Spinn-bundles. (We have chosen the spin structure on X' which is topologically equivalent to the one given on x.) For any fixed spinor representation f.1: Spinn --+ SO(M) we have associated spin or bundles S = PSPin(X) xJl M and S' = PSpin(X') xJl M, and the map (5.36) gives us a bundle isometry tfiJl:S--S'. (5.37) We now modify this isometry by setting (5.38) The resulting map 'P: S --+ S' is a bundle isomorphism which is conformal on each fibre. The basic result is the following: Theorem 5.24. Let I/J: r(S) --+ r(S) and I/J': r(S') --+ r(S') be the canonical Atiyah-Singer Dirac operators defined over the conformally related riemann-ian manifolds X and X' respectively. Then Corollary 5.25. Let I/J and I/J' be as in 5.24. Then dim(ker I/J) = dim(ker I/J'). (5.39) In other words the dimension of the space of harmonic spinors remains con-stant under pointwise coriformal changes of the riemannian metric. Note. Suppose we have a decomposition defined by a canon-ical volume element as in the next chapter. Then Corollary 5.25 applies to each of the operators I/J + and I/J -, that is, dim(ker I/J ±) = dim(ker(I/J') ±). Proof of Theorem 5.24. We have two metrics <.,.) and <.,)' defined on the same vector bundle T = TX = TX'. Let V and V' respectively de-note the associated canonical riemannian connections. It is straight-foward to verify that for vector fields V and W we have = VyW + (V·u)W + (W·u)V -<V,W)grad(u) where the gradient is taken in the metric <.,. ). (Check the axioms.) Sup-pose now that $ = {el' ... ,en} and tfi($) = {e'l> ... are local ortho-normal frame fields for < ., . ) and < ., . )' respectively, and let wji = <Vei, e j) and Wji = <V' e;, ej)' be the associated I-forms. One easily finds that for 134 11. SPIN GEOMETRY AND DIRAC OPERATORS any tangent vector V, WJi(V) = Wji(V) + (ei' u)<V,ej) -(ej· u)<V,e). The local tangent frame field $ = {e1, ••• ,en} determines a local frame field [I' = {a l' ... ,aN} for S. Similarly, $' = {e'l, ... determines a frame field [1" = {a'l," . for S' where aj = t/lJl(a) for each j. From (4.35) we see that the induced connections on Sand S' are related as follows: = aa + t(grad(u)' V-V, grad(U))aa}' Since grad(u)' V = -V·grad(u) -2<grad(u), V), we conclude the following. Lemma 5.27. Let VS and Vs' denote the riemannian connections on Sand S' respectively. Then Corollary 5.28. I/J' = t/lJl 0 {I/J + (n -l)grad(U)} 0 t/I;; 1 We now observe that for any constant, iX, and therefore I/J(eaUa) = eau( I/Ja + iX (eju)ep) = eaU(I/Ja + iXgrad(u) . a), 'I' 0 I/J 0 '1'-1 = e --2-ut/lu 0 I/J 0 e-2-ut/I;;1 n-1 (n-1) = t/lJl 0 (I/J + (n -l)grad(U)) 0 t/I;; 1 = I/J'. • §6. FUNDAMENTAL ELLIPTIC OPERATORS §6. The Fundamental Elliptic Operators 135 In this section we shall use the Clifford bundle and its modules to sys-tematically derive the fundamental elliptic operators in riemannian geom-etry, that is, the Euler characteristic operator, the signature operator, and the Atiyah-Singer A-operator. The basic construction is the following. Let S be a Dirac bundle with Dirac operator D over a riemannian manifold X, and suppose that S is Z2-graded. This means that there is a parallel decomposition (6.1) so that cti(X) . Si Si+ j for all i,j E Z2' From the definition (5.1) of the Dirac operator it is clear that D is of the form D=(;o (6.2) where and (6.3) Since D is self-adjoint, we see that DO and Dl are adjoints of one another. The ellipticity of D established in §5 implies that each operator Dk is also elliptic. In fact, the principal symbol of Dk at a cotangent vector is simply Clifford multiplication by that is, = Sk Sk+ 1 for k E Z2 (6.4) (see Lemma 5.1). Since . = we see that is an isomorphism for =I-O. It is a fact that over a compact manifold, the kernel and cokernel of an elliptic operator P are of finite dimension, and a basic invariant of P is its index which is defined as ind P = dim(ker P) -dim(coker P). (6.5) Since Dl is the adjoint of DO, there is an isomorphism ker(Dl) coker(DO), and so we have that ind DO = dim(ker DO) -dim(ker Dl). (6.6) From (6.2) it is clear that ker D = ker DO EB ker Dl. (6.7) In particular, if D is injective, then ind DO = O. Whenever X is oriented and even-dimensional, we recall (cf. (3.l1)ff) that there is an important method for introducing a Z2-grading on any Dirac bundle S. Let Wee be the complex volume element, given in terms of a posi-tively oriented orthonormal tangent frame (el' ... ,e2m) by (6.8) 136 n. SPIN GEOMETRY AND DIRAC OPERATORS where 2m = dim(X). This is a globally defined section of Cl',(X) = Cl',(X) ® C with the following properties. VW<c = 0, (6.9) = 1, (6.10) w<ce = -ew<c for any e E TX. (6.11) Property (6.9) follows easily from the derivation property (4.26) of V. (Fix x E X and choose e/s with (Vej)x = 0.). For the rest, see Proposition 3.3 of Chapter I. Suppose now that S is a Dirac bundle over X (complex if m is odd). Then S has a decomposition S = S+ EB S-(6.12) into the + 1 and -1 eigenbundles for multiplication by W<c. These bundles can be simply expressed as (6.13) From (6.9) we see that this decomposition is parallel, and from (6.11) we see that for any e E T X, (6.14) This means that after identifying SO with S + and Sl with S -, the decom-position (6.12) gives a d'rgrading on S. We now examine some important examples of this construction. EXAMPLE 6.1 (the Euler characteristic operator). Let X be a compact riemannian manifold and consider the basic case where S == Cl',(X) = Cl',o(X) EB Cl',l(X) (cf. (3.2)). By Theorem 5.7 we see that under the canonical isomorphism Cl',(X) A *(X), the operator DO : r(Cl', O(X)) --+ r(Cl', l(X)) corresponds to the operator Consequently, we have ind DO = dim Reven -dim Rodd = the Euler characteristic of X. EXAMPLE 6.2 (the signature operator). Let X be a compact, oriented riemannian manifold of dimension 4k and consider again the basic case S == Cl',(X) = Cl', +(X) EB Cl', -(X) §6. FUNDAMENTAL ELLIPTIC OPERATORS 137 where the Z2-grading is now given as above by the complex volume ele-ment Wee = (-l)kw (see (6.12) and also (5.26)ff). There is a corresponding decomposition ker D = ker D+ EB ker D-. Since Wee is parallel, it preserves ker D and the subspaces ker D ± are just the ± 1 eigenspaces under multiplication by Wee on ker D. That is, ker D± = (1 ± wdker D. Now, under the canonical isomorphism ct(X) A *(X) we know that ker D B = HO EB ... EB H4\ the space of harmonic forms (see Corollary 5.9). Furthermore, under this isomorphism, left multiplication by wee cor-responds to the Hodge *-operator, that is, for <p E AP(X), p(p-l) k+--wee'<p=(-l) 2 *<p (6.15) (see (5.35)). Consequently, for each p = 0, ... ,2k we have an isomorphism Wee: HP H4k-p. This, in turn, implies that the space H(p) = HP EB H4k-p, for p < 2k, has a decomposition H(p) = H+(p) EB H-(p) where the subspaces H±(p) = (1 ± wdH(p) are of the same dimension. Since ker D± = H± = H±(O) EB ... EB H:t(2k -1) EB (R2k)±, where (H2k)± = (1 ± wdR2k, we conclude that ind D + = dim(H2k) + -dim(H2k)-= sig(X). For this last statement we recall that the signature of X, denoted sig(X), is defined to be the signature of the quadratic form Q(<p,t/J) = Ix <p A t/J ([<p] u [t/J])([X]) on H2k H2k(X;IR). Since * Wee in dimension 2k and since Ix <p A *<p = 11<p112, we see that this signature is just the difference of the dimensions of the + 1 and - 1 eigenspaces of * on H2k. EXAMPLE 6.3 (the Atiyah-Singer A-operator). Let X be a compact rie-mannian spin manifold of dimension 4k and consider the complex spin or bundle $ee with Dirac operator I/J. We split $ee $t EB $i-under the 138 11. SPIN GEOMETRY AND DIRAC OPERATORS complex volume element as above. Then it is a consequence of the Atiyah-Singer Index Theorem in Chapter III that ind()b+) = A(X) where A(X) is a rational Pontryagin number of X called the i-genus. We shall examine this invariant in detail in §11 of Chapter Ill; however, some discussion of it is in order here. The A-genus has an important multiplicative property like that of the sig-nature. If X and Y are compact oriented manifolds. Then A(X x Y) = A(X) x A(Y). (6.16) The A-genus is, in general, not an integer. For example, for a compact 4-manifold X, it is a fact that 1 1 -A(X) = 8" sig(X) = 24 PI(X) (6.17) where Pl(X) is the first Pontryagin number of X. In particular, for the complex projective plane [P2(C), we have H2(IP2(1C)) 7L. and so the sig-nature is 1. It follows that A([P 2 (IC) ) -1/8. The index of an elliptic operator is, of course, an integer. Hence, we conclude the following result (cf. Atiyah-Hirzebruch [1]). The A-genus of a compact spin manifold is an integer. (6.18) Note also that from (6.17) we retrieve the basic fact that the signature of a spin 4-manifold must be a multiple of 8 (see Corollary 2.12 and the fol-lowing discussion). An important set of spin manifolds with non-zero A-genus is provided by the hypersurfaces V2n(d) of complex projective space [p2n + l(1C) (see Example 2.7). Recall that the manifold v2n(d) is spin if and only if the degree d is even. It is a fact that (cf. Lawson-Michelsohn [1]) A(V2n(d)) = 2 -2nd fI (d2 -(2k)2). (6.19) (2n + I)! 1 Thus, each of the spin manifolds V2n(2d), for d > n, has non-zero A-genus. Taking products of these gives further examples by (6.16). Each of the fundamental examples above gives rise to a family of as-sociated operators by the process of taking "coefficients in a bundle." This works as follows. Let S = SO EB SI be a 7L.z-graded Dirac bundle as before, and let E be any riemannian bundle with connection over X. Then the bundle S ® E = (So ® E) EB (Sl ® E) is again a 7L.z-graded Dirac bundle over X (see Proposition 5.10). §7. eCk-LINEAR OPERATORS 139 EXAMPLE 6.4. Let S = ct(X) = ct +(X) Ct -(X) be as in Example 6.2. Then for any bundle E over X we can construct the twisted signature operator Dt : qCt +(X) ® E) --+ qct -(X) ® E) whose index is sig(X;E) == {ch2E' L(X)}[X]. This formula is explained in detail in Chapter Ill. EXAMPLE 6.5. Let = $t $i be the complex spinor bundle of Ex-ample 6.3. Then for any bundle E over X we can construct the twisted Atiyah-Singer operator J/)t : r(st ® E) --+ r(Si ® E) whose index is A(X;E) == {ch E· A(X)} [X]. Again see Chapter III for details. §7. Ctk-Linear Dirac Operators There is a variation of the constructions given above, due to Atiyah and Singer, which has proved to be very important in riemannian geometry. To introduce the concept we return to a point mentioned earlier (in Example 3.7). Let PSPin(X) be the principal Spin.-bundle of an n-di-mensional spin manifold X. Then from the representation t: Spin. --+ Hom(Ct.,Ct.) given by left multiplication, we have the associated vector bundle (7.1) Since right multiplication commutes with t, we see that there is a right action of the algebra Ct. on the bundle which preserves the fibres. This action makes a bundle of rank-l Ct.-modules. The idea now is to construct elliptic operators and an appropriate index theory which take into account this action of ct •. We shall begin with the construction of such an operator in the basic case of Note first that the action of ct. on clearly commutes with Clifford multiplication by elements of ct(X). Since is associated to PSPin(X), it carries the canonical riemannian connection, and as such it is clearly a Dirac bundle over X. In fact, as a vector bundle is simply a direct sum o/irreducible (real) spinor bundles 0/ X. (This comes from the decomposition of ct. into irreducible modules under left-multiplication.) The right action of Ct. on is parallel in the riemannian connec-tion, i.e., for any section a E and any element qJ E ct., we have V(a' qJ) = (V a) . qJ. (This is evident since the holonomy in is left multiplication by elements of Spin •. It can also be seen directly from the methods of §4.) 140 n. SPIN GEOMETRY AND DIRAC OPERATORS From the definition (7.1) we see that the decomposition ct" = ct; gives rise to a parallel decomposition (7.2) which is not only a Z2-grading over the bundle Ct(X) but also over the free Ct,,-action. That is, we have (7.3) for all i, j E 1"2' Since is a Dirac bundle it carries a canonical Dirac operator which, with respect to the decomposition (7.2) is of the form ( 0 = 0 . (7.4) Furthermore, this operator commutes with the action of Ct". To see this, note that = L ejVe/(J(p) = L (ejVep)qJ = since multiplica-tion by qJ is parallel and commutes with multiplication by elements from ct(X). This operator is called the Ct,,-Iinear Atiyah-Singer operator of X. We now proceed as above and consider the restricted operator _ From (7.3) and the paragraph above we conclude the following: Lemma 7.1. The operator is a real, elliptic first-order operator which commutes with the action of Ct,,-l on = This construction is sufficiently important that we shall axiomatize it. DEFINITION 7.2. By a Ct,,-Dirac bundle over a riemannian manifold X we mean a real Dirac bundle over X, together with a right action ctk '-+ which is parallel and commutes with multiplication by elements of ct(X). This can be thought of as a Dirac bundle which carries "scalar multi-plication" by ctk• Notice that a Ct1-or ClrDirac bundle is just a com-plex or quaternionic Dirac bundle respectively. DEFINlTION 7.3. A Ctk-Dirac bundle is said to be Z2-graded if it carries a Zrgrading = as a Dirac bundle, which is simulta-neously a Z2-grading for the Ctk-action (that is, (7.3) is satisfied). Any Ctk-Dirac bundle has a canonically associated Dirac operator !t) which commutes with the Ctk-action. If is Z2-graded, then !t) de- §7. eCk-LINEAR OPERATORS composes as in (7.4), and we get an elliptic operator !t)0: r(5°) --+ r(51) which commutes with the action of Ctk-1• 141 (7.5) When X is compact, we can directly define an analytic index for such operators as follows. Since !t)0 commutes with Ctk-1, the kernel of !t)0 is a finite dimensional C£k-cmodule, and thereby ker!t)° deter-mines an element in the Grothendieck group 9Jlk-1 of such modules. Consider now the residue class of this element in 9Jlk-di*9Jlk, where i* is induced by the homomorphism i*: Ctk-1 Ctk determined by the in-clusion map i: IRk-1 4 IRk. Recall (from Chap. I, §9) that these quotient groups are naturally isomorphic to the KO-groups of a point. DEFINITION 7.4. Let 5 = 5° EEl 51 be a Z2-graded Ctk-Dirac bundle over a compact manifold. Then the analytic index of the Dirac operator '1:)0: r(5°) r(51), denoted by indk(!t)O), is the residue class [ker !t)0] E 9Jlk-di*9Jlk KO-k(pt). We recall that the groups KO-*(pt) are the same as the stable homotopy groups of the orthogonal group, that is, k == 0 (mod 4) k == 1 or 2 (mod 8) otherwise. (7.6) A standard argument shows that this index is constant under deforma-tions of the operator. One of the deepest aspects of the work of Atiyah and Singer is the computation of this index topologically (see 111.16). The applications of this result are among the most far-reaching in all of dif-ferential geometry. For this reason we devote the remainder of this section to a detailed examination of such operators particularly in the cases of fundamental interest in geometry. We begin with a remark which will be useful when studying the multi-plicative properties of the operator. Recall from Chapter I (Proposition 5.20) that there is a natural equivalence between the category of (ungraded) modules over ctk -1 and the category of Z2-graded modules over ctk• This equivalence induces a natural isomorphism (7.7) where as before 951k denotes the Grothendieck group of finite dimensional Z2-graded IR-modules over ctk• The map from the graded to the ungraded case is given by taking the even part. Given now a Dirac operator'!): r(5) r(5) as above, we see that ker(!t)) is a Z2-graded Ctk-module. The even part of ker(!t)) is exactly 142 11. SPIN GEOMETRY AND DIRAC OPERATORS This gives the following: ALTERNATIVE DEFINITION 7.4. Let 5 = 5° 51 be a Z2-graded ctk-Dirac bundle over a compact manifold. The analytic index of the Dirac operator of 5 is the residue class [ker E 951Ji*951k+ 1 KO -k(pt). Via the isomorphism (7.7) this index coincides with the index [ker given in Definition 7.4. This second definition is actually more natural and is important for understanding the multiplicative properties of the index. With this second definition we see that the Clifford index generalizes the classical one in that indo(·) = ind(·) = dimlJ;l(ker·) -dimlJ;l(coker·). To see this, note first that cto = IR and Ct1 = IC. A Zz-graded Cto-module is just a pair of real vector spaces VO V1. Now [V 0] = -[0 V] in 9510/i*9511 because V V V ® IC extends to be a graded Ct1-module. Consequently, = [ker ker [ker 0] -[ker 0] as claimed. REMARK 7.5. All of these constructions could be carried out in the complex category. One could consider complex Ctk-Dirac bundles, etc. Here the index will be valued in if k is even if k is odd. Unfortunately, this leads to nothing essentially new, so we have con-centrated our attention on the real case. Some examples are in order. The first and most illuminating one comes by taking the "Ctk-ification" of an ordinary elliptic operator. EXAMPLE 7.6. Let S = SO EEl Sl be an ordinary real Z2-graded Dirac bundle over a compact manifold X, and let be its Dirac operator. We now consider an irreducible real Z2-graded module V = vo V1 over the Clifford algebra ctb and take the tensor product where V is here considered as the trivialized bundle V x X X. This bundle is, in a natural way, a Zz-graded Ctk-Dirac bundle. The grading 5 = 5° 51 is given by 5° = (SO ® VD) (Sl ® V1) and 51 = (SO ® V1) (Sl ® VD). §7. eCk-LINEAR OPERATORS 143 Of course, multiplication by ctk takes place on the V-factor. The asso-ciated Dirac operator on is simply the extension of D, i.e., 5 = D ®Idv. Consequently, we have that and therefore / where bj == dimlJ;l(ker Dj). In passing to the quotient 9Jlk-di*9Jlk we see that [VD] = -[Vl] since VO V1 is a Ctk-module. (Caution: one must show that VO V1 can be made a Ctk-module in such a way that VO and V1 are invariant subspaces under the subalgebra Ctk-1 c ctk• This requires a case by case check modulo 8.). Hence, and since [VD] generates this group for each k (see 1.9) we conclude that if k == 0 (mod 4) if k == 1 or 2 (mod 8) otherwise. (7.8) This formula shows that, as one would expect, nothing essentially new can be found by this trivial construction. The interesting examples are those where the Ctk-structure is more intrinsic to the geometry. This is the case in the following: EXAMPLE 7.7 (the Kervaire Semicharacteristic). Let X be a compact oriented (riemannian) manifold, and take 5 to be the Clifford bundle Ct(X) = cto(X) ct l(X) (considered as a Z2-graded Dirac bundle as in Example 6.1). If X has dimension 4t + 1, then 5 is naturally a Ct1-Dirac bundle as follows. Consider the oriented volume form w = e1 .. ; e4t+ 1 of X. This form is parallel and satisfies w2 = -1. Hence, right multiplication by w in ct(X) makes ct(X) a Ct1-Dirac bundle. Since w is of odd degree, right multiplication by w interchanges CtO(X) and Ct1(X). Hence, ct(X) is Z2-graded as a Ct1-Dirac bundle. Hence, the operator = DO (d + d*)iAeven has an index in KO-1(pt) Z2' To compute this index we want to find the residue class ofker if9Jlo/i*9Jl1. Since ct1 C and cto IR, we easily identify 9Jlo as the Grothendieck group of equivalence classes of real finite dimensional vector spaces, and 9Jl1 as the complex analogue. Hence i*9Jl1 is given by spaces of even 144 11. SPIN GEOMETRY AND DIRAC OPERATORS dimension, and we clearly have == dimlJ;l(ker DO) (mod 2) == dimlJ;l(Beven) (mod 2) 2t == L b2iX) (mod 2) where bi(X) denotes the ith Betti number of X. In other words, is exactly the Kervaire semicharacteristic of X, a basic cobordism invariant of an oriented (4t + I)-manifold. Note that in order that the volume element w have square -1 and be of odd degree, one is restricted to dimensions 4t + 1. It should be pointed out that an exactly analogous construction fails for the signature complex (Example 6.2), since the subbundles Ct ±(X) = (1 ± w,dCt(X) are each invariant under right multiplication by any Clifford element. Hence, this is never a grading for right multiplication. We now pass to an example which illustrates this construction. THE FUNDAMENTAL CASE (the Atiyah-Milnor-Singer Invariant). Let X be a compact spin manifold of dimension n, and let = EEl be the Zz-graded Ctn-Dirac bundle given by (7.1). It is a conse-quence of the Atiyah-Singer theorem that indn == is a spin cob or-dism invariant of X, and in fact gives a graded ring homomorphism (7.9) This homomorphism coincides with the one defined homotopy-theoreti-cally in (3.22). We shall return to this in Chapter Ill. Because of their fundamental nature, it is useful to examine these Ctn-spinor bundles in some detail. We begin with the simplest case where n = 8k. Consider an irreducible real left module VSk for the Clifford algebra ctSk. Let VSk denote the right Ctsk-module obtained from VSk by simply multiplying by the transposed element, i.e., by setting v· <p == <pt • v for <p E ctSk and v E VSk. Then there is an isomorphism of bimodules (7.10) To see this we recall from Proposition 5.18 of Chapter I, that in all even dimensions we have the complex bimodule isomorphism (7.11 ) where denotes the irreducible complex Ct2k-module and where Ct2k Ct2k ®1J;l C. The assertion (7.10) follows from (7.11) since in dimen-sions 8k the complex case is simply the complexification of the real one. §7. Ctk-LlNEAR OPERATORS We conclude from (7.1) that in these dimensions 145 = PSPin X t ctSk = PSPin X aSk ® 1 (VSk ® YSk) (7.12) = (P Spin X aSk VSk) ® VSk = $(X) ® YSk where $(X) denotes the real spinor bundle of X, and where YSk now be-comes the constant (trivialized) VSk bundle. Recall now that VSk has two inequivalent Z2-gradings, obtained one from the other by interchanging the factors. Either one gives the same Zz-grading on the tensor product VSk ® YSk' and the bimodule isomor-phism (7.10) is a Z2-graded one. We are now exactly in the situation of Example 7.6. That is, = I/> ® Idy, where I/> is the Atiyah-Singer operator on spinors. We can there-fore read off from (7.8) that = ind(I/>°) = A(X). (7.13) The situation in dimensions 8k + 4 is very much similar. The principal difference is that in these dimensions the irreducible real module VSk+4 for CtSk+4 is also the irreducible complex module; that is, (7.14) (Recall that CtSk+4 is a quaternion, and hence also a complex, matrix algebra.) From (7.11) we know that ®<c CtSk+4 CtSk+4 ®1J;l C, which yields the real bundle isomorphism (7.15) This implies, as above, that there is an isomorphism of real Z2-graded Ctsk+4-bundles (7.16) where $dX) is the complex spinor bundle of X and is the tri-vialized bundle. Hence, we have that = I/><c ®<c Idy!:. Using (7.14) and the arguments above, it is not difficult to see that . ° 1 mdsk+ ) = 2" A(X). (7.17) Notice that implicit in equation (7.17) is the fact that in these dimensions the A-genus of a spin manifold is an even integer. A direct proof of this can be given by observing that the Atiyah-Singer operator in these dimensions is quaternion linear, and so the complex dimensions of its kernel and cokernel are even. This fact, in dimension four, is just Rochlin's Theorem 2.13. 146 n. SPIN GEOMETRY AND DlRAC OPERATORS We now examine this index in the interesting cases where it is an integer modulo 2. We begin with dimension 8k + 1 and the observation that from the classification of Clifford algebras we have (7.18) In fact, the element i in this algebra corresponds exactly to the volume form 0) = e1 ... eSk+ 1> which is clearly central and has 0)2 = -1. Thus, the decomposition in (7.18) gives the Z2-grading on ctSk+ l' In particular, we have and These isomorphisms show that VSk+1 VSk ® C VSk iVSk (7.20) and Combining this with (7.11), we find that I[; -I[; CtSk+1 = ctSk VSk ®I[; VSk = (VSk ® q ®I[;CVSk ® q ° -0 = (VSk+1 ® q ®I[; (VSk+1 ® q ° -0 = (V Sk+ 1 ® q ®1J;l V Sk+ 1 -0 = VSk+1 ®1J;l VSk+1' This implies as above that -0 $(X) $(X) ®1J;l V Sk+ 1 (7.22) and that I/> ® Idyo. It follows immediately that = HO ® vgk+ 1 where HO == ker(I/>°). Thus we have that (mod 2). (7.23) This index can be reinterpreted in more elementary terms as follows. Observe that from the discussion above, we have in these dimensions that the spinor bundle is the complexification of a real bundle, i.e., $(X) = $o(X) 0) $O(X). We can construct from the Dirac operator 1/>0: r($o(X)) r($l(X)), an operator P: r($o(X)) --+ r($o(X)) (7.24) by setting P == 0)' 1/>0. (7.25) This operator is elliptic and skew-adjoint. To prove the skew adjointness, we note that since 0) is parallel, commutes with I/> and satisfies 0)2 = -1, §7. eCk-LINEAR OPERATORS 147 e have (PO','t') = (w,p°O','t') = (w2,p°0',w't') = -(,p°O',w't') = -(O',,p°w't') = (O',w,p°'t') = -(O',P't'). It is an elementary fact that the parity of a real skew-adjoint Fredholm perator is conserved under deformations (cf. 111.10). Thus for a skew-djoint elliptic operator P on a compact manifold, the mod-2 index ind;!'2(P) == dim(ker P) (mod 2) (7.26) s well defined. The result (7.23) can be reexpressed as indSk+ = ind;!'2(P) (7.27) here P is given by (7.25). In the final case of dimension 8k + 2 there is a strongly analogous ituation. Here we have from the classification of Clifford algebras that CfSk+2 CfSk ® IHI VSk+2 VSk ® IHI and and Cfgk+2 CfSk ® C, (7.28) Vgk+2 VSk ® C. (7.29) Using (7.11) one can deduce from here that CfSk+2 VSk+2 ®1Hl Vsk+2 -0 VSk+2 ®<c VSk+2' It follows as before that -0 $(X) ®<c V Sk + 2 (7.31) with respect to which ,p ® Idv. Consequently, ker 1110 -0 ker(.., ) ®<c VSk+2, and so (mod 2). (7.32) Now since the representation VSk+ 2 is IHI-linear, the bundle $(X) carries a parallel quaternion structure, i.e., there are parallel endomorphisms I, J, K of $(X) which satisfy the standard quaternion relations: 12 = j2 = K2 = -1, IJ + J1 = 1K + K1 = JK + KJ = O. The Z2-grading on $(X) can be written in terms of these as We can then define a skew-hermitian operator P = J o,po on the bundle $O(X), and as before we have indsk+ == ind;!'2(P) where ind;!',(P) == dimdker P) (mod 2). (7.33) (7.34) 148 n. SPIN GEOMETRY AND DlRAC OPERATORS We shall now examine this mod 2 index of the Atiyah-Singer operator in low dimensions. The computations here are of some importance since, as we shall see, the map ind*: -4 KO -*(pt) is a ring homomor-phism, and the non-zero element '1 E KO -1(pt) has the property that: '1x and '12 x are not zero whenever x is an odd multiple of the generator in degree 8k. EXAMPLE 7.8 (the circle as a spin manifold). Consider the circle SI with a riemannian metric. The oriented orthonormal frame bundle is canon-ically diffeomorphic to the circle itself, i.e., PSO(SI) si, since there is exactly one oriented unit tangent vector at each point. A spin structure on SI is a 2-fold covering PSPin(SI) -4 PSO(SI) of the circle. As we noted in Chapter I, there are two such coverings, one connected and one with two components; and it is the non-connected covering Si x 7L2 -4 Si which is not spin-cobordant to zero. We shall refer to this as the interesting spin structure on Si. Notice that since C£1 C we have that = SI X C where the product structure gives the connection. Of course IR and the 7L2-grading on C£1 is the standard decomposition C = IR EB ilR. Thus, = SI X IR and = SI X iIR. Sections of are just complex-valued functions f(s) on the circle, and the Dirac operator is simply where s is arc-length on SI. .d ds (7.35) The kernel of -4 is the set of real-valued constant functions on SI. The dimension of this space as a C£ 0 IR module is one. Hence, we have that for the interesting spin structure on SI, ind1(SI) =I 0, (7.36) i.e., ind1(SI) is the generator of KO-1(pt) 7L2• EXERCISE. Show directly that ind1 = ° for the "uninteresting" spin struc-ture on SI. Show also that the connected sum of the intersecting spin structure with itself is the uninteresting one. EXAMPLE 7.9 (the torus as a spin manifold). Let T = Si X Si be the flat square torus. The oriented orthonormal frame bundle is canonically trivialized §7. C£k-LINEAR OPERATORS 149 The spin structure on T given by squaring the interesting spin structure on the circle is the covering: S1 Id x z2 1 PSpin(T) = T x T x S = Pso(T). Since C£z IHl and c£g C, we see as before that = T x IHl and = T x C. The kernel is just the complex-valued constant functions, and we conclude that indz(S1 x S1) =I 0 where S1 carries the interesting spin structure. EXERCISE. Compute indz for the remaining spin-structures on T. EXERCISE. Prove directly that indz(SZ) = o. (7.37) It is a good time to summarize what we have established so far. To each riemannian spin manifold X, we have associated the canonical bundle given in (7.1). This is a Zz-graded C£n-Dirac bundle and its Dirac operator has an index, which we denote indk(X) in KO-k(pt). We have proved the following. Theorem 7.10. Let X be a compact spin manifold of dimension n. When n == 1 or 2 (mod 8), let H = ker(ffi) denote the space of real harmonic spinors, that is, the kernel of the Atiyah-Singer operator on the irreducible real spinor bundle of X. Then {dim(:H (mod 2) . dimlHlH (mod 2) mdn(X) = !A:(X) A:(X) ifn == 1 (mod 8) ifn == 2 (mod 8) ifn == 4 (mod 8) ifn == 0 (mod 8) Furthermore, for the interesting spin structure on S1 and for its square on SI x S1, this invariant is non-zero. We shall now investigate the multiplicative properties ofind*. To begin we recall the ring structure on KO-*(pt) which was discussed in Chapter I, §9. We set '1 = the generator of KO -1(pt) zz} Y = the generator of KO-4(pt) Z x = the generator of KO -8(pt) Z (7.38) Then the multiplicative structure on KO -*(pt) is given by the following (ef. 1.9): (7.39) 150 n. SPIN GEOMETRY AND DIRAC OPERATORS where the grading is fixed by the requirement that deg(l1) = 1, deg(y) = 4 and deg(x) = 8. Our next result is that ind* is, in fact, a ring homomorphism. More specifically, we mean the following. The homotopy invariance of the index shows that indiX) is independent of the choice of riemannian metric on X. (Any two metrics are smoothly homotopic.) We consider then the free abe1ian group generated by spin-structure preserving diffeomor-phism classes of connected spin manifolds. The sum = EBn is naturally a graded ring under the direct product of manifolds. Theorem 7.11. The mapping ind* : ---+ KO -*(pt) is a surjective graded-ring homomorphism. Proof. The map ind* is additive by definition. To prove that it is multiplicative, we use the Alternative Definition 7.4 of the index. Let Xl and X 2 be compact riemannian spin manifolds of dimensions n1 and n2 respectively. Let -+ be the Atiyah-Singer operator for the C£nk-Dirac bundle = defined in (7.1), for k = 1,2. Let -+ be the corresponding object for X = Xl X X 2 with the product riemannian and spin structure. For this structure we have PSpin(X 1 X X 2) :=l PSpin(X 1) X 71.2 PSPin(X 2) where 71.2 acts by (-1, -1) on the product. From here one sees easily, using (7.1), that (7.40) the exterior 71.2-graded tensor product. This is a C£n, +n2 = (C£n, ® C£n2)-Dirac bundle, and one can straightforwardly verify that with respect to (7.40), the Atiyah-Singer operator of can be written as = ® Id2 + ex1 ® where ex 1: C£n, -+ C£n, is the automorphism extending -Id on /Rn,. Since + = 0, it follows that = ® Id2 + Id1 ® Each of the operators and is non-negative, self-adjoint and elliptic over a compact manifold. It follows from standard spectral theory that = ® Since = we conclude that (7.41) §7. C£k-LlNEAR OPERATORS 151 where the tensor product in (7.41) inherits the structure of a £:2-graded tensor product of £:2-graded Clifford modules from the structure of (7.40). This graded tensor product is exactly the multiplication in KO -* 9R*/i*9R* + l' Thus, we have established that and so ind* is a ring homomorphism. To see that ind* is surjective we need only show that it maps onto the set of multiplicative generators {l1,Y,X} given in (7.38). Example 7.8 shows that the circle with interesting spin structure maps onto 11. The K3-surface Y given in Example 2.14 is a compact spin 4-manifold whose A-genus is 2. Hence, ind4(Y) = tA(Y) = y. Finally, to produce a spin 8-manifold X with inds(X) = x, we must find one such that A(X) = 1. For this one could take X = Mg, the almost parallelizable 8-manifold of index 224 constructed in Kervaire-Milnor [1]. Alternatively, one could take the 4-disk bundle E over S4 with X(E) = 1 and P1(E)2 = 900. Then oE = S7, and the mani-fold X = E US7 DS, obtained by attaching an 8-disk along this boundary, is a closed spin 8-manifold with A(X) = 1 (see Milnor [7]). This completes the proof of Theorem 7.11. • REMARK 7.12. To prove the index theorem for this fundamental case, it remains only to prove that ind* is a spin-cobordism invariant. It will then descend to a ring-homomorphism ind*: KO -*(pt), which can be seen to agree with the map (3.22) directly. (For the torsion part, this requires some argument.) It is worth noting that all of the above could be carried out with coefficients in a vector bundle. Let X be a compact riemannian spin mani-fold of dimension n, and let E be a real vector bundle over X with an orthogonal connection. Then the bundle ® E is naturally a £:2-graded bundle with a Dirac operator which we shall denote by We shall denote (7.42) Now we also have the fundamental real spinor bundle $ over X and we can take the tensor product $ ® E in the spirit of Example 6.5 above. We denote the Dirac operator on $ ® E by J/JE' Then arguing exactly as we did above proves the following. Theorem 7.13. Let X be a compact spin manifold of dimension n, and let E be a real vector bundle over X with an orthogonal connection. When n == 1 or 2 (mod 8), let HE == ker(J/JE) denote the space of real harmonic E-valued 152 11. SPIN GEOMETRY AND DlRAC OPERATORS spinors. Then {dim(:HE (mod 2) . dimlHlHE (mod 2) mdE(X) = Hch E {ch E· A(X)} [X] ifn == 1 ifn == 2 ifn == 4 ifn == 0 (mod 8) (mod 8) (mod 8) (mod 8) A topological computation of this index can be given as follows. Embed xn c sn + 8k for k sufficiently large, and identify the normal bundle v of X with a tubular neighborhood of the embedding. The spin structure on X determines a unique spin structure on v (via Proposition 1.15 and the uniqueness of the spin stru'cture on sn+Bk). Let $+(v) and $-(v) denote the canonical real spinor bundles of v (whose dimension is 8k). Lift $±(v) to the total space of v by the projection n: v --+ X, and at each non-zero vector nE v consider the isomorphism I1n: n*$+(v) n*$-(v) given by Clifford multiplication by n. Then the difference element 'v == [n*$+(v), n*$-(v);I1] represents a class in the relative KO-group KO(v, v -X), where X c v is the zero-section. This is the KO-theory Thorn class of v. Given a real bundle E over X, we can consider the class 'v(E) = 'v· [n*E] E KO(v, v -X). Since v is embedded as a domain v c sn+ 8k, we have an excision isomor-phismj: KO(v, v -X) Ko(sn+ 8k, sn+Bk -X). Composing this with the natural map i:Ko(sn+Bk,S"+8k -X) --+ KO(sn+Bk) and applying Bott Periodicity f3:KO(sn+8k) KO(S") == KO-n(pt), we obtain a class dE(X) = f3 0 i 0 j(,.(E)) E KO-n(pt) One assertion of the Atiyah-Singer Cfk-Index Theorem, proved in 111.16, is that Theorem 7.14. indE(X) = siE(X). The "cobordism invariance" in this case asserts that this map ind = si determines a transformation si*: ---+ KO -*(pt) where denotes the spin-bordism of the space BO. REMARK 7.15. It is interesting to note that in dimensions 1 and 2 (mod 8), the index can be changed even by twisting with a flat bundle. (This is not §8. VANISHING THEOREMS AND APPLICATIONS 153 true of the usual index.) For example consider the non-trivial real flat line bundle t over the circle (the M6bius band). Using the analysis of Example 7.8, one easily finds that ® t t EB it. Since the kernel of i(djds) consists oflocally constant sections, we see that ker = {O} and indAS1) = o. Recall that ind1(Sl) "# o. This sensitivity to flat bundles is a reflection of the interesting fact that these (mod 2).-invariants are not local, i.e., they cannot be computed by universal formulas involving only the local data of the operator, as in the Gauss-Bonnet and Chern-Weil theorems. Another indication is that the (mod 2)-invariants are not multiplicative under coverings (see Atiyah-Singer [4]). §8. Vanishing Theorems and Some Applications Suppose X is a compact riemannian manifold and let D2 be the Dirac laplacian on the bundle Cf(X). One of the major results in riemannian geometry was the following discovery of S. Bochner. There exists a second, naturally defined laplacian V*V on Cf(X). It is self-adjoint, non-negative and has the same symbol as D2. The difference, D2 -V*V, which is nec-essarily an operator of order ;£ 1, is, in fact, of order zero and can be expressed in terms of the curvature tensor of X. Using harmonic theory, Bochner was thereby able to conclude the vanishing of certain Betti num-bers of X under appropriate positivity assumptions on the curvature tensor. Following Bochner's original paper, arguments of this kind have ap-peared repeatedly in real and complex geometry. They are known gene-rically as "Bochner's method." In this section we shall give a systematic derivation of Bochner-type formulas on general Dirac bundles. This will include the classical formu-las on Cf(X), although even here the algebra is vastly simplified by using Clifford multiplication. It will also include the Lichnerowicz formula for the Dirac laplacian on spinor bundles, and generalizations of this to spinors with coefficients in an arbitrary vector bundle. This latter result is very useful in the study of manifolds of positive scalar curvature (cr. Gromov-Lawson [1], [2], [3]). Everything will follow from a single elegant formula which applies to any Dirac bundle. We begin with the definition of the operator V*V. Let E be any riemannian vector bundle over X, and assume E has a riemannian con-nection with co variant derivative V. Then to any pair of tangent vector 154 n. SPIN GEOMETRY AND DIRAC OPERATORS fields V and W on X, we associate an invariant second derivative nE) --+ nE) by setting (8.1) (Here, V v W is the riemannian covariant derivative on X.) At any point x E X, the operator depends only on the values Vx and Wx, i.e., it is tensorial in these variables. This is evidently true for v" since it is a general property of the covariant derivative. That it is true for w" follows from the identity' -Vfv.v = Rv.w (8.2) where R is the curvature tensor of E. Equation (8.2) is an immediate con-sequence of the fact that VvW -VwV = [V,Wl Now given a smooth section cp of E, we see that cp is a section of T* ® T* ® E; that is, at each point it defines a bilinear form on the tan-gent space with values in E. The connection laplacian V*V: nE) ---+ nE) is defined by taking the trace, i.e., V*Vcp == (8.3) In terms of a local orthonormal tangent frame field (el' ... ,en) on X, n V*Vcp = -L V;j.ejCP' j= 1 It is easy to see that the symbol of V*V at a cotangent vector is = (8.4) and so V*V is elliptic. We shall noW show that it is also symmetric and Recall that the inner product (".) on nE) is defined by integration: (cp,l/I) = Ix <cp, 1/1). Similarly, we define (Vcp, VI/I) = Ix <Vcp, VI/I) (8.5) where <Vcp, VI/I) is defined in terms of local orthonormal tangent frames (el,· .. ,en) by the expression <Vcp, VI/I) = Lj <VejCP, Vejl/l). Proposition S.l. The operator V*V: nE) --+ nE) is non-negative and for-mally self-adjoint. In particular, (V*Vcp,l/I) = (Vcp, VI/I) (8.6) for all cp,1/I E nE) provided that one of cp or 1/1 has compact support. §8. VANISHING THEOREMS AND APPLICATIONS 155 If X is compact, then V*Vcp = 0 if and only ifVcp == 0 i.e., if and only if cp is globally parallel. Proof. Fix x E X and choose a local orthonormal tangent frame field (el' ... ,en) with the property that (Vej)x = 0 for all j. Then we have at the point x that <V*Vcp,l/I) = -<VejVejcp,l/I) (8.7) J = -2: {e/Vejcp,l/I) -<Vejcp,Vejl/l)} j = -div(V) + <Vcp, '111/1) where V is the tangent vector field on X defined by the condition that (V, W) = (Vwcp,l/I) for all WE T(X). The last line of (8.7) is proved as follows. At x, div(V) == <Vey,e) J = J Equation (8.6) now follows by integration of (8.7). The remainder of the proposition is a consequence of formula (8.6). • Using arguments similar to those given for Theorem 5.7, one can show that on a complete riemannian manifold the operator '11*'11 is essentially self-ad joint, i.e., it has a unique self-adjoint closed extension on L2(S). Further-more, the kernel of '11*'11 on L2(S) consists of the parallel sections of S, i.e., those which satisfy Va = O. If X has infinite volume, then no such sections exist except a = 0, since lIall is constant. We suppose now that S is any Dirac bundle over X, and we define a canonical section ffi of Hom(S,S) by the formula " ffi(cp) ==! 2: e{ ek' Rej.ek(CP) j.k= 1 (8.8) where (el' ... ,en) is any orthonormal tangent frame at the point in ques-tion, where Rv.w is the curvature transformation of S, and where the dot "." denotes Clifford multiplication. Theorem 8.2 (the general Bochner Identity). Let D be the Dirac operator and '11*'11 the connection laplacian for any Dirac bundle S. Then D2 = '11*'11 + ffi I. (8.9) 156 11. SPIN GEOMETRY AND DIRAC OPERATORS Proof. Fix x E X and choose a local orthonormal frame field (el' ... ,en) such that (Ve)x = 0 for all j. Then using (8.2) we have at x that D2 = L e{ Veiek' VeJ j,k = Lej'ek,VejVek j,k = L e j' ek . V;j,ek j,k = -L V;j,ej + L ej' ek' (V;j,ek -V;k,e) j j<k = V*V + m. • Recall that the Ricci transformation of T(X) = A l(X) is defined by the formula n Ric(cp) = -L Rej,,,,(ej) j= 1 (8.10) where R is the curvature transformation of T(X). This determines a bilinear form, called the Ricci curvature form, by setting Ric(cp,t/I) = <Ric(cp),t/I). (8.11) From the fundamental identity (4.44) for the Riemann curvature tensor, the form Ric is seen to be symmetric. A consequence of Theorem 8.2 applied to the bundle S = Cf(X) is the following Corollary 8.3. Let .11 be the Hodge laplacian and V*V the connection lap-lacian of the tangent bundle T(X). Then I .11 = V*V + Ric I· (8.12) Proof. Consider T(X) = N(X) c Cf(X). Since D2 = .11 it suffices to com-pute the right hand side of (8.9) for vectors cp EA l(X). Note that since [L,v] = [L,Ll] = 0 (cf. Lemma 5.18 and Corollary 5.21), both of the operators V*V and .11 preserve the subbundle A l(X). Hence, so does the operator m. Therefore, using the identities (4.43) and (4.44), we have m(cp) = t L e;ejRe;,eiCP) i,j §8. VANISHING THEOREMS AND APPLICATIONS 157 +-!-L eiej<Rei.e/CP), e)ej + -!-L eie/Rei.e/cp), e;)ei i,j i,j = -L <Rei,eicp), ej)ei = -L <Rq>,eie;), e)ei i,j i,j = Ric(cp), • Note. As pointed out above, we know from (8.9) that m(A I(X)) C A I(X). However, eiejek E N(X) if i =li =I k =I i. Hence, the third line of the com-putation above constitutes a proof of the first Bianchi identity (4.43). In Theorem 5.16 we saw more sophisticated examples of curvature identities that also followed from operator identities and Clifford multiplication. Corollary 8.3 has the following important consequence: Theorem 8.4 (Bochner). Let X be a compact riemannian manifold without boundary. If Ric > 0, then the first Betti number b1(X) is zero. The con-clusion also holds if Ric 0 and > 0 at one point. Proof. Suppose b1(X) == dim Hl(X;IR) > O. Then there exists a non-zero harmonic I-form cp E HI Hl(X;IR) by Theorem 5.15. By Corollary 8.3 and Proposition 8.1 we have that Ix Ric(cp,cp) = -(V*Vcp, cp) = _[[Vcp[[2. (8.13) If Ric 0, we conclude that Vcp == 0, i.e., cp is parallel. In particular, [[cp[[ is constant. Hence, if at some point we have Ric > 0, then S Ric(cp,cp) > 0 and we have a contradiction. • Note that this argument also proves that when Ric 0, every harmonic 110rm is parallel. Under the metric correspondence T* X TX, the parallel 1-forms become parallel vector fields. Thus we can conclude the following Theorem 8.5. Let X be a compact riemannian manifold of non-negative Ricci curvature. Then b1(X) equals the dimension of the space of parallel vector fields on X. In particular, b1(X) ;£ dim(X) with equality if and only if X is a fiat torus. Proof. Let k = b1(X). Then by the argument above, there are k linearly independent parallel vector fields on X. Parallel vector fields are linearly 158 H. SPIN GEOMETRY AND DIRAC OPERATORS independent if and only if they are linearly independent at each point. Hence, k ;£ dim(X), and k = dim(X) if and only if X has a globally parallel framing. That X must then be a flat torus can be seen as follows. We may choose parallel vector fields El, ... ,Ek which are pointwise orthonormal. Since [Ei,Ej] = V E;Ej -V EjEi = 0 for all i,j, these vector fields generate a locally free IRk-action. Since k = dim(X), we see that X is an orbit of this action, i.e., X IRk/A, where A is a lattice in IRk. The metric on X clearly agrees with the usual one on IRk. • Any parallel vector field generates an isometric flow, and its integral curves are geodesics. Thus even when k = bl(X) < dim(X) (and Ric 0), we get a locally free action of IRk by isometries on X with totally geodesic orbits. We can also consider the dual basis CPl" .. ,CPk of parallel I-forms. Integration of cP = (CPl"" ,CPk) gives a riemannian submersion J:X ..... Tk = IRk/A where A is the lattice in IRk generated by the "periods." That is, A = PE IRk: A = fr cP for some closed curve y in X}. By construction, the map J is a covering map on each orbit. With a little more work one can show that the universal covering space X of X splits as a riemannian product X = IRk X X 0 where X 0 is compact. The original manifold X is a (possibly twisted) riemannian product of Tk with X o' REMARK. The first statement in Theorem 8.4 was considerably improved by Myers [1]. Much stronger results for non-compact complete manifolds with Ric 0 were proved by Cheeger and Gromoll [2]. It is a deep theorem of Gromov [1], that there exist explicit a priori bounds, depend-ing only on dimension, for all the Betti numbers of a compact manifold of non-negative sectional curvature. Such bounds do not exist under the weaker hypothesis Ric 0 (except for degree 1) by examples of Sha and Yang [1]. As one might guess, theorems similar to 8.4 can be proved for the higher Betti numbers. For each p, there is a positivity assumption on the curvature tensor which guarantees that bP(X) = O. For detailed statements of these results the reader is referred to Bochner and Yano [1] and to Goldberg [1]. However, one of the more quotable results of this type can be proved rather easily using the Clifford formalism. We present this now. Recall that the curvature tensor <RV,.V2 V3, V4) is antisymmetric in (VI' V2) and in (V3' V4) and symmetric under the interchange of these pairs (see (4.44)). Hence, R can be considered as a symmetric endomorphism R: A 2(X) ..... A 2(X). We call this transformation the curvature operator and say that it is positive (or non-negative) if all of its eigenvalues are < 0 (or ;£ 0 respectively). Theorem 8.6 (Gallot and Meyer [1 ]). Let X be a compact riemannian n-manifold (without boundary) with the property that its curvature operator §8. VANISHING THEOREMS AND APPLICATIONS 159 s positive at every point. Then all the Betti numbers, bP(X) are zero for = 1, ... ,n -1, i.e., X is a real homology sphere. The same conclusion olds if the curvature operator is a and > a at some point. roof. It will suffice to prove that positivity of the curvature operator mplies that <91(cP),cP) > a for all non-zero cP E AP(X), p = 1, ... ,n -1. (8.14) he argument then proceeds as in the proof of Theorem 8.4. From the urvature identity (5.17) we have that <91(cP),cP) = L <eiejRei.e/CP),cP) i<j From Theorem 4.16 we know that the curvature transformation Rv.w: ct(X) -+ ct(X) can be written as Rv.w = t Li<j <RV.W(ei), e)adeiej' Hence, <91(cP),cP) = -t L <Rei.e/ek),et)<adeie/cp),adeke,(cp) (8.15) i<j k<t = -t L <R(eie),eket)<adeie/cp),adeke,(cp). i<j k<t Observe now that the elements {eieJi<j form an orthonormal basis of A2(X) c ct(X). The last expression in (8.15) is clearly independent of the choice of orthonormal basis. That is, we can write where is any orthonormal basis of A 2(X) c Cf(X). We choose a basis that diagonalizes R. Let be the eigenvalues of R and set Aa = Then (8.16) becomes <91(cP),cP) = L a where Aa> a for all (1.. This proves that <91(cP), cp) a and equality holds iff cp) = a for all E A 2(X). Thus, it remains only to prove the following: Lemma 8.7. Consider a form cP E AP([R") c ct([R") for 1;:;; p ;:;; n -1. If ad!cp) = afor all E A2([R"), then cp = a. Proof. Recall that the representation on AP([R") is just the standard representation of the Lie algebra 50(n) A2([R"). This lemma is therefore 160 11. SPIN GEOMETRY AND DIRAC OPERATORS equivalent to the well-known fact that these representations have no fixed vectors for p =1= 0 or n. However, since an elementary proof is possible we shall give it. Let el, ... ,en be the standard basis of [Rn and write cp = LIII=p aIel' By assumption, [eiej, cp] = 0 for all i < j. One can see easily that [eiej, el] = 2eie je 1 if both i,j I if both i,j E I otherwise. If i I and j E I, then eie je 1 = ± el v {i) -{j)' Therefore, [eiej, cp] = 0 implies that a1 = 0 whenever i E I or j E I but not both i,j E I. Applying this for all i < j shows that cp = 0, provided that p =1= 0 or n. This completes the proof of the lemma and the theorem. • We now take up the case of spinor bundles. From now on we assume that X is a compact spin manifold with a fixed spin structure on its tangent bundle. Let S be any spinor bundle for T(X) endowed with its canonical riemannian connection. Before stating the vanishing theorem in this case we recall one of the simplest invariants of the riemannian curvature tensor, the so-called scalar curvature. This is a function K: X -+ [R defined by setting K == trace(Ric) = -2 trace(R). In terms of an orthonormal tangent frame (el' ... ,en) at a point x E X, n K = -L <Rei.ej(ei)' ej). i,j= 1 (8.17) When X has dimension two, K coincides with the classical Gauss cur-vature function. Theorem 8.8 (A. Lichnerowicz [1 ]). Let X he a spin manifold and suppose S is any bundle of spinors over X endowed with the canonical riemannian connection. Let I/J denote the Atiyah-Singer operator and V*V the connec-tion laplacian on S. Then I I/J2 = V*V + i K I· (8.18) This formula has the following striking consequence. We say that a spin manifold X has no harmonic spinors, if ker I/J = 0 for any spinor bundle associated to T(X). Corollary 8.9. Any compact spin manifold of positive scalar curvature admits no harmonic spinors. In fact, the same conclusion holds if the scalar cur-vature is 0 and > 0 at some point. §8. VANISHING THEOREMS AND APPLICATIONS 161 Proof of Corollary 8.9. This follows from formula (8.18) as before. Suppose (J E qS) satisfies I/J(J = O. Then by integration of (8.18) we find fx KII(JW = -(V*V(J,(J) = -IIV(J112• IfK 0, then we must have V(J = O. Hence II(JII is constant, and if K(X) > 0 for any point x, then J KII(J112 > 0 and we have a contradiction. • Note that we have also proved the following: Corollary 8.10. On a compact spin manifold with K == 0, every harmonic spinor is globally parallel. Proof of Theorem 8.8. We need to compute the curvature term in equa-tion (8.9) for the canonical spinor connection. In Theorem 4.15 we es-tablished that for all V, WE Tx(X), the curvature transformation Rv,w: Sx -+ Sx is given by the formula Rv,w = t Lk,t <RV,W(ek),et)eket, where Rv,w: Tx(X) -+ TAX) is the curvature transformation of X and where (eb ... ,en) is any orthornomal basis of Tx(X). Consequently, using the identities (4.43) and (4.44) we see that: = t L i,j + ?: <Reie/eJ,et)eiejei + ?: <Reie/ej),et)eiejej}et 'J = t L <Reie/ei),et)eh i,j,t = -t L Ric(ej,et)ejet j,t =tK .• As a consequence of the Atiyah-Singer Index Theorem applied to the fundamental spin complex (Example 6.3), we have the following: Theorem 8.11. Let X be a compact spin manifold of dimension 4k. If X admits a metric of positive scalar curvature, then A(X) = O. More generally, applying the same argument to the Dirac operator 'fl of the bundle (7.1), we find that in any dimension, positive scalar curvature implies that the analytic index indn('fl°) must vanish (see Definition 7.4). 162 11. SPIN GEOMETRY AND DIRAC OPERATORS From the CtcIndex Theorem, this implies the following: Theorem 8.12. Let X be a compact spin manifold. If X admits a metric of positive scalar curvature, then d(X) = O. By Theorem 7.10 we know that this result implies Theorem 8.11 above. However, it gives new information in dimensions one and two (mod 8). Recall that d(X) is an invariant of the spin-cobordism class of X. The above results therefore explicitly present large classes of manifolds which cannot carry metrics of positive scalar curvature. In particular, from Theorem 2.8 we have the following striking result: Theorem 8.13 (N. Hitchin [1]). In every dimension n == 1 or 2 (mod 8) where n > 8, there exist compact differentiable manifolds which are homeo-morphic to the n-sphere but which do not admit any riemannian metric with positive scalar curvature. In fact, they admit no metrics such that K 0 but ;(=0. Such spheres are hardly Platonic. This theorem can be perhaps best appreciated in light of the current positive results. The standard metric on the standard n-sphere S" = {x E [R"+ 1: Ilxll = I} is of course the most uniformly positively curved manifold. Its curvature transformation is given by the formula -Rv.w = V /\ W, i.e., -Rv.w(U) = <V, U)W -<W, U)V (8.19) for all U,v,WE Tx(S"). Hence, Ric(V) = (n -1)V and K == n(n -1). The most symmetric exotic spheres (see Hsiang and Hsiang [1]) are the Kervaire spheres which have dimension 4k + 1. They can be constructed by taking the boundary of the manifold obtained by plumbing together two copies of the tangent disk bundle of S2k+ 1: D2k+l x D2k+! S2k+! §8. VANISHING THEOREMS AND APPLICATIONS 163 However, Brieskorn has proved that these manifolds can be described algebraically as follows (see Hirzebruch and Mayer [1 ]). For each integer d, consider the complex polynomial Pd(ZO,··· ,zn) = + zi + + ... + Z;+I' Let V(d) = {z E e+2:piz) = O}; s2n+3 = {z E Cn+2:llzll = 1}; and set (8.20) For n = 2k and d == 3 or 5 (mod 8), M2n + I(d) is a Kervaire sphere. In this thesis Hernandez [1] proved that the Brieskorn manifolds M2n+ I(d) carry metrics of positive Ricci curvature. Moreover, Gromoll and Meyer [2] have proved that a certain exotic 7-sphere carries non-negative sectional curvature which is strictly positive on an open subset. It is therefore something of a surprise that in dimension nine there exist exotic spheres which carry no metric of even positive scalar curvature! Manifolds which carry positive scalar curvature are not hard to find. Any homogeneous space G/H, where G is a compact Lie group, carries a metric with K O. Furthermore, it carries K > 0 unless G/H is a torus. More generally one has the following: Theorem 8.14 (Lawson and Yau [1]). Let X be a compact manifold which admits an effective differentiable action by a compact, connected, non-abelian Lie group. Then X admits a metric of positive scalar curvature. Corollary 8.15. Let X be a compact spin manifold such that d(X) =f. O. Then the only compact, connected Lie transformation groups of X are tori. In particular, this conclusion holds for any exotic sphere which does not bound a spin manifold. In dimensions 4k, it is a result of Atiyah and Hirzebruch [3] that a compact connected spin manifold X with A(X) =1= 0 does not even admit an SI-action! (See IV.3.) REMARK 8.16. We point out that the results above definitely require that X be a spin manifold. We know from (6.19) that the complex pro-jective spaces 1P2k(1C) have A(lP2k(lC)) = (_In-4k(2kk). (8.21) These spaces are all homogeneous; IPn(1C) = U(n + l)/U(l) x U(n -1). Hence, they have metrics with K > 0 and large non-abelian Lie trans-formation groups. However, as we saw in §2, 1P2k(1C) is not a spin manifold. Compact manifolds of positive scalar curvature are now rather well understood. A thorough discussion will be given in Chapter IV. 164 11. SPIN GEOMETRY AND DIRAC OPERATORS We conclude this section with an important generalization of the Lichnerowicz formula (Theorem 8.8) to the case of "twisted" spin or bundles. This result is also quite useful in applications as we shall see in Chapter IV. Let X be a compact riemannian spin manifold, and let S be a spinor bundle for X with the canonical riemannian connection. Let E be any vector bundle over X equipped with an arbitrary orthogonal connection. Then the bundle S ® E, equipped with the tensor product connection, is again a Dirac bundle over X (see Proposition 5.10). Here the Clifford multiplication takes place in the "S-factor". We now define a smooth, symmetric bundle endomorphism 9lE; S ® E -----+ S ® E by the formula n 9lE(0" ® e) == t L (ejekO") ® (R:j.eke) (8.22) j.k= 1 on vectors 0" ® e of simple type. Here RE denotes the curvature of the bundle E, and, as usual, (el' ... ,en) denotes an orthonormal tangent frame to X at the point in question. The sum in (8.22) is essentially the trace of a bilinear object defined on A 2 T X. Note. In the case that S is a complex spinor bundle, we may assume E to be complex and endowed with a unitary connection. The tensor product of S with E can then be taken over the complex numbers. Theorem 8.17. Let X be a riemannian spin manifold with scalar curvature K, and let S ® E be any twisted spinor bundle over X as above. Then the Dirac operator I/JE and the connection laplacian V*V of S ® E satisfy the identity: I/Ji = V*V + tK + 9lE I. (8.23) REMARK 8.18. Note that the operator 9lE depends linearly and uni-versally on the components of the curvature tensor RE of E. Thus, if Eis fiat, 9lE = 0, and if RE is small, then 9lE is correspondingly small by an estimate depending only on dimension. Proof. Recall that the covariant derivative of the tensor product con-nection on S ® E acts as a derivation, i.e., V(O" ® e) = (VsO") ® e + 0" ® (VEe) §8. VANISHING THEOREMS AND APPLICATIONS 165 where Vs, VE denote the covariant derivatives on Sand E respectively. The commutator of two derivations is again a derivation. This fact (or direct verification) shows that the curvature transformation of S ® E is also a derivation, i.e., R(O" ® e) = (RsO") ® e + 0" ® (REe) where RS and RE denote the curvature transformations of Sand E respecti vely. We now wish to compute the curvature term 91 in the general Bochner Identity (8.9). It is given by n 91(0" ® e) = t L e/!kRejoek(O" ® e) j.k; 1 n = t L ejek ® e + 0" ® (R:j,eke)} j,k; 1 = (t f ® e + t f (ejekO") ® (R:j,eke). j,k; 1 j,k; 1 Now from the Lichnerowicz calculation (see the proof of Theorem 8.8), we know that the first term in the last line above is just tK. Hence from (8.22) we conclude that 91(0" ® e) = tK(o" ® e) + 9lE(O" ® e), and the proof is complete. _ CHAPTER III Index Theorems In this chapter we shall present the analytic underpinnings of the subject of spin geometry. In particular we shall formulate and prove various forms of the Atiyah-Singer Index Theorem. This will include the classical theorem and its consequent cohomological formula for the index. It will also in-clude the Index Theorem for G-Operators, the Index Theorem for Families, and the Index Theorem for Ctk-Linear Operators. This last result is one of the deepest in the theory. It involves indices in KO-theory which are in general not locally computable on the manifold. Our exposition of this result differs somewhat from that which currently appears in the literature. There are now in existence many beautiful and illuminating proofs of the more classical index theorems which use the asymptotics of the heat kernel. This is a method that was pioneered by Gilkey and Patodi in the late 1960s. We have elected to present here instead the arguments which originally appeared in Atiyah-Singer [1 ]-[5]. This is in part because the Atiyah-Singer methods lead to the non-local results just mentioned (and these results are not accessible by heat equation techniques). It is also, however, because their arguments, which proceed in the spirit of Grothen-dieck, are really quite beautiful and simple. The essential idea is this. One observes that the index is an insensitive object, unperturbed by rather bru-tal changes in the analytic data. Furthermore, the index is a "functorial" object, which transforms nicely with respect to global operations such as the embedding of one manifold into another, the addition and multiplica-tion of operators, etc. Using an appropriate form of K-theory, one then for-mulates a topologically defined index possessing the same transformation properties as the analytic index. By performing manipulations allowed in the theory, everything can be reduced to the trivial case where the mani-fold is a point. Here the analytic and the topological indices are easily seen to coincide, and it follows that they must coincide in general. Our ambition has been to make the presentation in this chapter rea-sonably self-contained. All the requisite material on pseudodifferential op-erators is developed assuming only a knowledge of elementary Fourier analysis. Along the way a proof of the generalized Hodge Decomposition Theorem is given. Most of the material necessary for the derivation and computation of the cohomological formulas is also presented in detail. §1. DIFFERENTIAL OPERATORS 167 The exposition in this chapter owes much to the writings of Atiyah and Singer and also to Gilkey and Nirenberg. The reader is encouraged to con-sult the excellent literature on the subject of pseudodifferential operators and index theory which has appeared over the past twenty-five years. §1. Differential Operators This section presents the basic notion of a linear elliptic differential opera-tor over a manifold X. We begin by fixing notation. For an n-tuple of non-negative integers rx = (rxl' ... ,rxn)' we set Irxl = Lk rxb and for each E IRn we set = ••• In local coordinates (Xl' ... ,xn) on X we define the differentiation operators Da by i1alDa == a1a.1jaxa == ... Recall that for a smooth vector bundle E on X, the symbol qE) denotes the space of smooth (i.e., COO) cross-sections of E. DEFINITION 1.1. A differential operator of order m on X is a linear map P:qE) -+ qF), where E and F are smooth complex vector bundles over X, with the following property. Each point of X has a neighborhood U with local coordinates (x 1, ... ,xn) and local trivializations: Elu ..:. U x CP and Flu":' U x U, in which P can be written in the form: ala I P = L A<x(x)-a a X (1.1) where each Aa(x) is a q x p-matrix of smooth complex-valued functions and where Aa =1= 0 for some rx with Irxl = m. A real differential operator of order m is defined similarly with C re-placed by IR. Observe that if we make a change of the local trivializations of Elu and Flu by smooth maps gE: U -+ GLp(1C) and gF: U -+ GLq(1C) respectively, then in these new trivializations P has the form where the 4a 's are again p x q-matrices of smooth functions of X and where for Irxl = m. (1.2) If we make a change of local coordinates x = x(x) on U, then using the fact that for eachj, 168 III. INDEX THEOREMS we find that P again takes the form where Aa = L AP [irJa IPI=m ox P for lal = m (1.3) and where [oX/ox J: denotes the symmetrization of the mth tensor power of the Jacobian matrix ((oxk/ox). Equations (1.2) and (1.3) together imply that the coefficients {im A"}I"I =m represent a well-defined section a(P) of the bundle (0mTX) ® Hom(E,F) where 0 denotes symmetric tensor product. DEFINITION 1.2. The section a(P) E r((0mTX) ® Hom(E,F» is called the principal symbol of the differential operator P. Recall that for a vector space V, the space 0mV is canonically isomor-phic to the space of homogeneous polynomial functions of degree m on V*. Hence, for each cotangent vector E T:X, the principal symbol gives an element (1.4) If we fix local coordinates and trivializations as in Definition 1.1, we find that for = dXk = im L (1.5) lal=m It is now possible to present one of the fundamental concepts of this chapter. DEFINITION 1.3. Let P be a differential operator of order m over a mani-fold X. Then P is elliptic if for each non-zero cotangent vector E T* X, the principal symbol Fx is invertible. EXAMPLE 1.4. Let E = F be the trivialized line bundle and consider the Laplace-Beltrami operator Ll : COO(X) COO(X) of a given riemannian metric on X. In local coordinates (Xl" .. ,Xn) we have N -_1 f (JrJ gjk Of) -JrJ j,k= I OXj OXk n o2f = L gjk + lower order terms j,k=l UXjUXk §1. DIFFERENTIAL OPERATORS 169 'Lg jk dx j dXk is the metric tensor and where ((gjk)) = ((g jk)) -1. For given cotangent vector = dXk we find that = -L = is certainly invertible (as a linear map C q for #-o. EXAMPLE 1.5. Let S be a Dirac bundle over a riemannian manifold X (see II.5.2), and consider the associated Dirac operator D: r(S) r(S). It is straightforward to show that = where denotes "Clifford multiplication by Since = this map is certainly invertible for #-o. EXAMPLE 1.6. Let S be as in Example 1.5 and consider the Dirac Laplacian D2:r(S) r(S). Then one has that aiD2) = This shows, from the discussion in Chapter II, §5, that the Hodge La-placian on exterior p-forms is an elliptic operator. The proof of the fol-lowing statement is an easy exercise. Proposition 1.7. Let P: r(E) r(F), pi: r(E) r(F) and Q: r(F) r(L) be differential operators over X where P and pi have the same order. Then for all E T* X and for all t,t' E IR, one has that a + t' Pi) = ta + t' a and 0 P) = 0 This proposition says that the symbol is a rather nice object when con-sidered to live on the cotangent bundle. Given P: nE) r(F), we pull back the bundles to T* X via the projection n: T* X X and consider the principal symbol as a bundle map a(P): n* E ----+ n* F. (1.6) If P is elliptic, this map is an isomorphism away from the zero section, and we can assign topological data to the operator as follows. Fix a metric on X and set DX = E T* X: I}. Then via the construction given in Chapter I, §9, the symbol of P defines a class i(P) == [n*E,n*F;a(P)] E K(DX,oDX). (1.7) IMPORTANT FACT 1.8. Suppose X is a spin manifold of even dimension and that P is the Atiyah-Singer operator on complex spinors. Then the 170 Ill. INDEX THEOREMS class i(P), when restricted to any fibre, becomes the element '1 which gen-erates the Bott periodicity mapping in K(D2k, S2k-l) K(S2k) K(pt) (see 1.9). This means that on an even-dimensional spin manifold, the principal symbol of the Atiyah-Singer operator gives a K-theory orientation on the cotangent bundle, i.e., a generator of the Thorn-isomorphism (see Appen-dix C). Similarly, on a spin manifold of dimension 8k, the principal symbol of the real Atiyah-Singer operator gives a KO-theory orientation on the co-tangent bundle. §2. Sobolev Spaces and Sobolev Theorems Let E be a hermitian vector bundle with connection V on a compact riemannian manifold X. Given u E r(E) we have Vu E r(T* X ® E), and using the tensor product connection on T* X ® E, we have VVu E r(T* X ® T* X ® E). This process continues, and for any k we can de-fine the norm Ilullf == it Ix IYV 'v' • j times (2.1) calIed the basic Sobolev k-norm on r(E). An easy exercise shows the equiv-alence class of this norm to be independent of the choice of metrics and connection. The completion of r(E) in this norm is the Sobolev space Lf(E). It is straightforward to verify the folIowing: Proposition 2.1. A differential operator P: r(E) --+ r(F) of order m extends to a bounded linear map P: Lf(E) --+ Lf-m(F) for all k ;;; m. Ultimately we shall see that if P is elIiptic then these extensions have finite dimensional kernel and "co kernel" which consist of smooth sections and are independent of k. Our aim at present is to establish some analytical tools. This is best done using Fourier transform methods. To this end we select a good sys-tem of trivializations of our bundle E. To start we choose a finite cover-ing of X by closed coordinate balls y p : Up --+ jjn = {y E IRn : I yl 1}, f3 = 1, ... ,N. Over each ball Up, we choose a smooth trivialization of E Elu/I Up X cP which possesses a smooth extension to an open neighborhood of Ua• We further assume that the open balls of radius 1/J2 cover X, i.e., X = Bp where Bp == {p E Up: IyP(pW < t}. §2. SOBOLEV SPACES AND SOBOLEV THEOREMS 171 We now change each coordinate Yp to a local coordinate xp by setting 1 xp = ,\It -lypl2 Yp· Note that xp: !Rn and that xP(Rp) = Rn = {x E !Rn : Ixl < I}. Further-more, under the given trivialization over Up, any smooth section of E re-stricts to become a bounded function u:!Rn -+ cp. In fact the function IDau(x)l(l + Ixl),a, is bounded for any a. Let's now choose a smooth partition of unity {Xp}Z= 1 subordinate to the covering {Rp}Z= l' Any section u E nE) can now be written as u = L up where up == xpu. In our system of coordinate trivializations, each up be-comes a smooth function with compact support in the unit ball Rn. DEFINITION 2.2. Any system of local coordinates for X and local triv-ializations for E, together with a partition of unity, all chosen as above, will be called a good presentation of E. Good presentations of each of a family of vector bundles over X having the same local coordinates and the same partition of unity, will be called a good presentation of the family. Using a good presentation of E, we can reduce the study of nE) to the study of smooth CP-valued functions with compact support in Rn. Here we can apply the classical Fourier transform (2.2) whose elementary properties we summarize (see Taylor [2] as a basic ref-erence). We assume all functions to be CP-valued, and define the Schwartz space ff == {u E coo(!Rn): such that IDau(x)1 Ca,k(l + Ixl)-k on !Rn}. (Recall that Da == i-laI8Ial/8xa.) The Fourier transform defines an isomorphism (. r: ff -+ ff whose inverse is given by the following "Inversion Formula" u(x) = (2n)-n/2 f = = (PlanchereI's Formula) where (U,V)L2 == J <u,v) is the usual L2 inner product. (2.3) (2.4) (2.5) (2.6) DEFINITION 2.3. For SE !R and u E ff, the Sobolev s-norm is the norm lIulls given by the formula Ilull; == f(l + (2.7) The completion of ff in this norm is the Sobolev space L;. 172 Ill. INDEX THEOREMS REMARK 2.4. Let s be a positive integer. Then there are constants Cl and C2 so that cl(1 + 1 + + ... + IWs cz(1 + It fol-lows from formulas (2.4) and (2.6) that there are constants Cl and Cz so that Clllull; L fIDau(x)j2 dx Czllull;. (2.8) This means that the norm (2.1) (for the trivialized cP-bundle with the triv-ial connection) is equivalent to the norm (2.7). For any integer k 0, let Ck denote the space of k-times continuously differentiable functions on JRn equipped with the uniform Ck-norm, defined for u E Ck by == sup L IDau12. (2.9) 1Rl" Our first main result is the following: Theorem 2.5 (The Sobolev Embedding Theorem). For each real number s > (nI2) + k, there is a constant Ks such that Ilulb Ksllulls for all u E 9'. Consequently there is a continuous embedding L; c Ck for each such s. Proof. We begin with k = O. For x E JRn, formula (2.3) gives lu(x)1 (2n)-n/2 = (2n)-n/2 f (1 + + (2.10) (2.11 ) Since (1 + is integrable if2s > n we have by the Schwarz inequality that lu(x)j2 (2n)-n f (1 + f (1 + = K;llull;· Repeating this argument for each derivative Dau with lal < s -(nI2) and then summing, proves the result. • Observe that since (1 + (1 + if s' < s we have that Ilulls' lIulls Vs' < s. Hence there is a continuous inclusion L; c L;' for all s' < s. When re-stricted to functions with support in a fixed compact set, this inclusion is "compact." §2. SOBOLEV SPACES AND SOBOLEV THEOREMS 173 Theorem 2.6 (The RelIich Lemma). Let {uJi= 1 be a sequence of functions with support in Rn such that Ilujlls C for all j. Then for any Sf < s there is a subsequence which is Cauchy in the norm 11'lls and therefore converges in L;'. Proof. We begin by recalling another elementary fact concerning the Fourier transform. Let cp be a smooth ([>valued function with compact support in !Rn. Then for all integrable functions u, /"0-A Ad./"'-.. A A cpu = cp*u an qJ*u = cpu (2.12) where "*" denotes the convolution product given by cp*u(x) == J cp(x -y)u(y) dy. (2.13) Suppose now that supp u c Rn and cp == 1 on Rn. Then u = cpu and so U = cP*u. Taking derivatives gives the formula = -'1)u('1)d'1, and applying the Schwarz Inequality then gives J(1 + -'1)d'1 J(1 + 1'11)2s\u('1Wd'1 == where is the continuous function defined by the first integral. Applying (2.14) to the given sequence {UJ1=l shows that the sequence {Dauj} i= 1 is uniformly bounded on compact subsets of !Rn for any a. In particular the sequence {uj} 1= 1 is uniformly equicontinuous on compact subsets, and by the Arzela-AscoIi Theorem there is a subsequence which is uniformly Cauchy on compact subsets. Fix r > 0 and split the integral \\Uj -Uk\\;' = (t + -+ (1 + -For > r, we have that (1 + \W2S' (1 + r)-2(S-S'l(1 + and so the first integral is \\Uj -Uk\\;/(l + r)2(S-S'l 2Cjr2(S-s'l. Hence, for any given B > 0 we can make the first integral less than B/2 for all j and k by choosing r sufficiently large. The second integral is then bounded above by a constant multiple of sup -:5r Hence, by the previous paragraph there is a J so that the second integral is B/2 for all j,k J, and the sequence {Uj}1=l is Cauchy in L;, as claimed. _ 174 Ill. INDEX THEOREMS Combining the theorems above gives the following: Corollary 2.7. Let {Uj}1=l be a sequence in L; with supp uj c Bn and Ilujlls c for all j. If s > (nI2) + k, then there is a subsequence which con-verges to a function u E in the uniform Ck-norm. We denote by z· w = Lf=l ZjWj the standard C-bilinear pairing on cP. Theorem 2.8. The bilinear function (u,v) == S is a perfect pairing on L;'x that is, it identifies with the dual of L;,for any SE IR. Proof. For u,v E ff, we have (u,v) = S u@(1 + IWs, v@(1 + and so by the Schwarz Inequality, l(u,v)1 Ilullsllvll-s. Hence, the bilinear function has a continuous extension to L; x L:'s' In particular, for any v E we have that sup I (u,v) I IIvll-s· lIull.= 1 (2.15) It remains to establish equality. For this we choose u so that = + IW-2S. Then we find IW(1 + = IW(1 + IW-2S so that lIulls = Ilvll-s. Furthermore, (u,v) = SIW(1 + = Hence (u,v)/llulls = Ilvll-s and (2.15) can be made an equality. We have established the isomorphism = (L;)*. • Corollary 2.9. Let T: ff -+ ff and T*: ff -+ ff be linear maps such that (Tu, v) = (u, T*v) for all u,v E ff. If for some s E IR and some constant c, the map T satisfies the condition 11 Tu lis cllulls for all u E ff, (2.16) then T* satisfies the condition IIT*vll-s = cllvll_s for all v E ff. §2. SOBOLEV SPACES AND SOBOLEV THEOREMS 175 In particular, if T extends to a bounded linear map T: Lf -+ Lf for all positive integers k, then T* extends to a bounded map -+ all negative integers -k. Proof. Given u,v E ff we have I(T*v, u)1 = I(v, Tu)1 Ilvll-sllTulls cllvll-sllulls. Hence, by Theorem 2.8 we find IIT*vll-s = sup{I(T*v,u)l: Ilulls = 1} cllvll_s. • There is a stronger conclusion possible here which is proved by the interpolation methods of Calderon. Since we shalI only need 11·llk for k E 7l.. in our work here, we simply state the result. Theorem 2.10. Let T: ff -+ ff be a linear map such that (2.16) is satisfied for s = Sl and s = S2' Then T also satisfies (2.16)for all values of s between Sl and S2' CorolIary 2.9 gives the folIo wing key to transferring our local results to global results on a compact manifold. Proposition 2.11. Let A be a smooth matrix-valued function on !Rn so that IDaAI is bounded for all a. Then the map T: ff -+ ff given by Tu = Au extends to a bounded linear map T: L; -+ L; for all s E R Proof. By formula (2.6) we see that T*u = Atu where At(x) denotes the transpose of the matrix A(x). For any integer k 0, the maps T and T* are clearly bounded with respect to the classical norm = Ltk which is equivalent to Ilullk (see 2.4). The result now folIows immediately from 2.9 and 2.10. • For any open set Qc !Rn, let L;'n denote the 11' lis-closure of CO'(Q) == {u E ff: supp u c Q}. Proposition 2.12. Let Q, Q' be bounded open sets with smooth boundary in IRn, and let <1>: 0 -+ 0' be a diffeomorphism. Then the map T: CO'(Q') -+ CO'(Q) given by Tu == u 0 <1>, extends to a bounded linear map T: L;,n' -+ L;,n for all s. Suppose similarly that <1>: !Rn -+ !Rn is a diffeomorphism which is linear outside a compact subset. Then the map T: ff -+ ff given by Tu = u 0 <I> extends to a bounded map T: L; -+ L; for all s. Proof. We begin with the second statement. Note that (Tu,v) = (u, T*v) where T*v = j(<I»u 0 <1>-1 where j(<I» denotes the lacobian determinant of <I> -1. As above we easily see that T and T* are bounded for the norm 176 Ill. INDEX THEOREMS 11 11 Ilk for any integer k O. Hence, 2.9 and 2.10 apply to give the result. The first statement is proved similarly using the straightforward exten-sions of 2.8, 2.9 and 2.10 to the case of Qc [Rn. • Proposition 2.13. Let P = Llal Aa(x)Da be a differential operator of order m on [Rn whose coefficients are bounded as in Proposition 2.11. Then P:f/' -+ f/' extends to a bounded linear map P:L; -+ L;-mfor all s. -----Proof. Consider P = Da. Since IDaul2 = we find immediately that IIDaull; = f (1 + Ilull;+lal for any Q( and s. Applying the triangle inequality and (2.11) completes the proof. • We are now in a position to globalize. Let E be a smooth vector bundle over a compact manifold X and fix a good presentation for E with coor-dinates xp: Up -+ [Rn, f3 = 1, ... ,N, and with partition of unity {Xp} sub-ordinate to the covering {Bp = xpl(Bn)} (see 2.2). Since '[.Xp = 1, any u E nE) can be written as u = '[.up where up = Xpu. For s E [R, a Sobolev s-norm can be defined on nE) by setting (2.17) where Iluplls is defined in terms of the presentation: Elup [Rn X cP. The following is a direct consequence of Propositions 2.11 and 2.12 which assert the bounded effect of changes of coordinates and trivializations. Proposition 2.14. For any SE [R and any smooth vector bundle E over a compact manifold, the equivalence class of the norm 11'lls is independent of the good presentation chosen to define it. Furthermore, when s = k is a non-negative integer, the equivalence class of 11'llk is the same as that defined by (2.1) for any choice of metric or connection on E. This justifies our use of the symbol 11'lls without subscripts to indicate the presentation chosen. It is now straightforward to see that our main theorems, 2.5-2.8 and 2.13, can be globalized. We summarize the result here. Theorem 2.15. Let E and F be smooth vector bundles over a compact mani-fold X of dimension n. (1) For each integer k 0 and each s > (n/2) + k, there is a continuous inclusion L;(E) c Ck(E). Furthermore, every sequence {Uj}i=l which is bounded in the 11· lis-norm, has a subsequence which converges in the uniform Ck-norm. §3. PSEUDODIFFERENTIAL OPERATORS 177 (2) For any riemannian volume measure fl on X, the bi/inear map on qE) x qE*) given by setting (u,u*) = Ix u*(u) dfl extends to a perfect pairing L;(E) x L:'s(E*) for all s. (3) Multiplication TAU == Au by any element A E qHom(E,F» extends to a bounded linear map TA: L;(E) -+ L;(F) for all s. (4) Any differential operator P: qE) -+ qF) of order m extends to a bounded linear map P: L;(E) -+ L;_m(F) for all s. §3. Pseudodifferential Operators The concept of a pseudo differential operator has its roots in the following observation. Let P = IAa(x)Da be a differential operator on [Rn acting on functions u with, say, compact support. By Fourier Inversion (2.3) any such u can be written as u(x) = (2n)-n/2 I Applying P we find that Pu(x) = (2n)-n/2 I (3.1) where == L (3.2) is the (total) symbol of P. Replacing p by a more general function of x and defines a pseudodifferential operator. Note that in (3.2) the order of P corresponds to the degree of p as a polynomial in In the general case one must be careful with growth in the DEFINITION 3.1. Fix m E [R. A smooth (matrix-valued) function on [Rn X [Rn is said to be a symbol of order m iffor each IX,IX' there is a constant Caa, such that (3.3) for all Let Symm denote the space of these symbols. Proposition 3.2. To each p E Symm the formula (3.1) defines a linear opera-tor P: ff -+ ff. If p has compact x-support, this operator has a continuous extension P: L;+m -+ L; for all s. 178 Ill. INDEX THEOREMS Proof. If U E Y, then U E Y. For any integer N > 0, we have IxI2NPU(X) = (-It = (-It I The second integral is bounded by (3.3) and growth properties of U. Hence, PUEY. To prove the second part note first that integration by parts gives (a I dx = I ei<X,{> dx. Since p has compact x-support, (3.3) then implies that II :5; ct(1 + + IW-t for each t E Z +. It follows that '¥( '1;f II dxl (1 + -s-m(1 + 111l)s :5; ct(1 + + 111l)s(1 + -l1ltt :5; Ct(1 + -l1l)-t+lsl where Ct and Ct are constants that depend on t. In particular there exists a constant C such that and for all and 11. Using the pairing from (2.8) we now compute that (Pu,v) = I PU(l1) . V(l1) dll = I = II {f Setting = + and V(l1) = V(l1)(1 + 111lts, we find that I(Pu,v)l:5; :5; {II dll }}{II '¥( dll}} :5; Cllulls+mllvll-s· Applying duality and interpolation completes the proof. • The operators P given in Proposition 3.2 are called pseudodifferentiaJ operators of order m on /Rn, and the space of all such is denoted by '¥ DOm· J §3. PSEUDODIFFERENTIAL OPERATORS 179 We shall study some of the properties of these "local" operators before globalizing them to bundles on manifolds. Note that a pseudodifferential operator can have order -m < O. Such an operator is said to be smoothing of order m. A linear map T: Y -+ Y which extends to a bounded linear map T: Ls -+ Ls+m for all s and m is called an infinitely smoothing operator. Note that by the Sobolev Embed-ding Theorem 2.5, we have T(Ls) c Coo for all s. Two pseudodifferential operators P and P' will be called equivalent if P -P' is an infinitely smoothing operator. We want now to examine what happens to the symbols of pseudodiffer-ential operators under the operations of composition, of taking adjoints and of changing coordinates. This is referred to as the "symbol calculus." To this end we make the following definition. DEFINITION 3.3. Let P be a pseudodifferential operator with symbol p. Then p is said to have a formal development 00 L Pj j= 1 (each Pj E Symmj for some mj) if for each integer m, there is a K so that P -B= 1 Pj E Sym -m for all k K. (Hence, the corresponding operator is m-smoothing for all k K.) The utility of such formal developments is evident from the following result. Proposition 3.4. Any formal series Lf= 1 Pj' where Pj E Symmj and mj -+ -00, is the formal development of a pseudodifferential operator. This oper-ator is unique up to equivalence. Proof. By grouping terms we can assume mj+ 1 < mj for allj. Fix a smooth function q>: IR + -+ [0,1] such that q>(t) = 0 for t.:s; 1 and q>(t) = 1 for t 2. For any sequence of radii {rj}f= 1 with lim rj = 00 the symbol 00 = L j= 1 is well defined since the sum is finite for each Set = for e E IRn, and for each j define mj = (j + l)nll<plb Recall that for each a,a' andj there is a constant Ca.a.'j such that :::;; Ca.a.,il + Immrla.'I. Let Cj = max{ Ca.a.'j : lal .:s; j and la'l .:s; j} and choose rj > mj2jCj. Then for any k > j and for lal .:s; j, la'l .:s; j, we have :::;; Ci1 + :::;; Ci1 + + ;:::;; 12j(1 + mj 180 III. INDEX THEOREMS for all rj. Setting = q>(IWr), we see that ;j (1 + for all \oc\,\oc'\ j < k and for all It follows easily that p == I q>jPj E Symml and, moreover, that for all k k P -I PjE Symmk+l. j=l This proves the existence of the operator. Its uniqueness up to infinitely smoothing operators is obvious. • Before beginning the symbol calculus it is useful to note that up to equivalence any pseudodifferential operator can be made "local" in the sense that Pu always has support in a neighborhood of supp u. For this and many other basic calculations we shall need the following: Workhorse Theorem 3.5. Let be a smooth matrix-valued function on X X with compact x-and y-support. Fix m E and assume that for each oc,p,y, there is a constant Capy such that Capy(1 + Then the operator K:Y' --+ Y' given by (Ku)(x) == (2n)-n II (3.4) is a pseudodifferential operator whose symbol k has asymptotic development (3.5) Proof. Note that the y-integral in (3.4) is a Fourier transform. Using the rule Uv = a*v (and neglecting constants (2n)-n/2 which will take care of themselves), we find that. (Ku)(x) = II -= I I -U(IJ) dIJ I dIJ where a denotes Fourier transform in the second variable. The interchange of integration is allowed since for each integer t we have -:5; CAl + + -IJI) -t(1 + IIJI) -t, §3. PSEUDODIFFERENTIAL OPERATORS 181 and the right hand side is integrable for t sufficiently large. Formula (3.6) shows that K is pseudodifferential with symbol k(X,l1) = f = f ei(X,Oa(x,(" + l1)d,. For each integer t we have the Taylor expansion in the third variable: 'Ial a(x,',' + 11) = L + Rt(x,',' + 11) lal,;;t oc. where the remainder is given by the formula 1 11 Rix,',' + 11) = (t+lW+1 L I" 11'1 =t'+ 1 /.1. 0 Recalling that (. r denotes the F ourier transform in the middle variable, we find that f d, = f ei(X,o,a(J$,)(X,',l1)d' = f = Consequently, the symbol of K can be written in the form 'Ial k(X,l1) = L + rix,l1) lal;;;t' oc! and by Proposition 3.4 it will suffice to show that rix,l1) f ei(x,{> Rix,',' + 11) d, E Symm-(t+ 1) for each t. To prove this we first show that for each oc, P, and k there is a constant Capk such that the inequality Capk(l + 1'+l1l)m-IPI(l + IW-k is satisfied for all X,',l1. To establish this we first note that from our basic assumption on a we have +11)1 = I f + l1)e-i(y,O dyl = I + l1)e-i(y,O dyl Capy(ly_supp)(a)dY)(1 + le + 111)m-IPI = CaPy{l + le + lllt -IPI 182 III. INDEX THEOREMS After setting, = " the inequality follows easily. Using this inequality and the above formula for Rt, we calculate that + 11)1 = I(t + 1) L il +11)("(1 -t)t dtl III Jo CaPtk I: (1 + It' + lllr-(t+ 1)-IPI(1 + Im-kl(lt+ 1(1 -tt dt Captk(1 + 111l)m-v+ lHP1(1 + I'i}t+ l-k. From this inequality it follows that there are constants Cap so that Cap(1 + 111/)m-V+l)-IPI. Hence, rt E Symm-v+ 1) and the theorem is proved. _ Theorem 3.5 has the following easy consequences: Observation 3.6. If the function of Theorem 3.5 vanishes for all (x,y) in a neighborhood of the diagonal, then the corresponding operator K given by (3.4) is infinitely smoothing. Proof. By (3.5) we have k O. _ For A c /Rn and a > 0, we set Ae == {x E /Rn : distance(x,A) a}. An op-erator P: Y -+ Y is said to be a-local if for all u E CD supp Pu c (supp u)e' (3.7) Corollary 3.7. Given PE If'DOm whose symbol has compact x-support, and given any a> 0, there exists Pe E If'DOm which is eqUivalent to P and is a-local. Proof. Choose a smooth real-valued function IjJ on /Rn X /Rn such that ljJ(x,y) == 1 in a neighborhood of the diagonal, and ljJ(x,y) = 0 if Ix -yl e. Let p be the symbol of P. Then the operator (Peu)(x) == (2n)-n II (3.8) is clearly a-local. By Theorem 3.5 Pe is pseudodifferential with symbol Pe p. Hence, Pe is equivalent to P. -Corollary 3.8. Let X = (X 1 ,X2) be a pair of real-valued functions with com-pact support on /Rn. Then for any PElf' DOm the operator px given by (3.9) is also in If'DOm• §3. PSEUDODIFFERENTIAL OPERATORS 183 Proof. The operator pr. can be expressed in the form (3.4) with = • Theorem 3.9. Let P be a pseudodifferential operator and u a function in its domain (in L; for some s, say). Then for any open set U c /Rn, u Iv E coo ==;> Pu Iv E coo. Proof. Suppose ulv is smooth and fix x E U. Choose X = (Xl,X2) as above so that: x E supp Xl C supp X2; Xl == 1 near x; and X2 == 1 in a neighbor-hood of supp Xl' Since X2U E C;;" we have X1P(X2U) E Coo. Furthermore, by 3.6 we have that X1Pu -X1P(X2U) = X1P((1 -X2)U) E Coo. Consequently, Xl Pu E Coo and so Pu is smooth near x. • The reader can probably see the potential usefulness of the above corol-laries in trying to define and study pseudodifferential operators on general manifolds. These considerations motivate the following definition. DEFINITION. An operator PE 'f'DOm is said to have support in a com-pact set K, if supp(Pu) c K for all u E C;;" and if Pu = 0 whenever supp u n K = 0. The linear space of such operators is denoted 'f'DOK.m. We now begin the local symbol calculus. Theorem 3.10. Given PE 'f'DOK,? and Q E 'f'DOK,m with symbols p and q respectively, the composition P 0 Q E 'f' DO K,? + m has symbol with formal development il"l sym(P 0 Q) L -" od Given PE 'f'DOK,m, we define its formal adjoint p* by setting (Pu, Vh2 = (u, P*v)u for all u,v E Y with support in K. (3.10) (3.11) Theorem 3.11. Given PE 'f'DOK,m with symbol p, its formal adjoint p* E 'PDOK,m has symbol p* with formal development il"l p* D"D"pt L...od x (3.12) where (. Y denotes the transposed matrix. Theorem 3.12. Let <p: U -+ V be a diffeomorphism between open subsets of IRn. Thenfor each compact subset K c U, <p induces a map <p*: 'f'DOK,m -+ 184 Ill. INDEX THEOREMS '¥D0<pK,m by setting (3.13) Proof of Theorem 3.11. Choose U,V E COO with support in K, and note that (Pu, v)u = ff v(x) dx = fff v(x) dx = fff = (u,P*v)u Fix a real-valued function cp E Cgo such that cp == 1 on K, and note that since cpu = u, we can write (P*v)(y) = ff (3.14) This operator satisfies the conditions of Theorem 3.5 and therefore has a symbol p* with formal development where we use the fact that cppt = pt because p has x-support contained in K. • Proof of Theorem 3.10. Note that (PQu)(x) = f and so we need a reasonable expression for To find this, note that Q = (Q*)* and so from equation (3.14) (Qu)(x) = ff (3.15) for x E K, where q* = sym(Q*). Now (3.15) isjust an inverse Fourier trans-form. Hence, = f e -dy where r == (q*y. It follows that (PQu)(x) = ff §3. PSEUDODIFFERENTIAL OPERATORS 185 to which Theorem 3.5 applies. We conclude that PQ is pseudodifferential with formal development: sym(PQ) '" Ix = y i,a, ocl = aT ilPI + Iyl = L L --a p+y=a Ply! 'IPI ('IYI ) = L IPI L P • y y. ilPI '" L -P' P • where the last line is a consequence of Theorem 3.11. • Proof of Theorem 3.12. Write x = cP(x) and x = cP -l(X) == I/J(x). We note that _ _ ,1 d x -y = I/J(x) -I/J(y) = Jo dt I/J(tx + (1 -t)y)dt = '¥(x, y) . (x -y) where '¥(x,y) is a smooth matrix-valued function. Since '¥(x,x) = (ol/J/ox)x and I/J is a diffeomorphism, the matrix '¥(x,y) is invertible for (x,y) in a neighborhood (f) of the diagonal. We choose X E Cg'((f)) such that X == 1 in a (smaller) neighborhood of the diagonal. Let J = det(oI/J7ox) denote the Jacobian determinant of I/J, and note that [(cP*P)u](x) = [P(u 0 cP)](x) = II = II Let .ff denote the integrand in this last integral, and write .ff = X.ff + (1 -X).ff, By 3.6, the integral of (1 -X).ff represents an infinitely smoothing operator. In the integral of X.ff we can make a change of coordinates e = ['¥t(x,y)] -1, == 0(x,y) . , and find that, modulo infinitely smoothing operators, we have where a(x,y,O = X(x,y)J(y)ldet 0Ip(l/J(x), 0(x,yK)· 186 Ill. INDEX THEOREMS By the workhorse Theorem 3.5 we conclude that cp*P is pseudodifferential. • Applying formula (3.5) and recalling that X == 1 near the diagonal, we find that il"l I sym(cp*P) 0Ip(t/I(x),00 x=y == p( x(x), () (mod Symm-l) since 0(x,x) = [(ot/l/OX),]-l = (ox/ox), and Idet 0(x,x)1 = r lex). Equa-tion (3.16) states exactly that, modulo symbols oflower order, the symbol of a pseudodifferential operator transforms like a function on the cotangent bundle. More precisely, let the coordinates for the Fourier transform be considered as standard coordinates for I-forms w = L dXj on x-space. DEFINITION 3.13. Let PE '¥DOm have symbol PE Symm. Then the prin-cipal symbol of P is the residue class a(P) = [p] E Symm/Symm-l. Our dis-cussion above shows the following: Corollary 3.14. The principal symbol a(P) transforms under diffeomorphisms like a function on the cotangent bundle of [Rn. We are now in a position to consider global questions. Let X be a com-pact n-dimensional manifold, and let E and F be smooth complex vector bundles over X. We say that a linear map P: r(E) -+ r(F) is infinitely smoothing if it extends to a bounded linear map P : L;(E) -+ L;+m(F) for all s,m E [R. This implies by (2.15) that P(L;(E)) c r(F) (= COO-sections) for all s. It is a straightforward exercise to show that given a riemannian volume measure J.l on X, any infinitely smoothing operator can be written as an integral operator: Pu(x) = Ix K(x,y)u(y)dJ.l(Y) (3.17) where K(x,y) E HomdEy,FJ varies smoothly on X x X. DEFINITION 3.15. A linear map P: r(E) -+ r(F) is called a pseudodiffer-ential operator of order m if modulo infinitely smoothing operators P can be written as a finite sum P = LP" where each P" can be expressed in some system of local coordinates x,,: U" -+ [Rn and smooth bundle trivi-alizations as a pseudodifferential operator of order m with compact sup-port. The linear space of all such operators is denoted '¥DOm(E,F). Two such operators are called equivalent if they differ by an infinitely smoothing operator. Any differential operator P of order m is pseudodifferential. §3. PSEUDODIFFERENTIAL OPERATORS 187 REMARK 3.16. Given a riemannian metric on X and PE '¥DOm(E,F), e see from 3.7 that for any a> 0 there is an operator PE E '¥DOm(E,F) hich is equivalent to P and is a-local. (Of course, a differential operator O-local.) Using a good presentation of the bundles E and F (cf. 2.2), and patching gether local pseudodifferential operators with a partition of unity, one n manufacture interesting elements in '¥DOm(E,F) for any mE IR. Given a riemannian volume measure J.l on X, we can associate to any perator P: r(E) -+ r(F) a formal adjoint P*: r(F*) -+ r(E*) by setting Ix <Pu,v)dJ.l = Ix <u,P*v)dJ.l (3.18) rUE r(E) and v E r(F*). The following theorem is an immediate con-quence of the previous results of this section. heorem 3.17. Let E,F and G be smooth vector bundles over a compact anifold X andfix operators P E '¥DOm(E,F) and Q E '¥DOAF,G). Then the llowing statements hold: (i) P extends to a bounded linear map P: L;(E) -+ L;-m(F)for all s. (ii) For any open set U eX, Pulu is Coo. (iii) Q 0 PE '¥DOm+r(E,G). (iv) P* E '¥DOm(F*,E*) for any J.l. (v) A diffeomorphism cjJ: X -+ X induces a linear map cjJ*: '¥DOm(cjJ*E,cjJ*F) --'¥DOm(E,F) by the formula cjJ*[(cjJ*P)u] = P(cjJ*u). We now discuss some elements of the symbol calculus in the global setting. Let n* E and n* F denote the pull-backs of E and F respectively to the cotangent bundle n: T* X -+ X. DEFINITION 3.18. Let p be a smooth cross-section of the bundle Hom(n* E, n* F) on T* X. Then p is a symbol of order m if in a good presentation of E and F, p defines an element of Symm in each local co-ordinate chart of the presentation (where the variables ... are canonically identified with the coefficients of cotangent vectors in the basis {dXl' ... ,dxn}. It is easy to see that this definition is independent of the good presen-tation that is used. Wc let Symm(E,F) denote the vector space of all such symbols of order m. From Corollary 3.14 and Theorem 3.10 we have the following conclusion. 188 Ill. INDEX THEOREMS Theorem 3.19. Each PE 'JIDOm(E,F) has an associated "principal symbol" u(P) defined in the quotient space Symm(E,F)jSymm-l(E,F). It would be useful to be able to canonically construct for each ele-ment pE Symm(E,F) an associated operator PE 'JIDOm(E,F) with the same principal symbol as p. This can be done after introducing a riemannian metric on X and a connection V on E. We proceed as follows. Fix p > 0 sufficiently small that the exponential map expx: TxX -+ X gives a smooth embedding of the p-disk at each x. Fix a smooth "cut-off" function 1/1: [O,p] -+ [0,1] with 1/1 == 1 near 0 and 1/1 == 0 near 1. The riemannian metric determines a Lebesgue measure in each fibre of TX and of T* X, and we can define a "Fourier Transform" (. r: r(E) -+ r(n* E) as follows. For E we set == f e-i(v.oU(V)dV TxX (3.19) where U(V) == 1/I(lVllu(expx V) and where u(y) denotes the parallel trans-late of u(y) along the (unique, shortest) geodesic ray joining y to x, when distance (y,x) < p. For IVI p we set U(V) = O. The function lies in the Schwartz class of each fibre, Given a symbol p E Symm(E,F), we now define an operator P = p(V) as follows. For u E r(E) we set Pu(x) == (2n)-n fn,x (3.20) We leave as an exercise the proof that P E 'JIDOm(E,F). A symbol cal-culus for operators defined in this way has been worked out in Bokobza-Haggiag [1] and Widom [1]. If we take X to be flat euclidean space, and E = F to be the trivialized line bundles, and if we choose = L then a direct calculation shows that Pu(x) = L Aa(x)Dau(x). §4. Elliptic Operators and Parametrices Recall that a differential operator P:r(E) -+ r(F) over a compact mani-fold is called elliptic if its principal symbol is invertible at all non-zero cotangent vectors In this section we shall prove the fundamental result that modulo infinitely smoothing operators, an elliptic operator is invertible. To this end we consider the "local" case of pseudodifferential operators on [Rn which map Ck-valued functions to themselves. DEFINITION 4.1 An operator PE 'JIDOm with symbol p is said to be elliptic if there exists a constant c > 0 such that for all c the matrix §4. ELLIPTIC OPERATORS AND PARAMETRICES inverse of exists and satisfies c(1 + 189 (4.1) For example, an operator P whose symbol is of the form where p(t) is a polynomial with constant positive coefficients, is elliptic. REMARK 4.2. It is straightforward to verify that if P: r(E) --+ r(F) is an elliptic differential operator, then the local representations of P in a good presentation of E and F are elliptic in the sense of 4.1. Theorem 4.3. Let PE 'f'DOm be elliptic. Then there exists an operator Q E 'f'DO -m' unique up to equivalence, such that PQ = Id -S' and QP = Id -S (4.2) where S and S' are infinitely smoothing operators. Proof. Let p be the symbol of P and let c be the constant in Definition 4.1. Set qo(x, = -1 where x: IR + --+ [0, 1] is a smooth function with X(t) = 0 for t :s; c and X(t) = 1 for t 2c. • Lemma 4.4. qo E Sym-m. Proof. We must show that for each !Y. and f3 there is a constant CaP so that CaP(l + IW-m-IPI. Forrx = f3 = 0, this follows immediately from (4.1). For higher derivatives we first note that is estimated uniformly in x by derivatives of p -1. Taking derivatives of the equation pp-1 == p-1p == I (for c) and applying (3.3) and (4.1), we see that = Cj(l + IW-m-1 foreachj. Taking further derivatives, using (3.3), and applying straightforward induction complete the proof of the lemma. • Note that qop - I = pqo -1 = 0 for 2c. Consequently these func-tions lie in Symm for all m and the corresponding operators are infinitely smoothing. Unfortunately, pointwise multiplication of symbols does not give rise to composition of operators. We have instead the complicated formula (3.10): Placing qo in this formula, we find from the above observation that at least sym(QoP -1) E Symm-1. This suggests proceeding inductively to define a formal development (4.3) 190 Ill. INDEX THEOREMS where qk E Symm-k is defined by (4.4) By Proposition 3.4 there is an operator Q E 'f'DO -rn' unique up to equivalence, whose symbol has the formal development (4.3)-(4.4). Since the composition of any pseudodifferential operator with an infinitely smoothing operator is infinitely smoothing we are free to replace P and Q with any operators equivalent to them. In particular, by Corollary 3.7 we may assume P and Q to be I-local. Theorem 3.10 now applies to prove that QP -I is an infinitely smoothing operator. A completely analogous argument proves the existence of an operator Q' E 'f'DO -m such that PQ' -I is infinitely smoothing. Note, however, that Q '" Q(PQ') = (QP)Q' '" Q', that is, Q and Q' are equivalent. • The operator Q constructed in Theorem 4.3 is called a parametrix for P. Its existence proves many of the basic important facts concerning elliptic operators. Here is an example: Theorem 4.5. Let P E 'f'DOrn be an elliptic operator and choose u E L;, for some s. Then on any open set U £; [Rn, it is true that: Pu is Coo on U u is Coo on U. (4.5) Furthermore, if Pu = A.U for some A. E C and if m > 0, then u is smooth. Proof. Choose a parametrix Q with Id = QP + S as above. By 3.9, if Pu is smooth on U, then u = QPu + Su is smooth on U. If P is elliptic, so is P -A. Id for any A. E C provided m > O. Hence, by the above, if (P -A.)u = 0, then u is smooth. • We pass now to the global case. Let E and F be smooth vector bun-dles over a compact manifold X and consider an elliptic differential operator P: r(E) -+ r(F) of order m. Choose a good presentation of the bundles E,F with coordinates xp : Up -+ [Rn where 13 = 1, ... ,N, and with partition of unity {!/Jp} subordinate to the covering {Bp} where Bp = {p E Up : IxP(p)1 < I} (see Definition 2.2). In the 13th system P defines an elliptic operator Pp E 'f'DOrn for which there is a parametrix Qp E \P DO_m satisfying PpQp = Id -S'p and QpPp = Id -Sp, (4.6) where Sp and S'p are infinitely smoothing. By (3.7) we may assume that Qp and, therefore, also Sp and S'p are I-local. Now observe that for any I-local operator R in the 13th coordinate system, the operators !/J pR and §4. ELLIPTIC OPERATORS AND PARAMETRICES 191 Rt/lp have compact support in 2Bp {xp: Ixpl < 2}, and therefore define global operators in 'PDO*(E,F). (In particular, if u E r(E), we have t/lpRu = t/lpR(cpu) + t/lpR(1 -cp)u) == t/lfJR(cpu) where cp is any smooth cut-off function with support in Up and with cp == 1 on 2BfJ·) We now define global operators Q,Q' E 'PDO_m(E,F) and S,S' E 'P DO _ oo(E,F) by setting Q = L t/lpQfJ' Q' = L Qpt/lp, S = L t/lpSp, S' = L S'pt/lp. It follows immediately from (4.6) that PQ'u = L PpQ,lt/lpu) = L t/lpu-L: S'p(t/I pu) = u -S'u since L t/I p == 1. Similarly, one finds that QP = Id -S, and therefore also that Q '" Q(PQ') = (QP)Q' '" Q', that is, Q and Q' are equivalent. We have proved the following main result: Theorem 4.6. Let P: r(E) --+ r(F) be an elliptic differential operator of order m over a compact manifold. Then there is an operator Q E 'PDO -m(F,E), unique up to equivalence, such that PQ = Id -S' and QP = Id -S where Sand S' are infinitely smoothing operators. The operator Q is called a parametrix for P. (4.7) Notice that the equations (4.7) imply that PQP = P -S'P = P -PS and QPQ = Q -QS' = Q -SQ, and consequently that PS = S'P and QS' = SQ. (4.8) Furthermore, it is evident that Slkerp = Id and S'lker Q = Id. (4.9) REMARK 4.7. Theorem 4.6 carries over to pseudodifferential operators. An operator PE 'PDOm(E,F) is said to be elliptic if its principal symbol u(P) E Symm(E,F)/Symm-l(E,F) has a representative p which is pointwise invertible outside a compact set in T* X and satisfies the estimate :$; C(1 + for some constant C and some riemannian metric on X. A straightforward adaptation of the arguments above shows that Theorem 4.6 remains valid if the word "differential" is replaced by "pseudo-differential." This fact will not be used here, so we leave the details to the reader. REMARK 4.8. Given a differential operator P: r(E) .... r(F) of order m and a riemannian volume measure on X, let P*: r(F*) --+ r(E*) be the formal adjoint given by (3.18). Then P* is again a differential operator of order m whose principal symbol u(P*) is the pointwise transpose of u(P). In particular, P is elliptic if and only if P* is elliptic. 192 Ill. INDEX THEOREMS §5. Fundamental Results for Elliptic Operators In this section we shall prove some basic theorems for elliptic operators. These will include the fundamental elliptic estimates, the classical "Hodge Decomposition Theorem," the spectral decomposition for self-adjoint el-liptic operators, and estimates for the growth of eigenvalues, used later to establish the strong convergence properties of the heat kernel. Throughout this section E and F will denote smooth vector bundles over a compact n-dimensional manifold X. We shall prove our results here for differential operators but many of them carry over to pseudo-differential operators of positive order. To begin we recall some basic concepts concerned with a bounded linear operator T: H 1 --+ H 2 between Hilbert spaces. The kernel of T is the subspace ker(T) == {v E H 1: Tv = O}, and the range of Tis the subspace Im(T)== {TVEH2:VEH1}. The cokernel of T is the quotient space coker(T) == H 2/Im( T) by the closure of the range. The operator is called Fredholm if its kernel and cokernel are finite dimensional and its range is closed. Its index is then defined to be the integer ind(T) = dim(ker T) -dim( coker T). At the other extreme from Fredholm operators are compact operators. A bounded operator T: H 1 --+ H 2 is said to be compact if the image of each bounded sequence from H 1 has a subsequence which converges in H2• By the Sobolev Embedding Theorem, the inclusion L;.(E) c L;(E) for s' > s is compact. In particular any infinitely smoothing operator s: L;(E) --+ L;(E) is compact. The Fredholm operators are exactly those which are invertible modulo the compact ones. Lemma 5.1. Let T: H 1 --+ H 2 and Q: H 2 --+ H 1 be bounded linear maps such that QT = Id -Sl and TQ = Id -S2 where Sl and S2 are compact. Then T and Q are Fredholm operators. Proof· Since Sllker(T) = Id and Sl is compact, ker(T) must be finite dimen-sional. Taking adjoints, we find Q* T* = Id -and since is compact we conclude as above that dim(ker T*) = dim(coker T) < 00. It remains to prove that Im(T) is closed. By restricting to (ker T).L we may assume that T is injective. Let Vk = Tuk, k = 1,2, ... , be a sequence such that Vk --+ v in H2. We want to show that v = Tu for some U E H1• We note first that the sequence {ukH'O= 1 is bounded. Otherwise by passing to a sub-sequence, we can assume that IIukll --+ 00 and so T(uklllukil) = vJllukll .... O. Since QT = I -Sl and Sl is compact, we may assume by passing to a subsequence that lim(uklllukli) = lim Sl(ukllluklD = w where IIwll = 1. How-ever, by continuity Tw = 0, and since T is injective, w = O. We conclude that {uk}:'= 1 must be bounded. §5. FUNDAMENT AL RESULTS 193 Consider now the convergent sequence QVk = QTuk = uk -Sl(Uk) -+ Qv. Since {ukH'°= 1 is bounded and S 1 is compact we may assume, after passing to a subsequence, that Sl(Uk) -+ U<XJ. Applying T to the line above we find lim(Tuk -TS1Uk) = v -Tuoo = TQv. Hence, v E Im(T) and we have proved that Im(T) is closed. Therefore T (and by symmetry Q also) is Fredholm. • We now state our first main theorem: Theorem 5.2. Let P: qE) .... qF) be an elliptic operator of order m over a compact manifold X. Then the following is true: (i) For any open set U c X and any U E L;(E), pUlv E Coo (ii) For each s, P extends to a Fredholm map P: L;(E) -+ L; _ m(F) whose index is independent of s. (iii) For each s there is a constant Cs such that IIUlls C.(llulls-m + Ilpulls-m) for all U EL;. Hence the norms 11·lls and 11·lls-m + lip· Ils-m on L; are equivalent. Proof. Part (i) is a restatement of Theorem 4.5. For part (ii), note first that by Proposition 2.13, P extends to a bounded linear map P: L;(E) -+ L;-m(F). That this extension is Fredholm follows immediately from the existence of the parametrix (Theorem 4.6) and the lemma above. By part (i), ker P consists of smooth sections and its dimension is therefore inde-pendent of s. Similarly, the cokernel of P is isomorphic to the kernel of the adjoint 211 211 (5.1) L:'s+m(F*) L:'.(E*) which is easily seen to be the natural extension of the formal adjoint of P. Since P* is elliptic (see 4.8), dim(ker P*) is also independent of s. This proves part (ii). For part (iii), let Q E 'f'DO _m(F,E) be a parametrix as in Theorem 4.7. Then U = QPu + Su, and since S is infinitely smoothing, Ilulls IIQPulls + IISulls qllpulls-m + Ilulls-m)· • The above proof shows the following. Let P: qE) -+ qF) be an elliptic operator and P*: qF*) -+ qE*) its formal adjoint (defined using any 194 Ill. INDEX THEOREMS volume form on X). Define the index of P to be ind(P) == dim(ker P) -dim(ker P*). (5.2) CoroUary 5.3. The index of an elliptic operator P equals the index of any of its Fredholm extensions P: L;(E) --+ L;-rn(F). Part (iii) of Theorem 5.2 is calIed a "fundamental elIiptic estimate." Notice that if we solve the equation Pu = v, it alIows us to estimate the 11' lis-norm ofu in terms of the 1I'lIs-rn-norms ofu and v. It will be useful in later discussions to write down an important local consequence of this result. Theorem 5.4. Let P be an elliptic differential operator (of order > 0) defined on an open subset 0 of !Rn. Then for every compact subset K c 0 and every integer k 0, there is a constant C K,k such that for all solutions u of the equation Pu = 0, one has that (5.3) where ,,'IIK,Ck denotes the uniform Ck-norm on K and 11· IIel,L2 denotes the L2-norm on O. Proof. Choose cp E CO'(O) with cp == 1 on K. Observe that P(cpu) = cpPu + L: aix)(Dau)(x) where the sum is over lal < m = order(P) and where the coefficients aa depend only on P and cp. Assume Pu = O. Then the funda-mental elliptic estimate 5.2(iii) applies to give < + IIpcpullfl,L;_J Taking a sequence K c c 01 C C O2 cc· .. C C ON = 0, applying the argument repeatedly, and then using the Sobolev Embedding Theorem 2.15(1) completes the proof. • It is often useful, when studying an operator P: qE) --+ qF) to consider non-negative operators P* P and pp* where P*: qF) --+ qE) is the formal adjoint defined via bundle metrics. For this reason we shall assume from this point on that E and F are equipped with hermitian inner products and unitary connections, and that X is furnished with a riemannian metric. AII connections will be denoted by V. The Sobolev norms 11' Ilk with k E 7L + will be given explicitly by (2.1). Given an mth order differential operator P: qE) --+ qF) and bundle metrics as above, we define the formal adjoint of P to be the map P*: qF) --+ qE) such that (PU,V)L2 = (u,P*V)L2 for alI u E qE) and all v E qF). Integration by parts shows that P* exists and is also a differential operator §5. FUNDAMENTAL RESULTS 195 of order m. Furthermore, u(P*) = u(P)* and so P* is elliptic if and only if P is elliptic. A differential operator P: nE) --+ nE) is then said to be self-adjoint if P = P*. The Dirac operator of a Dirac bundle is always self-adjoint, and so, of course, are its powers (see 11.5.3.). An important example for rie-mannian geometry is the Dirac operator D on the Clifford bundle C£(X). Under the canonical isomorphism C£(X) A*(X) we have D d + d*, and so D2 dd* + d*d == Ll, the Hodge Laplacian (see 11.5.12). Theorem 5.5. Let P: nE) --+ nE) be an elliptic self-ad joint differential op-erator over a compact riemannian manifold. Then there is an L 2-orthogonal direct sum decomposition: nE) = ker P EB Im P (5.4) Proof. With respect to the decomposition L 2(E) = ker P EB (ker P).l any element u E nE) c L2(E) can be written as u = Uo + Ul with PUo = O. Since u and Uo are smooth, so is ul. Consider now the Fredholm map P:L2(E) --+ whose adjoint P* : L?;.(E) --+ L 2(E) (cf. 5.1) is the natural extension of P itself. We clearly have that (ker P).l = Im p* = P(L?;.(E)). Hence, we can write ul = PUl for ul E L?;.(E). Since Ul is smooth, so is ul by Theorem 5.2. • Consider the special case E = A*X and P = Ll = dd*+d*d (see 11.5.12 forward). Clearly, Im Ll = Im d + Im d*. However, since (d*cp, dt/l)L2 = (cp, d2t/1)u = 0 for all cp,t/I, this sum is U-orthogonal. Letting B* == ker Ll denote the harmonic forms we have the following: Corollary 5.6 (The Hodge Decomposition Theorem). Let X be a compact riemannian n-manifold. Then there is an L2-orthogonal direct sum decom-position of the smooth p-forms nAPX) = BP EB Im d EB Im d* (5.5) for p = 0, ... ,no Theorem 5.5 has another important consequence: Corollary 5.7. Let P: nE) --+ nE) be a self-adjoint elliptic operator over a compact riemannian manifold, and let H: nE) --+ ker P be the orthogonal projection given by the decomposition (5.4) Then there is an operator G: nE) --+ nE) of degree -m, called the "Green's operator," such that PG = GP = Id -H. Proof. The map P: Im P --+ Im P is an algebraic isomorphism and has an inverse p-l. Let G = p-l 0 (1 -H). G obviously extends to a bounded 196 III. INDEX THEOREMS map G: L;(E) --+ L;+m(E) which is a Hilbert space isomorphism on (ker P).l for all s. • We now consider the spectral theory for a self-adjoint elliptic operator P: nE) --+ nF). For A. E C we consider the l-eigenspace of P given by EA = ker(P -A.I), (5.6) and we say that A. is an eigenvalue of P if dim E;. > O. Theorem 5.8. Let P: nE) --+ nE) be a self-adjoint elliptic differential op-erator of order m > 0 over a compact riemannian n-manifold. Then each eigenspace of P is finite-dimensional and consists of smooth sections. The eigenvalues of P are real, discrete and tend rapidly to infinity in the fol-lowing sense. If d(A.) = dim( E8 E;.), (5.6) Then there is a constant c such that d(!\);£ cN(n+2m+2)/2m. (5.7) Furthermore, the eigenspaces of P furnish complete orthonormal systems for L2(E), i.e., there is a Hilbert space direct sum decomposition (5.8) Proof. The first statement follows immediately from the fact that P -A.I is elliptic. To see that each eigenvalue is real, note that if Pu = A.U, then A.llul12 = (PU,U)L2 = (u,Pu)u = J:HuI12. To prove the estimate (5.7) we proceed as follows. Given e > 0, we say that a subset A c X is 8-dense if for each point x E X there is a point a E A with dist(x,a) < e. For each e > 0, let N(e) be the minimal possible number of elements in an e-dense subset of X. One sees easily that for some constant C. N(e) ;£ Ce-n for all e > O. (5.9) Consider now a function u E E;. and note that for each integer k > 0, pku = A.ku. Hence by the elliptic estimates 5.2(iii) we see that there is a constant Ck, independent of u, such that (5.10) Set E(A.) = E8IAI "A E;. and write u E E(A.) as u = L a;.u;. where u;. E EA has Ilu;.11 = 1. Then pku = L a;.A.ku;., and since U;. 1-uJj for A. "# j.l, we have that I = L la;.121A.12k ;£ L la;.12A.2k = Consequently, by (5.10) we have that §5. FUNDAMENTAL RESULTS 197 for all u E E(1\.). If we assume that mk > (n/2) + 1, the Sobolev Embedding Theorem 2.15 gives us a constant such that sup IVul + 1\.k)llullo (5.11) x for all u E E(1\.). Suppose now that for a given e > 0 we have d(1\.) = dim E(1\.) > N(e). Then there will be an element u E E(1\.) with Ilullo = 1 such that U = 0 on an e-dense subset of points in X. It then follows from (5.11) that sup lul + N). x 1 However, this is impossible if + N)vol(X)2 < 1 since J lul2 = 1. We 1 conclude that d(A.) N(eA) where eA 1 == + N)vol(X)2 Consequently, by (5.9) we have d(1\.) CeAn ckNk• Choosing k = [(n + 2m + 2)/2mJ completes the proof of the estimate (5.7). It remains to prove (5.8). Let Vs denote the closure in L;(E) of the subs pace consisting of finite linear combinations of eigensections of P. Note that by 5.2(ii) the map P: L;(E) --+ L;_m(E) is an isomorphism on (ker P).l. Hence, P: V --+ V _ m is a Hilbert space isomorphism for all s. We want to show that Vii-= {O}. Let's assume Vii-of-{O} and consider p2 = inf{(p2u,u) = u E Vii-and lIullo = 1}. Note that is dense in Vii-and so p2 < 00. (This density follows from the fact that implies V2m is dense in Vo and is dense in Choose a sequence {Uk}kX)= 1 c Vii-with Ilukllo = 1 for all k and lim(p2uk,uk) = p2. Since + is uniformly bounded we know from Theorem 5.2(iii) and the compactness of the embedding L;'(E) c L6(E), that there exists a subsequence, also denoted {uk}k°= b such that Uk --+ U in L6(E). We claim that (5.12) for all v E vtm (V-2m)*' If not, then there exists v E with Ilvllo = 1 and ((P2 -p2)U, v) = -rx < O. For t E IR, consider the vectors uk(t) = Uk + tv and note that O ((P2 -p2)Uk(t), Uk(t))O -2rxt + t2 0 < < = 1 + 2t(u,v) + t2 for It I sufficiently small. This proves (5.12), which in turn proves that p2u = p2u. Since p2 is elliptic, the p2-eigenspace is finite dimensional and P-invariant. Consequently it contains an eigenvector of P in contradiction with the definition of Vii-. This completes the proof of Theorem 5.8. • 198 Ill. INDEX THEOREMS A self-adjoint operator P is said to be positive if (Pu,u)o 0 for all U E r(E). Theorem 5.8 can be reformulated in the positive case as follows: Corollary 5.9. Let P: r(E) -+ r(E) be a positive self-ad joint elliptic differ-ential operator of order m> 0 over a compact manifold. Then there is a complete orthonormal basis {Uk}k=1 of LME) such that for all k (5.13) where 0 ::; Al ;£ A2 ;£ ... -+ 00. In fact for some constant c > 0, for all k. (5.14) Proof. The first statement is clear. For the second note that d(Ak) = k. • §6. The Heat Kernel and the Index Let P: r(E) -+ r(E) be a positive self-adjoint elliptic differential operator of order m over a compact riemannian n-manifold X. In this section we shall explicitly construct the heat operator e-tP: r(E) -+ r(E), for t > 0, which is an infinitely smoothing operator with the property that if Ut = e-tPu, for some U E r(E), then Ut satisfies the equation d dt Ut + PUt = O. We shall define e-tP as an integral operator of the form (e-tPu)(x) = Ix Kt(x,y)u(y)dy (6.1) (6.2) where Kt(x,y): Ey -+ Ex is a linear map depending smoothly on x,y and t. (Hence, K is a smooth section of the obvious bundle over x X x X.) This kernel K is called the heat kernel for P and is defined as follows. Let {Uk}k= 1 be a complete orthonormal basis of L 2(E) consisting of eigen-sections of P with PUk = AkUk and 0 ;£ Al ;£ A2 ;£ ... -+ 00 as guaranteed by Corollary 5.9. We set 00 Kt(x,y) = L e-.l.ktUk(X) ® ut(y) (6.3) k=1 where v* E Ej denotes the element such that v*(u) = (u,v)y for all U E EY' Lemma 6.1. For any r 0 and any closed interval le (0,00), the series (6.3) converges uniformly in the C-topology on I x X x X. Proof. Fix a positive integer s with ms > (nI2) + r. By the Sobolev Em-bedding Theorem 2.15 and the Fundamental Elliptic Estimates 5.2(iii) we have constants c and c', depending only on s, such that each Uk satisfies §6. HEAT KERNEL AND THE INDEX 199 the inequality: IIUkllcr :;:;; c'llukllms :;:;; c(llukllo + IlpSUkllo) = c(1 + An where 11'llv denotes the function C-norm on X. By Corollary 5.9, Ak satis-fies the inequality Ak kY where y = 2m/n(n + 2m + 2). This implies that e -Akt :;:;; (e -kYt)( kSY) for all k > (S/t)l/Y. The uniform C-convergence of (6.3) now follows di-rectly from the convergence of the integral H' e -txxs dx. • This lemma has the following immediate consequence: Theorem 6.2. For each t > 0 the operator e-tP: r(E) --+ r(E) is infinitely smoothing. Furthermore, given U E L;(E), for any s, the section u(t,x) == (e-tPu)(x) is Coo on !R x X and satisfies the "heat equation," au/at = -Pu. Proof. The first two statements are an immediate consequence of the fact that K is Coo. To see that au/at = -Pu, note that a at KAx, y) = -L e-AktAkUix) ® ut(y) = -PxKt(x,y). • For a given operator P as above, we introduce the following important concept. DEFINITION 6.3. The trace of the heat kernel for P is the function which is defined and analytic for all t > O. Note that tracex(uk(x) ® ut(x)) = IUk(X)!2 and so the second equality follows from the fact that = 1 for all k. EXAMPLE. Let X be the flat cubical torus !Rn/2n;En, where !Rn carries the standard metric, and let P = -= -L a2/aof acting on functions. The normalized eigenfunctions are given by uN(O) = (2n)-n/2ei<N,B) for N = (N 1, .•• ,N n) E ;En. Note that -= INI2uN, and so Kt(O, 0') = L e-INI2t+i<N,B-B') Ne7L" 200 and Ill. INDEX THEOREMS 00 tr(etd) = L a(k)e-kt k=O where a(k) denotes the number of elements NE 7Ln with INI2 = k. For 11 = 1, we find -00 Let us return now to the case of a general elliptic differential operator P : r(E) -+ r(F) over a compact riemannian manifold X. We assume E and F are equipped with bundle metrics, and we consider the "Laplace operators": P*P:r(E) -r(E) and PP*: r(F) -r(F). Since (p* Pu, u)o = and (P P*v, v)o = we see that these elliptic operators are self-ad joint and positive. It is furthermore evident that ker P*P = ker P and ker PP* = ker P*. Consequently, we have (cf.(S.2» that ind(P) = dim(ker P* P) -dim(ker P P*) (6.5) On the other hand, the operators P* P and P P* have exactly the same sequence oJnon-zero eigenvalues. To see this, set EA = {u E r(E): P*Pu = AU} and FA = {vEr(F): P P*v = AV} for A E IR. Observe that if u E E A' then (PP*)(Pu) = P(P*Pu) = A(PU); that is, P(EA) c FA-Similarly, we find that P*(FA) c EA. Since P*P = A Id on EA, we conclude that P:EA FA is an isomorphism for all A "# O. Let 0 < A1 A2 A3 ... -+ 00 denote the common non-zero eigen-values of P* P and P P* (listed to multiplicity). Then taking the difference of the traces of the heat kernels, we get enormous cancellation(!): tr(e-tP•P) -tr(e-tPp.) = (dim Eo + L e-Akt) -(dim Fo + L e-Akt) = dim Eo -dim Fo. This proves the following: Theorem 6.4. LetP: r(E) -+ r(F) be an elliptic differential operator over a compact riemannian manifold, and let P* : r(F) -+ r(E) be defined via inner products in E and F. Then ind(P) = tr(e-tP•P) -tr(e-tPp.) Jor all t > o. Observe that as t -+ 00, the operator e-tP•p converges strongly to the orthogonal projection HE: L 2( E) -+ ker P. Consequently the direct sum §7. TOPOLOGICAL INVARIANCE OF THE INDEX 201 Dt =e-tP*P EFl (_e-tPP*) acting on nE EFl F) can be thought of as a "ho-motopy" of operators which, as t -+ 00, converges to HE EFl ( -H F), the "difference of harmonic projections." We see that for all t, Dt is of trace class and its trace, tr(Dt) = ind(P), is time independent. It is useful to con-sider the operator Dt at the other end of the "homotopy," as t \. 0. When deg(P) = 1, it turns out that as t \. 0, the heat kernel for P* P has an asymptotic expansion 00 tracexKt(x,x) '" L Pk(X)t(k-nl/2 k=O (6.6) Where pix) are densities on X which are locally and explicitly computable in terms of the geometry of X and P. Theorem 6.4 says that the index of P depends only on the coefficient Pn(x) for the operators P*P and PP*. Careful computations of these terms yields a proof of the classical Atiyah-Singer Index Theorem. §7. The Topological Invariance of the Index The index of an elliptic operator is a quite stable object. It remains con-stant during continuous perturbations of the operator and, in fact, it can be. shown to depend only on the homotopy class of the principal symbol. A proof of this fact is one of the main objectives of this section. In succeed-ing sections we shall consider elliptic operators with additional structure, such as operators which are Cfk-linear for some k or operators which commute with a given action of a compact Lie group. In each case we shall define a refined index and examine its elementary homotopy invari-ance properties. The index of an elliptic operator P: nE) -+ nF) is the index of any of its Fredholm extensions P: L;(E) -+ L;-m(F). For this reason we begin with a discussion of Fredholm operators. Let H 1 and H 2 be separable Hilbert spaces and let IL' = IL'(H 1, H 2) denote the Banach space of bounded linear maps from H 1 to H 2 with norm given by IITII = sup{IITvll: Ilvll 1}. It is elementary to verify that the subset IL' x c IL' of linear isomorphisms is open in the norm topology (cf. Palais [1]). We are interested here in the subset !Y = !y(H 1, H 2) of Fredholm operators from H 1 to H2 (see §5). To each TE !y, we have defined the index of T: ind T = dim(ker T) -dim( coker T). Proposition 7.1. The map ind:!y -+ 7L is locally constant on !y, and induces a bijection ind : 1to(!Y) 7L between the connected components of !Y and the integers 7L. 202 Ill. INDEX THEOREMS Proof. We begin the proof with a lemma which will be useful later on. Lemma 7.2. Fix To E and let Vc H 1 be a closed subspace offinite codi-mension with V n (ker To) = {O}. Then there is a neighborhood I1ll of To in IR such that for all Tin I1ll we have: (i) V n (ker T) = {O}, (ii) TV is closed in H 2, (iii) the subspace W == (To V)l. c H 2 projects isomorphically: H2/TV. Furthermore, the isomorphisms of (iii) assemble to give that: (iv) Thefamily UT E "11 (H2/TV) -+ 11ll, topologized as a quotient ofl1ll x H 2, is equivalent to the trivial bundle I1ll x W -+ 11ll. Proof. To each TE IR we associate the bounded linear map (7.1) given by setting T(w,v) = w + Tv. This correspondence T -+ 1" defines a continuous map IR -+ IR(WEFl V,H2) in the norm topology. Since To is an isomorphism, so is 1" for all T in a neighborhood I1ll of To. This estab-lishes (i)-(iv). • Corollary 7.3. is open in IR. Proof. Choose V = (ker To)l. in Lemma 7.2 and note that I1ll c by (i), (ii), and (iii). • Corollary 7.4. The index is constant on connected components of Proof. Fix To E set V = (ker To)l., and let I1ll be given by Lemma 7.2. It will suffice to show that ind T = ind To for all T E 1111. Fix T E I1ll and con-sider the (non-orthogonal) direct sum decomposition H 1 = (ker 1') EFl Z EFl V where Z == (ker T EFl V)l. (see 7.2 (i». By the Open Mapping rem, T induces an isomorphism between Z EFl V and the closed subspace TZ EFl TV. Setting coker T == CTH 1)1. = (TZ EFl TV)l., we obtain the fol-lowing facto ring of T: 'Ltl -T H2 III III ker T EFl Z EFl V --+ coker T EFl TZ EFl TV Dividing by V = (ker T 0)1. gives an isomorphism ker To ker T EFl Z. (7.2) (7.3) §7. TOPOLOGICAL INVARIANCE OF THE INDEX 203 Dividing by TV and using 7.2(iii) (with W = coker To) gives an isomor-phism coker To coker T Et> TZ coker T Et> Z. (7.4) It follows immediately that ind To = ind T. • We shall examine this argument again when the maps have more struc-ture. The proof of Proposition 7.1 is completed by the following: Lemma 7.6. Any two operators To, Tl E with the same index lie in the same connected component of fY. Proof. Since ind(T*) = -ind T, it suffices to consider the case where ind To = ind Tl O. To begin we note that any TE with ind T 0 is homotopic in fY to a surjective map. To see this, choose any linear surjec-tion L,' ker T -++ coker T = (Im T)l., and consider the homotopy T + tL, t O. We may assume therefore that To and Tl are both surjective. Set Kj = ker(Tj) and consider the decomposition H 1 = Kj EB KT for each j. We have the isomorphism B == TllTo:Kt Kt, and since Ko and K1 have the same finite dimension, we can choose an isomorphism A:Ko -+ Kl. The direct sum C = A Et> B:Ko Et> Kt -+ K1 Et> Kt is an automorphism of H 1 such that To = T1 C. Suppose we can find a contin-uous family of isomorphisms Ct: H 1 -+ H 1, 0 t 1, with Co = C and Cl = Id. Then Tt == Tl Ct connects To to Tl. Therefore we are done when we have proved the following: Lemma 7.7. The set [£X = [£X(H,H) of isomorphisms of Hilbert space is connected. Proof. We fix C E [£ x and show that C can be connected to the identity. To begin we put C in polar form C = U A where A is the positive square root of the positive self-adjoint operator C*C, and where (since U*U = A -1 C* CA -1 = A -1 A 2 A -1 = Id), U is unitary. Since the positive bounded self-adjoint operators form a convex set, we see that C is homotopic to U. Now U has a spectral decomposition of the form U = i5" ei;" d1t;" and can be written as U = eiT where T = )'d1t;". The homotopy Ut = eitT, 0 t 1, completes the proof of Lemma 7.7 and so also of Lemma 7.6 and Proposition 7.1. • Note. When H 1 = H2 == H, the composition of operators makes 1tofY a semi group. Interestingly, the map ind: 1tofY -+ 7L is a group isomorphism. To prove this we need only observe that ind(S 0 T) = ind(S) + ind(T) (7.5) 204 Ill. INDEX THEOREMS for all S,T E !Y. To begin let TC: H -+ (ker S).L be orthogonal projection and note that since T is homotopic to TC 0 T in !Y (by a linear homotopy) we can assume that Image(T) c (ker S).L. The assertion is now easily checked. Let us now return to the case of elliptic operators over a compact manifold X. A family Pt: qE) -+ qF),O :s; t :s; 1, of such operators is said to be continuous if in any good presentation of the bundles the coefficients of the local representations Pt = Aa(x,t)Da are jointly continuous in x and t. Under this hypothesis the order of the operators must be a constant, say m. Furthermore, for any s the map [0,1] -+ !y(L;(E), L;-m(F)) given by t f-+ Pt, is continuous in the norm topology. Consequently, ind(Pt) is constant in t by Proposition 7.1. Recall that two elliptic operators P o,P I: qE) -+ qF) over X are homo-topic if they can be joined by a continuous family Pt, 0 :s; t :s; 1, of elliptic operators. Our remarks above can be restated as follows: Corollary 7.8. The index of an elliptic operator on a compact manifold depends only on its homotopy class. An immediate consequence is this: Corollary 7.9. The index of an elliptic operator on a compact manifold depends only on its principal symbol. Proof· If P o,P I: nE) -+ qF) have the same principal symbol, then so does each element in the family Pt = (1 -t)Po + tPI. • This result suggests making deformations of the principal symbol. Re-call that the principal symbol of an elliptic operator is a section (J E r[(OmTX) ® Hom(E,F)] with the property that (J : E -+ F is an isomophism for all "# O. (7.6) We shall say that two such symbols (Jo,(Jt are regularly homotopic if there is a homotopy (Jt, 0 :s; t :s; 1, joining them such that satisfies (7.6) for all t. Theorem 7.10. The index of an elliptic differential operator on a compact manifold depends only on the regular homotopy class of its principal symbol. REMARK 7.11. Using Theorem 3.19 and the subsequent discussion there, one can generalize this result to elliptic pseudodifferential operators. Proof. In light of 7.8 and 7.9 it will suffice, given (Jt, to construct a family of operators Pt with (J(Pt) = (Jt. This is evidently possible locally given coordinates on X and trivializations of E and F. Patching together with a partition of unity over some good presentation of E and F does the job globally. • §8. INDEX OF A FAMILY 205 The question, manifest at this point, is whether the index of an elliptic operator can be computed directly from its principal symbol. A procedure for doing this was first given by Atiyah and Singer. It will be discussed in detail in § 13. §S. The Index of a Family of Elliptic Operators In this section we shall present the important concept, due to Atiyah and Singer, of an index for families of elliptic operators. In examining the topological invariance of this index we shall prove that the Fredholm operators on Hilbert space constitute a classifying space for K-Theory. Let A be a Hausdorff topological space, and let E and F be smooth vector bundles over a compact manifold X. The definition given in §7 of a continuous family Pt: r(E) -+ r(F), t E [0,1], of elliptic operators, can be extended directly by replacing [0,1] with A. (We shall call this a product family.) However, when A is homotopically non-trivial, it is important to allow the manifold and the operators to twist globally over A (in the spirit of the families of complex analytic objects considered by Kodaira and Spencer). For a smooth vector bundle E -+ X, let Diff(E;X) be the set of diffeo-morphisms of E which carry fibres to fibres linearly. Note the homo-morphism f3: Diff(E;X) -+ Diff(X) onto the diffeomorphism group of X. Let == Diff(E,F;X) be the subgroup of Diff(E EFl F;X) which maps E to E and F to F. This group acts naturally on the space Opm(E,F) of all differential operators P: r(E) -+ r(F) of order m (by setting g(P) = g2 0 P 0 g11 for g = (gl,g2»' DEFINITION 8.1. A continuous family of smooth vector bundles over X parameterized by the Hausdorff space A is a fibre bundle I! -+ A whose fibre is a smooth vector bundle E over X and whose structure group is Diff(E;X). From the homomorphism f3: Diff(E;X) -+ Diff(X) we get an associated fibre bundle q; -+ A whose fibre is X and whose structure group is Diff(X). Note that I! is just a vector bundle over the total space of q; which is smooth on each fibre. Furthermore, this smooth structure is changing continuously over A. DEFINITION 8.2. By a continuous pair of vector bundles over X param-eterized by A we mean a bundle I! EFl /ff' -+ A whose fibre is a split bundle E EFl F over X and whose structure group is = Diff(E,F;X). Associated to such a pair is the family of differential operators of order m from E to F. This is the bundle Opm(t!,/ff') -+ A associated 206 Ill. INDEX THEOREMS to the principal .@-bundle of It Et>:F by the action of .@ on Opm(E,F). A continuous section P of such a bundle whose fibre is elliptic for all a E A is called a family of elliptic operators parameterized by A. EXAMPLE 8.3. Let X be a compact manifold and consider its asso-ciated Jacobian torus lx == Hl(X;IR)/Hl(X;£:). Recall the isomorphisms: Hl(X;IR) Hom(nlX,IR) and Hl(X;£:) Hom(nlX;£:), Let X denote the universal covering of X. Then we define a flat complex line bundle Lover X x 1 x by taking the quotient L == X x Hl(X;IR) x C/n1(X) x Hl(X;£:) (8.l) where the action of n1X x Hl(X;£:) is given by cp(g,hlX,v,z) == (gx,v + h,e21tiV(g)z). (8.2) We think of L as a family of flat line bundles on X parameterized by lx· Suppose now that X is even-dimensional and oriented, and let S be any Dirac bundle over X with associated Dirac operator D. (For example, one could choose the Clifford bundle or the spinor bundle for some riemannian metric on X.) Then S ® L = (S+ ® L) Et> (S-® L) is a con-tinuous pair of vector bundles on X parameterized by 1 x. Using the given flat connection on L, the Dirac operator on S extends to a family of Dirac operators (8.3) for v E lx. For S = C£(X), this family was introduced by G. Lusztig [1] is his proof of the Novikov Conjecture for n1 £:m. Suppose now that P is a family of elliptic operators defined over a compact Hausdorff space A. Following Atiyah and Singer [3] we shall define an analytic index ind(P) in the group K(A). If the dimension of ker Pa (and therefore also of coker Pa) were locally constant on A, then this index would simply be the formal difference of finite dimensional vector bundles: ind P = [ker P] -[coker P] E K(A). (8.4) In general these dimensions are not constant, and so we must stabilize the picture. Let It, /F and q; be as above. For a E A, let Ita (and /Fa) denote the fibre of It (and /F respectively) at a. This is a smooth vector bundle on X and we denote by r(lta) its space of smooth cross-sections. Lemma 8.4. There exists a finite set of sections {Wl' ... ,wN} of /F over q; such that for each a E A the map Pa: CN Et> r(lta) --+ r(:Fa) given by Pa(tl,' .. ,tN,cp) = L tjwJ'l'a + Pa(cp), is surjectivefor all a. The vector spaces §8. INDEX OF A FAMILY ker Fa form a vector bundle over A, and the element [ker F] -[CN] E K(A) depends only on the operator P. 207 (8.5) Proof. From the local triviality of fibrations each point ao E A has a neighborhood U in which Pa, a E U, is a product family. For any s, this family extends to a continuous map P: U -+ jj(L;(E), L;-m(F», and we can apply Lemma 7.2. For W = ker(P:o) this lemma shows that the maps Fa: W EFl L;(E) -+ L;-m(F) given by Fa(w,cp) = W + Pa(CP) are all surjective in a neighborhood U' of ao. Theorem 5.2(i) implies that Wc r(F) and that the restriction Fa: W EFl r(E) -+ r(F) is also surjective for all a E U'. Corollary 7.4 shows that dim(ker Fa) is locally constant on U'. We now globalize this construction. Let {w1, ... ,wr} be a basis for W. Each Wj can be considered as a (constant) section of :F over U', or alternatively as a section of the vector bundle :F over the open set n-1(U') c PI (where n: PI -+ A is the fibration). Clearly each Wj can be extended to a global section Wj of :F over all of PI. Taking a finite covering of A by such neighborhoods, and taking {w1, ... ,WN} to be the union of the sections constructed from each neighborhood, we establish the first statement of the lemma. The local constancy of dim(ker P J follows by repeating the argument above with W replaced by the direct sum of Ws from each neighborhood. It remains to prove that the class (8.5) is independent of the choice of global sections W1, ... ,WN• Let ... ,WN' be another such choice, and the family of maps Fa: cN EFl cN' EFl -+ r(:FJ given by Fa(t,t',cP) == L tiwi(a) + L tjwj(a) + Pa(CP)· We have a commutative diagram n n ker Pa c CN EFl CN' EFl r(:Fa) 1 pr 1 CN' CN' where the map pr denotes projection. Since the map Fa is surjective one easily sees that the sequence o ----+ ker F ----+ ker F ----+ CN' ----+ 0 a a is exact. It follows that [ker Fa] -[CN+N] = [ker Fa] -[eN] in K(A). The argument applies to the operators CN' EFl r(CJ -+ to prove that [ker Fa] -[CN+N] = [ker -[CN], and the proof is complete. • 208 Ill. INDEX THEOREMS We are now authorized to make the following definition: DEFINITION 8.5. The analytic index of a family of elliptic operators P over a compact Hausdorff space A is the element ind(P) E K(A) defined by (8.5). As one might imagine, this index enjoys invariance properties analogous to those of the ordinary index. It depends only on the principal symbol of the family, and in fact only on the homotopy class of the principal symbol (all taken in the obvious sense; see Atiyah-Singer [3] for details). This invariance is related to a basic and interesting fact. Let H be a (infinite-dimensional) separable Hilbert space. Recall that any two such spaces are isomorphic. Furthermore, there is Kuiper's Theorem which states that the group If' x = If' x (H,H) of linear isomorphisms of H with the norm topology is contractible. Consider a family of elliptic operators P over A as above. This is a section of the bundle with structure group fl). We can fix s and complete each fibre in the Sobolev norms to get a bundle with fibre H lJ(L;(E), L;-m(F)). This is the bundle asso-ciated to the principle fl)-bundle by the homomorphism fl) --+ If' x (H, H). Since If'X(H,H) is contractible, this bundle is trivial as an If'X(H,H)-bundle in a homotopically unique fashion. Under the trivialization, the family P becomes a continuous map P: A --+ H = lJ(H b H 2) where H 1 = L;(E) and H 2 = L;-m(F). Note that the set 0: = lJ(H, H) of Fredholm operators on a Hilbert space H has a continuous associative semi-group structure given by the composition 0: x 0: --+ 0:. For any topological space A, this makes the space [A, 0:], of homotopy classes of maps of A into 0:, into an associative semigroup. Theorem 8.6. (cr. Atiyah [4]). For any compact Hausdorff space A there is a natural isomorphism ind: [A, 0:] --K(A) (8.6) It has the property that for any continuous map f : A' --+ A between such spaces, ind 0 f* = f* 0 ind (8.7) Consequently, 0: is a classifying space for K-Theory. This theorem completely generalizes Proposition 7.1 (where A = [0,1]) and is the foundation for proving the homotopy invariance of the index for families. Proof. Let T: A --+ 0: be a continuous map. By Lemma 7.2 and the compactness of A we know that there is a closed subspace V c H of §8. INDEX OF A FAMILY finite codimension so that for all a EA (i) V n (ker Ta) = {O}, (ii) Ta V is closed and of finite codimension in H, and (iii) HjTV == UaEA (HjTaV) is a vector bundle over A. One then defines ind(T) == [H/V] -[H/TV] E K(A), 209 (8.8) where [H/V] = [A x (H/V)] denotes the trivial bundle. We must show that this element is independent of the choice of V. Let V' be another such choice. Since V n V' is also a choice, we may assume V' c V. There is then an exact sequences of vector bundles: 0 ...... V/V' ...... H/TV' ...... H/TV ...... 0, which shows that [H/TV] -[H/TV'] = [V/V'] = [H/V] -[H'V']. Hence, (8.8) depends only on T. Given T: A ...... 0: and a map f: A' ...... A, the subspace V chosen for T also works for To f and we evidently have ind(T 0 f) = f*(ind T). Suppose now that T: A x I ...... 0: is a homotopy between To and T 1 where Tj = To ij, and ij: A ...... A x I is the inclusion iia) = (a,j). By the above paragraph ind(T) = ij(ind T) for j = 0,1, and it is a basic fact that i6 = it in K-theory. This proves the homotopy invariance of ind. It remains to show that (8.6) is an isomorphism. We show first that it is a homomorphism. Let T: A ...... 0: and T': A ...... 0: be continuous maps and choose Vc H for T as above. Note that T' is homotopic to prv 0 T' where prv:H ...... V is orthogonal projection. Hence, we may assume T' H S V. Let V' be a choice of subspace for T', and note that V' is also a choice for the composition ToT'. Therefore we have ind(T 0 T') = [H/V'] -[H/TT'V'] ind(T) = [H/V] -[H/TV] ind(T') = [H/V'] -[H/T'V']' T From the exact sequence: 0 ...... V/T'V' ...... H/TT'V' ...... H/TV ...... 0 of vec-tor bundles over A, we see that [H/TT'V'] = [H/TV] + [V/T'V'] = [H/TV] + [H/T'V'] -[H/V]. Plugging this into the equation above shows that ind(T 0 T') = ind(T) + ind(T') as required. To prove that ind is surjective we first recall from Chapter I, §9, that every element in K(A) can be written as ECk] -[E] where E is a vector bundle on A and Ck denotes the trivial k-plane bundle. For each integer k we define an operator Sk EO: of index k by fixing a complete orthonormal basis {e J.I= 1 of H and setting S {ej-k k(e) = 0 otherwise. ifj-k>O The constant map T == Sk on A has ind T = ECk] for k O. 210 Ill. INDEX THEOREMS Fix a vector bundle E over A and recall from Chapter I, §9, that for some N there is a continuous map f: E --. CN which is a linear injection on each fibre. Let pra: CN --. CN denote orthogonal projection onto f(E,,) and let pr; = Id -pra denote projection onto the orthogonal comple-ment. Then the map TE: A --. ® H, CN ® H) given by setting T J.a) == pra ® S -1 + pr; ® Id has the property that ind T E = -[E]. Choosing an isomorphism CN ® H H yields an isomorphism ® H, CN ® H) Since ind(Sk 0 TE) = ECk] -[E], we have proved that ind: --. K(A) is a surjective homomorphism. Our next step is the following. Let x denote the invertible elements in Lemma 8.7. If T: A --. has index zero, then T is homotopic to a map T' : --. x C Proof. Choose a subspace V c H as above so that ind T = [H/V] -[H/TV]. The hypothesis ind T = 0 means that for some k, we have A x [(H/V) Efl Ck] (H/TV) Efl Ck• This implies that if we replace V by a closed subspace of codimension k in V, we have the bundle isomorphism A x (H/V) H/TV. It is elementary to verify that there is a continuous map H/TV --. H which carries H/TaVisomorphically onto (Ta V).l for each a E A. Combining with the isomorphism above, we get a linear map T.l: A --. !l'(H/V, H) where for each a, T; is a linear injection of H/V onto (TaV).l. The direct sum T: == T; Efl Ta: (H/V) Efl V --H defines a continuous map T x : A --. x C This map can be connected to T in by the homotopy Tt = tT.l Efl T for 0 $; t $; 1. This proves the lemma .• U sing the theorem of Kuiper [1] that x is contractible, we conclude that any T: A --. of index zero is homotopic to the constant map T == Id. Hence ind is injective and Theorem 8.6 is proved. • One of the important results of Atiyah and Singer is the establishment of a topological formula for the index of a family P of elliptic operators. We shall present this formula in §15. As a final remark we point out that the arguments given above adapt easily to prove the following real analogue of Theorem 8.6. Let = denote the space of (real) Fredholm operators acting on real Hilbert space §9. THE G-INDEX 211 Theorem 8.8. For a'!y compact Hausdorff space A there is a natural isomorphism ind : [A, KO(A) (8.9) having thefunctorial property (8.7) above. Consequently, is a classifying space for KO-theory. An important consequence of 8.6 and 8.8 is the following: Corollary 8.9. For all k 0, there are isomorphisms: K(Sk) == K-k(pt) KO(Sk) == KO-k(pt). §9. The G-Index (8.10) (8.11 ) In this section we shall study operators which are preserved by the action of a compact Lie group G. To be specific we fix vector bundles E and F over a compact manifold X together with an action of G on the triple (X,E,F). This means a smooth action Jl: G x X -+ X of G on X together with smooth actions of G on both E and F which carry fibres to fibres linearly and which project to Jl. In this context we have the following notion: DEFINITION 9.1. A differential operator P: r(E) -+ r(F) is called a G-operator if P(gqJ) = gP(qJ) for all 9 E G and all qJ E r(E). A good example is given by the isometry group Gx of X acting on et(X) A * X. This action preserves both splittings et(X) = eto(X) Efl et l(X) and et(X) = et +(X) Efl C.e-(X) and commutes with the Dirac operator. Therefore both DO and D + are Gx-operators. Similarly, if X is spin, then either Gx or a two-fold covering of Gx acts on the spinor bundle of X and commutes with the Atiyah-Singer operator. The basic observation here is that if P is elliptic and a G-operator, then ker P and coker P are finite-dimensional representation spaces for G. This leads us to consider the representation ring (or the ring of virtual repre-sentations) R(G) of G. This can be defined as the free abelian group gen-erated by the equivalence classes of irreducible finite-dimensional complex representations of G. Since every finite-dimensional representation of G can be decomposed uniquely, up to equivalence, into a direct sum of ir-reducible ones, R(G) can also be defined as the Grothendeick group of all finite-dimensional representations. In other words, each element of R(G) can be expressed as a formal difference [V] -[W], where [V] and [W] 212 Ill. INDEX THEOREMS are equivalence classes of finite-dimensional representations of G, and where [V] -[W] = [V'] -[W'] if and only if V EE> W' is equivalent to V' EE> W. The tensor product of representations is distributive over the direct sum operation and makes R(G) into a commutative ring. EXAMPLE 9.2. Let G = Sl and let tm stand for the 1-dimensional repre-sentation: IPm(o)z = eim"z. Then R(Sl) is easily seen to be the ring of Lau-rent polynomials in t:R(Sl) Z[t,t-l]. DEFINITION 9.3. Let P be an elliptic G-operator on a compact manifold. Then the G-index of P is the element indG(P) = [ker P] -[coker P] E R(G). Note that when G = {1}, R(G) Z and we recover the usual index of P. The G-index can be specialized to individual elements of G. Given a complex representation p: G -+ GL(V) we define its associated character to be the function Xp(g) = trace(p(g)). This function completely determines the representation up to equivalence, and it has the properties that XP,+P2 = Xp, + XP2' X p, &JP2 = Xp,XP2' etc. (see Adams [1] for details). For this reason R(G) is alternatively called the character ring of G. Given a G-operator P as above and given 9 E G, we can define (9.1) This is the difference of the characters of the two representations, ker P and coker P, evaluated at g. Consequently indg is the specialization of indG to 9 as claimed. Note that for D == d + d* : Nven(x) -+ Ndd(X), and for 9 E Isom(X), the number indiD) is just the classical Lefschetz number of g. The general topological formulas for indG given by Atiyah, Singer, Bott and Segal represent generalizations of fundamental work of Lefschetz and Hopf. Our object at the moment is to establish some elementary stability properties of indG. As before, the regularity theory of elliptic operators implies that the spaces ker P and ker P* coker P remain unchanged if we pass to any Sobolev completion P: L;(E) -+ L;-m(F). Since G is com-pact, we may choose metrics on X, E and F which are G-invariant. The actions of G on qE) and qF) then extend to unitary representations on L;(E) and L;(F) which commute with P and P*. This leads us to examine the following concept. Let HI and H 2 be separable complex Hilbert spaces equipped with unitary representations G -+ U(Hj),j = 1,2, ofa compact Lie group G. Let O:G = O:G(H b H 2) denote the space of all G-equivariant Fredholm maps, i.e., all Fredholm maps T: H 1 -+ H 2 such that Tgv = gTv for all 9 E G §9. THE G-INDEX 213 and all v E H l' To each T E O:G we associate the index indG(T) == [ker T] -[coker T] E R(G) Proposition 9.4. The map indG: O:G R(G) is constant on connected com-ponents of O:G' Furthermore, if H 1 and H 2 are G-isomorphic, then indG induces an injection (9.2) Note. The map (9.2) will be a bijection provided that each finite-dimen-sional irreducible representation of G occurs with infinite multiplicity in H 1 = H 2' The map (9.2) is also an additive homomorphism since indG(T 0 S) = indG(T) + indG(S). This is proved exactly as above (see (7.5)). Proof. The argument for the first statement is identical with the one given for the first part of Proposition 7.1 above. It is only necessary to check that all subspac'es are G-invariant and that all maps (such as (7.2), (7.3) and (7.4)) commute with G. We now assume that H1 = H2 = H (as G-spaces). To prove the second statement we must show that any two elements To,T1 E O:G with the same G-index must lie in the same connected component of O:G' We begin by observing that any T E O:G can be deformed to one which is "relatively prime," that is, one for which no non-zero subspace of ker T is G-isomor-phic to a subspace of coker T = (Im T).l. Indeed, if such a G-isomorphism L exists, it can be extended to a map L: H H by defining it to be zero on the complementary subspace. The family T + tL, t 0, gives the de-sired deformation. We may assume now that To and T1 satisfy this "relatively prime" condition. Consequently, the hypothesis indGTo = indGT1 implies that there are G-isomorphisms: ker To ker T1 and coker To coker T1. We now observe that given two finite dimensional, G-invariant subspaces Vo, V1 c H which are G-isomorphic, there exists a G-isomorphism C:H H with C(Vo) = V1• In fact, C may be taken to be the identity on (Vo + v1l. The subs pace Vo (\ V1 is G-invariant and its complements in Vo and V1 respectively are G-isomorphic; so the existence of C is clear. It now follows that there exists a G-isomorphism C: H H which carries Im(To) isomorphically onto Im(Td. We define a second G-isomor-phism C' : H H by taking the given isomorphism ker To .; ker T 1 and extending by the map TI1CTo : (ker To).l .; (ker Td.l. We find that T 1 = CT o( Cr 1, and the existence of a homotopy from To to T 1 is an immediate consequence of the following: Lemma 9.5. The set If' (H, H) of G-isomorphisms of H is connected. 214 Ill. INDEX THEOREMS Proof. The argument here is identical to the one given for Lemma 7.7 above. One needs only to check that the entire construction is G-equivariant. • Arguing as in §7 we see that Proposition 9.4 has the following imme-diate consequence. Let P: r(E) --+ r(F) be an elliptic G-operator over a compact manifold X. Corollary 9.6. The G-index of P depends only on the homotopy class of P in the space of elliptic G-operators. In particular, indG(P) depends only on the principal symbol of P. Note that the action of G on X, E and F makes L == (omTX) ® Hom(E,F) into a G-bundle, i.e., a bundle with a smooth G-action which maps fibres to fibres linearly (and isomorphically). The principal symbol of P is an invariant section of L, i.e., o{P) E r G(L) where r G(L) == {o E r(L) : go = a for all 9 E G}. Two principal symbols of elliptic G-operators, 0'0 = O'(Po) and 0' 1 = 0'( PI), are said to be regularly G-homotopic if there is a regular homotopy Ut, 0 :$; t :$; 1, joining them (see (7.5) forward), such that Ut E r G(L) for all t. Theorem 9.7. The G-index of an elliptic G-operator on a compact manifold depends only on the regular G-homotopy class of its principal symbol. Proof. As in the proof of Theorem 7.10 we construct a family of elliptic operators Pt with O'(Pt) = at for 0 :$; t :$; 1. Averaging over G, by integrating Pt<p == fG(gPtg-1<p)dg = {fG9(Pt)d9}<P with respect to Haar measure on G, produces a homotopy of G-operators with O'(Pt) = Ut for all t (since Ut is G-invariant). The result now follows from Corollary 9.6. • §10. The Clifford Index In this section we shall discuss, in general terms, elliptic operators which are Ctk-linear for some k. Several important examples of such operators have been introduced and discussed in detail in Chapter II, §7. Motivated by these Ctk-Dirac operators, we make the following definitions. DEFINITION 10.7. By a eCk-bundle on a space X we mean a bundle of real, left Ctk-modules. This is a real vector bundle E over X together with a continuous map '1': CCk x E --+ E such that 'I' ,l) == 'I'(<p,"): E --+ E is a bundle endomorphism for all <p E ctb and the restriction ctk x Ex --+ Ex makes the fibre into a Ctk-module for each x E X. §1O. THE CLIFFORD INDEX 215 Note that Cfl-bundles and Cfrbundles are simply complex and qua-ternion bundles respectively. In general, a Cfk-bundle is thought of as having the algebra Cfk as "scalars." REMARK 10.2. A Cfk-bundle E will be called riemannian if it carries a bundle metric which is preserved under multiplication by each unit vector e E [Rk C Cfk. Starting with any metric and averaging over the Clifford group, as in 1.5.16, makes any Cfk-bundle riemannian. Note that if E is riemannian, then multiplication by any element W E [Rk C Cfk is fibre-wise skew-adjoint. Thus (10.1) for all Ul,U2 E r(E). Let us fix smooth Cfk-bundles E and F over a manifold X. DEFINITION 10.3. A differential operator P: r(E) -+ r(F) is said to be Cfk-linear (or simply a Cfk-operator) if <pP(u) = P(<pu) for all <p E Cfk and all u E r(E). Tensoring with Cfk makes any differential operator trivially into one which is Cfk-linear. More interesting examples, such as the Cfk-Dirac operators of 11.7, occur when there exists a Zrgrading. We say that a Cfk-bundle Eis Z2-graded if there is a bundle decomposition E = EO Et> El making each fibre into a Z2-graded Cfk-module. A Cfk-linear differential operator P: r(E) -+ r(E) on such a bundle is called Z2-graded (or simply graded) if with respect to the decomposition E = EO Et> El, it has the form (0 PI) P = pO 0 . (10.2) Note that pO:r(EO) -+ r(El) is Cfk_1-linear. When E and X are riemannian, the operator P is (formally) self-adjoint if and only if pI = (p0)*. We would like to define the analytic index of an elliptic operator of this type. For this we recall the groups 9Jlk-di*9Jlk 9Rk/i* 9Rk+1 KO-k(pt) (10.3) where 9Jlk and 9Rk denote respectively the Grothendieck groups of equiv-alence classes of Cfk and Zrgraded Cfk modules. (For detailed discussions of these, see 1.5.20, 1.9 and 11.7.) Recall also that KO-k(pt) KO-k+8(pt) for all k, and we have k I z, I :, I : I : I : I : I : I : I KO-k 216 Ill. INDEX THEOREMS Suppose now that P: r(E) --+ r(E) is an elliptic self-adjoint graded ctk-operator on a compact riemannian manifold X. Then ker P is a finite-dimensional Zrgraded and we have the following: DEFINITION 10.4. The analytic Clilford index of P is the residue class indkP [ker P] E 9Rk/i*9Rk+1 KO-k(pt). (10.4) Under the isomorphism (10.3) this element corresponds to the residue class indkP [ker pO] E Wlk-di*Wlk (10.5) where pO is given in (10.2). (The equivalence of these definitions is discussed in detail in Chapter 11. See 11.7.4.) The category of self-adjoint graded Ctk-linear operators is a natural and important one as we have seen by examples in Chapter 11. However, there is a twin category which has some advantages when studying sta-bility properties of the Clifford index. This is the category of graded op-erators which are skew-adjoint and Ctk-antilinear. A differential operator P: r(E) --+ r(F) between Ctk-bundles is said to be Ctk-antilinear if P(wu) = -wP(u) for all WE [Rk c ctk and all u E r(E). This is equivalent to the requirement that P(q)U) = CI:(cp)P(u) (10.6) for cp E ctk and u E r(E), where Cl: denotes the involution of ctk engendered by w --+ -wo Suppose now that E = EO Et> El is a riemannian Z2-graded Ctk-bundle over a compact riemannian manifold X, and let P: r(E) --+ r(E) be a Z2-graded elliptic differential operator which is Ctk-antilinear and formally skew-adjoint. With P written as in (10.2) above, this means that p1 = _(pO)*. We define the analytic Clifford index of such an operator to be indkP == [ker P] E 9RJi*9Rk+1 KO-k(pt) and note as before that this is equivalent to taking [ker PO] in Wlk-di*IDlk• Observation 10.5. There is a natural transformation between graded elliptic differential operators which are formally self-ad joint and Ctk-linear, and those which are formally skew-adjoint and Ctk-antilinear. It is given by associating to = (0 P1) P po 0 the operator -(0 P= pO §10. THE CLIFFORD INDEX 217 We shall now explore the topological invariance of these indices. To begin we recall from 5.9 above that the elliptic self-adjoint operator P above can be diagonalized on L 2(E) with finite-dimensional eigenspace and discrete eigenvalues Aj with limj IAjl = 00. Thus, P can be written as. P = L A/Ttj where 1tj: L 2(E) denotes orthogonal projection. In these terms the (essentially) positive operator p2 is written as p2 = L AJ1tj. Note that p2 is Ctk-linear and preserves the factors of the splitting L 2(E) = L2(EO) Et> L2(E1). We shall now define an associated Ctk-linear operator on L2(E) which preserves these factors by the formula 1 1 (1 + p2) -"2 == L (1 + AJ) -"21tj. The operator (10.8) is a Z2-graded self-adjoint Fredholm operator on L 2(E) = L2(EO) Et> L2(E1). We leave as an exercise the verification that this con-struction associates to a continuous family Pt, 0 ::;; t ::;; 1, of such elliptic operators, a continuous family Pt, 0 ::;; t ::;; 1, of Fredholm operators. The reader will note that if P is the companion of P in the sense of . 1 1 10.5, then the operator P = (1 + p2) -"2 P = (1 -p2) -"2 P is a Zz-graded Fredholm operator which is Ctk-antilinear and skew-adjoint. These con-siderations motivate the following definitions. Let H = HO Et> H1 be an infinite-dimensional separable real Hilbert space which is a graded module for the algebra Ctk• Assume in addition that for each unit vector e E [Rk C Ctk, the corresponding map e: H H is a skew-adjoint isometry. Such a space will be called a graded Hilbert module for eck• Examples are easily constructed by taking the tensor product H = H' ® Ctk = (H' ® Et> (H' ® CtD for some real Hilbert space H'. Let ijk c ijrw.(H, H) denote the subset of those Fredholm operators T:H H which are Zz-graded (i.e., T(HO) H1 and T(H1) HO), ctk-linear and self-adjoint. Similarly, let c ijrw.(H, H) denote the subset of graded operators which are Ctk-antilinear and skew-adjoint. (As should be obvious, an operator T E ijrw.(H, H) is called Ctk-Iinear if T(q)U) = <pT(u) for all <p E ctk and u E H. It is called Ctk-antilinear if T(<pu) = e>:(<p)T(u) for all such <p and u.) As in Observation 10.5 we have a natural homeomorphism ijk fik' We define the Clifford index of an element T E ijk (or fik) to be the residue class indkT = [ker T] E ID1k/i*ID1k+1 KO-k(pt). Our first main result is 218 Ill. INDEX THEOREMS the following: Proposition 10.6. The Clifford index KO-k(pt) is constant on connected components of lJk' Proof. Fix T E lJk and recall that since T is Fredholm, ° is an isolated point in the spectrum of T. Replacing T by an appropriate scalar multiple we may assume that the non-zero spectrum of T lies outside [ -2, 2], i.e., that T2 > 2Id on (ker T)1.. Choose a neighborhood U of T in lJk so that for all S E U, we have spectrum(S2) c [0, t) u (l,ro), an,d 11 T2 -S211 < 1/2. Fix S E U and let W c H be the range of the spectral projection of S2 onto [0, n We claim that orthogonal projection pr: H -+ ker T (= ker T2) restricts to give an isomorphism pr: ker T. (10.9) To begin, suppose v E W n (ker T)1. and note that «T2 -S2)V, v) (2 -t)llvl12 Ilv112. Since IIT2 -S211 < 1/2 we must have v = ° and so (10.9) is injective. Similarly any v E W1. n ker T satisfies «S2 -T2)V, v) IIvl12 and is therefore 0. Hence, (10.9) is surjective and the claim is proved. Observe now that W is a l'2-graded Ctk-submodule of H, and fur-thermore splits into graded submodules W = ker S EB V where V = (ker S)1. n W is S-invariant. One easily sees that the projection (10.9) preserves the graded module structure, and so we have a graded ctk-module equivalence ker S EB V ker T. It remains only to enhance the structure of V to that of a graded ctk + C module. This is done as follows. Let Sv: V -+ V denote the restriction of S to V. This is a symmetric, l'rgraded Ctk-linear map, and so also is 1 J = Note that j2 = Id. With respect to the decomposition V = VO EB V1 we can write J in the form J = (;0 The graded endomorphism of V given by -(0 J = JO is Ctk-antilinear and satisfies J2 = -Id. It therefore makes V into a graded ctk + 1-module as required. Hence indk T = indkS and the proof is complete. _ §1O. THE CLIFFORD INDEX 219 Consider now an elliptic differential operator P: r(E) -+ r(E) which is tk-linear and graded. Note that its principal symbol is also Ctk-linear nd graded. In this context a regular homotopy of symbols is required to reserve these additional properties. Arguing as in Theorem 7.10 and using he proposition above directly proves the following. Assume, as before, hat P is defined over a compact manifold and is self-adjoint. heorem 10.7. The Clifford index of P depends only on the regular homo-opy class of its principal symbol. Proposition 10.6 implies that indk induces a map on the connected omponents of lJk. In previous cases this map was a bijection, and the naiogous result is almost true here. However, we need some preliminary djustments. To begin, we shall assume for each k that H = HO EB Hl s a graded Hilbert module for Ctk+2 and is considered to be a graded tk-module under restriction to ctk c ctk+ 2. (This algebra inclusion s induced by the euclidean space inclusion IRk c IRk+2 as the first k-oordinates.) This assumption assures us, for example, that when k == mod 4), each of the two distinct irreducible graded Ctk-modules appear ith infinite multiplicity in H. For each k, we now define a new space ilk as follows. If k ¥= -l(mod 4), hen ilk == lJk If k == -l(mod 4), then we consider for each T E lJk' he associated operator w(T) == el ... ekTIHO where el'." ,ek is a fixed rthonormal basis of IRk c ctk• Note that w(T): HO -+ HO is a self-adjoint redholm operator. We now decompose lJk into three disjoint subsets ij:, lJi;, and ilk consisting respectively of those T's such that w(T) is ssentially positive, essentially negative, or neither. (A self-adjoint operator is essentially positive if it is positive on a closed invariant subspace of nite codimension.) Each of these subsets is open in lJk and hence is a union of connected components of lJk' We now show that when k == -l(mod 4), the space ilk is not empty. Recall that H is a module for Ctk+2 :::J ctk. Let eh' .. ,ek+2 be an exten-sion of our orthonormal basis above, and write the action of ek + 1 on H = HO EB Hl as ek+ 1 = -S*) o . Then the map ek+ 1 : H -+ H given by ek+l = (10.10) is Ctk-linear and self-adjoint. Furthermore W(ek+ d anticommutes with ek+2ek+l, and so we have ek+l E ilk' 220 Ill. INDEX THEOREMS The following striking result is due to Atiyah and Singer [5]: Theorem 10.8. For each k, the Clifford index induces a bijection indk: 1l:0«(Jk) --+ KO -k(pt). (10.11) Proof. From Proposition 10.6 we know that the map (10.11) is well de-fined. To show that it is surjective, let V = VO EEl VI represent an element of9.1lk/i*9.1lk+1 KO-k(pt), and let ek+1 be as in (10.10) above. Then the map ek+ I EB 0 is an element of (Jk(H EB V) and has kernel V. Since H EB V H, this shows that the map (10.11) is surjective. To prove injectivity, it is convenient to first divest ourselves of the grading. Recall that the endomorphisms e? == ekel' .. , == ekek-I make the subspace HO into an ungraded module over Ctk-I Let denote the space of all skew-adjoint, Ctk_cantilinear Fredholm op-erators on HO. There is a natural map (10.12) defined by setting TO == ek TIHo. It is easily checked that this map is a homeomorphism. Passing from graded to ungraded modules as in (10.4) and (10.5), we find that [ker TO]. Assume now that we are given two operators S,T E 'iJk with indkS = indk T. It will suffice to show that SO and TO are homotopic in Our first step is to observe that we may assume SO to be a Hilbert space isometry on (ker SO)1. (and similarly for TO). This is accomplished by putting the operator in polar form and deforming away the "radial" part as in the first step of the proof of Lemma 7.7. Properties of skew-adjoint-ness and Ctk _I-antilinearity are preserved. The operator SO now satisfies (SO)2 = -Id on (ker SO)1., and setting == SO makes (ker SO) into a Ctk-module. The same remarks apply to TO of course. Observe now that our hypothesis indkS = indk T implies that there exist ungraded V and W together with a ctk _I-module isomorphism ker SO EB V ker TO EB W. (10.13) We now claim that the module V can be realized as an SO-invariant subspace of (ker SO).1. This means simply that V is isomorphic to a ctk-submodule of (ker SO)1. where acts by SO as above. When k ¥= -l(mod 4), all irreducible Ctk-modules are equivalent, and our claim is obvious. When k == -l(mod 4), there are two equivalence classes of irre-ducible Ctk-modules and we must show that each of them appears with in-finite multiplicity in (ker SO)1.. Recall that these two representations are distinguished by whether the central element ill = e? ... acts by 1 or -1. There is a splitting (ker SO)1. == M = M+ EB M-where M± = §1O. THE CLIFFORD INDEX 221 (1 ± w)M, and we want to know that dim M+ = dim M-= ro. However, w = e? ... = e? ... _ 1 SO = ± w(S) where the sign depends only on k, and so the desired property follows from our assumption that w(S) is neither essentially positive nor essentially negative. The same argument, of course, shows that W may be realized as a TO-invariant submodule of (ker TO)-1. We now consider the two orthogonal decompositions: It is clear from the discussion above that V' and W' are isomorphic as Ctk-modules. This means simply that there is a Ctk-linear Hilbert space isomorphism L: V' =. W' such that SO = L -1 TO L on V'. By the obvi-ous homotopy we may assume that SO = 0 on Vand TO = 0 on W. Ex-tending L by the isomorphism (10.13) gives an isomorphism L: HO -+ HO such that SO = L -1T°L. The theorem now follows from the fact that the group of Ctk-linear isometries of HO is connected. This fact is proved by suitable adaptation of the argument given for Lemma 7.7 above. _ REMARK 10.9. The mapping (10.12) above gives a natural transforma-tion between graded and ungraded operators. All the previous discussion of this section could be so transformed. Consequently we find that there exists a parallel index theory for real elliptic operators which are skew-adjoint and (ungraded!) ctk _ cantilinear. Let us examine some examples. Let P be the operator. If k = 1, we find that ind1P = dim[J;l(ker P)(mod 2). (Thus, every real skew-adjoint elliptic operator has a well-defined index in £'2!) If k = 2, then ker P is a ct1 C module and we find that ind2P = dimdker P)(mod 2). If k = 4, then ct3 !HI EB !HI and the bundle E on which P is defined splits as E = E+ EB E-where E± = (1 ± w)E and w = ele2e3' By antilinearity, P(E±) c r(E+) and we split Pinto P+ and P-= -(P+)* as before. Note that ker P = ker p+ EB ker P-and that ind4P = 0 if and only if dimlHJ(ker P+) = dimlHJ(ker P-). (This is because the ± spaces are the two distinct Ct3-modules.) It follows easily that ind4P = dimlHJ(P+) -dimlHJ(P-) = the index of p+ considered as a quaternionic operator. We conclude this section with a discussion of a nice result of Atiyah and Singer which states that the spaces ilk form a "spectrum" (in the sense ofhomotopy theory). Together with the periodicity phenomena in Clifford modules and Theorem 8.8, this will give a new proof of Bott periodicity. To begin we notice that for each k 1 there is a natural inclusion ilk + 1 ilk 222 HI. INDEX THEOREMS and a distinguished element ek+1Eilk given by (10.10). We extend this by convention to k = 0 as follows. Let il 0 denote the space of all real Fredholm operators on HO, and let il 1 4 ij 0 be the map which associates to T E ill the operator TO = el TIHO. (This identifies ill with the space of all skew-adjoint operators in ilo.) Set el = Id E ilo. Let nilk denote the loop space of ilk defined here to be the set of all continuous paths y: [O,n] -+ ilk with y(O) = ek + 1 and y(n) = -ek + l' This space carries the compact-open topology. It is homotopy equivalent to the space of all paths which both begin and end at ek + l' The following is the main result of Atiyah-Singer [5] and is presented here without proof. Theorem 10.10 (Atiyah and Singer). For each k 0, the map ilk + 1 --+ nilk which assigns to T E ilk+ 1 the path from ek + 1 to -ek + 1 given by (cos t)ek+ 1 + (sin t)T, is a homotopy equivalence. o t n, This gives us a generalization of Theorem 8.8 above. Theorem 10.11. For any compact Hausdorffspace A andfor any k there is a natural isomorphism indk:[A,ty,,] --+ KO-"(A) with the functorial property (8.7). Hence, 8" is a classifying space for the functor KO-". Proof. The case k = 0 is just Theorem 8.8 above. For higher k we have [A,tyA:J [A, n"8oJ [I:1cA,8oJ KO(I:"A) KO-"(A) where I:A:A de-notes the k-fold suspension of A. • Theorem 10.12 (Bott Periodicity). For each k 0 there is a homeomot-phism 81: 81:+s, Therefore, by 10.10 there is a homotopy equivalence BA: '" OSB" which implies that KO-A:(A) KO-k+S(A) for any compact H ausdorjJ space A. Proof. Let H = HO E9 Hl be a graded Hilbert module for ct" as above, and let V = VO E9 VI be an irreducible graded (real) module for Cts. The graded tensor product H ® V is a module for ctA: ® Cts ctt+8' §10. THE CLIFFORD INDEX 223 There is an isomorphism V 1R16 which yields an explicit identifica-tion CCs 1R(16). The fundamental results of Chapter I, §4, give an isomor-phism CCk+ s CCk ® 1R(16) (ungraded tensor product), where elements of the form 1 ® <p act by Id ® <p on H ® v. Let !l'k(H) denote the Banach space of bounded operators on H which are CCk-linear and graded. Define a map 'P: !l'k(H) -+ !l'k + s(H ® V) by setting 'P(T) = T ® Id. This map is continuous and injective. Further-more, we claim that it is surjective. To see this, fix T E !l'k+s(H ® V), and for each x E H consider the map Tx: V -+ H ® V defined by Tx(v) = T(x ® v). Since TA<pv) = <pTAv) for all <p E 1R(16), we conclude that dim(Image Tx) is either 0 or 16. It follows that Image(Tx) = t ® V for a unique I-dimensional subspace t c H. Consequently, T(x ® v) = T(x) ® v for a unique element T(x) E H. The map T: H -+ H is easily seen to be bounded, CCk-linear and graded. Clearly 'P(T) = f. We have shown that 'P is surjective and therefore by the Open Mapping Theorem that 'P is a homeomorphism. It is easy to check that 'P carries the self-adjoint Fredholm operators in !l'k(H) onto those in !l'k+s(H (8) V). This gives the desired homeomorphism IYk IYHS' • Kuiper's results [1] show that the group of isometries of HO is contractible. This shows that the construction of the homeomor-phism above is canonical up to homotopy. Note that for each k ;;:; 0 we have isomorphisms KO-k(pt) (: JrO((Jk) !. @di*@k+l where rJ. is given by 10.11 and where f3 is the Clifford index defined in 10.4. It is shown in Atiyah-Singer [5] that the map rJ. 0 p-l coincides with the isomorphism given by the Aityah-Bott-Shapiro construction (cf. 1.9). There is a natural ring structure in KO-* where the multiplication is induced by the tensor product of operators. This is defined as follows. Let Hk and H{ be graded Hilbert modules for Ctk and Ct{ respectively. The graded tensor product Hk ® H{ is then a module for Ctk ® Ct{ = CtkH (see 1.5). Representing by IYk+AHk ® H{, Hk ® Ht), we define a map (10.14) by requiring that for elements v E Hk and WE H{ of pure degree with respect to the grading, (S ® T)(v ® w) = (Sv) ® W + (_l)deg "v ® (Tw), where S E IYk and T E IY{. One checks easily that S ® T is Ctk-linear and CCrlinear, and therefore Ctk+IAinear. The inner product on Hk ® H{ is given by <v ® w, Vi ® w') = <v,v')<w,w'), and S ® T is clearly self-adjoint. Since Sand T interchange even and odd factors, so does S ® T, 224 Ill. INDEX THEOREMS and one computes that (S ® T)2 = S2 ® Id + Id ® T2. From this one sees that S ® T is Fredholm and that there is an identification ker(S ® T) = (ker S) ® (ker T). (10.15) The graded tensor product of modules induces a multiplication in m* = EBkmk' which descends to the quotient KO -*(pt) EB(mdi*9Jlk+ 1). Hence, (10.15) gives the following: Proposition 10.13. For SE jJk and T E jJt, one has that indkH(S ® T) = indk(S) ® indAT). It is straightforward to check that the map (10.14) above preserves the subsets ilk' that is, it restricts to give a continuous mapping ®: ilk x ilt ijkH· Applying the isomorphism of 10.11, we get a multiplication KO-k(A) x KO-t(A) KO-k-t(A) defined for any compact Hausdorff space A and for all k, t O. In fact, for any pair of such spaces A and B, we have a transformation KO-k(A) x KO-t(B) KO-k-t(A x B) given by associating to f: A -+ ilk and g: -+ ilt the map (fg)(a,b) = f(a) ® g(b). (When A = B the multiplication is obtained by restricting to the diagonal.) These transformations coincide with the ones convention-ally defined in KO-theory (see I.9). REMARK 10.14. Recall that the homeomorphism (10.12) identifies ill with the space jJskew of all skew-adjoint Fredholm operators on real Hilbert space. Consequently, Theorem 10.11 shows that 3'skew is a clas-sifying space for the functor KO -1. Similarly, the assignment T -+ Ih T IHo associates to each element T E ij7, a Ct6-linear map of the ungraded Hilbert module H6 for Ct6. Note that Ct6 1R(8) and we can take H 6 to be the product H 6 = H ® 1R8 with 1R(8) acting on the right-hand factor. Any 1R(8)-linear map of H6 is then of the form A ® Id. This identifies jJ7 with the space jJsymm of all self-adjoint Fredholm operators on real Hilbert space, and Theorem 10.11 shows that jJsymm classifies the functor KO-7. §11. MULTIPLICATIVE SEQUENCES 225 REMARK 10.15. The entire discussion of this section can be carried through in the complex case. One considers complex Clifford algebras, complex graded modules, complex Hilbert spaces, etc. All of the analogous fundamental results remain true. One essentially recaptures the "classical" index theory discussed in §§7 and 8. However, a new fact which emerges is that the functor Kl is classified by the skew-adjoint Fredholm operators on complex Hilbert space. The analogue of Theorem 10.10 (also proved in Atiyah-Singer [5]) leads to a proof of Bott Periodicity for the unitary group. §11. Multiplicative Sequences and the Chern Character In this section we present some fundamental constructions in K-theory and the theory of characteristic classes. These constructions will be needed later when we discuss the cohomological formula for the index of an elliptic operator. Throughout the section cohomology groups will be taken with rational coefficients, although much of what we do carries over to more general coefficient rings. There is a principle underlying much of what we do here. Roughly stated it asserts that for computational purposes every complex vector bundle is a direct sum of line bundles. Moreover, if the bundle is the com-plexification of a real bundle, the non-trivial line bundles occur in complex conjugate pairs. To make this precise we need the following result, often referred to as the "Splitting Principle": Proposition 11.1. Let E be a complex vector bundle over a manifold X. Then there exists a manifold YE and a smooth, proper jibration n : YE -+ X such that (i) The homomorphism n* : H*(X) -+ H*(YE) is injective. (ii) The bundle n*E splits into a direct sum of complex line bundles: n* E t 1 EB ... EB t n (11.1) Proof. Let p: IP(E) -+ X denote the projectivization of E, i.e., the bundle whose fibre at x is the projective space IP(Ex) of all complex lines in Ex. The bundle p*E contains a line bundle defined tautologically at a line t c Ex to be t itself. Using some fixed hermitian metric in E, this gives us a tautological splitting p*E = t EB tJ. The homomorphism p*: H*(X; if) -+ H*(IP(E); if) is injective by the Leray-Hirsch Theorem C.l4 in Appendix C. 226 Ill. INDEX THEOREMS We now repeat the process for the bundle tJ. and continue inductively to complete the proof. • There is a direct analogue of the proposition and its proof for the case of real vector bundles. We shall not state this. However, we do want to signal the following "hybrid" result. Proposition 11.2. Let E be an oriented real vector bundle of dimension 2n over a manifold X. Then there is a smooth proper fibration n: YE --+ X such that n*: H*(X) --+ H*(YE) is injective and the bundle n*(E ® q splits into complex line bundles: (11.2) where t; denotes the "inverse" or "complex conjugate" bundle to tj (see below). In fact, there is a splitting (11.3) into oriented real 2-plane bundles such that Ek ® C = tk EB lkfor each k. Proof. Suppose, to begin, that dim[J;l(E) = 2. Fix a metric in E and let J: E --+ E be the map which rotates each fibre by nj2 in the positive di-rection. We let YE = X and note that E®c=tEBl where at x E X and are the + i and -i eigenspaces of J ® C. Note incidentally that as com-plex bundles we have the bundle isomorphism (11.5) For the general case we fix a metric in E and consider the bundle p: G(E) --+ X whose fibre at a point x consists of all oriented 2-dimen-sional subs paces of Ex. Then there is a canonical splitting p*E = El EB Et where El --+ G(E) is the tautological oriented 2-plane bundle whose fibre at PE G(E) is P itself. The argument given for Theorem C.l4 adapts im-mediately to prove that the homomorphism p*: H*(X; Z) ---> H*(G(E); Z) is injective. Repeating the process for the bundle Et and proceeding in-ductively, we construct the desired splitting bundle YE' • REMARK 11.3. An analogous result holds when dim[J;l(E) = 2n + 1. One must add a trivial line bundle onto the decompositions (11.2) and (11.3). §11. MULTIPLICATIVE SEQUENCES 227 For many purposes it is permissible to add a trivial real line bundle to E and work directly in the even-dimensional case. NOTE 11.4 (conjugate bundles). Recall that if E is a complex vector bundle, then its conjugate bundle E is obtained from E by redefining scalar multiplication. The new scalar multiplication by tEe is the old scalar multiplication by t. For any given hermitian metric (.,.) in E, the map v -+ «Jv(·) == (., v) identifies E with E* == HomdE, C). If E = Eo ® C is the complexification of a real vector bundle Eo, then there is a complex bundle isomorphism E E. NOTE 11.5 (line bundles). The set !l'(X) H1(X; Sl )ofequivalence classes of complex line bundles on a manifold X has a natural commutative multi-plication given by the tensor product. Since 1 ® t t* ® t trivial, the multiplication is invertible, and !l'(X) is a group. The first Chern class c1:!l'(X) -+ H2(X;Z) is a group isomorphism (see Appendix A, Example A.5). The Splitting Principle above can be applied directly to characteristic classes. OBSERVATION 11.6 (splitting the Euler class). Let E be an oriented real vector bundle of dimension 2n, and let X(E) E H2n(x; IQ) denote its Euler class. If E decomposes into a sum of oriented 2-plane bundles E = El EB ... EB En, then since X(E EEl E') = X(E)X(E') we can write (11.6) where Xk = X(Ek) for each k. If we now complexify and write E ® C = t1 EB 11 EEl ... EB tn EB 1n then we see from (11.5) that for each k. Using the Splitting Principle, formula (11.6) can, in fact, be used to define the Euler class. OBSERVATION 11.7 (splitting the total Chern class). Let E be a complex vector bundle, and denote by c(E) = 1 + c1(E) + ... + ciE) the total Chern class of E. Recall that c(E EB E') = c(E)c(E'). Hence, if E decomposes as a sum of line bundles E = t 1 EB ... EB t n, then n c(E) = TI (1 + xk) (11. 7) k=l where Xk = C1(tk) for k = 1, ... ,no In particular, we find that j = 1, ... ,n (11.8) where (Jj denotes the jth elementary symmetric function of Xl' ... ,Xn• 228 Ill. INDEX THEOREMS If E does not split as a sum of line bundles, we may lift it to the space YE where it does. The map H*(X) --+ H*(YE) is injective and identifies Cj with O"iXl' ... ,xn) as in the previous case. Note that any symmetric polynomial expression in the x/s can be rewritten as a polynomial in 0" l' ... ,0" n (i.e., in Cl' .•. ,cn). This will enable us to define characteristic invariants in a particularly useful way. As a simple application note that since c1(1) = -c1(t), we have (11.9) for all j. OBSERVATION 11.8 (splitting the total rational Pontrjagin class). Let E be a real oriented vector bundle of dimension 2n, and recall that the rational Pontrjagin classes of E are defined by piE) = ( -1)ic2iE ® q, j = 1, ... ,no (Since E ® IC (E ® q, the Chern classes of odd degree are zero by (11.9).) The total (rational) Pontrjagin class is defined to be p(E) = 1 + Pl(E) + ... + Pn(E). It has the property that p(E EB F) = p(E)P(F). If E ® IC decomposes as a direct sum, E ® IC = tt EB 11 EB··· EB tn EB 1n' then c(E ® q = n c(tk)c(lk) = n (1 -xD, and we find that n p(E) = n (1 + k=l where Xk = C1(tk) as before. In particular, we have pj(E) = O"j(xI, ... ,x;) for each j. (11.10) (11.11) Let Q[[x]r denote the set of formal power series in x with rational coefficients and with constant term 1. It is easily seen that Q[ [x] r is a group under multiplication. Fix an element f(x) E Q[[xJ], and for each nE Z+ consider the formal power series in n indeterminates given by f(x1) ... f(xn). This is evidently symmetric in the x/s, and so it has an expansion of the form f(x1)·· • f(xn) = 1 + F1(0"1) + Fz(O"I> 0"2) + F3(0"1,0"2, 0"3) + ... where O"k(Xl, ... ,xn) == I Xi! ... Xik il < ... <ik for 1 ::s:; k ::s:; n denotes the kth elementary symmetric function in Xl' ... ,xn, and where F k is weighted homogeneous of degree k, i.e., for all t E Q. §11. MULTIPLICATIVE SEQUENCES 229 Each of the polynomials Fk(a1, ... ,ak) is well defined and independent of the number of variables Xj' This is easily seen by adding more variables and using the obvious fact that a k(X 1, ... ,Xn,O, ... ,0) = a k(X l' ... ,xn) if k n, and ak(x1, ... ,Xn,O, ... ,0) = ° if k > n. The sequence of polynomials {Fk(a1,'" ,ak)}k'=l is called the multi-plicative sequence determined by the formal power series f(x). It has a universal multiplicative property which we shall now describe. Let B be a commutative algebra with unit over Q, and assume that B has a direct sum decomposition B = BO EB B1 EB B2 EB ... with the prop-erty that Bk. Bt s; Bk+ t for all k,t 0. For example, B could be H2*(X;Q) or H4*<X;Q) for a space X. It could also be the polynomial ring Q[x] with Bk == QXk. Given such an algebra B, let denote the set of all formal sums 1 + b1 + b2 + ... where bk E Bk for each k. Note that the finite sums (those with only a finite number of non-zero terms) actually belong to Band form a set which is closed under multiplication. This extends to by defining (1+b1 +b2+·· .)(1+c1 +c2+·· .)=1+(b1 +c1)+(b2+b1c1 +c2)+···, that is, by defining the nth term of the product to be L bkcn-k. Every element of has a multiplicative inverse, and so is an abelian group. As an example, note that if B = Q[x], then = Q[[ x Jr. Fix a multiplicative sequence {Fia1"" ,ak)}k'=l' Then to each alge-bra B as above we associate a map F: by assigning to b = 1 + b1 + b2 + ... E the element (11.12) Lemma 11.9. The map F: is a group homomorphism, i.e., F(bc) = F(b)F(c) for all b,c E Proof. In the polynomial algebra B == Q[X1' ... ,xnJ consider the element a = (1 + Xl)" . (1 + Xn) = 1 + a1 + ... + an E By definition of {Fd we have that F(a) = f(x1)'" f(xn) = 1 + F1(a1) + Fz(a1,a2) + .... We now increase the number of variables. Let B == Q[X1" .. ,xn+mJ, and consider the subalgebras B' = Q[x1, ... ,xnJ and B" = Q[Xn+ 1, ... ,xn+ml Let a,a',a" be the corresponding elementary products in each case. Then we have and F(a) = f(x1)'" f(xn+m) = F(a')F(a"). 230 Ill. INDEX THEOREMS The general result now follows easily from the algebraic independence of the a/so • It is easy to see that the universal multiplicative property 11.9 char-acterizes the sequence. The formal power series is recaptured by taking F(1 + x) = f(x) in the algebra Q[xJ. The concept of a multiplicative sequence is due to F. Hirzebruch, who established their importance in the theory of characteristic classes. BASIC CONSTRUCTION 11.10 (multiplicative sequences of Chern classes). Let {Fk} be a multiplicative sequence associated to the formal power series f(x) E Q[[xJr. To each complex vector bundle E over a space X, we associa te the total F -class F dE) == F(c(E» E H2·(X;Or. Since c(E Et> E') = c(E)c(E'), this class has the property that F dE Et> E') = F dE)F dE'), (11.13) for any two complex vector bundles E and E' over X. If we decompose E = tt Et> ••• Et> t" according to the Splitting Principle, then F dE) = f(xt) ... f(x,,) (11.14) where Xj = Ct(t) for eachj. EXAMPLE 11.11 (the total Todd class). Associated to the formal power series d(-x 112 t x) = 1 _ e JC = 1 + "2 x + 12 x +"". is the mUltiplicative sequence {Td",} called the Todd sequence. The total Todd class is denoted by Tdc. Its first few terms are: 1 Tdt(Ct) ="2 Ct d 1 2 T 2(CtoC2) = 12 (C2 + Ct) d . 1 T 3(Ct,C2,C3) = 24 C2Ct" If X is a compact complex manifold of dimension n, and if E = TX, then the number Td(X) == Td,,(TX)[X] (where [X] denotes the fundamental class of X in H 2"(X; 0» is called the Todd genus of X. BASIC CONSTRUCTION 11.12 (multiplicative sequences of Pontrjaji& classes). Let {Ft} be a multiplicative sequence associated to the formal power series f(x). To each real vector bundle E over a space X, we asso- §11. MULTIPLICATIVE SEQUENCES ciate the total F-class F(E) = F(p(E» E H4*(X; Qt. 231 Given two such bundles E and E' over X, we have p(E EEl E') = p(E)p(E') and so F(E EEl E') = F(E)F(E'). (11.15) Assume E is oriented and of dimension 2n, and decompose E ® C as t1 EB 11 EEl ... EB tn EEl l" according to the Splitting Principle. Then F(E) = f(xi) ... f(x;) where Xk = Cl (tk) for each k. (11.16) EXAMPLE 11.13 (the total A-class). Associated to the formal power series A Fxl2 1 7 2 a(x) = sinh(FxI2) = 1 -24 x + 27 . 32 . 5 x + ... is a multiplicative sequence {Am} called the A-sequence. The first few terms of the sequence are Given a real bundle E, the total A-class of E is the sum A(E) = 1 + A1(PIE) + Az(PIE,pzE) + ... If we write E (8) C = I1 EEl 11 EEl ... EEl In EEl 1" as above, then from (11.16) we sce that A(E) = fI . xp 1 slllh(x)2) (11.17) where of course PjE = O")XT, ... ,x;). Closely related to the A-sequence is the A-sequence {Am} determined by the power series a(x) = a(16x). One easily sees that Am = 16m Am for each m. The Todd class and the A-class are intimately related. Proposition 11.14. For any oriented real vector bundle E it is true that TddE (8) C) = A(E)z. 232 Ill. INDEX THEOREMS Proof. We may assume dim E is even (see Remark 11.3) and we consider a formal splitting E ® C = t1 EB 11 EB ... EEl t n EEl In" Then by definition n x. (-x.) TddE®C)=n1 lx, j=l -e ) -e) where Xj = c1(t). Multiplying by eXj/2e-xj/2 in the denominator gives TddE ® C) = fI [ x'/2 Xj _X'I2J2 j= 1 e) -e ) = fI j= 1 sinh(xi2) = [i(E)Y • EXAMPLE 11.15 (the total L-class). Associated to the formal power series _ JX 1 2 t(x) = = 1 + -3 x -45 x + ... tanhJX is a multiplicative sequence {Lm} called the Hirzebruch L-sequence. The first few terms of the sequence are Given a real bundle E, the total L-class of E is the sum L(E) == 1 + Lt(P1E) + LiP1E,P2E) + ... If we write E ® C = t1 EEl 11 EEl ... EEl tn EEl 1n according to the Splitting Principle, then from (11.28) we see that n x. L(E) = n l, j= 1 tanh Xj (11.18) where PjE = oixi, ... Closely related to the L-sequence is the I.-sequence {Lm} determined by the power series l(x) = t(x/4). One easily sees that Lm = 4mI.m. For a real oriented bundle E of dimension n, we have I.(E) = fI xi2 j= 1 tanh(xi2) (11.18') §11. MULTIPLICATIVE SEQUENCES 233 Suppose we fix a multiplicative sequence {F k} as above. Then for each ifferentiable manifold X we define the total F-c1ass of X to be F(X) = F(TX) E H4*(X;Q) n particular, the examples above give us a total A-class A(X) and a total -class L( X). If we furthermore assume that X is compact, oriented and of dimension 7, then we define the F-genus of X to be the rational number F(X) obtained y evaluating F(X) on the fundamental homology class [X] E Hn(X; Q) of he manifold. In other words: if n = 4k if n =f= 0 (mod 4) ( 11.19) rom (11.15) one easily deduces that the F-genus is multiplicative in the ense that F(X x X') = F(X)F(X') ( 11.20) or any pair of compact oriented manifolds X and X'. Suppose now that Y is a compact oriented manifold with boundary ]Y = X. Note that TYix = TX EEl (trivial), and so p(Y)ix = p(X). Con-equently, F(Y)ix = F(X) and, since X is homologous to zero in Y, the F-genus of X must be zero. This proves that the F-genus of a manifold depends only on its oriented cobordism class. In fact the F-genus gives a ring homomorphism (11.21) from the oriented cobordism ring into Q. Two important examples here are the A-genus and the L-genus. An important result of F. Hirzebruch says that for any compact oriented 4k-manifold X, L(X) = sig(X). This can be proved by direct verification on a set of generators for the ring (8) Q (cf. Hirzebruch [1]). It also follows from the Atiyah-Singer Index Theorem. Note in particular that L(X) is always an integer. This is not true for A(X). The formulas above show, for example, that A(1P'2(1C)) = -(1 18)L(1P'2(1C)) = -1 18. Nevertheless, it will follow from the Index Theo-rem that A(X) is an integer when X is a spin manifold. It is a fact, inciden-tally, that the A-genus is always an integer. We now discuss the Chern character. Let E be a complex vector bundle of dimension n over a manifold X, and via the Splitting Principle express 234 Ill. INDEX THEOREMS the total rational Chern class of E formally as n c(E) = 1 + Cl + ... + Cn = Il (1 + xd k=1 so that Ck = (Jk(X l' ... ,xn). Consider the expression The term of degree k in this expression is just the symmetric polynomial 1 n chkE=-L Xk k! j= I J (11.23) which can be rewritten as a universal polynomial expression in the ele-mentary symmetric functions Cl' ... ,cn-In particular, ch E = n + ch I E + ch2 E + ... is a well-defined element of H2*(X; 0). It is called the Chern character of E. Note that if t is a complex line bundle, then ch(t) = ecI(t) (11.24) where Cl (t) denotes the first Chern class of t. The importance of the Chern character lies in the fact that it respects the (semi) ring structure on the set of vector bundles. Proposition 11.16. The Chern character has the following properties for any pair of complex vector bundles E and Ef over X: (i) ch(E EEl Ef) = ch(E) + ch(E') (ii) ch(E ® E) = ch(E)ch(Ef). Proof. Consider formal splittings n m c(E) = Il (1 + Xk) C(Ef) = Il (1 + xj) k=1 j= 1 where as above the Chern classes are the elementary symmetric functions of the x's. Then we have corresponding splittings n m c(E ® E) = Il (1 + xd Il (1 + xi) k=1 j=1 m n c(E ® Ef) = Il Il (1 + Xk + xj) j=1 k=1 The first is obvious. The second is a consequence of the basic fact that for complex line bundles c1(t ® tf) = c1(t) + c1(t'). (Compare Note 11.5 §11. MULTIPLICATIVE SEQUENCES or (A.7) in App. A.) By definition we now have n m ch(E EEl E') = L eXk + Lex} = ch(E) + ch(E') k=l j=l 235 Corollary 11.17. For any compact H ausdorff space X, the Chern character descends to a ring homomorphism ch: K(X) ----+ H2*(X; Q). REMARK 11.18. It is a result of Atiyah and Hirzebruch [2] that if, say, X is a finite complex, then the associated map ch: K(X) ® Q -+ H2*(X; 0) is an isomorphism. They show, moreover, that this map extends to a ring isomorphism ch: K*(X) ® Q .:; H*(X; Q) which carries Kl(X) ® Q onto HOdd(X;Q). We now examine some basic constructions in K-theory. To make cal-culations we shall always assume our bundles to be a direct sum of line bundles. This is justified by the Splitting Principle 11.1 and 11.2 provided the answer is independent of the splitting. We assume throughout that X is a manifold or a finite simplicial complex. CONSlRUCTION 11.19 (the exterior power operations). Let E be a com-plex vector bundle of dimension n over X and for each k, 1 ::;; k ::;; n, con-sider the bundle NE. This operation on vector bundles has the property N(E E9 E') = L (NE) ® (NE'). (11.25) i+j=k To extend the operation to K-theory we consider the ring K(X)[[t]] of formal power series with coefficients in K(X). Assigning to a vector bundle E the element At(E) = L [NE]tk (11.26) k=O gives a map with the property that At(E E9 F) = At(E)At(F) (11.27) by (11.25). From the universal property of K(X) the map (11.24) extends to a group homomorphism At: K(¥) ----+ The k-component of this map is called the kth power operation on K(X). 236 Ill. INDEX THEOREMS For vector bundles, )ot(E) is a polynomial and Am(E) is a well-defined element of K(X) for all mE Z. For example, AM') = 1 + t[t] for any line bundle t. For general elements of K(X) the above statement is false. Note, for example, that Al-et]) = (1 + I = L (-t)m[tr. Fix a vector bundle E and consider a formal splitting E = tl EEl ... EEl tn with xk = CI(tk). From (11.27) we have that At(E) = Il )-t(tk) = Il (1 + t[tk])' and so n ch(lotE) = n (1 + teXk). (11.28) k=l In particular, we conclude that n ch(L lE) = ch(NvenE -NddE) = Il (1 -eXk). (11.29) k=1 CONSTRUCTION 11.20 (the Adams operations). Closely related to the power operations are a family of ring homomorphisms I/Ik: K(X) ---+ K(X) called the Adams operations. For a line bundle I, they are defined by setting, ( 11.30) The extension is then determined by the Splitting Principle. Write E = tl EEl ... EEl in and define I/Ik(E) = EEl ... EEl We see that I/Ik(E EEl E') = I/Ik(E) EEl I/Ik(EI),} I/Ik(E (8) E') = I/Ik(E) ® I/Ik(E'). (1l.31) The first is obvious. For the second, note that I/IkW': () (8) (I tj)] = I/Ik[I ti ® tj] = I (ti (8) I'l = I 17 (8) (lY = (I m (8) (I tf) = I/Ik(I/J (8) I/Ik(I tj). Note that ch I/IkE = I ekxj = Pkch E where Pk: H2*(X; 0) ---+ H2*(X; 0) is defined by setting Pk == km on H2m(x; 0). The Adams operations transform naturally under the homomorphism f* : K(X) ---+ K( Y) induced by a continuous map f: Y ---+ X. It is an interesting exercise to show that I,:o= I (-t)kl/lk(E) = -t[(d/dt)AlE)]/ AlE). CONSTRUCTION 11.21 (the Clifford difference element). Let E be a real oriented riemannian vector bundle of dimension 2n, and let WE == ine I ... e2n be the oriented unit volume element in the complex Clifford bundle Ct:(E) = CqE) ® c. Since = 1, we have a splitting Ct:(E) = Ct: +(E) EEl CC(E) where Ct: ±(E) == (1 ± wE)Ct:(E). We then define the Clifford difference element b(E) == [Ct: +(E)] -[Ct: E K(X). (11.32) §II. MULTIPLICATIVE SEQUENCES 237 Suppose E' is another such bundle. From the fact that Ct(E EEl E') = Ct(E) ® Ct(E') and WE a> E' = WEWE" one sees easily that b(E EEl E') = b(E)b(E'). (11.33) The proof of the Splitting Principle 11.2 shows that we may consider E to be a direct sum of 2-plane bundles E = El EEl ... EEl Em and E ® C = t1 EEl 11 EEl ... EB tn EEl 1. where Ej ® C = tj EEl Consider therefore the case where dim E = 2. We have E ® C = t EEl 1 for some complex line bundle t. In fact, if at a fixed point x E X we choose an oriented orthonormal basis (e1,e2) of Ex, then tx = C . (e1 -ie2) and 1x = C . (e1 + ie2). Clearly Ct(E) = C· 1 EEl C . WE EEl t EEl 1, and since WE = ie1e2 we find that and Since the bundles C . (1 ± WE) are trivial, we find that for dim E = 2, b(E) = [1] -[t]. Applying the Splitting Principle and (11.33), we conclude that for E = El EB ., . EEl En, n b(E) = n ([4] -[tk]) k=l and so n ch[<5(E)] = n (e-Xk -eXk) (11.34) k=l where Xk = C1(tk) for k = 1, ... ,no From (11.6) we know that X(E) = Xl ... x". Consequently, one verifies that ch[<5(E)] = X(E) fI (e-Xk -eXk) k= 1 Xk " (eXk/2 _ e-Xk/2) = (-l}"x(E) f1 (eXk/2 + e-Xk/2) k=l Xk = ( _ 2)"X(E) fI (sinh(X,J2))2 ( x,J2 ) k=l Xk/2 tanh(xJ2) From (11.17) and (11.18') we conclude the following: Proposition 11.22. For any'oriented real vector bundle E of dimension 2n, one has the relation I ch[<5(E)] = (-2)"x(E)L(E)i(E)-2·1 238 Ill. INDEX THEOREMS CONSTRUCTION 11.23 (the spin or difference element). Let E and WE be as in Construction 11.21 and suppose E carries a spin structure. Let SdE) be the complex spinor bundle for E and consider the decomposition SdE) = EB Si(E) where Si(E) = (1 ± wE)SdE). We define the spinor difference element to be s(E) == -[Si (E)] i= K(X). (11.35) It follows easily from the material of Chapter I that if E' is another such bundle, then s(E EB E') = s(E)s(E'). (11.36) As above we may now restrict attention to the case dim E = 2 where E ® C = t EB 1. Recall from (11.5) that as oriented 2-plane bundles E t. The fact that E is spin is equivalent to the fact that X(E) = c1(t) is even, i.e., that t has a square root t1/2. The spinor bundle is given by SdE) = t1/2 EB 11/2 One verifies that = 11/2 and Si(E) = t1/2, and so s(E) = [11/2] -[t1/2J. Passing to the general case E = El EB ... EB En via the Splitting Principle as above, we then have n s(E) = n ([liI2] -[til2 ]), k=l and so n ch[ s(E)] = n (e -xkl2 -eXkl2) (11.37) k=l This proves the following: Proposition 11.24. For any real spin vector bundle E of dimension 2n, one has that I ch[s(E)] = (-ltX(E)A(E)-l I §12. Thorn Isomorphisms and the Chern Character Defect We present here some material relevant to understanding the general form of the Index Theorem. It is not necessary, however, for understanding the cohomological formula for the index in the basic cases. Let X be an oriented n-dimensional manifold which is not necessarily compact. Let denote the cohomology of the complex of rational §12. THOM ISOMORPHISMS 239 singular cochains with compact support on X. (A cochain c has compact support ifthere is a compact subset Kc X such that c(a) = ° for any chain a which does not meet K.) Poincare duality states that there is a canonical isomorphism (12.1 ) for p = 0, ... , n (see de Rham [1]). If Yis another such manifold of di-mension m, and if f: Y --+ X is a continuous map, then for each p with p m -n there is a linear map .t;: (12.2) called integration over the fibre (or the Gysin homomorphism). It is de-fined by setting .t;(u) = where f* is the usual map induced on homology. An important example is provided by the following. Let E be an oriented vector bundle of fibre dimension k over X. Consider the maps and where 1t is the bundle projection and i denotes inclusion as the zero sec-tion. Since these maps are homotopy equivalences, 1t* and i* induce iso-morphisms on homology with 11:*i* = Id. Consequently, the maps and are isomorphisms for all p. Since 1t!i! = = Id, we see that 11:! = (i!)-l. (12.3) DEFINITION 12.1. The map i!: --+ is called the Thorn iso-morphism of E for compactly supported cohomology. Note that if X is compact, then = HP(X). Furthermore, if DE de-notes the disk bundle of E, then by excision and Lefschetz duality we have Hn_p(E) Hn-iDE) Hk+P(DE,iJDE). (12.4) We thereby recover the Thorn isomorphism in its more conventional form (cr. Milnor-Stasheff [1 ]). We also have the following basic result: Lemma 12.2. If X is compact, then for all u E H*(X) = one has i*i!(u) = X(E) . u (12.5) where X(E) is the Euler class of E. Proof. We shall only outline an argument (for full details, see Milnor-Stasheff [1]). Let Z = i*[X] = be the class of the zero-section in 240 HI. INDEX THEOREMS Hn(E). The equation X(E) = i*i!(l) = is one of the standard characterizations of the Euler class. To wit, if c E Hk(X), then (X(E),c) = = (l,Z n i*c), that is to say that X is given by intersection in E with the zero section. For a general class u E HP(X), we fix c E Hk+p(X) and note that (X(E)' u, c) = (X(E), n c) = (1, Z n n c)) = (1, n i*c) = (E2i1i*E2xu,i*c) = (i!u, i*c) = (i*i!u, c) • In K-theory there is a group KcplX) analogous to the group discussed above. It consists of homotopy classes of triples [E, F; (J] where E and F are complex vector bundles over X and where (J is a bundle isomorphism from E to F defined outside some compact subset of X (see 1.9). If X is a compact manifold, then Kcpt(X) = K(X). If U c X is an open subset of any manifold, then there is a natural inclusion homomor-phism Kcpt(U) -Kcpt(X). Suppose now that n: E -X is a complex vector bundle of rank k over a manifold, and let i: X -E be the inclusion as the zero section. Then, as proved in Appendix C, Theorem C.8, there is a natural Thorn iso-morphism i!: Kcpt(X) -KcplE) of the form i!(u) = A-1 . n*u where A-1 = [n* (J] and where (J is defined at each non-zero vector e in E by setting (Je = e /\ -(e*)L. (The element e* is the dual of e under some fixed hermitian metric.) If X is compact, then A -1 = i!( 1) is a well-defined element of Kcpt(E). When X is not compact, the product A-1 . n*u can still be shown to be a well-defined element of Kcpt(E) (cr. Karoubi [2]). Roughly speaking, n*u has compact support in the "X -directions" and A _ 1 has compact support in "fibre-directions". If we restrict the element A-1 to the zero section, we recover the element A. -1 (E) E K*(X). This gives the following. Lemma 12.3. If X is compact, then for all E K(X) = Kcpt(X), one has = L l(E) . (12.6) where A._1(E) = [NveoE] -[NddE] E K(X). §12. THOM ISOMORPHISMS 241 Assume now that X and Y are smooth manifolds and that f: X <=-+ Y is a smooth proper embedding. Assume furthermore that the normal bundle N to f(X) is equipped with a complex structure. (Hence, dim Y -dim X is even.) Under these circumstances we can define a natural mapping it: Kcpt(X) Kcpt(Y) by taking the Thorn isomorphism i!: Kcpt(X) Kcpt(N) followed by the map Kcpt(N) Kcpt(Y) obtained by identifying N with a regular neigh-borhood of X in Y. Observe now that for any proper embedding f: X 4 Y of manifolds, the normal bundle to the associated (proper) embedding f*: TX <=-+ TY has a canonical complex structure. This normal bundle is just the pull-back to TX of N EB N where N is the normal bundle to X. The first factor is thought of as lying in "manifold-directions," the second in "fibre-directions." The complex structure is given by T = Consequently, for any proper embedding of manifolds f: X <=-+ Y, there is an associated map (12.7) which is of fundamental importance in defining the topological index of an elliptic operator. In the last section we defined the Chern character ch: K(X) Heveo(x). This homomorphism has a direct extension ch: Kcpt(X) ----+ to the case of compact supports. For any given complex vector bundle n: E X, we have defined Thorn isomorphisms: and and it is natural to ask whether i!ch = ch i!. This is not true in general and the resulting "correction term" is of basic importance. We assume from this point on that the manifold X is compact. Then to each complex vector bundle n: E X we associate the class (12.8) Note that for any E K(X) we have n!ch = n!ch(q 1) . = J[![ ch i!(1) . ch = [n!ch i!(1 )Jch and so, since n! = (i!) -1 on H;pt(E), (i!) -= '1'(E)ch (12.9) That is, '1'(£) is just the "commutativity defcct" mentioned abovc. 242 Ill. INDEX THEOREMS Now it is not difficult to check that :1:(E) is natural, i.e., f*:1:(E) = :1:(f* E) for any continuous map between manifolds. Consequently :1:(E) is a characteristic which we shall compute. Note that i!:1:(E) = ch il(1) = ch A_1. Applying i* and Propositions 12.2 and 12.3, we find that X(E):1:(E) = i*i!:1:(E) = i*ch A-1 = ch i*A_1 = ch L1(E). This compu-tation can be carried out for the universal bundle E over the classifying space BUn whose cohomology ring is a polynomial ring generated by the Chern classes Cl, ... ,Cn• Here we are authorized to write the equation :1:(E) = ch L l(E) X(E) (12.10) If we split c(E) = n (1 + Xj) formally as in §11, we find from equations (11.6) and (11.29) that n 1 -e"'k :1:(E) = n--k= 1 Xk From (11.11) and (11.9) we can rewrite this as :1:(E) = (-ltTddE)-l (12.11) Assume now that E is a real oriented riemannian vector bundle of even dimension over X. Then one can define the basic element (12.12) where /le == e· denotes Clifford multiplication bye. It is evident that = (5(E), the difference element considered in 11.21. Arguing as above we see that: X(E)1!!ch = i*i!1!lch = i*ch = ch (5(E). Applying the calculations of Proposition 11.22 then proves the following. Propositon 12.4. For any oriented real vector bundle E of dimension 2n on X, one has If we now assume that E has a spin structure, we can construct the element (12.13) where again /le = e· denotes Clifford multiplication bye. Clearly, i*s(E) = s(E) = the element discussed in Construction 11.23. Arguing as above and applying the calculation of Proposition 11.24, we find the following: Proposition 12.5. For any real spin vector bundle E of dimension 2n on X, one has §13. THE ATIYAH-SINGER INDEX THEOREM 243 These last two propositions extend easily to the case of coefficients. Given any element u E K(X), one immeidately verifies that 1trch[ 6(E) . 1t*u] = ( -2tch U • L(E).i(E) -2 1trch[ s(E)' 1t*u] = ( -1 )"ch U • 1(E) -1 (12.14) (12.15) Indeed, note that 1t!ch(t5(E)1t*u) = 1tr[ ch t5(E)ch 1t*u] = 1t1[ ch t5(E)1t*ch u] = [1trch t5(E)]ch U. If u = [E'] is the class corresponding to a complex vector bundle E' over X, then t5(E) ·1t*u [1t*C£ +(E) ® E', 1t*Ce-(E) ® E'; IL], s(E)·1t*u [1t*S,t"(E)® E', 1t*Si(E)®E'; ILl (12.16) These elements t5(E) and s(E) are fundamental. Using them, one can define Thorn isomorphisms in K-theory for E as follows: Proposition 12.6. Let 1t : E -+ X be an oriented real vector bundle of dimen-sion 2n on X. Then the map given by i?(u) == t5(E) . 1t*u is an (additive) isomorphism. If E is spin, then the map if: K(X) Kcpt(E) is an (additive) isomorphism. given by if(u) == s(E) . 1t*u For a proof of this proposition and a discussion of related results, the reader is referred to Appendix C. §13. The Atiyah-Singer Index Theorem We present here the topological formulas of Atiyah and Singer for the index of an elliptic operator on a compact manifold. We begin with the general K-theoretic formula for which we give a detailed proof. Then, using material derived above, we shall rewrite the formula in cohomo-logical terms and work out the details in some important special cases. Let X be a compact differentiable manifold of dimension n and consider an elliptic operator P: nE) -+ nF) where E and F are smooth complex vector bundles over X. Recall from §1 that the principal symbol CT(P) of P defines a class a(P) == [1t* E, 1t* F; a(P)] E Kcpt(TX) (13.1) where 1t: TX -+ X is the tangent bundle of X. (We have identified Kcpt(TX) with K(DX,oDX) since DXjoDX is naturally homeomorphic to the one point compactification of TX.) Choose now a smooth embedding 244 Ill. INDEX THEOREMS f: X <=-+ into some euclidean space, and consider the induced map It: Kcpt(TX) (13.2) defined in (12.7). Follow this by the homomorphism q!: K(pt) 7l.. (13.3) where q : --+ pt is the canonical "scrunch" map taking to a point. Note that = EB = eN and q: eN --+ pt can be considered as a vector bundle. Viewed in this way, the map q! is just the inverse of the Thorn isomorphism i! : K(pt) --+ Kept(eN). (It is also just the Bott periodic-ity map. For an alternative view, consider the embedding = --+ S2N = U {oo} and the induced map --+ K(S2N) K(pt).) Applying scrunch-shriek to 1; gives the index. DEFINITION 13.1. The topological index of P is the integer top-ind(P) == qJ..u(P). (13.4) One must verify that this definition is independent of the choice of f. To begin consider j = j 0 f where j: <=-+ +N' is a linear inclusion. The induced map jl: --+ is just the Thorn isomor-phism for the bundle eN+N' --+ eN, and one easily checks that if/I. = qJI where if: + N' --+ pt. If we are given two embeddings f 0: X 4 and f1: X <=-+ then the embeddings jofo: X <=-+ [RNo+Nl and j1 0 f1: X --+ + N " defined as above, are isotopic. That is, Ft = tj 1 0 f1 + (1 -tHo 0 fo, 0::; t::; 1, is a smooth family of embeddings. Applying the homotopy invariance of Kept completes the proof that (13.4) is independent of the choice off. One of the basic results in mathematics is the following: Theorem 13.2 (The Atiyah-Singer Index Theorem [1]). For any elliptic operator P over a compact manifold, one has ind(P) = top-ind(P), that is, the topological and analytic indices of P coincide. Proof. For the purposes of the proof we introduce a special class of operators. Let E and F be (smooth) corn lex vector bundles over a com-pact riemannian manifold X. An operator PE l.JIDOm(E, F) is called clas-sical if its principal symbol is homogeneous of degree m in ( outside of some compact subset of T* X, that is, P is classical if there is a constant c so that = tm(J for all ( E T* X with 11(11 c and for all t 1. If X is not compact, we define the classical operators to be those which have this property over every compact subdomain of X. The set of all such operators will be denoted 'PCOm(E, F). §13. THE ATlYAH-SINGER INDEX THEOREM 245 Given an operator P E 'l'COm(E, F) we can consider its asymptotic prin-cipal symbol A (P) _ I' -lm -m-00 t defined for all in aDX == E T* X : = I}. This gives us an exact sequence o 'l'DOm_1(E,F) 'l'COm(E,F) nHom(n*E,n*F)) 0 (13.5) where n: aDX -+ X is the bundle projection. The surjectivity of <7 is seen as follows. Given a section s E nHom(n* E,n* F)), extend s smoothly to all of T* X so that it is homogeneous of degree m in for 1. Given local trivializations of E and F over a coordinate chart U on X, one easily con-structs an operator P in U with principal symbol s. Let {Uj} be a finite covering of X by such charts and let {xI} be a partition of unity subor-dinate to this covering. Then the operator P = L XjPXj E 'l'COm(E, F) has principal symbol (J(P) = s, and the surjectivity is proved. Our first main step in the proof will be to show that the analytic index makes sense at the symbolic level and, in fact, gives a well-defined homo-morphism ind: KcPt(T* X) -+ 7l.. We begin with a technical lemma which will be useful later on. For this lemma, X is assumed to be a manifold which is not necessarily compact but is of finite topological type. Lemma 13.3. Let n: B -+ X be a smooth, real vector bundle over X. Then every element in Kcpt(B) can be represented by a triple of the form (n* E, n* F; (J) E 2'\(B)cPt where E and F are vector bundles on X which are trivial outside a compact set, and where (J: n* E -+ n* F is homogeneous of degree 0 on the fibres of B (wherever it is defined). Note that outside a compact subset of X, (J is defined everywhere on the fibres. At such points the homogeneity implies that (J is in fact con-stant on the fibres. Proof. We know from Chapter I, §9 that any element in KcplB) can be represented by a triple (Eo, F 0; (Jo) where (Jo: Eo -+ F 0 is a bundle equiv-alence defined outside a compact subset K c B. There exists a bundle Et on B so that the sum Eo EB Et is trivial, and we can replace (Eo, F 0; (Jo) with the equivalent triple (E,F;B) == (Eo EB Et,Fo EB Et;(J EB Id). Then there exist trivializations 246 Ill. INDEX THEOREMS so that a = ,p 1 0 'E. (Let, £ be the assumed trivialization, and set, F = ,£00'-1.) Choose now a compact domain Q c X so that K c Bin. Set E = i* E and F = i* F where i: X <=-+ B is the zero-section, and let, E and, F denote the restrictions of the trivializations 'E and '1' to E and F respectively. We claim that over B there exist bundle isomorphisms and (13.6) which are compatible with the given trivializations over BI(x-n), i.e., which have the property that and at points of Bb-n). These isomorphisms are constructed as follows. Let h : B x [0,1] --+ B be the homotopy defined by h(b,t) = tb, and set tf = h* E and :F = h* F. Note that tflB x {O} = n* E, tflB x {1} = E, :FIB x {O} = n* F :FIBx{1} = F Introduce connections on tf and :F which extend the canonical flat con-nections (compatible with the trivializations) over Bh-n). Parallel trans-port along the curves b x [0,1] gives the desired maps (13.6). The bundle map (J = iF 0 a 0 iE 1: n* E --+ n* F is an isomorphism which is defined on B -K and constant on the fibres of B -n-l(Q). Fix r> ° so that K c {b E Bin: Ilbll r}. We now redefine (J in the set where Ilbll r so that it is homogeneous of degree zero (by setting = for = rand t 1). This gives the desired triple and completes the proof of the lemma. • Suppose now that X is a compact manifold and choose an element U E Kcpt(T* X). Represent u by an element (n* E, n* F; (J) as in Lemma 13.3. Fix an integer m. From the discussion above we know that we can choose an (elliptic) operator PE \{JCOm(E, F) whose asymptotic principal symbol is exactly (J, and so in particular u(P) = u. We now set ind u == ind P (13.7) and show that this definition is independent of all the choices involved. We know from §7 that ind P depends only on the homotopy class of its principal symbol. Now if P' E \{JCOm(E, F) satisfies 8(P') = 8(P), then u(P') and u(P) are homotopic (reI 00). Therefore, ind P is independent of the choice of P with a given asymptotic principal symbol. It is also indepen-dent of the homotopy class of the representative (n* E, n* F; u) of u. To see this suppose that (n* E', n* F'; u') is another such representative and that there exists !m element a = (E, F; a) E !t'l(T* X x [0,1 ])cpt whose restric- §13. THE ATIYAH-SINGER INDEX THEOREM 247 tion to T* X X {k} is (n* E(k), n* F(k); ark») for k = 0,1. The argument used for Lemma 13.3 applies here to prove that a can be replaced with an element of the form (n* E, n* F; 8) where E and F are bundles on X x [0,1] and where 8 is homogeneous of degree zero outside a compact set. Given mth order operators P and P' associated as above to these representatives, one can easily use the element (n* E, n* F; 8) to construct a homotopy between them and thereby show that ind P = ind P' as claimed. Suppose now that we have two distinct representatives Uk = (n* Ek, n* Fk; ak) for our class u, and that we have chosen associated zero-order operators Pk where k = 0,1. By the definition of the equivalence defining L1(T*X)cpt K(T*X)cpt this means that there exist elementary complexes ek = (n*Gb n*Gk; Id), k = 0,1, so that ao EB eo and a1 EB e1 are homotopic. We can choose associated operators = Pk EB Id for ak EB ek, and one easily sees that ind Pie = ind Pk for each k. Hence, we have ind Po = ind P l' and so ind u is well defined at least if we choose operators of order m = O. The definition is also independent of the choice of the order m. To see this, suppose we are given an elliptic operator P E 'l'COm(E, F). Choose a metric and a unitary connection on E, and let V*V be the associated laplacian on E (see II.8.3). Then for any integer t, we consider the com-position Pt = P 0 (1 + V*Vl/2 E 'l'COm+AE, F). It is easily seen that 8(Pt) = 8(P) and since (1 + V*vt/2 is invertible, that ind Pt = ind P. It fol-lows that ind u is well defined using operators of any order. We have now shown that for any compact manifold X, the analytic index (13.7) gives a well-defined homomorphism: ind : KcPt(T* X) 71. (13.8) Our task now is to prove that this coincides with the homomorphism top-ind defined above. This will be accomplished if we can establish the following two properties: Property 1. In the special case where X = T* X = pt, the homomorphism ind : K(pt) --+ 71. is the identity. Property 2. If X and Y are compact manifolds and f: X <=-+ Y is a smooth embedding, then ind(u) = ind(};u) for all u E KcPt(T* X), where J; is the homomorphism (12.7). To see that these properties suffice to prove the theorem, we first choose an embedding f: X <=-+ SN and let j: pt <=-+ SN denote the inclusion of a 248 Ill. INDEX THEOREMS point. By Property 2 we have ind(u) = ind(J;u) = ind(j!-1J;U), and by Prop-erty 1 we know that ind oj!-l = q!. We conclude that ind(u) = q,J;(u) = top-ind(u). Property 1 is easily established. Each element in K(pt) can be repre-sented in the form [C'] -[CS] where (C',CS) E !l"t(pt) is a pair of vector spaces. An elliptic operator on this pair is just a linear map P: C' -+ Cs, and we see that ind P = r -s. Property 2 is more difficult to establish. Following Atiyah and Singer [1], we split this up into more easily established properties. The first one shows that ind is well defined over open manifolds of finite topological type. The Excision Property 13.4. Let (9 be an open manifold, and let f:(9 X and 1':(9 X' be two open embeddings into compact manifolds X and X'. Then ind 0 J; = ind 0 f; on KcPt(T*(9). Proof. Fix u E KcPt(T*(9). By Lemma 13.3 we know that u can be repre-sented by a triple (n* E, n* F; (J) where E and F are bundles over (9 which are trivial outside a compact subset of (9 and where (J is homogeneous of degree 0 outside a compact subset of T*(9. In particular, outside a compact set Q c (9 there are trivializations rE:EI(l"-n) ((9 -Q) x cm and rF:FI(II-n) ((9 -Q) x cm (13.9) with respect to which = (Jx = (rF);l 0 (rE)x at all points T*((9 -Q). This means that over T*((9 -Q) the morphism (J comes from a bundle map (Jo: E -+ F over the base. Moreover, with respect to the trivializations (13.9), (Jo becomes the identity mapping, i.e., (JO(Zl' ... ,zm) = (Zl' ... ,zm) at all points x E (9 -Q. Recall that a bundle map ao E nHom(E, F)) is just a differential operator of order zero. We now choose a zero-order elliptic operator PE 'I'COo(E, F) which has symbol (J(P) = (J outside a compact set in T*(9 and which is the operator (Jo = Id in (9 -Q. Such an operator clearly exists. Suppose now that we are given an open embedding f: (94 X. Using (13.9), we extend the bundles E and F trivially over X -f((9), and we extend the operator P to be the identity there. This defines an elliptic operator J;p on X with the property that [a(J;P)] = J;[ (J(P)] = J;u. (13.10) Clearly any element in ker(J;P) has support in Q and hence belongs to the subspace ker P (under the natural embedding ker P c ker J;p given §13. THE ATlYAH-SINGER INDEX THEOREM 249 by extending by zero). Hence, dim(ker J;P) = dim(ker P). The same re-marks apply to the adjoint (J;P)*. Assuming that X is compact, we con-clude from this and from (13.10) that ind(J;u) = ind(J;P) = dim(ker P) -dim(ker P*). Since the right hand side is independent of j, our assertion 13.4 is proved . • The Multiplicative Property 13.5. Let X and Y be compact manifolds. Then for all elements U E Kcpt(T*X) and v E Kcpt(T*Y) we have that ind(u'v) = (ind u)(ind v) (13.11) Proof. Naively the argument goes as follows. We represent u and v as above by first-order elliptic operators P: r(E) -r{F) and Q: r(E') -r(F') over X and Y respectively. We introduce metrics and define a "graded tensor product" D:r«E®E') E9 (F®F'» -r«F®E') E9 (E®F') by D = (P ® 1 -1 ® Q*) 1®Q p*® 1 (13.12) Note that E ® E', etc., here denotes the exterior tensor product over X x Y. The operators P ® 1, etc. are uniquely determined by requiring that for qJ E r(E) and tjJ E r(E') we have (P ® 1)( qJ(x) ® tjJ(y» = (PqJ(x» ® tfJ(y). Using the fact that P® 1 and 1 ® Q commute, one easily computes that D*D = (p*p® 1 + I®Q*Q 0 )) o pp* ® 1 + 1 ® QQ* (13.13) . DD* = (Pp* ® 1 + 1 ®Q*Q 0 ) o P*P®1 + I®QQ* Note that: D*DqJ = 0 => (D*DqJ,qJ) = (DqJ,DqJ) = 0 => DqJ = O. Hence, ker D*D = ker D. Furthermore, since D*D is diagonal, it suffices to com-pute ker D*D separately on each summand, E ® E' and F ® F'. Given qJ E r(E ® E'), we see that: D*DqJ = 0 => (p*PqJ, qJ) + (Q*QqJ, qJ) = 0 => IIPqJW + IIQqJI12 = 0 => PqJ = QqJ = 0 (where R ® 1 or 1 ® R, whichever is appropriate). Note that ker P n ker Q ker P ® ker Q. 250 Ill. INDEX THEOREMS Continuing in this fashion we deduce that ker D = ker D* D (ker P ® ker Q) E9 (ker P* ® ker Q*) coker D ker D* = ker DD* (ker p* ® ker Q) E9 (ker P ® ker Q*) are therefore in K(pt) [ker DJ -[coker DJ = ([ker PJ -[coker PJ)([ker QJ -[coker QJ). In particular we have ind D = (ind P)(ind Q). (13.14) Now the principal symbol of D is exactly the (outer) tensor product of the symbols of P and Q. Therefore, naively we have established (13.11). However, there is one technical flaw in the argument. This is the fact that for PE 'PC01(E,F), the operator P ® 1 E 'PD01(E ® E', F ® E')doesnot in general belong to the class 'PC01(E ® E', F ® E') because its principal symbol is not homogeneous outside a compact set in T*(X x Y). It is only homogeneous outside a uniform neighborhood of the "T*Y-axes" in T*(X x Y). This flaw is repaired as follows. We shall construct a continuous fam-ily of operators (P ® 1), E 'PC01(E ® E', F ® E') for e > 0 such that lim".o (P ® 1), = P ® 1, where this limit is taken in the space of bounded linear maps from Li(E ® E') to L6(F ® E'). Applying the construction to each entry in (13.12) will give us a family of elliptic operators D, in 'PCOl such that lim".o D, = D as bounded (Fredholm) maps between Sobolev spaces. From the local constancy of the index (see 7.3.) we have ind D, = ind D for all e > O. On the other hand it will be evident from the construction that given any compact subset K c T*(X x Y), there exists a constant eK > 0 so that a(D,) == a(D) on K for all e eK' It follows by excision that [a(D,)J = [a(D)J = U· v for all e sufficiently small. Hence, ind(u' v) = ind(D,) = ind(D) = (ind P)(ind Q) = (ind u)(ind v), and the property will be established. It remains to construct the operator (P ® I),. This is done by multi-plying the symbol of P ® 1 by a function of the cotangent variables E T*X x T*Y. This function is constructed as follows. Fix a Coo function cjJ: + -+ [0, 1 J such that cjJ(t) = 0 for t 1 and cjJ(t) = 1 for t 2. Then for e > 0 and for r,s 0, set !/Iir,s) = 1 -cjJ(eJr2 + S2)cjJ(eS/r). In multiplying the symbol of P ® 1 by one can use a good co-ordinate presentation or some symbol calculus. The choice of method is not critical. It is a straightforward and worthwhile exercise to check that the resulting family (P ® I), has the properties claimed above. This com-pletes the proof of 13.5. • §13. THE A TIYAH-SINGER INDEX THEOREM 251 We shall actually need the multiplicative property in the more general ntext of twisted products, i.e., fibre bundles. However, we only need to sider the special case of sphere bundles which arise from vector bundles y adding a section at infinity. More specifically, let n: P -X be a prin-cipal O,,-bundle over a compact manifold X and consider the associated bundles v = P xo .. Z = P x 0 .. S" (13.15) where 0" acts on by the standard representation and acts on Sft by extending this representation to the one-point compactification on (That is, 0" acts on S" by trivially extending the standard representation to 1 = x 1 and then restricting to the unit sphere.) We define a product (13.16) as follows. Choosing a metric in Z we get a splitting T*Z = n*T*X liB T(Z/X) where T(Z/X) = T*Z/n*T* X denotes the tangent spaces along the fibres of the projection n: Z -X. This splitting gives us a multipli-cation KcPt(T*X) ® KcptT(Z/X) ---+ Kcpt(T *Z). (Given a direct sum of vector bundles E liB E' on X, the map on Kcpt(E) ® Kcpt(E') is defined by first taking the outer tensor product on 252 Ill. INDEX THEOREMS E X E' over X X X and then restricting to the diagonal.) Combining this with the composition KoJT*sn)cPt -+ KoJP X T*sn)cPt -+ Kcpt(P xOn T*sn) = Kcpt(T(Z/X)) gives the desired multiplication (13.16). The associated bundle construction, which associates to a linear repre-sentation p: On -+ ON the vector bundle Vp = P x P /RN, extends naturally to a homomorphism (13.17) The ring KcPt(T* X) is naturally a K(X)-module. Therefore, via (13.17) it becomes an R(On)-module. We are now in a position to state the main property. The Multiplicative Property for Sphere Bundles 13.6. Let Z be an Sn-bundle defined as above over a compact manifold X. Then ind(u· v) = ind(u' indonv) for all u E KcPt(T* X) and v E KoJT*sn)cpt. Proof. The proof of this fact follows very much the argument given for 13.5. We represent u and v by first-order elliptic operators P and Q re-spectively. (Q is an On-equivariant operator on On-bundles.) Using local trivializations of the bundle, we cover P by a finite number of product neighborhoods {Vj x 0n}f=l' We lift the operator P back over each pro-duct and glue together with a partitLon of unity with respect to {Vj} on X, to get an On-invariant operator P on P. We now consider the tensor product operator jj on P X sn defined as in (13.12) (with P replaced by P). This is an On-operator and can be pushed down to give an operator D on the quotient Z. Notice that "pushing down" is equivalent to restricting jj to the subspace of sections coming from the base. It is easily checked that a(D) represents the class u . v where the multiplication is that defined in (13.16) above. It remains to compute the analytic index of D in terms of ind P and indon Q. We shall work upstairs with the operator jj restricted to sections coming from the base. Using (13.13) and the arguments which follow it, we see that: ker jj = ker jj* jj = (ker(P ® 1) n ker(l ® Q)) E9 (ker(P* ® 1) n ker(l ® Q*)) with an analogous statement for ker jj* = coker jj. Since the operators P ® 1 and 1 ® Q commute, we can carry out the computation in steps, that is, we first pass to the kernel of 1 ® Q (or 1 ® Q* whichever is rele- §13. THE ATlYAH-SINGER INDEX THEOREM 253 vant) and consider the operators ]5 ® 1 and ]5* ® 1 acting there. For example, on qE ® E') the space ker(l ® Q) consists of those sections iP which when restricted to each factor {p} x sn C P X sn, lie in the space E,,(p) ® ker Q. To say that iP comes from a section on the base means that iP satisfies the transformation law: iP(pg-l ,gx) = PgiP(p,x) for 9 EO", where P is the natural representation of 0" on ker Q. This means precisely that iP corresponds to a section qJ over X of the bundle E ® ker Q where ker Q is the vector bundle on X associated via the principal bundle P to the representation p. Therefore, in passing to ker(l ® Q) and ker(l ® Q*) the operator D descends to an operator on X on the form Pk + Pt. where Pt: qE ® ker Q) --+ r(F ® ker Q), and Pk*: r(E ® ker Q*) --+ r(F ® ker Q*). It follows that ind D = ind Pk -ind Pt. = ind[P ® (ker Q -coker Q)] = ind[ u . indoft v] as claimed. _ From the properties we have established, matters can be easily reduced Jo computing some simple cases. However, it is unavoidable that one must Compute t!te index of some operator at some point. We do this now. Lemma 13.7. Consider the n-sphere S" to be an On-manifold under the re-of the standard representation on R" liB R => S" (i.e., by rotations ;4bout an axis). Let i: pt '-+ S" denote the inclusion of one of the two fixed-of the action. Then indoft(itl) = 1 E R(O,,) '''00}. Consider the operator DO:ct° -+ Cfl :with respect to the stan-fdard metric on S". Recall that DO is just the de Rham-Hodge operator d + d* : A even -+ A odd. This is an Oil-operator and from Hodge Theory (11.5) we see easily that indoJDO) = [HO] + (-l)"[H"]. The action of 011 on no = {constant functions} is always trivial. The action on H" = R{the volume n-form} is trivial if and only if n is even. Therefore, we have ind DO = {2 . On 1-, ifn is even if n is odd where represents the non-trivial 1-dimensional representation on On. We leave the details of the computation of the symbol class of DO to the reader. One finds that in Koft(S") ° {2it(1) [u(D )] = (1 -,)it(1) ifn is even if n is odd. Combined with the above, this completes the proof of the lemma. _ 254 Ill. INDEX THEOREMS We now complete the proof of the Index Theorem. We have shown that it will suffice to establish Property 2, namely that ind = ind 0 J; for embeddings f: X 4 Y. By the Excision Property we may replace the com-pact manifold Y with a tubular neighborhood of X in Y, which is diffeo-morphic to the normal bundle of X in Y. Consequently it will suffice to prove that ind u = ind(J;u) for all U E KcPt(T* X) where V is a vector bundle over X and f: X 4 V is the inclusion of the zero-section. Again by the Excision Property we may compactify V by passing to the associated sphere bundle as in (13.l5). We then apply the Multiplicative Property for Sphere Bundles 13.6, with v = i,l. Using Lemma 13.7 we find that: ind(u· i,l) = ind(u· indo)i,l)) = ind(u). How-ever, by definition J;u = U· i,l, and the proof is complete. • The remainder of this section will be devoted to deriving certain co-homological formulas for the topological index. The most general one is the following: Recall that for any manifold X, the tangent bundle TX is canonically an almost complex manifold since T(TX) = n*TX E9 n*TX n*TX ® C. This gives TX a canonical orientation as a manifold. A positively oriented basis of T(TX) is of the form (e1>Je1>e2,Je2, ... ,en,Jen) where el, ... ,en is a basis of n*TX and J carries the "horizontal" to the "vertical" factor. With this orientation we can evaluate any element u E on the fundamental class [TX] of the manifold. The result is denoted by u[TXJ. Theorem 13.8. Let P be an elliptic operator over a compact manifold X of dimension n. Then I ind P = (-Inch n(P)· A(X)2}[TX] I where A(X) denotes the total A-class of X pulled back to TX. (13.18) Proof. We consider first the scrunch map q: TfRN = fRN E9 fRN = CN pt. Consider this as a complex bundle and let i: pt 4 CN be the inclusion as the origin. Fix an element u E Kcpt(CN) and apply the defect formula (12.9) with u == Recalling that q, = (i,)-1, that 3(CN) = 1, and that ch: K(pt) HO(pt) is an isomorphism, we find that: q,ch u = q,u. Since q, on is just integration over the fibre, we find that q1u = ch u[TfRNJ. (13.l9) Consider now a real vector bundle p: v X and let i: X v denote the inclusion as the zero-section. Taking derivatives gives the bundle §13. THE ATlYAH-SINGER INDEX THEOREM 255 : Tv -+ TX with zero-section i: TX -+ Tv. One sees easily that the bundle : Tv -+ TX is equivalent to n*v EB n*v = n*v ® Co While TX is not ompact, the map i is proper and equation (12.9) can be shown to hold or elements of Kcpt(TX). Thus for any (I E Kcpt(TX) we have Plch il(l = 3(v ® C)ch (I. valuating on the fundamental class and recalling that PI is integration ver the fibre give the formula: (ch i!(I)[Tv] = {3(v ® C)ch (I}[TX]. (13.20) Consider now an embedding f:X '-+ with normal bundle v. We ·dentify v with an open tubular neighborhood of f(X) in Similarly we ave an open embedding (13.21) as a tubular neighborhood of f(TX). Given any (I E Kcpt(TX), the class il(l has compact support in Tv and naturally extends to under the open inclusion (13.21). This extended class is, by definition, the element 1(1 E given in (13.2). In particular we have ch(il(l) [Tv ] = (13.22) Combining (13.19)-(13.21) gives ind P = {3(v ® C)ch (I(P)}[TX]. It remains to identify 3(v ® C). For this we note that since v is the normal bundle to X, we have TX EB v = (trivial). Since 3 is multiplicative, this means that 3(v ® C) = 3(TX ® C)-I. Applying formula (12.11) we see that (13.23) We have used here that TX ® C is self-conjugate. For the final step we invoke Proposition 11.14. • Integrating over the fibre immediately gives the following. Theorem 13.8 (the cohomological formula for the index; Atiyah-Singer [2]). Let P be an elliptic operator on a compact oriented n-manifold X and let (I = (I(P) E KcpiTX) denote the symbol class of P. Then (13.24) where n: TX -+ X is the bundle projection. 256 Ill. INDEX THEOREMS n(n+ 1) Note. The factor (-1 )-2 -compensates for the difference between the orientation on TX induced by the one on X, and the canonical orientation described above. We now consider two important special cases. Theorem 13.9. Let X be a compact oriented manifold of dimension n = 2m and consider the signature operator D+: 1((;,e+(X)) -> 1(C,e-(X)). Then ind D + = L(X) = sig(X) More generally, if E is any complex vector bundle over X, then the index of Di:r(C,e+(X) ® E) -> r(C,e-(X) ® E) is given by ind(Di) = {ch2E· L(X)}[X] (13.25) where by definition ch2E = I2kchkE. k Proof. Clearly we have that n(D+) = b(TX) where b is defined in (12.12). By Proposition 12.4, 7t!ch b(TX) = (-2)mi,(x)i(X)-2. Applying the for-mula (13.24) above, we find that ind D+ = 2mL(X) = if m is even if m is odd. (A direct identification of ind D + with the signature of X was carried out in 11.6.2). For the more general case we apply the formula (12.14) and (12.16). We conclude similarly that ind(Di) = 2m{ch E . L(X)}[X]. Writing this out and using the fact that Lt = 2 -U Lt, we find that 2m{ch E'i,(X)}[X] = L {2mchkE'LAX)}[X] = L {2m-UchkE·LAX)}[X] = L {2kchkE' LAX)}[X] since the sum is over (k,t) with 2t + k = m. This proves (13.25). • Similar arguments give the following in the spin case: Theorem 13.10. Let X be a compact spin manifold of dimension n = 2m and consider the Atiyah-Singer operator $)+: r($t(X)) -> 1($.["(C)). Then ind $)+ = A(X). More generally, if E is any complex vector bundle over X, then the index of $)i: r($t(X) ® E) -> r($.["(X) ® E) is given by ind($)i) = {ch E· i(X)}[x]. (13.26) §13. THE ATlYAH-SINGER INDEX THEOREM 257 Proof. Note that n(D+) = s(TX) where s is defined in (12.13). By Proposi-tion 12.5, n!ch s(TX) = ( -l)mA(X) -1. Applying formula (13.24) above, we find that ind I/J+ = A(X)[X] = A(X). For the more general case we apply formulas (12.16) and (12.15) to con-clude that nlch[ s(TX) ® E] = ( -l)mch E . A(X) -1. Therefore ind(Di) = {ch E· A(X)} [X] as claimed. _ REMARK 13.11. For a compact spin 2m-manifold X, formula (13.26) is equivalent to the full Index Theorem. This is seen as follows. Any elliptic zero, Po == (1 + p* P) -1/2 P, which has the same index and the same symbol class n E Kcpt(TX). By the Thorn isomorphism 12.6, n can be written in the form n = s(TX) . n*u for some u E K(X). (We use here that X is spin.) Writing u = [E] -[F] for vector bundles E and F on X, we see that n = ® E,n*$c ® E; J.L] -® F, n*$c ® F; J.L] i.e., at the level of the principal symbol, P is equivalent to the difference of two Atiyali-Singer operators with coefficients. This means essentially that PEEl I/J; is homotopic to I/J;'. Therefore, ind P = ind I/J;' -ind I/J; = {(ch E -ch F)' A(X)}[X] = A(X)}[X]. However, {12.15) gives nln = nl(s(X)n*u) = (-l)mch U· A(X)-l, and ind P = {nln' A(X)2}[X] as claimed. For non-spin manifolds, one can argue similarly by using the signature operator with coefficients. One interesting corollary of the index formulas above is the following: Theorem 13.12. On an odd-dimensional compact manifold, the index of every elliptic differential operator is zero. Note that this result does not remain true for pseudodifferentialoperators. Proof. Consider the diffeomorphism c: TX -TX given by c(v) == -v and note that if dim X is odd, then c*[TX] = -[TX]. Let P be an elliptic differential operator of degree m with principal symbol n(P). Since n_!..P) = (-l)"'n!,.P), we see that c*n(P) = (-ltn(P). Since n(P) and -n(P) are regularly homotopic (by n(t,P) = e1titn(P), 0::;; t ::;; 1), they define the same elements in K-theory, and we conclude that c*n(P) = n(P). Applying for-mula (13.18) now gives ind P = -{ch n(P) . A(X)2}[TX] = -c*{ch n(P) . A(X)2}C*[TX] = -{ch c*n(P) . A(X)2}C*[TX] = -{ch n(P) . A(X)2}( -[TX]) = -ind P. _ 258 Ill. INDEX THEOREMS Many important elliptic operators on a manifold X arise in the follow-ing way. Suppose the structure group of X can be reduced to a compact, connected subgroup G C S02m (where dim X = 2m), and that P: r(E) ..... r(F) is an elliptic operator where E and F are vector bundles associated to unitary representations, PE and PF respectively, of G. To apply the index formula (13.24) to P we must compute n1ch t1(P) where n: TX ..... X is the bundle projection. To do this we pass to the universal case. Let n: T ..... BG denote the universal 2m-plane bundle associated to the inclusion G C S02m, and let it,F be the complex bundles over BG associated to the representations PE and P F respectively. Let t1 = [n* E, n* F; (J] E Kcpt(T) be any elliptic symbol from it to F. Then from (12.5) and the fact that nl = (il)-l, we have that x(T)n1ch t1 = i*i1n1ch t1 = i*ch t1 = ch it -ch F. The algebra H*(BG;rQ) always embeds in the polynomial algebra H*(BT;rQ) where T c G is a maximal torus. Therefore, if X(T) =I-0, we can write: n!ch t1 = (ch if -ch F)/XCT). Pulling back to X by the classifying map for TX (with its G-structure) gives the following corollary to Theorem 13.8. Theorem 13.13. Let X be a compact 2m-manifold with structure group G C S02m as above. Let P: r(E) ..... r(F) be an elliptic operator where E and F are associated to unitary representations of G. Suppose that the image of the Euler class X under the map H2m(BS02m) ..... HZm(BG) is not zero. Then the characteristic class (ch E -ch F)/X(TX) E H*(X; rQ) is well defined, and ind P = (_1t{Ch E -;h F . A(X)2} [X]. X(T ) EXAMPLE 13.14 (The Riemann-Roch-Hirzebruch Formula). Let X be a compact complex manifold with a hermitian metric and let E be a holo-morphic hermitian bundle over X. The Dolbeault complex A D.D ® E A D.1 ® E ... A D.m ® E converts to an elliptic operator 8+8· AD,even ® E AD,odd ® E (13.27) where 8* denotes the adjoint of 8. Theorem 13.13 can be applied with G = Um c SOZm and with P = 8 + 8*. If we consider T == TX as an m-dimensional complex vector bundle, then AD,* and we see that ch AD,even -ch AD,odd = ch L1(T). From (12.10) and (12.11) we see that ChA_1(T)/X(T) = (-1tTddT)-1. From Proposition 11.14 we have A(X)2 = ® q = TddT EB T) = TddT)TddT). Plugging into §14. FIXED POINT FORMULAS 13.13 immediately gives the following: ind(a + a*) = {ch E· TddX)}[X] where TddX) = TddT) is the total Todd class of X. 259 Observe that ker P = ker p* P = ker(aa* + a*a) = {<p E A O,even ® E: a<p = a*<p = O}, and similarly coker P = ker P* = {<p E A O,odd ® E: a<p = a*<p = O}. Applying the Hodge Decomposition Theorem (11.5.6) gives the following: Theorem 13.15. Let H*(X; E) denote the kth cohomology group of the Dolbeault complex (13.27) over the compact complex manifold X. Then EXAMPLE 13.16. Let X be a compact oriented riemannian manifold of dimension 2m, and consider the operator DO: r(C£o X) -. r(C£ 1 X) given in 11.6.1. We leave as an exercise to the reader the verification that ind DO = x(X). §14. Fixed-Point Formulas for Elliptic Operators The proof of the Index Theorem outlined above carries over, almost without change, to the cases of G-operators, C£k-linear operators, families of operators, etc. The main point is always to find the right K-theoretic setting in which to work. In this section we consider the case of G-operators and the associated G-index Theorem. We assume throughout that X is a compact G-manifold, i.e., a manifold equipped with a given smooth action Jl: X x G -. X of a compact Lie group G. By a G-bundle on X we shall mean a complex vector bundle E -. X with a G-action which carries fibres to fibres linearly and projects to Jl. The Grothendieck group of equivalence classes of such G-bundles (ef. 1.9) is called the equivariant K-theory of X and is denoted KG(X), Equivariant K-theory has the same properties as ordinary K-theory if one restricts to the category of G-spaces and G-equivariant maps. Hence, given a G-operator P on X, one can pass through the same constructions as above (using a G-equivariant embedding X 4 IRN and the Thorn iso-morphism) to define the topological G-index top-indG(P) in KG(Pt) R(G). 1'Iaeorem 14.1 (Atiyah and Singer [1]). For any elliptic G-operator P on a compact G-manifold; one has that indG(P) = top-indG(P) 260 Ill. INDEX THEOREMS The proof of this "G-index Theorem" follows precisely the arguments outlined above for the basic case where G = {e}. In the case that G acts trivially on X, there is a cohomological formula for the index which is deduced in analogy with the basic case. An impor-tant fact is that when G acts trivially on X, there is a natural isomorphism KG(X) K(X) ® R(G) (14.1 ) determined as follows. Every finite-dimensional representation V of G can be written in the form EBi HomG(l'I, V) ® V; where the direct sum ranges over the set {V;} of equivalence classes of irreducible representations of G. Similarly, if G acts trivially on X, then any G-bundle E can be written as E = EB HomG(E;,E) ® Ei i (14.2) where Ei denotes the trivial bundle Ei = X x J.i. This association induces the isomorphism (14.1) (see Segal [1]). Composing this isomorphism with ch @ Id gives a homomorphism chG: KG(X) ---+ H*(X; 0) ® R(G). For non-compact spaces, such as TX, this extends to K-theory and coho-mology with compact supports. For each g E G there is a homomorphism Xg: R(G) -+ C determined by setting Xg(p) == trace(p(g» for each finite dimensional representation p: G -+ Hom(V, V). Composing with chG gives a homomorphism (14.3) which, for an element u = I Ui ® ri E K(X) ® R(G) is written chgu = I(chu;)xkJ The isomorphism (14.1) together with the arguments for the basic case given in § 13 show the following: Proposition 14.2. Let X be a compact n-manifold on which G acts trivially, and let P be an elliptic G-operator on X with symbol class a = a(P) E KG,cpt(TX). Then one has that indG(P) = (-1)"{ chGa . A(X)2}[TX]. (14.4) In particular, for each g E G, (14.5) In the case that G acts non-trivially on X, the formula (14.5) can be replaced by one which involves the data of the action in a neighborhood of the fixed-point set. The result is a grand generalization of the classical Lefschetz Fixed-Point Formula for maps of finite order. The key to the computation is a certain "localization" theorem of Atiyah and Segal [1]. §14. FIXED POINT FORMULAS 261 An excellent summary of the arguments involved and a large collection of illuminating examples are given in the book of Shanahan [1]. This exposition, together with the highly readable original literature, are recom-mended to the reader interested in full details. We shall present the result here in concise form. To begin, we recall that for each g E G, the fixed-point set of 9 is defined to be the set Fg=={XEX:gX=X}. Since G is compact, we know by averaging that G acts as isometries for some riemannian metric on X. An elementary argument using the expo-nential map then shows that for each g E G, the set Fg is a smooth closed submanifold of X (see Kobayashi [1]). In general, however, Fg is not con-nected and the dimensions of the components can vary. We wish to derive a cohomological formula for indg(P), where P is an elliptic G-operator. For this purpose, we may replace G by the closure of the cyclic subgroup generated by g. This is a compact abelian Lie group for which 9 is a "topological generator." With this assumption we see that F g = {x EX: g' x = x for all g' E G} == F G, the fixed-point set for the entire group. This is a trivial G-space; however, the normal bundle N of Fg is a non-trivial real G-bundle. The normal bundle to the induced embedding is the complex G-bundle n* N ® C (where n: TF g --+ F g is the bundle pro-jection). The element L l(N d == L l(n* N ® q is the Thorn class for this bundle and is used, as in (12.7), to define a homomorphism i!: KG.cpt(TFg) -----jo KG.cpt(TX) (14.6) The fundamental result of Atiyah and Segal [1] is that after localizing at g, i.e., after introducing formal in verses for all elements rE R(G) with Xg(r) #-0, the map (14.6) becomes an isomorphism with inverse given by '-1 i* I, =---. L1(Nd This leads to the following. Theorem 14.3 (The Atiyah-Segal-Singer Fixed-Point Formula). Let X be a compact G-manifold, where G is a compact Lie group, and let P be an elliptic G-operator on X with symbol class a = a(P) E KG,cPt(TX). For each element 11 e G, the "Lefschetz number" indll(P) == trace(glker p) -trace(glcoker p) is given by the formula . d{ chll(i*a) z}[ ] mdJP) = (-1) chJL1(Nd)' FII) TF, (14.7) 262 Ill. INDEX THEOREMS where Fg denotes the fixed-point set of g, where L l(N d denotes the Thom class of the complexified normal bundle of F g' and where d denotes the di-mension of Fg (an integer-valued function which varies from component to component). Note that G is not assumed to be connected; in particular, Theorem 14.3 applies when G is a finite group. Some of the most important appli-cations of the result come from this case. This theorem also gives rise to a number of intriguing relations between topology and elementary num-ber theory (see Hirzebruch-Zagier [1 ]). The most basic example of a G-operator comes from the de Rham com-plex. Let G act by isometries on a compact riemannian manifold X, and consider the Dirac operator DO: C£O(X) -+ C£l(X). (Recall that this is ex-actly the operator d + b: A;:veo(x) -+ This operator is G-equiv-ariant and for each 9 E G, the number indg(DO) is just the classical Lefschetz number of g, i.e., indg(Do) = L(g) == trace(gIHcven) -trace(gIHodd). The symbol class of DO is just the class a(Do) = L1(n*TX ® C). Re-stricting to TFg we have n*TX ® C = (n*TFg ® C) EB (n*N ® Tc EB No and so i*a(DO) = A-l(Tc EB Nd = A-l(TdLl(Nd. It follows that chg(i*a(Do)) = chg(L1(Tc))chg(L1(Nc)), and formula (14.7) becomes In particular, if 9 has isolated fixed points, then L(g) = card(Fg). This result uses strongly the fact that 9 is contained in a compact group of diffeomor-phisms. The general Lefschetz formula applies to any diffeomorphism f with isolated non-degenerate fixed points, i.e., points where det(/ -df) 1= O. In this case points are added with the weight factor sign[ det(I -df)], so that the Lefschetz number becomes equal to the algebraic sum of the fixed points. One might ask whether the fixed-point formula (14.7) can be extended to cover geometric automorphisms of an elliptic operator whil:h do not lie in a compact group. Such a formula was given by Atiyah and Bott [3,4] for the casc of automorphisms with isolated non-degeneratc fixed points. To give the reader some feeling for the fixed-point formula, we shall work out the details for the signature operator D +: nef +(X)) ---> nee -(X)) and the Atiyah-Singer operator ffi + : nr (X)) ---> ['($ -(X)) (where X is spin). Let us suppose that X is a compact oriented riemannian manifold and that g: X ---> X is an orientation preserving isometry. Let N denote the normal bundle of the fixed-point set Fg, and note that thc differential dg of 9 gives a bundle isometry dg: N ----+ N. (14.8) §14. FIXED POINT FORMULAS 263 Fix x E Fg and note that since g is an isometry, we have g(expxv) = expx(dg'v) for all v E N x' It follows that dg' v =f. v for each v =f. 0, since otherwise expx(tv) would lie in Fg for all t E R Since g lies in a compact abelian Lie group, we know from elementary representation theory that there is an orthogonal decomposition Nx = NxCrr) EB EBO<9<" Nx(8), where dgxk(,,) = -Id, and where the space N x(8) splits into 2-dimensional subspaces in which dgx rotates every vector by 8. This decomposition is "constant" on each component of Fg• This follows from (14.2) or by simply observing that parallel translation along any curve joining x to y in Fg gives a g-equivariant isometry of N x(8) with Ny(8). (This follows without difficulty from the fact that g is an isometry.) Consequently, we conclude that the bundle N admits a decomposition N = N(rr) EB EB N(8) (14.9) 0<9<" where dg acts on N(rr) by multiplication by -1, and where for each 8, o < 8 < rr, N(8) is a complex bundle in which dg acts by multiplication byei9• REMARK 14.4. Note that the bundle N(rr) may not be orientable but it carries the same orientation class as F g' i.e., for every loop y c F g' the orientation of Fg changes along y if and only if the orientation of N(rr) also changes along y. (This is because X is orientable and each N(8), 0< 8 < rr, is complex and hence also orient able.) For any real vector bundle E, we introduce the characteristic class L,,(E) == X(E)L -1(E) (14.10) where L(E) is the total L-class of Hirzebruch and where X(E) is the Euler class of E. If E is not orientable, then X(E) lives in cohomology with twisted coefficients. For any complex vector bundle E, and any 8, 0 < 8 < rr, we define L9(E) to be the total class associated to the multiplicative sequence of Chern classes with formal power series coth(x + tie). Therefore, if E = t 1 EB ... $ tk is a formal splitting into complex line bundles with c1(tj) = Xj' then k L8(E) = TI coth(xj + tie) j= 1 With these definitions we can state the following. (14.11) We recall now that if X is oriented and of even-dimension, then there is a canonical splitting ct(X) = ct + (X) $ ct -(X) and the Dirac con-struction gives the "signature operator" D+: qct + (X» -+ r(ct -(X». This operator is preserved by the group G of isometries of X. For each g E G we denote 264 Ill. INDEX THEOREMS to emphasize that this invariant depends only on X and g. If dim X = 4k and g = 1, then sig(X,g) = sig(X). Theorem 14.5. (The G-signature Theorem). Let g: X ---+ X be an orienta-tion-preserving isometry of a compact oriented 2m-dimensional manifold X. Then sig(X,g) = Le(N(8))· L(Fg)} [Fg] (14.12) where N = EBN(8) is the decomposition of the normal bundle to Fg given in (14.9). Note. The cohomology class in (14.12) factors into a product of an ordi-nary cohomology class with x(N(n)). As observed in Remark 14.4, N(n) and Fg have the same orientation type. Hence, the pairing in (14.12) is well defined. Proof. We begin with a remark on dimensions. Since g is orientation preserving, we see that dimlJ;!(N(n)) is even. Hence we may write dim Fg = 2d, dim N(n) = 2r, dimIJ;!N(8) = 2s(8) where d,r and s(8) are integers which vary from component to component on Fg• Recall that a(D+) = t5(TX), and therefore by the multiplicativity (11.33) of 15 we have i*t5(TX) = t5(TFg EB N) = t5(TFg)t5(N). By pushing forward to Fg and using Proposition 12.4, the Fixed-point Formula (14.7) becomes (14.13) It remains to evaluate the term chg(t5(N)A_l(N ® q-l). Since every-thing is multiplicative, we can consider the factors N(O) separately. Via the Splitting Principle it suffices to compute in the case where v is an oriented real 2-plane bundle on which g acts by rotation by O. We write v ® c = t EB 1 where t is a complex line bundle, and we set x = cl(t) = X(v). Then we have t5(v) = 1-t and A _ 1 (v ® q = (1 -t)( 1 -1). From the fact that chg(t) = ch(t)eiO = ex+iO, we find that e-x-i(} _ ex+i8 chg(t5(v))Ll(v ® q-l = -:-:( 1 -eX 'v)(1 _ e X iO) = coth t(x + iO). §14. FIXED POINT FORMULAS 265 This leads us to introduce the multiplicative sequence of Chern classes Le associated to the formal power series coth t(x + i8). The above computation shows that chg(t5(N(8»Ll(N(8) ® C)-I) = Le(N(8». This computation is equally valid when 8 = n provided that N(n) is orientable. In fact, under this assumption we take a formal splitting N(n) = VI Ei1 ... Ei1 Vr into oriented 2-plane bundles, and with Xj = X(v) we find that r chg(t5(N(n»)A'-I(N(n) ® C)-I) = n coth t(Xj + in) j= 1 r = n tanh(xp) j= 1 = fI Xj fI [ x)2 J-l j= 1 2 j= 1 tanh(x)2) = i(N(n) )L(N(n))-1 where i(N(n)) == 2 -rX(N(n)). Assembling these calculations, we find that sig(X,g) = Le(N(8))' L(Fg)} [FgJ where we have set I." = i' I. -1. Multiplying the appropriate homoge-neous component of this cohomology class by 2d allows us to remove the hats from the L's and gives us the desired formula (14.12). • REMARK 14.6. This formula undergoes only minor modification if one takes coefficients in a complex G-bundle E. Specifically, if D;: qct + ® E) -+ qct -® E) is the twisted signature operator, then setting sig(X,E, g) == indg(Di), we have sig(X,E,g) = {ch E· Z}[FgJ (14.14) where Z IS the co homological expression appearing in formula (14.12). The applications of Theorem 14.5 are numerous and varied (see Atiyah-Singer [2J, Shanahan [lJ, and Hirzebruch-Zagier [lJ for example). We mention here just a few corollaries. Note to begin, that the expression L" = X • L -1 involves the Euler class. Consequently we have the following. CoroDary 14.7. Let X, D+, g, etc., be as in Theorem 14.5. If dim N(n) > dim Fg, then sig(X,g) = O. Note that when g is an involution, (i.e, g2 = Id), we have N = N(n). In this case the corollary states that dim Fg < -!dim X :::-sig(X, g) = O. This statement has a pretty generalization. 266 HI. INDEX THEOREMS Corollary 14.8 (Atiyah-Singer [2J). Let X be a compact oriented 4m-mainfold, and let g: X -+ X be an orientation preserving involution with fixed-point set Fg. Let Fg' Fg denote the closed oriented manifold obtained by intersecting Fg with (a generic displacement of) itself. Then we have sig(X, g) = sig(F 9 • F g). The right hand side of the formula is independent of the transversal displacement of Fg used to define F,· Fg. It depends, of course, only on the oriented cobordism class of Fg . Fg. Proof. Since g2 = Id, we have N = N(n). One can verify easily that {L(F,)L-1(N)X(N)}[FgJ = {L(Fg)L-l(N)}[Fg' FgJ = {L(F,· F,)}[Fg' F,] = sig(F,' Fg). (We use here that N!Fg'Fg is the normal bundle to F,' F, in F,.) Applying (14.12) completes the proof. _ The proof of the following corollary is left as an exercise for the reader. Corollary 14.9 (Atiyah-Singer [2]). Let X be a compact connected oriented 4-manifold, and suppose that g: X -+ X is a difJeomorphism of odd orde; (necessarily orientation preserving). Suppose F, consists of oriented embedded sUrfaces S 1> ••• , Sft and that for each k, dg rotates the oriented normal bundle to Sk through an angle Ok' Then where Sk' Sk denotes the homological self-intersection number of Sk' Our second basic set of examples will come from considering the Atiyah-Singer operator l/J+: r($t) -+ r($c) acting on spinors over a com-pact even-dimensional spin manifold X. We shall assume that G is a compact Lie group acting by orientation preserving isometries of X. Note that there is an induced action of G on the bundle Pso(X) of oriented orthonormal tangent frames. Recall that the spin structure is a 2-fold covering Pspin(X) -+ Pso(X) which is non-trivial on the fibres. It corre-sponds to an element u E Hl(PSO(X); Z2)' DEFINITION 14.10. The action of G preserves the spin structure of X if it lifts to an action on the bundle PSpin(X). An individual isometry g: X -+ X is said to preserve the spin structure of X if the closed subgroup G c Isom(X) generated by g preserves this structure. Note· that if g preserves the spin structure, then g*u = u, where u E Hl(PSO(X); Z2) is the element corresponding to the structure. If G is connected, then it follows from elementary covering space theory that either G or a 2-fold covering group of G preserves the spin structure. In particular, if n1 G = 0, then G preserves every spin structure on X. §14. FIXED POINT FORMULAS 267 Whenever G preserves the spin structure, it acts on the bundle of spinors and commutes with the Atiyah-Singer operator I/J+ above. For each g E G one defines Spin(X,g) == indg(I/J+), and the Fixed-Point Theorem gives us a cohomological formula for this invariant. Its expression involves the sequences of Chern classes Ae asso-ciated to the formal power series [2 sinh J( x + 0)] -1 for ° < 0 < 7r. If E is a complex vector bundle with formal splitting E = t1 EB ... EB tk into line bundles with Cl (tj) = x j' then 1 k 1 k 2(Xj+ie) (E) = 2-k n = n e Ae j=lsinht(xj+i8) In the special case where 0 = 7r we define a characteristic class A,,(E) for any oriented real 2k-dimensional bundle E as follows. Let E = El EB ... EB Ek be a formal splitting into oriented 2-plane bundles, and set Xj = X(Ej). Then A,,(E)==2-kn. 11 . =(2i)-kn 1 j=l stnhz(xj + m) j=l cosh(xi2) Theorem 14.11 (The G-spin Theorem). Let g : X -+ X be a spin structure preserving isometry of a compact even-dimensional spin manifold X. Then where N = EBN(O) is the decomposition of the normal bundle to Fg given in (14.9), and where (): F g -+ {O, I} is a locally constant function which depends on the action of g on the spin structure. REMARK 14.12. In general the calculation of the sign function () is a complicated and subtle affair. We refer the interested reader to Atiyah-Bott [3J,[ 4J and Atiyah-Hirzebruch [3J for details. Proof We proceed much as in the proof of the G-signature Theorem. Re-call that a(D+) = s(TX) and therefore by the multiplicativity (11.36) of s we have i*a(D+) = s(TFg EB N) = s(TFg)s(N). Pushing forward to Fg and using Proposition 12.5 converts the Fixed-Point Formula to . + {Chg(S(N)) tndg(D ) = Chg(A_1(N ® C)) . A(Fg) [FgJ. Computing as before, we consider the case of an oriented 2-plane bun-dle v with v ® c = t EB l. Here we have that s(v) = {1/2 -t1/2 and 268 rn. INDEX THEOREMS A_l(V ® q = (1 -t)(1 -1), and therefore that chg(s(v)Ll(v ® q-l) = ± [et(X + iO) _ e -t(X + i6)] -1, if g acts on v by rotation by (). It follows directly that chg(s(N(()))Ll(N(()) ® q-l) = ±Ao(N(())) for each (), 0 < () ::;; n. This completes the proof. -The G-Spin Theorem can be enhanced by taking coefficients in a G-bundle E. The formula changes exactly as in the G-signature case (cf. Remark 14.6). §15. The Index Theorem for Families Consider now a family of elliptic operators P on a compact manifold X parameterized by a compact Hausdorff space A. In §8 it was shown that this family has a well-defined analytic index ind P E K(A). On the other hand the constructions of §13 can be generalized to define a topological index in K(A). The main result of this section asserts that these two indices coincide. The topological index of the family P is defined as follows. Let n ::?r --+ A denote the underlying family of manifolds (recall that this is a bundle with fibre X and structure group Diff(X)), and let T:?r --+ A denote the associated family of tangent bundles (so that (T:!{)a = T(:?r a) where :?r a == n-l(a)). It is not difficult to see that for sufficiently large m we can find a map f::?r --+ A x IRm which restricts to a smooth embedding fa: :?r a 4 {a} x IRm for each a E A. This induces a map Tf!l' 4 A x TlRm which for each a E A, restricts to an embedding T:?r a 4 {a} x TlRm with normal bundle Na EEl Na Na ® c. Here Na denotes the normal of fa(f!l' a) C IRm pulled back to T:?r a' We now define a map J;: Kcpt(T:?r) --t Kcpt(A x cm) by taking the composition Kcpt(T2l') --t Kcpt(N ® q --t Kcpt(A x cm) where the first map is the Thorn isomorphism, and where the second map is induced via an embedding N ® C 4 A x TlRm A x cm, which is con-structed fibrewise by identifying N a ® C with a tubular neighborhood of in la} x TlRm as before. The projection q: A x cm --+ A induces a Thom isomorphism (or Bott periodicity map) §15. INDEX THEOREM FOR FAMILIES 269 The composition q! 0 it: Kcpt(TEt) --+ K(A) is independent of the choice of the embedding f. DEFINITION 15.1. Let P be a family of elliptic operators on a compact manifold X parameterized by a compact Hausdorff space A as in §8. Let a(P) E Kcpt(TEt) be the class determined by the principal symbol of the family. Then the topological index of P is the element top-ind(P) = q!it(a(P)) E K(A). Straightforward generalization of the arguments given in § 13 proves the following: Theorem 15.2 (The Atiyah-Singer Index Theorem for Families [3]). For any P as above one has that ind(P) = top-ind(P). Applying the Chern character and arguing as in §13 establishes the fol-lowing cohomological form of this result. Theorem 15.3 (Atiyah-Singer [3]). If P is as above, then ch(ind P) = ( _1)"(n; 1) n!{ ch a(P) . A(T21)2} where n = dim X and where n: TEt --+ A is the natural projection. Suppose now that n : Et --+ A is a family of riemannian manifolds, i.e., a family of manifolds as above together with a family of riemannian metrics introduced continuously in the fibres. Corollary 15.4. Let n: Et --+ A be a family of compact oriented riemannian manifolds of dimension 2m, and let qCt + (TEt)) --+ qct -(TEt)) be the associated family of signature operators. Then = 2mnl{L(TEt)}. More generally, if we take coefficients in afamity of hermitian vector bundles 11 with connection, then the associated operator : qCt + (TEt) ® 11) --+ qCC(TEt) ® 11) has index given by = 2m1tl{ch 11 . L(TEt)]. Proof. The symbol of is a(£l}+) = 8(TEt). Using 12.4 to push forward over the projection no: TEt --+ Et, we find that (noMch a(£l}+)· A(TEt)2} = (-2tL(TEt)A(TEt)-2A(TEt)2 = (-2)mL(TEt). Projecting on to A then gives the first formula. The second formula follows similarly from the fact that a(£l};) = a(Tf!l') . • 270 III. INDEX THEOREMS It is an interesting exercise to apply this formula to the Lusztig family (8.3). Carrying through the computations above with 0 replaced by s gives the following. Corollary 15.5. Let n::?l' A be a family of compact oriented spin mani-folds of dimension 2m, and let : r(Y+ ® C) r(Y-® C) be the family of Atiyah-Singer operators twisted by a family of coefficient bundles C. Then All of these results go through for families of G-operators. One replaces K by KG in this case. §16. Families of Real Operators and the eCk-Index Theorem In discussing the Index Theorem thus far we have considered only com-plex differential operators. Given a real operator, say P, defined using real vector bundles, one can always complexify and apply the index theorem in the form above. Since dimffi!(ker P) = dimdker P ® q (and similarly for P*), we see that the ordinary real and complex indices of p coincide. Thus, in the basic case no information is lost under complexification. This is not true, however, if one passes to the index theorem for families. The index of a family of real operators takes its value in the group KO(A), and the complexification map KO(A) K(A) is not always injective. Note for example that KO(S") 71.2 for n = J(mod 8), but K(S") = {O} in these dimensions. For this reason Atiyah and Singer established a separate index theorem for families of real operators. It is a more subtle and profound result than one might naively expect. The constructions and arguments outlined above for complex families go through essentially unchanged in the real case, provided one employs the appropriate "K -theory." It is here that mattcrs become interesting, for the appropriate theory is not KO-theory, the straightforward theory of real bundles. It is the more general K R-theory which is defined on any space with involution and reduces to KO-theory when the involution is trivial. Recall (cf. I. 10) that if X is a space with involution f:X X, then KR(X) is the Grothendieck group of pairs where E is a complex vector bundle over X and where fE: E E is an involution which covers f and is C-antilinear on the fibres. The introduction of this theory is motivated by the simple fact that the principal symbol of a real operator is in general not real. Consider, for example, the operator alae: C')(Sl) C"(Sl) whose symbol is = i';. §16. THE Cfk-INDEX THEOREM 271 Recall also that for any real Dirac operator D, one has that (J = (cf. Example 1.5). The appearance here of the complex number i cannot be ignored. It is essential in the calculus of pseudo-differential operators and must be retained if the proofs discussed above are to carry over. Suppose now that P: qE) --+ qF) is a real differential operator between real bundles E and F over a compact manifold X. In local terms, we have that P = L A"(x)8Ial/8xa plus lower order terms, where the Aa are real matrix-valued functions. Consequently for any tangent vector we have = L from which it follows that u = (16.1) This is the key to defining the symbol class of a real operator. DEFINITION 16.1. Given a compact manifold X, consider the tangent bundle n: TX --+ X to be equipped with the canonical involution f: TX --+ TX defined by f@ = Given any real bundle E --+ X, consider n*(E ® C) to be equipped with the antilinear involution defined by com-plex conjugation. Then for any real elliptic operator P: qE) --+ qF) the Real symbol class of P is defined to be the element [n*(E ® C), n*(F ® C); u(P)] E KRcpt(TX). (16.2) Here n*(E ® C) and n*(F ® C) are Real bundles on the Real space TX, and (16.1) says that u(P) is an isomorphism of real bundles outside the zero section of TX. Hence, (16.2) naturally defines an element in KRcpt(TX.) In general if one wants to remember that a given bundle is the com-plexification of a real bundle, one carries along the associated complex conjugation map. The new point here is that when the bundle is pulled back over TX we think of this conjugation as covering the involution 1-+ With this little refinement everything works as one hopes. To define the topological index of a real elliptic operator P we first choose an embedding f: X c... IRm. The associated embedding TX c... TlRm is compatible with the involutions i.e., is a mapping of Real spaces. If N is the normal bundle to X in IRm, then n*N Efl n*N n*N ® C is the normal bundle to TX in TlRm. We consider this to be a Real vector bundle on TX as in Definition 16.1 above. For such bundles, the Thom isomor-phism holds for KR-theory. (It is defined exactly as it was for K-theory in §12. We need only note that the de Rham element t\-1 of a Real bundle is itself Real. See Atiyah [2].) We can now define a map it: KRcpt(TX) KRcpt(TlRm) as before by composing the Thorn isomorphism with the map induced by the inclusion of the normal bundle as a tubular neighborhood of TX in TlRm. This inclusion can be easily chosen to be compatible with the involu-tions. 272 Ill. INDEX THEOREMS If :?l' --+ A is a family of smooth manifolds over the compact space A, this construction extends, by using local triviality of the family, to give a map J;: KRcpt(T:?l') KRcpt(A x TlRm). We now identify TlRm IRm EEl IRm cm by associating to x + Clearly the involution on TlRm becomes complex conjugation on cm. The fundamental (1, I)-Periodicity Theorem (1,10.3), says that there is a natural isomorphism q!: KRcpt(A x cm) KR(A) for any compact (Real) space A. As before, the composition qJ 0 J; can be shown to be independent of the choices involved in the construction. DEFINITION 16.2. Let P be a family of real elliptic differential operators on a compact manifold X parameterized by a compact Hausdorff space A. Let a(P) E KRcpt(T:?l') be the symbol class of the family (where, as before, :?l' --+ A is the underlying family of manifolds). The topological index of the family P is defined to be the element top-ind(P) = qJJa(P) E KR(A) KO(A). (Note that A carries the trivial involution.) With this definition, the arguments for the Index Theorem discussed above go through easily to prove the following. Theorem 16.3 (Atiyah and Singer [3], [4]). Let P be a family of real elliptic operators on a compact manifold parameterized by a compact Haus-dorff space A. Let ind(P) E KO(A) be the analytic index of the family de-fined as in 8.5 by replacing complex objects with real ones. Then ind(P) = top-ind(P). It should be remarked that the index theorem for families is a useful tool, much more powerful than the standard index theorem. A number of applications are given in the next chapter. One important consequence of this result is the derivation of a topo-logical formula for the Clifford index discussed in §1O. No such formula appears in the current literature even though the derivation was known to Atiyah and Singer. We shall present the details here. Assume from this point on that E = EO EEl El is a real, Z2-graded C£k-bundle over a compact riemannian manifold X. Assume E carries a bundle metric for which Clifford multiplication by unit vectors in IRk is orthogonal. Let P: qE) --+ qE) be an elliptic self-adjoint operator and assume that P is C£k-linear and Z2-graded. In 10.4 we defined an analytic index §16. THE Ctk-INDEX THEOREM 273 indkP E KO-k(pt) for P in terms of the C£k-module ker P. We shall now give a topological formula for this index. To do this we construct the following family [!P of elliptic operators parameterized by IRk. We assume that P has degree zero by replacing P with (1 + P* P) -1/2 P. We recall from 10.2 that with respect to the splitting E = EO Efl El we can write P as P = (;0 where pi = (p0)*. The product family [!P on IRk x X is now constructed by assigning to each v E IRk the operator defined by the restriction to EO of the operator [!Pv==V+P (16.3) (16.4) where "v" denotes Clifford multiplication by v. Note that there is a "con-jugate family" of operators rjJ v defined by .rjJ v == V -P. Since P com-mutes with Clifford multiplication, we have that (16.5) In particular, is invertible for all v "# O. Since the invertible operators on Hilbert space form a contractible set, we could easily pass to a family parameterized by Sk. However, the calculations will be more transparent if we treat [!po as a family with "compact support" in IR\ whose index lies in KOcpt(lRk) KO-k(pt). The main result is the following: Theorem 16.4. Let P be an elliptic self-adjoint graded C£k-operator on a compact manifold. Then indk(P) = top-ind(g'Jo) where g'J is the family over IRk defined by (16.4). Proof. Set KO == ker po c r(Eo) and Kl == ker pi coker po c r(E1). By Theorem 5.5 there are U-orthogonal direct sum decompositions and (16.6) where pO: VO -=. Vi is an isomorphism. Since P commutes which Clifford multiplication, we have that V' KO = KI for all v "# 0 in Furthermore, since multiplication by v/llvll is U-orthogonal, we have v . VO = Vi for all such v. It follows that the family = v + pO: r(Eo) r(EI) decom-poses, with respect to (16.6), as a direct sum of two operators. By (16.5) 274 Ill. INDEX THEOREMS the first summand is an isomorphism for all v E !R\ and thus can be ignored for the purposes of computing the index. The second summand is just KO Kt This operator is independent of variables on X. Its index equals the index of [!po and is given by the element [KO,K1; v] E KOcpt(!Rk) KO-k(pt). Under the Atiyah-Bott-Shapiro isomorphism KO-k(pt)'; IDlk/i*IDlk+t> this corresponds exactly to the element represented by the Zrgraded ctk-module ker P = KO Efl Kt, i.e., it corresponds exactly to indkP. _ Theorem 16.4 can be applied to give topological formulas for the index of any of the graded ctk Dirac operators discussed in Chapter 11, §7. In particular this includes the Atiyah-Singer operator whose index is a basic invariant which we shall now discuss in detail. Let X be a compact spin manifold of dimension n, and recall that X carries a canonical graded ct" Dirac bundle == PsPiiX) x (Ct", whose associated Dirac operator is called the Atiyah-Singer operator (see equation (11.7.1) forward). This operator has an index E KO-"(pt) which coincides, by 16.4, with the index of the family !iJ on R" x X defined by setting !iJv,x = v + To compute the topological index of this family we must understand its symbol class a(!iJ) E KRcpt(!R" x T X). For this we note first that 1t: !R" x TX --+ X is a Real bundle over X whose fibre at x E X is !Rn X TxX with involution (v,e) 1-+ (v, -e). The fibre of the bundle at x is the Clifford algebra Ct(TxX) Ct". Vectors e E TxX act by left Clifford multiplica-tion as usual, and vectors v E !R" act by right multiplication. The principal symbol of!iJ is the map cr(!iJ): --+ defined by = v + ie (= Rv + where == ® e is treated as a Real bundle on !R" x T X with in-volution given by conjugation. The symbol class. cr(!iJ)] when restricted to any fibre !R" x TxX !Rn X !R" e", becomes exactly the element, §16. THE Cl:k-INDEX THEOREM 275 which, as we have seen in 1.10.12 is the generator of the group KRcpt(Cn) 71.. (It corresponds exactly to the deRham element A_l under the usual isomorphism ce* A*.) Consequently the symbol class == n*$b; E KRcpt(lRn x T X) is a Thorn class or "orientation class" for the bundle n: [Rn X T X -4 X in KR-theory. Under the Thorn isomorphism i,: KR(X) -4 KRcpt(!Rn x TX) we have = q1) (16.7) Choose now a smooth embedding f: X 4 !Rn + Sf and let N denote its normal bundle. This induces a smooth embedding f: TX 4 T!Rn + Sf with normal bundle n* N EEl n* N n* N ® C, and we get the following com-mutative diagram of bundle maps: X N p The vertical arrows are Real bundles. Taking zero-sections and embedding the normal bundles as tubular neighborhoods give the following commu-tative diagram of embeddings: !Rn X T X cb. !Rn X (n* N ® q 2: !Rn X en + Sf iU X c j iU N (16.8) We now recall from 1I.1.15 that N has a canonically induced spin structure. Since dim[J;!N = 8f there is a Thorn isomorphism j!: KO(X) -4 KOcpt(N) defined by setting Nu) = (p*u) . s(N) where s(N) == [S+(N), S-(N); 11] is defined as in (12.13) using the real spinor bundle of N (see Appendix C). We could view this as an isomorphism in KR-theory where the spaces X and N carry the trivial involution. There is a completely analogous Thorn isomorphism KRcpt(!Rn X TX) -4 KRcpt(!Rn x (n*N ® q) defined using the spin structure on n*N and the Real bundle n*SdN) = n*S(N) ® c. One easily checks that and where k, and k, denote the natural maps induced by the open inclusions 276 III INDEX THEOREMS k and k. Consequently, (16.8) leads to a commutative diagram KR ([Rn X TX) KR ([Rn X Cn+8t) -cpt cpt i'l i} KR-n(pt) (16.9) J, KR(X) KRcpl[Rn+8t) where f.. == k,j" = and where q! and if, denote the Bott periodicity isomorphisms. The topological index of the family f$ is defined to be the element Applying (16.7) and (16.9) we conclude that top-ind(f$) = ii,fti,(1) = q!f..(1). (16.10) DEFINITION 16.5. For a compact spin manifold X of dimension n, the Atiyah-Milnor-Singer invariant is the element sI(X) == q!f..(l) E Ko-n(pt) where f: X 4 [Rn+8t is a smooth embedding for some t, and where ql and f.. are defined as above. This definition is easily seen to agree with the one given in II.3.22. In particular, we see that .liJi'(X) depends only on the spin cobordism class of X. Combining (16.10) with Theorem 16.4 gives the following main result: Theorem 16.6. Suppose X is a compact spin manifold of dimension n, and let [($) ---> [($) he the canonical Atiyah-Singer operator. Then = .sI(X). From Theorem 11.7.11 and the spin-cobordism invariance of .li(X), we see that si: n:tn ---> KO -*(pt) is a graded ring homomorphism. Suppose now that E is any real vector bundle on X. Furnish E with any orthogonal connection and introduce the tensor product connection on $ ® E. This is again a graded Cfn-Dirac bundle and has an associated Dirac operator Let'fE = v + be its associated family. Onc easily checks that = i,([E]). Passing through the arguments above one finds that top-ind(tt'E) = qJ;([E]). This proves the following result: DEFINITIO]\; 16.7. Let X be a compact spin manifold of dimension n. For each real vector bundle E over X, the associated Atiyah-Milnor-Singer invariant is the clement .ci(X;E) == qJ;([E]) E KO-n(pt) where qJ;:KO(X) ---> KO-"(pt) is defined as above. Theorem 16.8. Suppose X is a compact spin manij(Jld of'dimension n, and let E he a real vector hundle over X. Let [($ ® E) ---> [($ ® E) he the §17. HEAT AHD SUPERSYMMETRY 277 Cen-linear Atiyah-Singer operator with coefficients in E (defined using any orthogonal connection on E.) Then = .si(X;E). §17. Remarks on Heat and Supersymmetry Since the basic work of Gilkey and Patodi it has been known that index formulas for classical elliptic operators (such as the Atiyah-Singer opera-tor with coefficients in a bundle) can be derived from the asymptotics of the heat kernel. As explained at the end of §6, it is necessary to make detailed computations of the density Pn occurring in the asymptotic ex-pansion trace K,(x,x) '" L Pk(X)t!k-n)/2 as t '" O. Expositions of this can be found in Gilkey [2] and Atiyah-Bott-Patodi [1]. In 1979 W. Allard showed us a simple and elementary calculation of these densities obtained by working with certain canonical operators on the underlying principal bundle and using the natural filtration of the Clifford algebra. (His resulting proof of the classical Index Theorem is completely elementary.) In 1982, E. Witten found a different approach to these formulas through considerations of symplectic geometry and supersymmetry. There is a fixed-point formula for S1-actions on (finite dimensional) symplectic mani-folds due to Duistermaat and Heckman [1]. Witten considered the anal-ogue of this formula for the canonical S1-action on the free loop space of a manifold. Using this together with some ideas from supersymmetry he outlined a proof of the index theorem for the Atiyah-Singer operator (see Atiyah [13]). His ideas have engendered a series of interesting papers on the subject, notably by L. Alvarez-Gaume, E. Getzler, N. Berline and M. Vergne, and J. M. Bismut. It should be pointed out however that none of these methods applies to prove the index theorem for families or the Cek-index Theorem (in their strong forms). These theorems in general involve torsion elements in K-theory which are not detectable by cohomological means. Moreover, it is known that the Z2-invariants appearing in the Cek-index Theorem are not computable from local densities, for this would imply a certain multi-plicativity under coverings which examples show not to be true. CHAPTER IV Applications in Geometry and TopoloBY In this chapter we shall use the results of our previous discussion to derive a series of consequences in differential topology and geometry. Some of the theorems we shall derive have other, completely different proofs, and some (to date) can only be proved by these means. Nevertheless we hope to make clear that spin geometry and the Index Theorem give a unified approach to a wide range of geometric problems. The applications presented here fall into several categories: integrality and divisibility theorems for characteristic numbers, immersions of mani-folds and the vector field problem, group actions on manifolds with posi-tive scalar curvature, Kiihler geometry, pure spinors and basic twistor geometry, the theory of calibrations, and the study of riemannian metrics with reduced holonomy. A few of these results are already mentioned in previous chapters in order to add spice to the exposition there. On the other hand there are applications given in previous chapters which do not appear here. Notable among them are the results of §§7 and 8 in Chapter 1, the new curvature identities 5.15 --16 of Chapter IT and their application to the Homology Sphere Theorem (11.7.6) of Gallot and Meyer. The first part of this chapter is concerned with applications of the Index Theorem to differential topology. Recall that the ordinary index of an elliptic is an integer. However, the Atiyah-Singer formula computes this index in terms of topological invariants which are in general just rational numbers. The fact that these rational numbers are always integers under certain hypotheses, plays a significant role in topology. Its importance stems from the underlying elliptic operators. Two general problems to which such integrality theorems often apply are: finding the smallest codi-mension for immersing a manifold into euclidean space, and finding the largest number of pointwise linearly independent vector fields on a mani-fold. We shall show how to prove results in these areas by directly con-structing the relevant operators via Clifford multiplication. Using more sophisticated forms of the Index Theorem, one can prove interesting results about smooth compact group actions on spin mani-folds. For example, applying the G-Index Theorem to the Atiyah-Singer operator shows that on spin manifolds admitting an SI-action the A- APPLICA nONS 279 genus is zero. Applying the C£k-Index Theorem to the C£k-linear Atiyah-Singer operator and invoking some basic riemannian geometry yields a more refined result about S3-actions. All of this is done in §3. Sections 4, 5 and 6 are devoted to the study of riemannian manifolds of positive scalar curvature. For compact simply-connected spin mani-folds, the KO-index of the Atiyah-Singer operator gives a nearly complete set of invariants for deciding whether or not there exists a metric of posi-tive scalar curvature. This is discussed in §4. In §5 the analogous question is discussed for manifolds with non-trivial fundamental group. Here there is an interesting interaction between spin geometry and the fundamental group which is mediated by an appropriately twisted Atiyah-Singer operator. The ideas developed here carry over to give results about the existence of complete metrics with positive scalar curvature on non-compact spin manifolds. For this it will be necessary to develop a "rela-tive version" of the Index Theorem over open manifolds. This is discussed in §6. The techniques of spin geometry can be applied to say something about the topology of the space of all riemannian metrics with positive scalar curvature on a given manifold. This is done in §7. On an even-dimensional riemannian manifold there is a concept, due to E. Cartan, of pure spinors. These are related directly to almost complex structures on the manifold and to the Penrose twistor construction. This together with theorems on integrability are examined in §9. A Kahler manifold is a riemannian manifold with a parallel almost complex structure. (Such a structure is always integrable, i.e., the under-lying manifold is always complex.) When X is a Kahler manifold, the Gifford bundle C£(X) has a very pretty decomposition. There are dif-ferential operators and such that = = 0 and + = D (the Dirac operator on C£(X». There are also O-order operators !R and fl such that (!R,fl, [!R,fl]) sI2(C) and !R + fl = L (cf. (11.5.20». From these one can define certain Clifford cohomology groups and establish within them all the classical identities of Kahler geometry (see §8). One can also decompose holomorphic Dirac bundles and define spinor coho-mology groups. A wide variety of vanishing theorems are then proved using Clifford formalism in §11. These are applied to prove the existence of compact manifolds with SPm-holonomy. There is an intimate relationship between riemannian manifolds with reduced holonomy, calibrations, and spin geometry. This is presented in detail in §10 with special emphasis on the exceptional cases of G2 and Spin? manifolds. In the last section we shall examine the role played by spinors in a proof of the Positive Mass Conjecture in general relativity. 280 IV. APPLICA nONS §1. Integrality Theorems The point of this section is to use the Index Theorem to prove the integrality and divisibility of certain rational characteristic numbers. These results were known before the Index Theorem and, in fact, they provided part of the motivation and insight for its proof. We begin with the following classical result of Atiyah and Hirzebruch [1]. Theorem 1.1. Let X be a compact spin manifold of dimension 4k. Then A(X) is an integer. Furthermore, if dim(X) == 4(mod 8), then A(X) is an even integer. Proof. The first statement follows from the fact that on a spin manifold, the A-genus is the index of the Atiyah-Singer operator. The second state-ment follows similarly from Theorem 6.16 of Chapter II but can be deduced directly as follows. Notice from the classification of Clifford al-gebras (Theorem 1.4.3), that in dimensions 4 (mod 8), the complex spin or representations are actually quaternionic. The connection in the spinor bundle is always such that multiplication by quaternion scalars is paral-lel. Hence, the kernel and cokernel of the Dirac operator are quaternion vector spaces, and their complex dimension is even. _ For manifolds which are not spin, the A-genus is very often not an integer. Recall that for a 4-manifold the signature is always 8 times the A-genus (see II.6.17). Thus, Theorem 1.1 generalizes the following result of Rochlin: Corollary 1.2. The signature of a compact smooth spin 4-manifold is a mul-tiple of 16. It was pointed out in Chapter II (see 2.12ff) that on a spin 4-manifold X, the intersection form is even. That is, x U x == O(mod 2) for all x EO H2(X; 2:'). This implies easily that the signature of X is a multiple of 8 (see Milnor-Husemoller [IJ, p. 242). The importance of smoothness in Rochlin's Theorem was recently underlined by Mike Freedman's profound results [1 J which proved, among other things, the existence of a topo-logical spin 4-manifold of index 8. Another integrality result comes from our discussion ofSpinc-manifolds. Theorem 1.3. Let X be a compact orientable manifold and suppose c EO H2(X; 2:') is a class such that c == w2(X) (mod 2). (Hence, X is a Spinc-manifold.) Then the rational number is an integer. §2. IMMERSIONS AND VECTOR FIELDS 281 Proof. This number is the index of an elliptic operator (see App. D). • Another immediate consequence is the following classical theorem of Bott [1], [2]: -Theorem 1.4. Let E be a complex vector bundle over the 2n-sphere s2n. Then the top Ch ern class of E is divisible by (n -1)!, that is, 1 [ 2 (n _ 1)! cn(E) S n] E 71.. Proof. Consider the twisted signature operator D+ :(Ct+ ® E) -r(Ct-® E) over s2n. Then the index of this operator is ind(D+) = {ch E· L(S2n)} [s2n] = (ch E)[S2n] = chn(E)[S2n] (n 1)! ciE)[S2n], since all the Pontrjagin classes of s2n are zero and since H2k(S2n) = 0 for k "# O,n. Of course, the index is an integer. • This result is useful for many things, among them the study of immer-sions of manifolds into [R". Here, however, there is a direct approach using spin geometry. §2. Immersions of Manifolds and the Vector Field Problem In this section we shall present a unified approach to the following two problems. Let X" be a compact manifold of dimension n and without boundary. I. Find the smallest integer q such that there exists an immersion X" '++ [Rn + q. 11. Find the largest integer q such that there exist q pointwise linearly independent vector fields on X In the first case we always have that q n -1 by the classical work of Whitney. Furthermore, using Stiefel-Whitney classes and the product for-mula w(T)w(N) = 1, one can sometimes find very good lower bounds for q (see App. B, for a discussion of the product formula). Nevertheless, for better results it is sometimes useful to use real characteristic classes or KO-theory. This is the approach we shall take. 282 IV. APPLICA nONS The techniques presented here generalize fundamental ideas of Atiyah [9]. They appeared in Lawson-Michelsohn [1]. A second approach, also using spin geometry, can be found in Mayer [1]. We shall assume throughout that X is a compact oriented manifold of dimension n. For simplicity we let T denote the tangent bundle of X. We begin with the first problem. Suppose that X admits an immersion X <++ IRn+q. Denote by N the normal bundle to this immersion, and note that TtfJN=r (2.1) where r denotes the trivial (n + q)-plane bundle. Let <".) and V be the metric and connection on r induced from euclidean space. Using the de-composition (2.1) we introduce on r a new riemannian connection V, called the projected connection, as follows. For a section <p = (V,v) E r(T tfJ N), we set (2.2) where ( )T,( t denote pointwise orthogonal projection onto T and N respectively. The connection thus induced on T is the canonical riemann-ian one for the induced metric. The difference V -V on r is the second fundamental form of the immersion. We now consider the bundle Ct(r) as a bundle of left-modules over C£(T) under the obvious inclusion C£(T) c C£(r). This makes C£(r), with the connection induced from V on r, into a Dirac bundle over X (cf. 11.4.8). If n = 4k, then we can split the bundle C£(r) via the parallel volume form w = e1 ••• en, and get the elliptic operator D+: r(C£ +(r)) ------. r(C£ -(r)). We shall prove below (Lemma 2.5) that ind(D+) = 2QA(X) (2.3) (2.4) where A(X) = 2n A(X) is the A-genus of X. It is a characteristic number which is always an integer. Notice now that while C£(r) is not in general trivial as a bundle of left C£(r)-modules, it is trivial as an abstract vector bundle. In fact, we can find n + q pointwise orthonormal sections 81"" ,8n+q ofr. These sections generate a finite multiplicative subgroup G == {1} U {8i 1 ••• 8ipL < ... < ip in the algebra of sections r(C£(r)). Evidently, the group algebra IR . G is just the Clifford algebra C£n+q' We now consider the group G acting on C£(r) by right multiplication Ry(<p) = <p . g. This action does not quite commute with the Dirac opera-tor, so we take the average -_ 1 '\' -1 D = -I -I L.. Rg 0 D 0 Rg G gEG (2.5) §2. IMMERSIONS AND VECTOR FIELDS 283 and note that for all g E G. (2.6) One can easily see that each commutator D 0 Rg -Rg 0 D is a differ-ential operator of order zero. Hence, the averaged operator jj has the same principal symbol as D. Of course right multiplication commutes with left multiplication, so the bundles C£ ±(r) == (1 ± w)C£(r) are clearly G-invariant. It follows that jj is also of the form -(0 D = jj+ with respect to the splitting C£(r) = C£ +(r) EB C£ -(r), and the operator jj+ has the same symbol as D+. From the topological invariance of the index we conclude that ind(jj+) = ind(D+) The main step now is to observe that since jj+ commutes with G, the kernel and cokernel of jj+ are !RG C£n+q-modules, and so the index of D+ is divisible by a large power of 2. There is one further observation which sharpens the result. Recall that C£(r) carries a 2:'z-grading C£(r) = C£o(r) EB C£l(r) determined as the ± l-eigenbundles of the parallel automorphism IY. (which extends v H -V on r). This grading clearly carries over to C£ +(r) and Cr(r), and makes each of these a bundle of 2:'z-graded modules under right Clifford multiplication. Furthermore, since IY.(D<p) = -D(IY.<p), for any <p E r(C£(r)), we see that ker D+ and ker D-are each 2:'z-graded under right multi-plication by !RG = C£n + q' This proves the following: Theorem 2.1. Let X be a compact oriented manifold of dimension n = 4k. If there exists an immersion X <B !Rn + q, then 2q -1 A(X) == O(mod an +q) where 2an+q is the dimension of an irreducible real2:'2-graded C£n+q-module. See the table below for values of an+q• Note that by Proposition 1.5.20, an+q is the dimension of an irreducible, real ungraded module over C£n+q-l' As a quick illustration we note that A([pz(C)) = 2 and that 2q == 0 (mod a4+q) only for q 3. Hence, [pz(C) immerses only in codimension 3, the range guaranteed by the Whitney Immersion Theorem. Moreover, since the A-genus is an oriented cobordism invariant, the conclusion is much more general. In dimension four, A(X) = 2 sig(X) and we have the 284 following: IV. APPLICATIONS Corollary 2.2. If a compact oriented 4-manifold X can be immersed into [R6, then the signature of X is even. These results can be somewhat strengthened by two tricks. The first trick allows us to deal with embeddings. Note that C£(r) = C£(T EB N) = C£(T) ® C£(N) where ® denotes the graded tensor product of Clifford bundles. Clearly we have C£ ±(r) = [(1 ± w)C£(T)] ® C£(N) = C£ ±(T) ® C£(N). Suppose now that the dimension of the normal bundle q == 0 (mod 4), and let WN = e1 •.. eq be the normal volume element. Then WN is parallel in the projected connection and commutes with all tangent vectors. Hence, wN commutes with the Dirac operator D. We can define bundles C£ ± ±(r) = C£ ±(T) ® C£ ±(N) = (l ± w)C£(T) ® (1 ± wN)C£(N) (2.7) and we get Dirac operators (2.8) where "." can be either + or -. Evidently, D+ = D+ + EB D+ -, and so ind(D+) = ind(D+ +) + ind(D+ -). (2.9) Furthermore, we shall prove below the following fact: Lemma 2.3. If the Euler class X(N) of the normal bundle is zero, then (2.10) Carrying out the averaging process above, we may replace all D's by ih. (Note that each of C£ ± ±(r) is invariant under right-multiplication by C£(r), and in particular by G.) It is an elementary fact that for an embedded submanifold, X(N) = 0 (see Milnor-Stasheff [1 ]). Consequently, we have the following: Theorem 2.4. Let X be as in Theorem 2.1 and suppose q == 0 (mod4). If there exists an embedding X '-+ [Rn+q, then 2q-2A(X) == 0 (mod an+q). The same conclusion holds for any immersion X <++ Rn+q for which the Euler class of the normal bundle is zero. Our second trick is to simply take coefficients in a vector bundle over X. This is technically quite easy and has the effect of introducing all of the rational K-theory of the space into the problem. We get conditions which are not simply in terms of characteristic numbers, but essentially involve the entire co homology ring. We proceed as follows. §2. IMMERSIONS AND VECTOR FIELDS 285 Let E be a vector bundle over X. Here E may be real, complex or quaternionic, and we assume it is equipped with an appropriate inner product and connection. Then we take r = T EB N as above and consider the bundle C€ (r) ® E with the tensor product connection. This remains a bundle of right and left modules over X. It is easy to check that all of the constructions above can be carried out with C€ (r) replaced by C€ (r) ® E. We need only compute the index of D +. We shall state the result here, and give the proof below. Lemma 2.5. The index of D+:r(Ce+(r) ® E) ---> qcr(r) ® E) is given by the formula (2.10) where A(X) is the total A-class of X, and where ch2E is defined as follows. Let ch E = 1 + chI E + ch2E + ... E HH(X; Q), denote the Chern charac-ter of E. Then for any t E [R;, cheE == L (chkE)tk k;,: 0 In the case that X(N) = 0, = 1 ind(D+) where ind(D +) is given by formula (2.10). (2.11 ) (2.12) We note that when E is real, we set ch E = ch(E ® q, and when Eis quaternionic, we set ch E = ch([ E],d where [-],c denotes restriction of scalars from IHl to C. To state our results we shall need the following table of numbers which is easily computed from the data given in Chapter I, §5. Let 2ak, 2bk and Ck denote the dimension over K of an irreducible 2:'2-graded K-module for cek where K = [R;, C and IHl respectively. Note that by Proposition 1.5.20, the numbers ak, hk and 1Ck correspond to the dimensions of irreducible ungraded modules for I' For all values of k we have that ak+8 = 16ak, hk+8 = 16hk, Ck+8 = 16ck· The values for k 8 are given in the following table. n I 2 3 4 5 6 7 8 an 1 2 4 4 8 8 8 8 hn 1 1 2 2 4 4 8 8 cn 2 2 2 2 4 8 16 16 286 IV. APPLICATIONS The method given above proves the following general result: Theorem 2.6. Let X be a compact orientable manifold of dimension n, and suppose there exists an immersion X <f+ [p;n+q. Thenfor any complex bundle E over X, Furthermore, if q is even and the normal Euler class X(N) = 0, then 2q-2{ch2E . A(X)} == ° (mod bn+q). (2.13) (2.14) When n == ° (mod 4), there are the following refinements: If E = ER ® C for a real bundle ER, or if E = [EH1:!or a quaternionic bundle EH' then (2.13) holds with bn+q replaced by an +q and cn+q respectively. If, furthermore, q == ° (mod 4) and X(N) = 0, the analogous improvements can be made in (2.14). Proof. Again the argument goes as follows. Let"C = T EB N and consider the Dirac bundle C£("C) ® E. Average the Dirac operator over the group in r(C£("C)) generated by the global orthonormal sections el' ... ,en+q of "C. Then the kernel and co kernel of the operator are Z2-graded modules over the algebra C£n+q, generated by el, ... ,en+q. (The Zz-grading comes from the even-odd grading of C£("C). In the complex case there are volume forms in every even dimension: Wc = ime1 ... e2m)' Applying the formula from Lemma 2.5 gives the first part of the theorem. Splitting the operator with the normal volume form and applying (2.12) gives the second part of the theorem. _ We point out that the improvements in this method obtained by intro-ducing coefficients are substantial. For example, they give results in every even dimension. The congruences (2.13) and (2.14) give a relatively simple machine for calculating lower bounds on embedding and immersing dimensions. For the complex and quaternionic projective spaces. and these methods give the best results of this kind for every n (see Lawson-Michelsohn [1] for details). Similar non-immersion theorems were found by K. H. Mayer [1]. There were earlier, slightly weaker results of this type due to Atiyah and Hirzebruch [1], whose methods did not involve elliptic operators. Proof of Lemma 2.5. We note that the operator in question is just the twisted signature operator (cf. 11.6). More precisely, since C£ ±(T EB N) = Ct ±(T) ® Ct(N), we see that we can write D+ as D+ :r(Ct+(T) ® E) -r(Ct-(T) ® E) §2. IMMERSIONS AND VECTOR FIELDS where E == E ® C£(N). It follows that ind(D+) = {ch2E' L(X)} [X], 287 (2.15) from the basic formula for the signature complex (see 111.13.9). Now ch2E = ch2(E ® C£(N)) = ch2(E)chiC£(N)), and since T EB N = r = [R;n+q, we see that C£(T)C£(N) = Cln+q = R2n+q (trivial bundles). It follows that (2.16) To compute this last expression, we apply the Splitting Principle and express the total Chern class of T ® C formally as n/2 c(T® q = n (1 + xk)(1 -xk) k= 1 so that the Pontrjagin classes are Pk(T) = O"bi, ... ,X;/2) where O"k denotes the kth elementary symmetric function. In this language, we have n/2 X L(X) = n k k= 1 tanh(xd Recall that C£(T) ® C At(T ® q and that therefore n/2 ch(C£(T)) = n (l + e-Xk)(1 + eXk) k=! n/2 = 2n n COSh2(Xk/2). k=l Consequently, and for any t E [R;, n/2 chlC£(N)) = 2q n cosh -2(Xkt/2). k= ! It follows that n/2 X chiC£(N))L(X) = 2q n . h( k h( ) k= 1 SIn xdcos xk n/2 2x = 2q n k k = 1 sinh(2x k) == 2qA(X), 288 IV. APPLICATIONS since the A-class is the multiplicative series given by the formal power series 2x/sinh(2x). Formula (2.10) now follows immediately from (2.15). For the second half of the lemma we make the following observation. Let E be a complex vector bundle with dimcE = 2m. Write c(E Q9[J;l q = n (1 -xf) so that Pk(E) = O'k(xi, ... ,x!) as above. Then we have 2m ch(Ct + E) -ch(Ct -E) = n (eXk -e-Xk) k=l = 2mx1 ••• Xm fi {_eXk--::-__ e-_Xk} k= 1 2Xk = 2mX(E)IX(E) where IX is a multiplicative sequence. It follows that ind(D+ +) -ind(D+ -) = {(ch2Ct + N -ch2CC-N)' ch2E' L(X)}[X] = {X(N)' C}[X] for some class C E Heven(X;Q). Hence, if X(N) = 0, we have ind(D+ +) = ind(D + -) = Mnd(D+) (cf. (2.9)). This completes the proof. • Using the techniques developed above, we shall now analyze the second problem mentioned at the outset of this section. Suppose our manifold X carries q pointwise linearly independent vector fields e1, ••• ,eq• We may assume el' ... ,eq to be pointwise orthonormal with respect to a riemannian metric on X. Let us suppose for the moment that X is of dimension n = 4k (and is orientable). Then we have the signature operator D+: r(Ct +(X) --. r(cr X). As above, we may average D over the finite group G c r(Ct X) generated by el' ... ,eq• That is, we set -_ 1 " -1 D = -IlL... Rg 0 D 0 Rg G geG where Rg denotes right Clifford multiplication by g. This multiplication clearly preserves the sub-bundles Ct ±(T) = (1 ± w)Ct(T), and we achieve an operator jj+: r(Ct+(T) --. r(Cr(T)) which is G-equivariant and has the same symbol as D +. As above we find that the kernel and cokernel of jj+ are Z2-graded modules over the Clifford algebra Ctq IRG. Since ind(jj+) = ind(D+) = sig(X), we have the following: Theorem 2.7 (Atiyah [9] Frank [1 ]). Let X be a compact oriented manifold of dimension 4k. If X carries q pointwise linearly independent vector fields, then sig(X) == 0 (mod 2aq) If, furthermore, q == 0 (mod 4), then sig(X) == 0 (mod 4aq). §2. IMMERSIONS AND VECTOR FIELDS 289 As an example, note that if X carries 3 such vector fields, then sig(X) == o (mod 8). If it carries 5, then sig(X) == 0 (mod 16). In general, if X carries 2m such vector fields, then sig(X) == 0 (mod 2m). Proof of Theorem 2.7. The first statement follows from the argument above. For the second statement, note that there is an orthogonal split-ting T(X) = To EB Tl where Tl == span(el' ... ,eq). Let w be the oriented volume element of To. Since dim To == 0 (mod 4), we have that WZ = 1 and w commutes with el' ... ,eq• By averaging !?fi as above, over the group 7Lz generated by Rw, we may assume 15 commutes with Rw' Since = 1, we see that Rw has ± 1 eigenbundles on each of et + and CC. They are given by et+± = et+(l ± w) and ec± = CC(1 ± w). Each bundle et ± ± is G-invaiiant, and we obtain two G-equivariant ellip-tic operators and with the property that 15+ = 15++ EB 15+-. We know, therefore, that ind(15+) = ind(15+ +) + ind(15+ -). A straight-forward computation, in the spirit of that given for the second part of Lemma 2.5, shows that ind(15++) -ind(15+-) = 2q/zX(To)[X] = O. Thus, ind(15+ +) = ind(15+ -) =! ind(15+) and since the kernel and cokernel of 15+ + are 7Lz-graded Ctq-modules, the result follows as before. • When Atiyah first presented these arguments (in [9]), he mentioned that they could not achieve the additional power of 2 (when q == 0(4)) obtainable by other means. However, the last argument given above suc-ceeds in getting the final 2 and gives the best known results of this type. Theorem 2.7 is very pleasant because Isig(X)I is a homotopy invariant. It is also an oriented cobordism invariant. However, for any given mani-fold which carries a q-frame field, there is much more information avail-able. For this we simply need to take coefficients in a vector bundle E. Applying the arguments given above directly to Ct(X) ® E proves the following result (cf. Lawson-Michelsohn [1]). Theorem 2.8. Let X be a compact oriented manifold of even dimension, and suppose that X carries q pointwise linearly independent vector fields. Then for every complex vector bundle E over X, {chzE' L(X)}[X] == 0 (mod 2bq) (2.16) 290 IV. APPLICA nONS If, furthermore, q is even and chqj2 E = 0, then {ch2E . L(X)}[X] == ° (mod 4bq) (2.17) When dim(X) = 4k, there are the following refinements. If E = ER ® C for a real bundle ER, or ifE = [EH]cfor a quaternion bundle EH' then (2.16) holds with bq replaced by aq or cq respectively. If, furthermore, q == ° (mod 4) and chqj2 E = 0, the analogous improvement can be made in (2.17). Proof. The only point which needs attention here is the proof that ind(i) + +) = ind(i) + -) when chqj2E = 0. This follows from the equation ind(i) + +) -ind(i) + -) = 2qj2{ch E· X(To)}[X], which can be verified by straightforward calculation. • It is an interesting observation of Atiyah [9] that all of the above carries over without change to prove the following generalization of Theorems 2.7 and 2.8. Theorem 2.9. Let X be a compact oriented manifold whose tangent bundle can be decomposed into oriented subbundles where for k = 1, ... ,q. Then all the conclusions of Theorems 2.7 and 2.8 hold for X. Proof. Let ek denote the oriented volume element for the plane field Tk, k = 1, ... ,q. Since these planes are mutually orthogonal and each has dimension one (mod 4), the elements el"" ,eq E qct(X)) generate a finite group G with IR· G = Ctq• The arguments now proceed exactly as before. • As a final application of these methods we consider the refined Clifford index for Ctk-Dirac bundles introduced in §7 of Chapter n. Let X be a compact oriented manifold of dimension 4k + 1. Recall that the Kervaire semi-characteristic of X is then defined to be the residue class 2k u(X) == L b2J{X) (mod 2) j=l == dim Heveo(x) (mod 2) == h(X) (mod 2). Theorem 2.10 (Atiyah [9]). Let X be a compact oriented (4k + I)-manifold. If X carries an oriented tangent p-plane field for some p == 2 or 3 (mod 4), then u(X) = 0. §3. GROUP ACTIONS 291 Proof. By assumption there exists a splitting T(X) = To EB T1 where dim(To) == 2 (mod 4), and where To and T1 are oriented. This decomposi-tion can be assumed orthogonal for some riemannian metric on X. Let Wo be the oriented volume form for To, and note that w6 == -1. We consider now the Euler characteristic operator DO: qCtO(X)) --. qCt l(X)). Since dim(To) is even, each Ctk(X) is invariant under right multiplication by Wo. Averaging the Dirac operator over the group Z4 generated by Rwo makes ker DO an Rwo -invariant space, i.e., ker DO be-comes a ct1 = 1:::-module. In particular, ker(Do) Heven EBk Hlk(X;IR) has even dimension (over IR), and so O'(X) = o. • §3. Group Actions on Manifolds This section is concerned with smooth group actions. Recall that a Lie grou p G is said to act effectively on a manifold X if there is a differentiable map <l>: G x X --. X so that the corresponding map cp(g) == <l>(g,') gives a continuous and injective homomorphism cp : G '-+ Diff(X). Here Diff(X) denotes the group of C"-diffeomorphisms of X in the standard COO topology. There are two basic groups we shall consider here, namely, Sl == {z E C : Izl = I} S3 == {q E IHl : Iql = I} (3.1) (3.2) where multiplication is induced from multiplication in the field. There are isomorphisms Sl SOl U 1 and S3 Spin3 SU 1 SP1' These two groups are basic for the following reason. We say that a Lie group G has an S" subgroup (for k = 1 or 3) if G contains a Lie subgroup isomorphic to a finite quotient of Sk. (The only possibilities are Sl when k = 1 and S3 or S3/Zl when k = 3.) Proposition 3.1. Let G be a compact connected Lie group (of dimension> 0). Then G contains an Sl-subgroup. In fact the Sl-subgroups are dense in G. Moreover, if G is non-abelian, then it contains an S3-subgroup. Proof. These statements follow directly from the elementary theory of compact Lie groups (see Adams [1 ]). To prove the first, one notes that every element of G is contained in a maximal torus, where Sl-subgroups are dense. To prove the last statement one takes the subgroup correspond-ing to the subalgebra spanned by a root space and a root vector. • DEFINITION 3.2. We say that a manifold X admits an S"-action (k = 1 or 3) if there exists an effective action of a finite quotient of Sk on X. 292 IV. APPLlCA nONS Two main results concerning connected group actions on spin mani-folds are the following: Theorem 3.3 (Atiyah and Hirzebruch [3]). Let X be a compact spin mani-fold which admits an Sl-action. Then A(X) = o. Theorem 3.4 (Laws on and Yau [1]). Let X be a compact spin manifold which admits an S3-action. Then .si(X) = O. Recall that .si is the generalization of A to KO-theory given in (11.3.22). In dimensions 4k, the two invariants are essentially the same. However, in dimensions 1 and 2 (mod 8), the invariant .si takes values in 7Lz and can be non-trivial while, of course, A = o. REMARK 3.5. The stronger hypothesis of Theorem 3.4 is required for its stronger conclusion. We can see this as follows. Let XB be a compact spin 8-manifold with A(XB) = 1 (see the paragraph before 11.7.12). Equip Sl with the interesting spin structure (cf. 11.7.8), and take the product Y = X X Sl with the product spin structure. Then Y admits an Sl-action, but .si(y) # o. The two theorems above could be combined into the following state-ment by using Proposition 3.1. Theorem 3.6 Let X be a compact spin n-manifold with .si(X) # O. If n == 1 or 2 (mod 8), then the only compact, connected Lie transformation groups of X are tori. If n == 0 (mod 4), then the only such group is the trivial one. As we have noted before (cf. 11.2.18), the spin manifolds with non-zero .si constitute half of the exotic spheres in dimensions 1 and 2 (mod 8). This gives the following: Corollary 3.6. Let be an exotic n-sphere which does not bound a spin manifold (n == 1 or 2 (mod 8)). Then the only compact connected Lie sym-metry groups of are tori of dimension [!(n + 1)]. Proof. The kernel of the surjective homomorphism .si: 0n --. 7Lz, for n == 1,2 (mod 8), is the subgroup of homotopy spheres which do not bound spin manifolds. This follows from the work of Milnor [7] and Adams [3]. If n is even, any toral transformation group Tk must have a fixed point set, and the induced linear action on the normal spaces must be effective. Therefore, 2k n. If n is odd and the fixed-point set is non-empty, a similar argument applies. For the general case, see Ku [1]. • Using other techniques, people have investigated the non-existence of torus actions on exotic spheres. For example, using results of R. Schultz §3. GROUP ACTIONS 293 [1] it can be shown that there are three exotic lO-spheres whose largest connected transformation group is Si or Si x Si. These spheres are indeed very unsymmetric! It is interesting to note that this asymmetry is contagious. The .s:i-invariant is additive with respect to the connected sum operation #. Thus, if X is a spin n-manifold with .s:i(X) = 0 and if L is an exotic n-sphere as above (n == 1 or 2 (mod 8)), then .s:i(X # L) = .s:i(L) #-0 and X # L only admits symmetry groups of toral type. This is particu-larly striking when X = GIB is homogeneous. The manifold (GIB) # L is homeomorphic to GIB, but has almost no symmetries. A good example is the complex projective space X = [p>4k + 1((:). Another interesting phenomenon takes place under coverings. Fix an exotic sphere L" as above, and let d be its order in the group 0" of homotopy spheres. Consider the homogeneous space X = (S3/7Ld) x S"-3. The manifold Y = X # L" is quite inhomogeneous, however, its universal (d-fold) covering space is y = (S3 X S"-3) # (L" # ... # L") = S3 X S"-3, d times a symmetric space. This instability under finite coverings is not true of the A-genus. Proof of Theorem 3.3. The argument is based on the Atiyah-Segal-Singer Fixed-Point Theorem or, more specifically, the G-Spin Theorem (111.14.11). It proceeds as follows. Let X be an even-dimensional compact spin mani-fold with an Si-action. We may assume by averaging that the action pre-serves the riemannian metric. By passing, if necessary, to a 2-fold covering group, we may assume that the Si-action preserves the spin structure, i.e., that it lifts to an action on the bundle PSPin(X). This produces an Si-action on the canonical complex spinor bundle $e, which projects to the given one on X. This action commutes with the Atiyah-Singer operator J/J, i.e., g 0 J/J 0 g -1 = J/J for all g E Si. The volume form is Si-invariant, and so the splitting $e = Et> $e is preserved. Thus the operator J/J+: --. q$e) becomes an Si-operator, and we have defined an Si-index: (3.3) where R(Sl) denotes the representation ring of Si. There is a natural isomorphism of R(Sl) with the Laurent polynomial ring (3.4) obtained by taking the trace of the representation. Here t = ei6 corre-sponds to the complex-valued function, or "character," on Si given by 294 IV. APPLICATIONS the natural inclusion SI c C. Under the isomorphism (3.4) the index defined in (3.3) becomes a Laurent polynomial which we denote by Note that for any g E SI, we have iJp+ (g) = tr(glker Jp+) -tr(glker (3.5) Now the Theorem (III.14.11) gives a formula for iJp+(g) of the form iJp+(g) = where Fg is the set of g. For almost all g E Si, in fact, for all non-torsion elements, the subgroup generated by g is dense. In this case F 9 = F = the fixed-point set of SI, and the expression varies only with "normal rotation angles" as follows. The normal bundle to F has an SI_ invariant decomposition N = EBmE2 N m where each N m carries the struc-ture of a complex bundle and where SI acts on N m by scalar multiplication by gm. For any given k-dimensional complex vector bundle E we define the characteristic function 1 1 k (2eFJ = j=l teX) -I where t is taken to be an indeterminate and where. as usual, the x/s arc formal roots of the total Chern class of E. Because of the square roots, the expression (3.6) is defined only up to sign. Now for Y E Si C IC a non-torsion elemcnt as above, the G-Spin Theo-rem states that where the signs on each component of F are determined by the action of Si on the spin structurc. It follows that we have an equality of rational functions §3. GROUP ACTIONS 295 From (3.6) it is evident that the expression of the right is zero for t = ° and t = 00. Hence, the Laurent polynomial i1.>+(t) must vanish identically. In particular, from (3.5) we have that i1.>+(1) = dim(ker 1/)+) -dim(ker 1/)-) = A(X) = 0. This completes the proof. • The above argument actually proves the following stronger result. Theorem 3.7 (Atiyah-Hirzebruch [3]). Let X be a compact spin manifold of even dimension. For every non-trivial SI-action which preserves the spin structure on X, the index This result on SI-actions has an interesting refinement observed by P. Landweber, R. Stong and S. Weinberger. To state it we must differentiate between actions of even and odd type. An SI-action on a spin manifold X is said to be of even type if it lifts to an action on PSPin(X). Otherwise it is said to be of odd type. (In the latter case the action of the 2-fold covering group lifts to PSPin(X).) These notions can be reformulated by considering a lifting of the element -1 E SI cC to PSPin(X). The action is of even type if and only if -1 has order 2 on PSPin(X). The action is of odd type if and only if -1 has order 4 on PSpin(X). Suppose that the element -1 acts freely on X. Then one sees easily that the SI-action is even if and only if the quotient XjZ2 is a spin manifold. If the fixed-point set X71.2 of -1 is not empty, then • 71.2 _ {o (mod 4) codlm(X ) = 2 (mod 4) if the action is even, if the action is odd. (3.7) We see this as follows. Recall that - 1 acts on its normal bundle N by scalar multiplication in the fibres. Since the action of -1 is orientation-preserving (it is connected to the identity in Diff(X)), the rank of N is even. Furthermore, there is a Z2-equivariant diffeomorphism of N with a tubular neighborhood of X71.2 in X. From this we see immediately that the action on X is even if and only if the action on the normal sphere bundle to X71.2 (with the induced spin structure) is even. Here the Z2-action is free and bi the paragraph above the question comes down to whether the quotient of the normal sphere is a spin manifold. By Remark 11.2.4 we know that real projective (n -I)-space is spin if and only if n == o (mod 4). This proves (3.7). • Theorem 3.8 (Landweber and Stong [1]). Let X be a compact spin mani-fold of dimension 4k with an SI-action. If the action is of odd type, then sig(X) = 0 296 IV. APPLICATIONS Proof. A result of Hattori and Taniguchi [1 J states that for any SI-action on an oriented manifold, sig(X) = sig(XSI) where XSI is the fixed-point set of the action with an appropriately chosen orientation. Consider now the inclusions: XS1 c X",2 eX, and note that (X",2( = XS1. By a result of Edmonds [lJ, the manifold X",2 is orientable. Therefore, applying the argument again shows that sig(X) = sig(X",2). However, by (3.7) we have that dim(X",2) == 2 (mod 4), and so sig(X",2) = O. • This result is analogous to the fact that on manifolds which admit orientation reversing diffeomorphisms, all the Pontrjagin numbers are zero. For Si-actions on spin manifolds which are free outside the fixed-point set, there are additional characteristic classes which vanish (see Landweber and Stong [lJ). This leads into the fascinating area of elliptic genera (cf. Chudnovsky and Chudnovsky [1 J). Pro()f of Theorem 3.4. Recall that any spin manifold X which carries a metric of positive scalar curvature has .J#(X) = 0 (see 11.8.12). Thus Theo-rem 3.4 is a corollary of the following more general result. • Theorem 3.9 (Lawson and Yau [IJ). Any compact spin manifold which ad-mits a non-trivial S3-action, carries a riemannian metric of positive scalar curvature. When the action is free, the proof of this theorem is rather easy. In this case, the manifold is the total space of a smooth principal S3-bundle X M = X/S3. Introduce an invariant splitting TX = 1/" EEl :Ye where 'P" denotes the field of 3-planes tangent to the orbits. The bundlej/" has a natural trivialization j/" X X [R3 given as usual by choosing an ortho-normal basis el,e2,e3 of TdS3) and defining sections Ej of f by EAx:) = d/dt (exp(tej)' x)t=o' We fix a metric on M and for each I: > 0 we construct a metric g, on X as follows. We declare f and :Ye to be everywhere orthogonal; we lift the given metric on M to:Ye via n; and we set (Ei' E) == r,2()ij. This uniquely determines the metric ge In this metric the orbits of S3 are all totally geodesic and isometric to the euclidean sphere of radius 1:. Applying the fundamental equations of O'Neill [1 J for a riemannian submersion, one easily sees that the scalar curvature K, of the metric y, is of the form KJX) = 6/1:2 + a(x) + 1:2b(x) for continuous functions a and h on X. Taking r, sufficiently small completes the proof for this case. The general case, where the action of S3 is not free, is more difficult and is handled as follows. Consider the diagonal action of S3 on X x S3. This action is free with quotient X. (Of course, this orbit space map §4. COMPACT MANIFOLDS WITH K > 0 297 x X S3 X is different from the "product" projection pr: X x S3 X.) We now fix an S3-invariant metric of X, and using this metric we con-struct, for each t: > 0, a metric ge on X x S3 in the manner above. The metric ge is, in fact, S3 x S3-invariant and thus makes the product pro-jection pr: X x S3 X into a riemannian submersion. Detailed calcula-tions show that for t: sufficiently small, this submersed metric has positive scalar curvature. The estimates in this calculation are most delicate at the fixed point set. Theorem 3.9 actually provides a wealth of non-existence theorems for S3-actions. We clearly have the following: Corollary 3.10. Any compact manifold which cannot carry a riemannian metric of positive scalar curvature has no S3-actions. In the next section we shall produce large families of manifolds which do not carry positive scalar curvature. One consequence of this will be a proof of the fact (due originally to Browder and Hsiang [1]) that if a spin manifold X admits an S3-action, then all the "higher A-genera" of X are zero. §4. Compact Manifolds of Positive Scalar Curvature One of the most important applications of the Atiyah-Singer operator in riemannian geometry is in the study of manifolds of positive scalar cur-vature. Recall that the scalar curvature of a riemannian manifold X is a function K:X IR defined at each point x by averaging all the sectional curvatures at x. Specifically we define n K(x) = I <Re"ej(ej), e) (4.1) i,j= 1 where {el' ... ,en} is an orthonormal basis of TxX and R is the Riemann curvature tensor. Note that K can also be written as K(X) = trace(Ricx) (4.2) where Ricx denotes the Ricci transformation at x. Thus, the condition of positive scalar curvature, i.e., K > 0, is the weakest of the various "positive curvature" assumptions. There has already been some discussion of manifolds with K > ° given in §8 of Chapter 11. We recall the basic results proved there. There is a graded ring homomorphism (4.3) which in dimensions 4k is a multiple of the A-genus. This homomorphism is defined in (3.22) of Chapter If. It coincides with the KO-index of the 298 IV. APPLICATIONS Atiyah-Singer operator, and can be computed in terms of the space of harmonic spinors (see Theorem 11.7.10). Applying the Bochner-type for-mula (11.8.18) gives the following result of Atiyah, Hitchin, Lichnerowicz and Singer. Theorem 4.1. Let X be a compact spin manifold with K > 0, then d(X) = o. This theorem implies, in particular, that a spin 4k-manifold with non-vanishing A-genus cannot carry a metric of positive scalar curvature. There are many examples of such manifolds. For example, let V2k(d) denote a non-singular complex hypersurface of degree d in [jJ>2k+ 1 (C). Then V2k(d) is spin if and only if d is even. (See 11.2.7.) However, we have that 2k 1 nk . A(V (d» = 22k(2k + I)! j= -k (d -2), (4.4) so that for all even d > 2k, and all k 1, the 4k-dimensional spin manifold V2k(d) cannot carry K > 0, and therefore cannot carry metrics of positive Ricci or positive sectional curvature. It is a consequence of the Calabi Conjecture, proved by S. T. Yau [2], that each of the hypersurfaces V2k(d) for d ;;:;; 2k + 1 carries a metric of positive Ricci curvature. This shows that the spin condition in Theorem 4.1 is necessary. It also shows, together with Theorem 4.1, that the poly-nomial p(d) == A(V2k(d» must have zeros for d = 2,4, ... ,2k.1t is interesting that with this information and some general considerations, it is rather easy to compute formula (4.4) for p(d). An important phenomenon occurs in the borderline case V2k(2k + 2) where Yau proves that there exist metrics of zero Ricci curvature. In this case we see that A = 2. It follows from Corollary 11.8.10, that in any Ricci flat metric, V2k(2k + 2) carries at least two parallel spinor fields. Such fields automatically reduce the holonomy group of the metric. We shall pursue these considerations further in §10. We now return to the discus-sion of Theorem 4.1. Recall that the invariant d is non-zero also in dimensions 1 and 2 (mod 8). Here it takes values in 71.2, Among the spin manifolds on which it is non-zero, are half of the exotic spheres in dimensions 1 and 2 (mod 8). (A general discussion of positive curvature metrics on exotic spheres is given in 11.8.) This fact has the following consequence: Corollary 4.2. In dimensions 1 and 2 (mod 8), every compact spin manifold is homeomorphic to a manifold which cannot carry positive scalar curvature. Proof. Let X be a spin k-manifold with k = 1 or 2 (mod 8), and let L be an exotic k-sphere with d(L) #-O. Since d(X # L) = d(X) + d(L), we may apply Theorem 4.1 to either X or X # L. • §4. COMPACT MANIFOLDS WITH K > 0 299 To keep these non-existence theorems in perspective It IS useful to examine manifolds which are known to carry positive scalar curvature. The classical examples of spaces with curvature 0 are the homogeneous spaces with normal metrics given as follows. Let G be a compact Lie group with Lie algebra g, and let (".) be any AdG-invariant inner prod-uct of g. Then for any closed subgroup He G, there is a G-invariant metric naturally introduced on the coset space X = G/ H. The sectional curvatures of this metric are always 0, and apart from the case of the flat torus, the scalar curvature is always> O. (For details, see Kobayashi-Nomizu [2] or Cheeger-Ebin [1].) These homogeneous spaces represent special cases of the general pheno-menon of Theorem 3.8. Any compact manifold which admits a compact connected non-abelian Lie transformation group, carries a metric with K >0. There is a general surgery procedure for constructing metrics with K > O. Recall (cf. Milnor [8]) that a surgery of codimension q on an n-manifold X is a modification of X of the following type. Let Ln-q c X be a smoothly embedded (n -q)-sphere with a trivialized tubular neighborhood. By this we mean a neighborhood U ofLn-q together with a diffeomorphism f: U --> sn-q x Dq such thatf(Ln-q) = sn-q x {O}. The surgery operation now consists of removing the neighborhood U sn-q x Dq and replacing it with the product Dn-q+ 1 X Sq-1 by gluing in the obvious, canonical way along the boundary sn -q X sq -1. Proposition 4.3 (Gromov-Lawson [2] and Schoen-Yau [2]). Let X be a manifold which carries a metric of positive scalar curvature. Then any manifold obtained from X by surgery in codimension 3, also carries a metric of positive scalar curvature. The method of proof in Gromov-Lawson [2] is an elementary con-struction, the proof in Schoen-Yau [2] uses techniques of partial differ-ential equations. Combining this result with techniques of the h-cobordism Theorem (cf. Milnor [8]) and facts concerning the cobordism ring it is possible to prove the following result: Theorem 4.4 (Gromov-Lawson [2]). Let X be a compact simply-connected manifold of dimension 5. (A) If X is not spin, then X carries a metric with K > O. (B) If X is spin, and is spin cobordant to a manifold with K > 0, then X carries a metric with K > O. Remark. Contrasting part (A) of this theorem with Corollary 4.2 makes it evident that the question of positive scalar curvature is intimately related to spin geometry. This relationship will soon be examined in detail. 300 IV. APPLICATIONS Proof of Theorem 4.4. Suppose X is spin and spin cobordant to a mani-fold X 0 which carries K > O. By a sequence of surgeries (on embedded circles) we can make X 0 simply-connected while preserving the spin corbordism class. By Proposition 4.3, the new X 0 will also carry K > O. Our cobordism assumption means that there exists a compact spin manifold W with aw X 1I X 0 (as spin manifolds). As above, we can kill 1!l(W) by surgery. Moreover, since dim W = n ;:;: 6 and W is spin, every 2-sphere embedded in W has a trivial normal bundle (cf. 11.2.11), and we can kill 1!2(W) by surgery. It follows that the groups Hn-k(W,XO) Hk(w,X) are zero for k = 1,2, and torsion-free for k = 3. By the basic theory of S. Smale (cf. Milnor [8]), there exists a smooth function f: W--> [0,1] with the following properties: (i) flxo == 0 and fix == 1, (ii) all critical points of f are non-degenerate and lie in the interior of W, (iii) all critical points of f have index ;£ n -3. It now follows from the fundamental theorem of Morse Theory (cf. Milnor [6]) that X can be obtained from X 0 by surgeries in codimension ;:;: 3. We then apply Proposition 4.3. When X is not spin, the argument is similar. From detailed knowledge of the oriented cobordism ring we know that there exists a compact oriented manifold W with aw = XII Xo, where Xo carries K > O. We may assume that 1!l(XO) = 0 as before. We now proceed to kill1!l(W) by surgery. We then have H2(W) 1!2(W), and the second Stiefel-Whitney class gives a homomorphism w2: 1!2(W) ---+ 71.2, (4.5) Since dim W;:;: 6, the group 1!2(W) is generated by embedded 2-spheres, and W2 exactly measures the non-triviality of the normal bundle to these embeddings. Thus, we can kill the subgroup ker(w2) C 1!2(W), The homo-morphism (4.5) then becomes injective. Observe now that W2 restricts to be the second Stiefel-Whitney class of X, which is non-zero since X is not spin. It follows that (4.5) is an isomorphism and that the map H 2(X 0) 1!2(X 0) --> 1!2(W) H 2(W) is surjective. In particular, H k(W,X 0) = 0 for k = 1,2. Consequently Hk(W,X) = 0 for k = 1,2, and H\W,X) is torsion free. The argument now proceeds as before. _ Let X be a compact manifold of dimension ;:;: 5. Theorem 4.4 states that there is an obstruction to the existence of a positive scalar curvature metric on X only if X is spin, and, furthermore, that obstruction depends only on the spin cobordism class of X. This leads us to consider the set 'lJn of spin cobordism classes IX E with the property that some manifold in IX carries positive scalar curvature. We claim that 'lJ* == EB 'lJn is an ideal in §4. COMPACT MANIFOLDS WITH K > 0 301 Closure under addition (which corresponds to disjoint union) is clear. Suppose now that IX E \.l3n and choose any class f3 E Represent IX by a spin manifold X with a metric gx with Kx > O. Represent f3 by a spin manifold Y with some metric yy. Then for all e > 0, the product metric g = egx + gy on X x Y has scalar curvature K = e -1 Kx + Ky which is >0 for all sufficiently small e. This shows that f3. IX E \.l3n+m, and \.l3* is an ideal as claimed. Consequently, setting {o* == we get a graded ring homo-morphism (4.6) Note that in each dimension n, {on is a finitely generated group. Theorem 4.4 states the following. Corollary 4.5. The classes 0* constitute a complete set of invariants for de-termining whether or not a simply-connected manifold (of dimension 5) can carry a metric of positive scalar curvature. The fundamental Theorem 4.1 implies that there is a factoring of graded ring homomorphisms . {o* / nstn 1 n )-;--. KO-*(pt) and it seems likely that n is an isomorphism, i.e., that 0 si. (We know n ® IQ to be an isomorphism; see Gromov-Lawson [2].) If this were so, then the d-class would be exactly the obstruction to the existence of posi-tive scalar curvature metrics in the simply-connected case. Since = 0 for n = 5, 6 and 7, Theorem 4.4 has the following con-sequence. Corollary 4.6. Every compact simply-connected manifold of dimension 5, 6, or 7 carries a metric of positive scalar curvature. We shall next take up the question of positive scalar curvature on non-simply-connected manifolds. Here the situation can be quite subtle, as the following example, due to L. Berard-Bergery [2], illustrates. Let L be an exotic 9-sphere such that d(L) =1= 0, and consider the spin manifold X == (lP7(1R) X S2) # L, with n1X 71.2. This manifold does not carry positive scalar curvature (since d(X) = d(L) =1= 0), however its two-fold universal covering manifold does. To see this, one uses the fact that 71.2 EB 71.2 (cf. Stong [1]) which shows that the universal covering X == (S7 X S2) # L#L S7 x S2. 302 IV. APPLICATIONS §5. Positive Scalar Curvature and the Fundamental Group We now take up the case of manifolds with non-trivial fundamental group. To set the stage we note that the product of any manifold with S2 carries a metric with K > o. Consequently, in dimensions 6, the presence of positive scalar curvature places no restriction on the isomorphism class of the fundamental group. (This is also true in dimensions 4 and 5; cf. R. Carr [1].) Nevertheless, if one takes into account the interaction of the fundamental group with the geometry of the manifold, then tremendous restrictions arise. The interaction we are seeking lies in the geometry of the covering spaces. In particular, we shall be interested in manifolds with covering spaces which are "large in all directions, or put another way, whose uni-versal covering spaces contain big "cubes." To give an idea of what this should mean, we examine a notion due to Gromov. Let r denote the standard metric cube in /Rn. We shall say that a riemannian n-manifold X contains a cube of size L if there exists a continuous map f: r -+ X such that dist(f(L)j(L')) L for each pair of opposite faces L,L' of the cube r. We can then say that a riemannian manifold X has big covering spaces, if for any L > 0, there is a covering space which (in the lifted metric) contains a cube of size L. For a compact manifold this concept is independent of the metric chosen (since any two metrics are uniformly commensurate). A good example of a manifold with big covering spaces is a torus Tn. This idea can be usefully generalized. However, for the purposes here it is better to "dualize" the concept by mapping out of the manifold rather than into it. DEFINITION 5.1. A Cl-map f: X -+ Y between riemannian manifolds is e-contracting (for a given e > 0) if Ilf*vll ;;:;; allvll for all tangent vectors v to X. This means that for any piecewise smooth curve y in X, one has length(f(y)) ;;:;; slength(y). Such a-contracting maps are not hard to find (consider the constant maps). Suppose, however, that X and Y are compact oriented manifolds of the same dimension. Then the existence of an a-contracting map f: X -+ Y of non-zero degree means that in a strong sense X is "bigger than Y" on an order of at least l/a. Now every oriented n-manifold admits maps of degree 1 onto the n-sphere. Thus, we let sn(l) denote the standard riemann-ian n-sphere of constant curvature 1, and we replace the above notion of big covering spaces with the following (cf. Gromov-Lawson [1 ],[3]): DEFINITION 5.2. A compact riemannian n-manifold is said to be en-largeable if for every a > 0 there exists an orientable riemannian covering space which admits an a-contracting map onto sn(l) which is constant at infinity and of non-zero degree. §5. K > 0 AND THE FUNDAMENTAL GROUP 303 If for each e > 0, there is a finite covering space with these properties, we call the manifold compactly enlargeable. Note. A map is constant at infinity if it is constant outside a compact set. The degree of such a map f: X ...... sn is defined as fxf*OJ deg(f) = r OJ Jsn (5.1) where OJ is an n-form on sn with non-zero integral. The degree can also be defined as usual in terms of regular values of f. The square flat torus Tn = /Rn /zn is certainly enlargeable since the uni-versal covering space has the required mappings for all e > O. This torus is, in fact, compactly enlargeable. We see this as follows. For each k E Z+, the lattice (k' Z)" c zn gives a kn-fold covering torus tn = /Rn /(k . Z)", which admits the (n/k)-contracting map to sn(l) of degree 1 pictured below. f * Theorem 5.3. The following statements hold in the category of compact manifolds: (A) Enlargeability is independent of the riemannian metric. (B) Enlargeability depends only on the homotopy-type of the manifold. (C) The product of enlargeable manifolds is enlargeable. (0) The connected sum of any manifold with an enlargeable manifold is again enlargeable. (E) Any manifold which admits a map of non-zero degree onto an en-largeable manifold is itself enlargeable. Proof. It is evident that (E) => (B) => (A) and that (E) => (0). To prove (E) we consider two compact oriented riemannian n-manifolds X and Y, and a map F: X ...... Y of non-zero degree. By compactness there exists a c > 0 so that IldF11 c on X (i.e., F is c-contracting). Given e > 0, there is a 304 IV. APPLICATIONS riemannian covering space p: Y -+ Y which admits a (a/c)-contracting map f: Y -+ sn(1) which is constant outside a compact set KeY and of non-zero degree. Taking the fibre product of p and F gives a covering space p' : X -+ X and a proper mapping F: X -+ Y so that the diagram -F -X ----+ Y p'l lp commutes. Since F is a lifting of F, we have Ilvill c on X. Hence, the composition f 0 F : X -+ sn(1) is a-contracting. Since F is proper, we see that f 0 F is constant outside the compact set F-1(K). It is easy to see that: deg(f 0 F) = deg(f)deg(F) =1= O. Hence, X is enlargeable as claimed. To prove (C), we fix a degree-l map cjJ: sn(1) X sm(1) -+ sn+m(1) and let c = suplldcjJll. This map is chosen to be constant on the set (sn(1) x {*}) u ({*} x sm(l)), where each "*" denotes a distinguished point in the sphere. Suppose now that we are given (a/c)-contracting maps, f: xn -+ sn(l) and g : xm -+ sm(1), which are constant (= *) at infinity and of non-zero degree. Then the map cjJ 0 (f x g): xn X ym ---+ sn+m(1) is a-contracting, con-stant at infinity and of non-zero degree. From here the argument is straightforward. _ Theorem 5.3 says that the category of enlargeable manifolds is closed under products, connected sums (with anything), changes of differentiable structure, etc. The category is also quite rich. It contains, as basic building blocks, many families of important K(n,l)-manifolds. Theorem 5.4 (Gromov-Lawson [1]). The following manifolds are enlarge-able: (A) Any compact "solvmanifold", i.e., a compact manifold dijfeomorphic to G/r where G is a solvable Lie group and r is a discrete subgroup. (B) Any compact manifold which admits a metric of non-positive sectional curvature. (C) Any "sufficiently large" 3-manifold (X is sufficiently large if there is a compact surface of positive genus LeX with n1L -+ n1X injective.) Proof. We begin with (B). It is easy to see that a manifold X of non-positive curvature is enlargeable. By the Cartan-Hadamard Theorem (cf. Milnor [6]), the exponential map at any point of the universal covering X is a diffeomorphism whose inverse exp; 1: X -+ TxX is distance de-creasing onto TxX [Rn with its euclidean metric. Composing with a fam-ily of a-contracting maps fr. : [Rn -+ sn( 1) as above completes the proof of (B). §5. K > 0 AND THE FUNDAMENTAL GROUP 305 To prove (A) we invoke the basic fact that every compact solvmanifold X admits a fibration n: X -+ Tk over a torus of positive dimension with a (compact) solvmanifold X 0 as fibre (see Raghunathan [1] for example). We introduce the standard flat metric on Tk == IRkjZk, and we fix a metric g on X. By replacing g with tg for t > 0 sufficiently large, we can assume that n is distance-decreasing. Consider now the covering X -+ X induced by lifting n over the cov-ering IRk -+ Tk. The induced fibration n: X -+ IRk is again distance de-creasing. Furthermore, there exists a diffeomorphism qJ : X ----+ IRk X X 0 such that n = pr 1 0 qJ where pr 1 : IRk x X 0 -+ IRk is the projection map. To see this when k = 1, lift a non-vanishing vector field from Tl to X and integrate to get the coordinate transverse to the fibres. When k > 1, apply induction. Suppose that B > 0 is given. Consider the ball B = {x E IRk: Ilxll :-;:;; njB} and fix a map f : IRk -+ Sk(1) which is B-contracting, constant on IRk -B and of degree 1 (as pictured in the figure above). Choose a metric go on X 0 so that the composition -cp pr2 X ----+ IR x X 0 ----+ X 0 is distance decreasing on the compact subset n-1(B) c X. (A sufficiently small multiple of any metric will do.) We may assume that X 0 is en-largeable by induction on dimension. Hence there exists a covering no: Xo -+ Xo and an B-contracting map g:io -+ sn-k(l) which is constant at infinity and of non-zero degree. Let q: X -+ X be the covering induced by lifting qJ over Id x no, so there is a commutative diagram (below). G x IRk X Xo Xo sn-k(1) q 1 lid x 1 X _«1_ IRk X Xo Xo F 1 1 306 IV. APPLICATIONS We now take the "smash product" sn(1) exactly as we did in the proof of Theorem 5.3(C). This map is 4e-con-tracting, constant at infinity and of non-zero degree. This completes the proof of part (A). For part (C), the reader is referred to Gromov-Lawson [1]. • Note that the tori Tn = Sl X ... X Sl, n 1, play a special role in this theory. For example, they belong to each of the three classes of manifolds in Theorem 5.4. Furthermore, there is the following "stability" phenomenon: If X is an enlargeable manifold, so is X x Tn for any n 1. (5.2) If a manifold X carries a metric with K > 0, so does X x Tn (5.3) for any n 1. There is an (adversary) relationship between enlargeability and positive scalar curvature, and the mediating agent is the Atiyah-Singer operator. Here, however, the operator has coefficients in a vector bundle. The first main result is the following: Theorem 5.5 (Gromov-Lawson [1], [3]). An enlargeable spin manifold X cannot carry a metric of positive scalar curvature. In fact any metric with K 0 on X must be fiat. Note. Here the spin assumption is not vital. It suffices to know that an appropriate covering space is spin. Note. Results of this type were first proved in dimensions -:;;;. 7 by R. Schoen and S. T. Yau [1], [2] using minimal surface techniques. Theorems 5.4 and 5.5 together produce the following nice "exclusion theorem." Corollary 5.6. A compact manifold which carries a metric with sectional curvature -:;;;. 0 (or < 0), cannot carry a metric with scalar curvature > 0 respectively). Proof of 5.5. For clarity's sake we shall present here a prooffor the case of compactly enlargeable manifolds. The more general result requires some tools from analysis (the Relative Index Theorem) and will be proved with more general results in the next section. Let X be a compactly enlargeable n-manifold, and suppose X carries a metric with K Ko for a constant Ko > O. From (5.2) and (5.3) we may assume that X has even dimension 2n. (If not, replace X by X X Sl.) §5. K > 0 AND THE FUNDAMENTAL GROUP 307 Choose a complex vector bundle Eo over the sphere s2n(l) with the prop-erty that the top Chern class ciEo) =1= O. This is certainly possible, since on s2n the Chern character ch E = dim E + 1 cn(E) (n -I)! gives an isomorphism ch: K(s2n) -+ H*(S2n; Z) (cf. Atiyah-Hirzebruch [2]). We now fix a unitary connection 'lED on Eo and we let RED denote the curvature 2-form. Let a > 0 be given and choose a finite orientable covering X -+ X which admits an a-contracting map f: X -+ S2n(l) of non-zero degree. Using f, we pull back the bundle Eo, with its connection, to X. This gives us a bundle E == f* Eo with connection 'lE == f*'lEo. We then consider the com-plex spinor bundle $e of X with its canonical riemannian connection, and consider the Atiyah-Singer operator I/J on the tensor product $e ® E (as in 11.5.10). We know from Theorem 11.8.17 that there is a "Bochner formula" (5.4) where mE(O" ® a) == Lj<k (ejekO") ® (R;j,eka) depends universally and lin-early on the components of the curvature tensor RE of E. The operator mE is a symmetric bundle endomorphism of $e ® E, and if its pointwise norm everywhere satisfies the inequality IlmEII < tKo, then we will have > 0, and the index of the operator I/Ji : ® E) ---+ q$c ® E), (5.5) (5.6) will be zero. To achieve this estimate we first note that there is a constant kn depending only on dimension, such that (5.7) We then observe that the curvature of E is the pull-back of the curvature of Eo, and therefore (5.8) To prove (5.8), we fix a point x E X and let {I/Il,'" ,I/In(n-l)d be an ortho-normal basis of A2TxX which diagonalizes the symmetric bilinear form {3(I/I,I/I') == <f*I/I,/*I/I'). This means that <f*I/Ij,/*I/Ik) = AJc5jk where, since f is a-contracting, we have IAjl ;;:;; a2 for all j. It follows that there exists an orthonormal basis {I!! l' ... -1 )/2} of A 2 Tr(X)S2n such that f*I/I j = j' for all j. We then compute that IIREII; = L = L = L ;;:;; a411REo112. 308 IV. APPLICATIONS Combining (5.7) and (5.8) shows that the positivity condition (5.5) is sat-isfied for all e < JKo/4kn. The index of the operator (5.6) must then vanish. However, this index is given by index(lPt) = {ch E· A(X)} [X] (5.9) = {(d + (n I)! Cn(E)). A(X)} [X] - 1 -= dA(X) + (n _ 1)! cn(E)[X] 1 -(n -1)! cn(f*Eo)[X] 1 -(n -1)! f*(cn(Eo))[X] (n I)! deg(f)cn(Eo)[S2n] ¥ o. This contradiction completes the proof of the first statement ofthe theorem. To prove the second statement, we call upon the following result which appears in Kazdan-Warner [1]: Theorem 5.7 (1. P. Bourguignon). Let X be a compact manifold which car-ries no metric with K> o. Then any metric with K 0 on X is Ricci fiat. To complete the argument we now apply the following corollary of the Splitting Theorem of Cheeger and Gromoll [2]. Proposition 5.8. A Ricci fiat enlargeable manifold is fiat. Proof. Let X be a (compact) enlargeable n-manifold with a Ricci fiat metric. The Splitting Theorem implies that the universal cover of X splits as a riemannian product [Rm x Y of a euclidean space with a compact simply-connected (Ricci-f1at) manifold Y. We claim that dim(Y) = o. If not, set (j == diameter(Y) > 0, and choose a covering X of X which admits a (n/4(j)-contracting map f: X -4 sn(l) which is constant at infinity and of non-zero degree. This covering is a quotient X = ([Rm X y)/r by a subgroup r of the deck group. The deck group acts freely and properly discontinuously by isometries which preserve the factors. By passing to a subgroup of finite index we may assume that r acts freely and properly discontinuously on [Rm. Then X is a fibre bundle p: X -4 [Rm /r over the quotient manifold with fibre Y. Since f is (n/4(j)-contracting, we see that each fibre is mapped into a ball of radius n/4 in sn( 1). Hence, there is a continuous map fo : [Rm /r -4 sn(1) §5. K > 0 AND THE FUNDAMENTAL GROUP 309 so that dist(fo(p(X)),f(X)) nl2 for all x E X. (Choose a section of the disk bundle whose fibre at y E IRm/r is the convex hull of f(Yp) = f(p-l(y)).) This map can be chosen to agree with f at infinity. By pushing along the unique geodesic arc joining f(x) to fo(p(x)), we construct a homotopy from f to fo 0 p which is constant at infinity. Since dim(Y) > 0, it follows that deg(f) = deg(fo 0 p) = 0 contrary to assumption. Therefore, we conclude that dim(Y) = 0, and both the proposition and the theorem are proved. _ The argument given for Theorem 5.5 actually proves much more than is claimed. To use the full force of the argument we only need to readjust our definitions. In doing this we shall unify these results with the previous ones concerning the A-genus. We begin by generalizing the notion of the degree of a map. Letf: X -4 Y be a smooth map between compact oriented manifolds, and suppose dim X -dim Y = 4k O. The inverse image f-l(y) of a regular value yE Yis a compact 4k-manifold whose oriented cobordism class is indepen-dent of y. To see this, join two regular values y and y' by a smoothly embedded are y, and wiggle f, away from f-l(y) u f-1(y'), so that it becomes transverse to y. The inverse image of y provides a cobordism between r l(y) and f-l(y'). It now follows that the A-genus A(f-l(y)) is independent of y. We call this number the A-degree of f. The A-degree can be computed by a formula of type (5.1). Let w be a volume form on Y with non-zero integral, and let Ak(X) be a de Rham representative (a closed 4k-form) of the kth-component of the total A-class of X. Then Ix f*w 1\ Ak(X) A-deg(f) = Iy w (5.10) Notice that the A-degree makes sense when X is not compact, provided that f is constant at infinity. We now generalize the notion of enlargeability by replacing the word "degree" in 5.2 by the word "A-degree". DEFINITION 5.9. A compact riemannian n-manifold is said to be A-nlargeable if given any e> 0, there exists a riemannian covering space hich admits an e-contracting map onto sm(1) (for some m == n (mod 4)) hich is constant at infinity and of non-zero A-degree. The basic facts re as before. heorem 5.10. The statements (A), (C) and (D) of Theorem 5.3 remain true ith the word "enlargeability" replaced by the word "A-enlargeability." 310 IV. APPLICATIONS Proof. The proof of part (C) of Theorem 5.3 goes through after making the obvious linguistic changes. Parts (A) and (D) are easy. • Of course, an enlargeable manifold is A-enlargeable. Moreover, an ori-entable manifold of non-zero A-genus is also A-enlargeable since the con-stant map to SO has non-zero A-degree. More generally we have the following: Corollary 5.11. A compact manifold which admits a mapping of non-zero A-degree onto an enlargeable manifold is A-enlargeable. In particular, the product of an enlargeable manifold and a manifold of non-zero A-genus is A-enlargeable. We now have a unification of many of the previous results. Theorem 5.12 (Gromov-Lawson [1], [3])). An A-enlargeable spin manifold cannot carry a metric with K > O. Infact, any metric with K 0 on such a manifold is Ricci fiat. Proof. The proof of Theorem 5.5 goes through with no essential changes. The main point is that the computation (5.9) now becomes ind(ffii) = {( d + (m 1)! Cm(E))' A(X)} [X] (5.11) - 1 -= dA(X) + (m _ 1 )! {cm(E) . AiX)} [X] 1 --(m _ 1)! {f*cm(Eo)' Ak(X)}[X] 1 2 (m _ 1)! (A-degf)cm(Eo)[S m] ¥ O .• This theorem implies, in particular, that a spin manifold of the form X x Y where X is enlargeable and A(Y) ¥ 0, cannot carry positive scalar curvature. In a sense, it interpolates between the two previous results which concerned enlargeability and A separately. A good interpretation of Theorem 5.12 can be made in terms of the higher A-genera. These are defined for any compact differentiable mani-fold X as follows. Fix a K(n,l)-space K and a map f: X -4 K (corre-sponding to a homomorphism niX -4 nIK). For each cohomology class u E H*(K; Z) H*(n; Z), the higher A-genus associated to u is the number Af)X) == {f*u· A(X)} [X] §5. K > 0 AND THE FUNDAMENTAL GROUP 311 where, as usual, A(X) is the total A-class of X. We could, of course, pass to the "universal" space K x = K( n 1 (X), 1) and the map Ix: X -4 K x through which every I: X -4 K(n,l) can be factored. These "universal" higher A-genera are simply indexed by the cohomology classes of the group nt(X). EXAMPLE 5.13. When the space K = K(n,l) is a compact oriented m-manifold and when u E Hm(K; Z) is the fundamental cohomology class, then it is evident that: Af,u(X) = (A-deg)(f). (5.12) Observe now that many fundamental examples of enlargeable manifolds (such as solvmanifolds, manifolds of non-positive curvature, and suffi-ciently large prime 3-manifolds) are manifolds of K(n,l)-type. Theorem 5.12 states that for a spin manifold X of positive scalar curvature, the higher A-genera, coming as in (5.12) from maps to an enlargeable mani-fold, must be zero. It is at the moment unknown whether every compact K(n,l)-manifold is enlargeable; however, any counterexample to this statement is likely to be quite complicated. Assuming it is so, one might be led to conjecture the following: Conjecture 5.14. For any compact spin manifold with positive scalar curva-ture, all the higher A-genera must vanish. There is much evidence for this conjecture if one restricts to "geometric" K(n,l)-spaces. At very least, the conjecture serves as a rough map of the wilderness. There is, in fact, a general theory which allows one to transform this conjecture to a more general conjecture in the K-theory of C*-algebras. Again it involves the Atiyah-Singer operator. To each discrete group r there is a canonically associated C*-algebra, denoted C*(r) and called the C*-algebra of r. Over any manifold with fundamental group r we can construct the flat C*(r)-bundle -4 X corresponding to the representation of r on C*(r) by right multiplication. This is just the associated bundle == X Xr C*(r) (5.13) here X is the universal covering space of X (a principal r-bundle). Note hat is a bundle of left C*(r)-modules. Each fibre is free of rank one. At this point, our discussion becomes rather general. A guide to the ormidible technical details required to make it rigorous is found in J. osenberg [I]. Suppose the manifold X is spin and consider the twisted tiyah-Singer operator (5.14) 312 IV. APPLICATIONS where carries its canonical flat connection. The kernel and cokernel of J/J + are (after possibly a compact perturbation of J/J +) finitely generated pro-jective C*(r)-modules. The formal difference gives an element in the alge-braic K-group, Ko(C*(r)). This element depends only on the homotopy class of the operator J/J + and is called the analytic index of 1) + • Mishchenko and Fomenko [1] have given a formula for this index of the Atiyah-Singer type: ind(J/J+) = {ch A(X)} [X] E Ko(C*(r)) ® QJ (5.15) The claim is that for "reasonable" fundamental groups r, this index carries all the higher A-genera of the manifold X. To prove this it is necessary to study a certain universal map (5.16) defined by using the analogous bundle over K(r,l). (Here K* denotes K-homology theory extended to infinite CW-complexes.) Our assertion is true whenever this map is injective after tensoring with QJ. Observe now that since the bundle X is flat, the curvature term \RR in the Bochner formula (5.4) is zero. This implies (by Rosenberg [1]) that if X carries positive scalar curvature, then ind(J/J+) = o. Conse-quently, Conjecture 5.14 is true whenever the map er ® QJ is injective. The injectivity of er has been established by Kasparov [1] for any r which arises as a discrete subgroup of a connected Lie group. Readers may have noticed the direct parallel between the higher A-genera and the Novikov higher signatures, or "higher L-genera." (Such classes can be formulated for any multiplicative sequence.) S. P. Novikov has conjectured that these higher signatures are, like the signature itself, homotopy invariants of a manifold. A proof of this conjecture for certain fundamental groups has been given by G. Lusztig [1] using a family of Dirac operators of the form Dt+ : (Ct +(X) ® Et) qcr(X) ® Et) where Et is a family of flat bundles over X induced from K(r,l). Much more generally, one can consider the operator D+:qCt+(X) qct-(X) in analogy with (5.14). The assertion here is that the index ind(D+) = {ch L(X)}[X] E K*(C*(r)) carries all the higher signatures of X. This again will follow from the in-jectivity of the map er ® QJ. For this reason, the conjecture concerning er is called the Strong Novikov Conjecture. We shall end this section with some remarks on a possible classification of manifolds with 7rl = r which admit positive scalar curvature. The homomorphism d: KO *(pt) induces a transformation of general- §6. COMPLETE MANIFOLDS WITH K > 0 313 ized homology theories. Thus, setting h*(r) == h*(K(r,l)), we get a trans-formation (5.17) Given any compact spin n-manifold X with niX r, there is a canonical mapping X -4 K(r,l), inducing an isomorphism on nl• This map deter-mines an element [X] E Again as a guide to the forest, one could conjecture that as in the case where r = {el, the classes d([X]) constitute a complete set of invariants for the existence of positive scalar curvature on X. There is some evidence for this. The analogue of Theorem 4.3 has been proved by Rosenberg [2] and Miyazaki [1]. They show that the question indeed only depends on the fundamental class [X] E in the spin case. In the non-spin case, it depends only on the class [X] E and often, for example when niX 7l.. or 7l..P' one can show that such X always carry K > o. Unhappily, the conjecture that d = 0 in the presence of positive scalar curvature fails for torsion groups r. Rosenberg [2] shows that every spin 5-manifold with nl 7l..3 carries K > 0, but that Im d t= 0 in this case. Nevertheless, for torsion-free groups r there is reasonable evidence for the following: CONJECTURE. For torsion free groups r, the classes d([ X]) (from (3.17)) constitute a complete set of obstructions to the existence of positive scalar curvature on compact spin manifolds X with niX r. §6. Complete Manifolds of Positive Scalar Curvature We now consider the question of the existence and structure of complete metrics of positive scalar curvature on non-compact manifolds. We shall touch only a few results. The reader is referred to Gromov-Lawson [3] for a much more extensive discussion. The first important concept here is the following: DEFINITION 6.1 A bundle with compact support on a manifold X is a vector bundle E -4 X which is trivialized at infinity together with a con-nection which is compatible with that trivialization. This means that outside a compact subset of X, there is a given iso-morphism of E with the product bundle. This isomorphism identifies the given connection with the canonical flat one on the product. The con-nection will always be assumed to be orthogonal or unitary (depending on whether E is real or complex). Suppose now that X is a complete riemannian spin manifold of even dimension n = 2k. Let $1[;, denote the spinor bundle of X with its canon-ical riemannian connection, and let E -4 X be any bundle with compact 314 IV. APPLICATIONS support. Then the Atiyah-Singer operator J/JE on L 2-sections of $1[;, ® E is essentially self-adjoint and satisfies ker J/JE = ker J/Ji, (see Theorem 11.5.7). This operator satisfies the pointwise formula (6.1 ) derived in 11.8.17. Since E is flat at infinity, this reduces to the equation J/Ji, = V*V + tK outside a compact subset of X. Using this fact and some standard arguments from analysis, we prove the following. We say that a function K is uniformly positive on a set A if K Ko, for some constant Ko> 0, on A. Proposition 6.2. Suppose X has uniformly positive scalar curvature outside of a compact set. Then the Dirac operator J/JE on L 2($1[;, ® E) has a finite-dimensional kernel and a bounded Green's function. Corollary 6.3. The operator J/Jt : ® E) ---+ U($c ® E) and its ad joint J/J i have finite-dimensional kernels. Hence the index ind(J/Ji) = dim(ker J/Ji) -dim(ker J/Ji) is a well-defined integer. (6.2) (6.3) Proof of6.2. Let Kc X be a compact set outside of which E is trivialized and tK Ko for some Ko > O. Let c > 0 be a constant such that c Id -(tK + 9\E) over K. Then for any element cP E ker(J/JE)' formula (6.1) implies that V*Vcp = -tKCP -9\E(cp). Integrating by parts, and es-timating gives the inequality Let II'IIA denote the U-norm over a set A c X. If we assume IIcpl11 (= + IlcpI11-K) = 1, then (6.4) implies that IIVcpl11 + (6.S) Ko + c Ko + c Consider now the uniform Cl-norm Ilcplb,K over K. By Theorem III.S.4 we know that there exists a constant C > 0 such that (6.6) for all cp E ker(DE)' Let us fix a number e> 0 and consider and e-dense subset of points I in K. If dim(ker DE) d, then there exists an element cp E ker DE with Ilcpllx = 1, such that cp(x) = 0 for j = 1, ... ,d. It §6. COMPLETE MANIFOLDS WITH K > 0 315 then follows from (6.6) that the pointwise norm of cp is uniformly less than Cc on K, i.e. Ilcpllco.K Cc. For c sufficiently small, this violates (6.5), and we conclude that dim(ker DE) is finite. The invertibility of the Green's function will not be used directly here. We refer to the reader to Gromov-Lawson [3] for the proof ofthis fact. • Proof of 6.3. With respect to the bundle splitting $1[;, ® E = ® E) EB ® E) the operator J/JE can be written (pointwise) as where J/Jf : q$t ® E) q$i ® E). Passing to L 2-sections, we get an orthogonal decomposition U($I[;, ® E) = ® E) EB U($c ® E), and J/JE continues to interchange factors as above. Consequently ker J/JE = ker J/Ji EB ker J/Ji and the finite dimensionality of ker J/Jf is clear. Since J/JE is self-adjoint, the operators J/Jf are adjoints of one another, and so ker J/Ji coker J/Ji. • Of course we will need a version of the vanishing theorems from the compact case. Proposition 6.4. Let X and E be as above, and suppose there is a constant c> 0 so that tK + 9\E cId. (6.7) on X. Then ker J/JE = 0, and in particular, ind(J/Jt) = o. Proof. As in the proof of 11.5.7, the completeness of X allows us to inte-grate by parts. Thus formula (6.l) implies that IIJ/JECPlli = IIVCPlli + fx«tK + 9\E)(cp),cp) cllcplli for all cp E U($ ® E), and it is evident that ker J/JE = o. • In order to apply the arguments of the previous section, we need some-thing to play the role of the Atiyah-Singer Index Theorem. This will be done by a Relative Index Theorem. We will present here only the special case that we need. More general versions of this result have been proved by J. Cheeger and by H. Donnelly [3]. Theorem 6.5 (Gromov-Lawson [3]). Let X be a complete even-dimensional spin manifold whose scalar curvature is uniformly positive at ir!/inity. Let Eo and E[ be two bundles with compact support on X, and assume that 316 IV. APPLICATIONS dim Eo = dim El. Then the difference of the indices of the Dirac operators J/Jio and J/Ji, is a topological invariant, given by the formula ind J/Ji, -ind J/Jio = {(ch El -ch Eo)· A(X)}[X]. (6.8) where, via the trivializations at infinity, the class ch El -ch Eo has com-pact support on X. Via Chern-Weil Theory (see Kobayashi-Nomizu [1].). the classes ch Eo, ch El' and A(X) can be represented canonically in terms of the curva-ture forms of the given connections. Formula (6.8) can then be rewritten as (Recall that Eo and El are flat outside a compact set.) Another method to compute the relative index is as follows. Chop off the manifold X outside the support of Eo and El to obtain a compact manifold X with boundary. Let Y be the double of X and let E be the bundle on Y given by Eo on one piece and El on the other piece with Eo and El glued together at the "seam" oX by the trivialization. Then the right hand side of (6.8) becomes {ch E· A(Y)}[Y]. An important special case of Theorem 6.5 is where El = E is some bundle with compact support on X and where Eo = X X Ck is the triv-ialized bundle of dimension k = dim E. Note that J/JEo = J/J Et> ... Et> J/J (k-times) where J/J is the usual Atiyah-Singer operator. We then define the reduced Chern character of E to be Ch E = ch E -ch Ck = ch1E + ch2E + ... (6.9) The formula (6.8) then becomes ind(J/Ji) -k ind(J/J+) = {Ch E· A(X)} [X]. (6.10) For the proof of Theorem 6.5 the reader is referred to Gromov-Lawson [3]. Completion of the proof of Theorems 5.5 and 5.12. Armed with Theorem 6.5 we are able to extend the arguments of the last chapter from the "compactly enlargeable" to the "enlargeable" case. Let X be a compact riemannian manifold with K Ko for a constant Ko > O. Let e > 0 be given and suppose X is a spin covering space which admits an e-con-tracting map f: X -4 s2m(1) of non-zero A-degree. Fix a bundle Eo with connection over s2m(1) with cm(Eo) l' 0 and set E = f*Eo with its induced connection. Note that E is a bundle with compact support on X, and so we can construct operators J/J and J/JE as above. §6. COMPLETE MANIFOLDS WITH K > 0 317 Since K Ko > 0, we have ind(ffi+) = O. However, for e sufficiently small, the inequality (6.7) will be satisfied, and we conclude that ind(ffit) = o. Consequently, by (6. 10) we have 0= {Ch E· A(X)} [X] = {cm(E)· A(X)} [X] contrary to assumption. _ = {f*cm(Eo) . A(X)}[X] = (A-deg)(f)cm(Eo)[S2m] Theorem 6.5 allows us to prove a number of results about complete metrics on non-compact manifolds. We begin by focusing on some impor-tant facts. The first fact is that in all the arguments of §5 we only used the fact that the mappings f were uniformly contracting on the curva-ture 2-form of a connection. This leads to the following notion: DEFINITION 6.6. A smooth map f : X -4 Y between riemannian mani-folds is said to be 8-contracting on 2-forms or (8,A Z)_ contracting if for all elements E A2TX, or equivalently if sup Ilf*l/IlI ;;:; e sup IIl/IlI x y for any differential 2-form l/I on Y. The strength of this is that to be (e,A 2)-contracting the map needs only to be contracting in (n -1) of the directions at each point, where n = dim(X). For example, the linear map L: [Rn -4 [Rn with matrix 1 o o IS (e,A 2)-contracting but not e-contracting. All of the discussion of §5 remains valid if the hypothesis "e-contracting" is replaced by "(e,A 2)-contracting". The real power of this observation will be seen in the non-compact case. DEFINITION 6.7. An orientable riemannian manifold X is called (8,AZ)-hyperspherical if there exists an (e,A 2)-contracting map f: X -4 sm(1) which is constant at infinity and of non-zero A-degree. 318 IV. APPLICA nONS The basic argument given above proves the following result. Theorem 6.8 (Gromov-Lawson [3]). There is a constant Cn > 0 so that any spin n-manifold carrying a complete metric with K 1, cannot be (BmA2)-hyperspherical. The number Bn can be explicitly estimated for each n. We now introduce a topological property of manifolds. DEFINITION 6.9. A manifold X is called weakly enlargeable if for each riemannian metric on X and each B > 0, there exists an orientable covering space of X which is (A 2,B)-hyperspherical in the lifted metric. Any (compact) enlargeable manifold is weakly enlargeable. However, there are many non-compact examples. Proposition 6.10. If X is enlargeable, then X x IR is weakly enlargeable. Proof. Fix a metric g on X x IR and consider the composition X x IR --> IR --> SI(I) where the first map is projection and where the second map is constant (= *) outside the interval (-1,1), and of degree 1. Call this com-position fJ.. Since X is compact, fJ. is b-contracting for some b 1. Consider also the projection X x [ -1,1] --> X, and choose a metric go on X so that this map is I-contracting from the given metric g on X x [-1,1]. Fix now an B > 0 and choose a riemannian covering (X,go) of (X,go) which admits an (B/b)-contracting map f: X --> sm(I) which is constant (= *') at infinity and of non-zero A-degree. Extend the map f trivially to X x IR and note that for the lift g of the original metric g, this map is (B/b)-contracting over X x [ -1,1]' Fix now a I-contracting map cr:Sm(l) x SI(1) --> sm+l(l) which is con-stant on the set (sm(1) x {*}) u ({*'} x SI(I)). Then the composition (6.11) is (B,A 2)-contracting with respect to the metric g, constant at infinity and of the same A-degree as f. • We now have, for example, that any manifold of the form xn x IR, where xn is compact and carries non-positive curvature, is weakly enlargeable. A basic case is that of Tn x IR. The property of weak enlargeability is contagious. Proposition 6.11. Let U be an open submanifold of X such that the map n I U --> nIX is injective. If U is weakly enlargeable, so is X. §6. COMPLETE MANIFOLDS WITH K > 0 319 Proof. This is an immediate consequence of the definition of weak en-largeability. _ Corollary 6.12. Any manifold X which contains a (compact) enlargeable hypersurface Xo with njXO nIX injective is weakly enlargeable. Proof. Let U be a tubular neighborhood of Xo and apply 6.10, 6.11. _ As an example, let X = Tn -A where A is any closed subset such that An Tn-l = 0 for some linear hypertorus Tn-j c Tn. This example gives substance to the following: Theorem 6.13 (Gromov-Lawson [3]). A weakly enlargeable manifold can-not carry a complete metric of positive scalar curvature. REMARK. J. Kazdan [1] has proved a version of Theorem 5.7 for non-compact complete manifolds. This strengthens Theorem 6.13 by adding the statement: Any complete metric with K ° on a weakly enlargeable manifold must be Ricci flat. Proof. Note that if the scalar curvature were uniformly positive, the result would follow easily from Theorem 6.8. To achieve the uniform positivity of K we shall multiply by a large 2-sphere. The estimates then become more delicate. Let X be weakly enlargeable and suppose X carries a complete metric g with K > 0. For any given B > 0, we can find a covering X of X which admits an (B,A 2)-contracting map f: X sm(l), constant at infinity and of non-zero A-degree. In fact, we may assume this map to be (B,A 2)_ contracting with respect to the (not necessarily complete) metric g' == Kg. Hence, the map f is pointwise BK(X) contracting on 2-forms with respect to the (lift of the) metric g. We now take the riemannian product X x S2(r) and consider the com-position (6.12) where (J is a fixed "smashing" map as above (cf. (6.11)). Call this composi-tion F. We want to derive a pointwise estimate for the contraction of F on 2-forms. Since f has compact support on X, it is c-contracting on tangent vectors for some c > 0. The dilation map on S2(r) is of course (l!r)-con-tracting on tangent vectors and (1/r2)-contracting on 2-forms. We may assume (J to be essentially I-contracting. It follows that at any point 320 IV. APPLICA nONS (X,y), the map F is e(x)-contracting on 2-forms, where e(x) == max {BK(X), c/r, 1/r2}. The map F is supported in a compact set K. Set Ko = inf{K(x): (x,y) E K some y} > 0, and choose r> 1 so that max{c/r, 1/r2} BKO. Then the map F will be BK(x)-contracting on 2-forms at every point (x,y). We now proceed as before to pull back a fixed bundle with connection from sm+2(1), via the map F. The curvature term (6.7) will be of the form 1 2 E 4 K(X) + r2 + m (6.13) where E = F*(Eo) is the induced bundle. There is a constant y depending only on dimensions so that pointwise on X x S2(r). However, since F is pointwise BK(x)-contracting on 2-forms, we see that IlmE11 YBK(X)IIREOlloo pointwise, where IIREolloodenotes the CO-norm over sm+2(1). We may as-sume that B satisfies B < trllREolloo since these constants are fixed at the beginning. The curvature term (6.13) then satisfies the inequality 1 2 E 2 4 K(X) + r2 + m r2 ' on all of X x S2(r). Hence, Proposition 6.4 applies to the twisted Atiyah-Singer operator J/Ji., and we conclude that index(J/Ji.) = 0. Of course, index(J/J+) = ° for the standard Atiyah-Singer operator. We now apply the Relative Index Theorem as before to complete the proof. To do this, it is necessary that the bundle Eo over sm+ 2(1) have a non-trivial Chern character, and therefore that m be even. If m is odd, however, we may simply replace S2(r) in the construction above, with S3(r), and everything goes through. We now complete the argument. Let k = 2 or 3, so that m + k = 2N, for some integer N. Then since ind(J/Ji.) = ind(J/J+) = 0, Theorem 6.5 together with formula (6.10) implies that ° = {cll E· i(X X Sk)}[X x Sk] = {cN(E)·i(X X Sk)}[XX Sk] = {F*CN(Eo}· i(X x Sk)}[X x Sk] = (A-deg)(F}cN(Eo)[S2N] = (A-deg)(f)cN(Eo)[S2N] contrary to assumption. _ §6. COMPLETE MANIFOLDS WITH K > 0 321 EXAMPLES. As a consequence of Theorem 6.13 we know that there is a large collection of manifolds which cannot carry complete metrics with K > O. This includes X x IR for any enlargeable manifold X. It also in-cludes manifolds of the form T" -C where C is any closed subset which misses some geodesic subtorus T"-l of codimension one. (For example, let C be finite.) One can also replace (T", T" -1) here by any compact pair (X", X" -1) where X" carries a metric of non-positive sectional curvature in which X" -1 is totally geodesic. The property of not admitting complete metrics with K > 0 remains after taking products of these manifolds and then taking connected sums with countable families of arbitrary spin man-ifolds. It persists also after "boundary" connected sums with any open spin manifold. The "exclusion theorem" 5.6 of the last chapter has a nice generaliza-tion to the open case. Corollary 6.14. A manifold which carries a complete hyperbolic metric of finite volume cannot carry a complete metric with positive scalar curvature. Proof A hyperbolic manifold X of finite volume has ends of the form N x IR where N has a finite covering by a nilmanifold and n 1 N --. n 1 X is injective. Since N x IR is weakly enlargeable, so is X by Proposition 6.11, and Theorem 6.14 applies. _ By introducing one further trick we are able to "localize" the results above. To do this we recall the notion of a warped product. Let Xl and X 2 be riemannian manifolds with metrics gland g 2 respectively, and let f: Xl --. IR + be a smooth function. The warped product of X 1 and X:z with warping function f is the cartesian product manifold X = Xl X X 2 with riemannian metric (6.14) It is an elementary exercise to compute that for such a warped-product metric, the scalar curvature is K = K1 + )2 {K2 -2nfV2f -n(n -1)IIVfI12} (6.15) where Xj is the scalar curature of Xj and n = dim(X 2)' If we let X 2 = S"(I), the euclidean n-sphere of radius 1, then on X = Xl x S"(I) we have n(n -1) ( 2) V2f K = K1 + P 1 -IINII -2n y' (6.16) By choosing f small and constant over a compact set, we can make K positive there. On a compact manifold X 0 we can then slowly change f to suit our purposes. This is the basic idea in proving the next result. 322 IV. APPLICATIONS Theorem 6.15 (Gromov-Lawson [3]). Let X be a non-compact, connected spin n-manifold containing a compact, connected, orientable hypersurface X 0 c X. Suppose that X 0 separates X into two components. Suppose fur-ther that there is a map F: X --. Y, onto an enlargeable (n -I)-manifold which, when restricted to X 0, has non-zero degree. Then X carries no com-plete metric with IRicl constant and with K uniformly positive outside a compact subset. Proof. Suppose that X carries a complete metric with K 2 outside a compact set. Let X + and X _ denote the two connected components of X -X o. Since X is non-compact, at least one of these components, say X +, is unbounded in the given metric. Define a function p: X --. IR by setting { dist(x, X 0) p(x) = . -dlSt(X, X 0) for x EX + forxEX_. By the assumption on Ric there is a smoothing p of P with IIV plI 2 and with IV2pI C for some constant C 2. Fix a number Po so that K 2 outside the compact set K = {x E : Ip(x) I Po}. Let en be the number given by Theorem 6.8, and choose R > 0 so that 1/ R < en" Let Kinf = inf{K(x): x E K}, and choose r, 0 < r < 1, so that l/r2 > 1 + IKinfl. Choose now a Coo function <p : IR --. IR so that {<P(t) = r <p(t) = R o I<ptl e 1<p"1 e where e is the constant for It I Po for It I > Po + 2R/e (6.17) e = r/3C. We take the warped product of X with S2(1) using the functionjon X given by f(x) = <p(p(x)). By (6.16) the scalar curvature K of this metric satisfies: K IKinfl + 2/r2 > 1 over K x S2, and K > 2 + 6e2C2r-2 > lover (X -K) X S2. Hence, K > 1 on all of X x S2. We shall now find a covering of X x S2 which is (Bn,A 2)-hyperspherical, in contradiction to Theorem 6.8. To begin we fix a number b > Po + 2R/e and we consider the compact set Q = {x EX: b p(x) b + 4n}. Fix a metric on Y. Let b = sup{IIF *"x: x E Q} and choose a riemannian cover-ing n: Y --. Y which admits an (en/b)-contracting map y: Y --. S" -1(1) which is constant at infinity and of non-zero degree. Taking the fibre §6. COMPLETE MANIFOLDS WITH K > 0 product of F and n gives a commutative diagram -j -323 where it is a covering map and F is proper. Let g = g 0 F and let h = pop 0 it: X --. Sl(1) where p: IR --. Sl(1) is the degree-l map given by collapsing everything outside the interval (b,b + 4n) to a point. (This map is (t)-contracting). Let G denote the composition where (J is a "smashing" map which collapses the axes sn-l(l) X {*} u {*} x Sl(l) to a point. The map G is a constant outside a compact set contained in Q = it-l(Q). Since h is I-contracting and g is en-contracting, it follows that G is, at every point, .J(1 + e;)-contracting and en-contracting in the hyperplane ker(h*). We now lift the warped metric to X x S2 and let ii denote the composition where (J' is again a "smashing" map. Since the warping function j = f 0 it = constant = R on the support of G, and since l/R < em we see that ii is again, like G, 1-contracting and "en-contracting in a hyperplane" at each point. In particular, the map ii is en-contracting on 2-forms. To complete the proof we must show that ii is of non-zero degree. The main point here is the following. Let t be any regular value of p and consider the compact manifold Xt == p-l(t). Clearly Xt is homologous to Xo == p-l(O), and so by the hypothesis on F, its restriction gives a map F: Xt --. Y of non-zero degree. The rest is straightforward. Set Xt = it-I(Xt) and note that the lift F is proper on Xt and has the same degree as F. Now the degree of g x h equals the degree of g restricted to a regular level set of h, and deg(g) = deg(g)deg(F) i= O. Hence, deg(G) = deg(ii) i= O. • There are many interesting applications of Theorem 6.15. For exam-ple, let X be an open n-manifold and X 0 c X a compact hypersurface. 324 IV. APPLICA nONS Suppose Xo disconnects X and let Iff be an unbounded component of X -Xo with alff = Xo (see diagram below). X I \ \ \ DEFINITION 6.16. We say that is a bad end if there exists a map F: Iff --. Y onto an enlargeable manifold Y so that Flxo is of non-zero degree. Theorem 6.17 (Gromov-Lawson [3]). Suppose X is a spin manifold with a bad end Iff. Then there is no complete metric on X which has IRicl bounded and K uniformly positive on Iff. Note. Actually, only the end Iff needs to be spin. Proof. Let X be given a complete metric. Consider the double []I(Iff) = Iff uXo Iff with a metric which agrees, outside a neighborhood of the "seam" alff = X 0, with the given one. This manifold satisfies all the hypotheses of Theorem 6.15, and therefore cannot have IRicl bounded and K uniformly positive outside the compact neighborhood of X. • Theorem 6.18. A compact 3-manifold of K(n, I)-type cannot carry a metric with positive scalar curvature. In fact, no compact 3-manifold which can be written as a connected sum with a K(n, I)-manifold, can carry positive scalar curvature. Proof. Let X be a compact K(n, 1) 3-manifold. We may assume that X is orientable and therefore spin. Choose an embedded curve y c X which is not homotopic to zero, and consider the covering space X --. X cor-responding to the cyclic subgroup of n 1 X generated by [y J. There is a lifting of y to an embedded curve ji c X which generates nlX, Note that X is a K(nlX, I)-manifold. In particular, n1X 7L and the inclusion ji c X is a homotopy equivalence. (It is not possible that n1X = 7Lm since H2k(K(7Lm• 1); 7Lm) 7Lm for all k.) Suppose now that X carries a metric g with K > O. Then g lifts to a complete metric g on X of uniformly positive scalar curvature with IRicl bounded. §6. COMPLETE MANIFOLDS WITH K > 0 325 Choose a tubular neighborhood U of )I, and set C == X -U. We claim that is a bad end. To prove this, we first show that the inclusion 4 induces an isomorphism on HI. This is an easy consequence of the Mayer-Vietoris sequence for X = U u which gives 0= H z(X) ---+ H ---+ H EB HI U ---+ HI X ---+ O. (Recall that the inclusions)! cUe X are homotopy equivalences.) By general theory the surjective homomorphism n 1 C --. H 1 (C) 7L EB 7L is represented by a continuous map F: --. SI X SI = K(7L EB 7L, 1). Re-stricted to Si x SI this map induces an isomorphism on ni' and therefore has non-zero degree. Thus, is a bad end, and the first state-ment of Theorem 6.18 is proved. For the second statement consider a compact oriented 3-manifold Z = X # Y where X is as above. Let s: X # Y --. X be a map given by collapsing the Y-summand to a point * E X. Choose}, c X -{*} and pass to the covering n: X --. X with the lifted curve )I c X as above. Taking the fibre product of sand n gives a commutative diagram XiiY where it is a covering map and s is proper. The map s simply consists of collapsing a family of"Y-summands" to the discrete family of points n-1(*). The neighborhood U of)! can be chosen to lie in X -n-1(*) where sis a diffeomorphism. Thus we can lift U back to X7rY and consider the end == X7iY -S-1(U). By restriction we have a proper degree-one map --. onto the end constructed above so that s: ..::. is a diffeomorphism. Composing with the map F: --. Si X Si above shows that the end is a bad one, and Theorem 6.17 applies as before. • Recall that every compact orientable 3-manifold has a "prime decom-position" X = Ll # ... # Lm # (SI X SZ) # ... # (SI X Sz) # Kl # ... # Kn where the manifolds L; have finite fundamental groups (in S04) and where each Kj is a K(n, 1 )-manifold (see Milnor [4] and Hempel [1 ]). The theorem above says that if X carries positive scalar curvature, then there are no K(n,l)-factors in the prime decomposition of X. There are alternative proofs of this fact using minimal surfaces (see Gromov-Lawson [31, Schoen-Yau [1 ]). If standard conjectures in the theory of 3-manifolds were true, this result combined with Proposition 4.3 would settle the question of which 3-manifolds carry K > O. 326 IV. APPLICATIONS Arguing as in the proof of Theorem 5.5, we see that a compact 3-mani-fold X with K 0, which does not carry K > 0, must be flat. There are six such manifolds up to diffeomorphism. (See Calabi [1].) Results for non-compact 3-manifolds M can be obtained by these methods. We say that M contains an incompressible surface if there is an embedding L 4 M of a compact surface of genus> 0, such that the induced homomorphism JrI(L) --. JrI(M) is injective. We say that M carries a small circle if there is an embedding Si 4 M whose class is of infinite order in H I(M; Z) and such that the class of the small normal circle is of infinite order in H I(M -SI; Z). Thus, SI x [R2 carries a small circle, but SI x S2 does not. From 6.12 and 6.13 we see that any 3-manifold which carries an incompressible surface cannot carry a complete metric with K > ° (a result due originally to Schoen and Yau [6]). Furthermore, any 3-manifold which carries a small circle cannot carry a complete metric with K 1. In particular, the manifold Si x [R2 carries no such metric (see Gromov-Lawson [3]). §7. The Topology of the Space of Positive Scalar Curvature Metrics Fix a compact m-dimensional manifold X and let Jt{X) denote the space of all riemannian metrics on X. Note that uIt(X) is a convex cone in the linear space r(T* X ® T* X) and is therefore contractible. It is acted on naturally by the group Diff(X) of diffeomorphisms of X. We consider here the subs pace c uIt(X) of those metrics which have positive scalar curvature. This subspace is stable under Diff(X), and as we have seen, could possibly be empty. The first results on the topology of were due to Hitchin [1] who defined for spin manifolds X a homo-morphism (7.1) for each n 1 as follows. Suppose lJ. E Jrn _ 1 is represented by a map f: sn -I --. Since uIt(X) is contractible, we can extend this to a map f: Dn --. uIt(X) where sn-I = aDn. (This extension process realizes the isomorphism Jrn -I -.:. Jrn(uIt(X), ).) Fix a spin structure on X and associate to each metric fey) the canonical Cfm-linear Atiyah-Singer operator 1ly+ for that metric. This gives a family of elliptic operators over Dn which at each point of aDn are invertible (since the scalar curvature is positive). The invertible operators form a contractible space, so we can deform f to be constant on sn -I. Taking indm of the resulting family gives an element h(f) E Ko-m(Dn, sn-I) Ko-m(sn) Ko-m-n(pt) which de-pends only on the homotopy class off. Hitchin is able to show that his homomorphism (7.1) is often non-trivial by the following means. Fix a metric y E without isometries and consider the embedding iy: Diff(X) 4 .o/'(X) given by iy(g) = g*(y). §7. SPACE OF METRICS WITH K > 0 327 Composing with h leads to a homomorphism 'Trn_1(Diff(X)) Ko-n-m(pt) whose value on an element u E 'Trn_1(Diff(X)) is d(Zu) where Zu sn is the fibre bundle with fibre X constructed by clutching together two copies of Dn x X along sn-l X X via u. Using deep results from topology, Hitchin is able to construct sufficiently many non-trivial examples to prove the following: Theorem 7.1 (Hitchin [1 ]). Let X be a compact spin manifold of dimension n with i= 0, then i= 0 i= 0 if n == 0 or 1 (mod 8) if n == 0 or -1 (mod 8) Note that these classes do not survive to the quotient A slightly different approach to the study of the topology of was given in Gromov-Lawson [3]. The cleanest statements require the Rela-tive Index Theorem 6.5. However, the basic idea is based on the following elementary construction. Fix y E and suppose Y is a compact manifold with ay = X. We say that Y extends to Y if there exists a metric y E which coincides with the product metric Y + dt2 on a collar neighborhood U ;:::: X x (0,1] of the boundary. This extendability depends only on the homotopy class of y in To see this, consider a smooth family of metrics Yt E t E IR, which is constant for t 0 and for t 1. It is easily checked that the metric Ytle + dt2 on X x IR has K > 0 for all suf-ficiently large constants c. Adding the collar X x [0, c] (with this metric) to Y shows that Yo extends to Y if and only if Yl does. Proposition 7.2. Suppose that X is a spin manifold. Let {yJaEA C .o/'(X) be a family of metrics, and suppose that there exists an associated family P;'}aEA of compact spin manifolds with a 1';, = X such that Ya extends over Yafor each a E A. If for all a i= [3, then Ya is not homotopic to Yp in alia i= [3. Proof. If Ya is homotopic to Yp in then Ya extends over both 1';, and Jp. The extended metrics fit together t..o give a metric of positive scalar curvature on 1';, u (-Yp), and therefore A(1';, u (-Jp)) = o. • The "extension pairs" (Ya,1';,) discussed above are not difficult to con-struct. There are two good sources. Lemma 7.3. Let 'Tr: V X be a riemannian vector bundle of (jib re) dimen-sion k over a compact manifold X, and let D(V) == {v E V: IIvll = 1}. If k > 2, then there exists a metric Y E which extends over D(V). 328 IV. APPLICATIONS Proof. Choose an Ok-invariant metric Yo on IRk which has K 1 and is isometric to the standard product metric on sk-t x [1,(0) outside the unit disk. Choose an orthogonal connection on Vand let ;Yt' denote the corresponding field of horizontal planes on V. Choose any metric Y t on X and lift this metric to ;Yt' via n. Introduce the metric Yo on the fibres of V. Using the formulas of O'Neill [1], the sum Yt + t2yo is seen to have positive scalar curvature for all t > 0 sufficiently small. • More elaborate arguments prove the following in any manifold Y: Theorem 7.4 (Carr [1]). The boundary of a regular neighborhood V of any smoothly embedded finite subcomplex of codimension > 2 in Y, admits a metric of positive scalar curvature which extends over U. EXAMPLE 7.5. Let V --+ S4 be a real vector bundle of dimension four with Euler number X and Pontrjagin number Pt. It is elementary that if X = ± 1, the manifold Lv = aD(V) is a homotopy sphere. Milnor showed that Lv = S7, the standard 7-sphere, if and only if pi == 4 (mod 896). Let f;k be the bundle with X = 1 and Pt = 4 + 896k. Calculations in Milnor [7] show that 8 A(f;k u D ) = k for each k. From Lemma 7.3, for each k we can construct a metric Yk on S7 which extends over f;k. By Proposition 7.2 these metrics are mutually non-homotopic in and so we have that is infinite. Since no(Diff S7) is finite, we also have that is infinite. The analogous comments apply to each of the non-standard Milnor spheres Lv. EXAMPLE 7.6. Fix an integer n > 1 and let Y denote the compact 4n-manifold with boundary constructed by plumbing together eight copies of the tangent disk bundle of s2n according to the Dynkin diagram for E8. Let Yk denote the boundary connected sum of k-copies of Y. It is ele-mentary to see that L = ay is a homotopy sphere. By the finiteness of the group of homotopy spheres, there is an integer m = m(n) such that aY"m = L # ... # L (km-times) S4n-t for all k. Using Theorem 7.4, we can construct for each k a metric Yk E which extends over Y"m. On the other hand, as shown in Carr [1], if k #-k'. (7.2) §7. SPACE OF METRICS WITH K > 0 329 In fact, since Y"m U ( -Y,,'m) is a (2n -I)-connected 4n-manifold, its A-genus and its L-genus are both certain universal (non-zero) multiples of Pn, the nth Pontrjagin number. It suffices therefore to show that the signature =I-0, Since sig(Y) = 1, we see that sig(Y"m U (-Y,,'m)) = (k -k')m. This establishes (7.2) and from Proposition 7.2 we conclude that 1)) and -1 )/Diff(S4n-1)) are infinite for all n > 1. From this we can deduce the following: Theorem 7.7. Let X be any compact spin manifold of dimension 4n -1 7 with =I-0, Then is infinite, Proof. Let Zk = (X x [0,1]) q Ykm where q denotes boundary connected sum. Note that aZk = X Il (X # S4n-1) X Il X. Fix a metric y E and take connected sum with the metric Yk constructed above on the component "X # s4n-1" (cf. 4.3). This metric extends over Zk' (Essen-tially this extension is constructed by taking the connected sum of the product metric on X x [0,1] with that on Y"m and modifying it; see Carr [1] for example.) Since Zk U (-Zk') = (X x Sl) # (Y"m U ( -y"'m)), we have A(Zk U (-Zd) = A(Y"m U (-Y,,'m)) =I-° and 4.3 applies. • The arguments given above can be nicely encapsuled by using the Relative Index Theorem. Let X be a compact spin manifold of dimen-sion 4n -1. For each pair of metrics gO,gl E we define a relative index i(gO,gl) E 7!.. as follows. Introduce on X x IR a metric g which equals the product metric go + dt2 on X x (-00,0] and gl + dt2 on X x [1,(0). Set i(go,gd == ind(ffi+) where ffi +: r(r) -+ r($-) is the Atiyah-Singer operator on X x IR. By Tpeorem 6.5 this index is independent of the choice of g in X x [0,1 J. If go is homotopic to gl in then we can choose g to have K> ° (as above) and so i(gO,gl) = 0. Thus, i(gO,gl) measures components in It is shown in Gromov-Lawson [3] that i(gO,gl) + i(gl,g2) + i(g2,gO) = ° for all gO,gl,g2 E Suppose further that X = ay. Then given g E we introduce a complete metric on Y -a Y which is the product g + dt2 on the boundary collar ay x [0,(0) = X x [0,(0). Using the Atiyah-Singer operatorfor this metric we define an invariant: 330 IV. APPLICATIONS which by 6.5 is independent of the extension and which has the property that i(g,Y) = 0 if g extends over Y. By the Relative Index Theorem we have that i(g,y) -i(g, Y') = A(Y u (-Y')). Using Browder-Novikov Theory and proceeding as above, one can detect non-trivial elements in for higher values of j. §8. Clifford Multiplication and Kiihler Manifolds Until now we have paid little attention to complex manifolds. This is not because the topic is irrelevant to spin geometry. In fact combining com-plex and spin structures yields the richer study of Spine manifolds, which are discussed in Appendix D. In this chapter we restrict attention to the fundamental case of Kahler manifolds. If Clifford multiplication enters naturally into riemannian geometry and leads to basic identities, then in Kahlerian geometry it should enter in an even more interesting way. This is indeed true. For Kahler manifolds there is a rich algebraic formalism which relates Clifford multiplication and the complex structure. We shall sketch here the principal results. We refer the reader to Michelsohn [1] for corn plete details. Let X be a 2n-dimensional manifold equipped with an almost complex structure, that is, equipped with a bundle automorphism J: T X --> T X such that J2 = -Id. DEFINITION 9.1. A riemannian metric <.,. ) on X is said to be Kiihlerian if J is pointwise orthogonal, i.e., (jV,JW) = <V, W) for all V,W E TxX at all points x, and if VJ = 0 where V denotes the canonical riemannian connection. Note. It is a basic fact that if X admits a Kahler metric, then the al-most complex structure is integrable, i.e., it comes from a system of holomorphically related coordinate charts on X. Suppose now that X carries such a metric and let D:r(C£(X))--> r(C£(X)) denote the associated Dirac operator on the complexified Clif-ford bundle C£(X) == C£(T X) ®Iffi C of X (considered as a real manifold). The first interesting fact is that there is a natural decomposition where and are first-order operators which are formal adjoints of one another = and satisfy = 0, = O. (8.1) §8. KAHLER MANIFOLDS 331 Similarly, the zero-order operator L defined in Chapter 11 can be expanded into a pair of operators 2 and fl> which are formal adjoints of one another and which canonically generate an sI2(IC)-subalgebra. In partic-ular, if we set :Ye = [2, fl>], then the three endormorphisms of IC£(X) satisfy the identities [:Ye,2] = 22, These operators are defined by setting where rlficp = L e/Ve/P !!JJcp = 8/Ve/P j } 2ep = -L ejep8j j fl>ep = -L 8jepej j ej = (ej -iJe) and 8j = (ej + iJe) (8.2) for any local orthonormal frame field of the form e1,Je1, ... ,en,Jen. These complex vector fields satisfy the "supersymmetry" relations eiej + = 8;8j + 8li = 0 e;8j + 8jei = -(jij The endomorphism J: T X -..... T X extends to IC£(X) as a IC-linear map which is a derivation with respect to Clifford multiplication. If we set ,$ == -iJ, then the commuting endomorphisms ,$ and :K are each diagonalizable with integral eigenvalues, and we can define the sub-bundles 1C£p,q(X) == {ep E IC£(X): ,$(ep) = pep and :K(ep) = qep} The representation theory for sI2(1C) is used to show that these bundles are non-zero exactly for pairs (p,q) E 7L x 7L with Ip + ql nand p + q == n (mod 2). This gives us the bigraded "Clifford diamond" below. q n p -n 332 IV. APPLICATIONS with "raising" and "lowering" operators: and with the differential operators r!fi and acting diagonally r!fi:qctp,q) -----+ qCtp+1,q+l), -----+ qCtP-1,q-l). The bigrading is nicely related to multiplication. One has that CtP,* . CtP',* c CtP+p',* for all p,p', and furthermore that {Ctp,q. Ctp"q, = {O} if q -q' #-p + p', and Ctp,q· CtP',q-(P+P'l c Ctp+p"q-p, for all p,p',q,q'. These facts follow easily from the identities fcp = wcp -cpw; ;:;e cp = wcp + cpw where by definition 2iw = I ej' J ej for any orthonormal tangent frame {el,Jeb' ., ,en,Jen} as above. We now examine the differential operators. It is easily seen that when restricted to any diagonal line in the Clifford diamond, r!fi gives an elliptic complex ... qCtr1,q-l) qCtp,q) qCtP+ 1,q+ 1) ... DEFINITION 8.2. The quotient ;:;e*,*(X) == ker r!fi jIm r!fi is called the Clifford cohomology of X. The general Hodge Decomposition Theorem gives us the following. We define a r!fi-laplacian d by and consider the associated harmonic spaces Hp,q(X) == (ker d) n qCtp,q(X)). Theorem 8.3. If X is compact, then ;:;e*,*(X) is finite dimensional and there are natural isomorphisms for all p,q. REMARK 8.4. If W is a holomorphic vector bundle over X, then the algebraic formalism above carries over easily to ct(X) ®<c Wand leads to Clifford cohomology groups ;:;ep,q(X;W) with coefficients in W. If X is compact, Hodge Theory applies to give finite dimensional harmonic spaces Hp,q(X;W) isomorphic to ;:;ep,q(X;W) for each p,q. §8. KAHLER MANIFOLDS 333 We assume from this point forward that X is compact. Our main point now is that the Clifford formalism of Chapter I leads easily to interesting structure in CIifford cohomology. The simplest operation is that of com-plex conjugation c: C£(X) -+ C£(X). One easily checks that c 0 f 0 c = -f and that c 0 ;Yt' 0 c = -;Yt'. This leads to the following "Serre Duality" Theorem: Theorem 8.5 (Michelsohn [1 ]). Complex conjugation in C£(X) induces iso-morphisms ;Yt'p,q(X) ;Yt'-P. -q(X) for all p,q. More generally, there are isomorphisms ;Yt'p,q(X;W) ;Yt'-P·-q(X;w*) for any holomorphic vector bundle W over X. Let *: C£(X) -+ C£(X) denote the C-linear antiautomorphism given by the transpose. Recall that for generators V1, ••• ,vp E TxX, we have *(v1 ••• vp) = vp' .. V1• Theorem 8.6 (Michelsohn [1 ]). The transpose antiautomorphism induces isomorphisms for all p,q. Clifford multiplication leads to the following interesting duality. Let (.,.) denote the usual L 2-inner product on sections. Theorem 8.7 (Michelsohn [1]). The C-bilinear pairing f3 defined on Hp,q(X) x HP. -q(X) by f3(cp,ljI) = (cp '1jI,1) is non-degenerate. In all of the above we have considered C£(X) as a left module over itself. Considering C£(X) as a right module produces a different Dirac operator (as in n.S) and a corresponding decomposition = + These are related to r!fi and by the antiautomorphism *: = * 0 r!fi 0 * and = * 0 0 *. This leads to cohomology groups ;Yt'*'*(X;Wr, and there are iso-morphisms 334 IV. APPLICATIONS for any holomorphic vector bundle W. The operators r!fi and {f); enter naturally when considering the sI2(C}-structures Proposition 8.8. Let W be a holomorphic vector bundle over X. Then the following relations hold on Ct(X) ® w: r!fi fl' + fl'r!fi = 0 r!fi fi> + fi> r!fi = r!fi = r!fi {f); fi> + fi> f!}; = 0 {f); fl' + fl' f!}; = {f); = {f); = 4d Theorem 8.9 (Michelsohn [1]). The cohomology groups admit an intrinsic filtration {O} S s s ... S where == {[tp] E fl'ktp = O}. Furthermore, the representation of the Lie algebra preserves the subspaces Jp,q(X;W) == ker(d) (\ (\ rCtp,q(X;W) = Hp,q(X;W) (\ Hp,q(X;Wr In the basic case when W is trivial it is shown that d = d and so we have the following: Theorem 8.10. The Lie algebra acting on rCt(X) preserves the subspace ker(d) = H*'*(X). Hence, there is a canonical sI2(C}-structure on the Clifford cohomology of X. The induced operators on cohomology are of the form = q fl':Hp,q(X) -----+ HP,q+2(X) fi>:Hp,q(X) -----+ HP,q-2(X). DEFINITION 8.11. A class tp E Hp,q(X) is called primitive if fl'tp = O. Theorem 8.12 (Michelsohn [1]). For all q > 0 and all p, the qth power fi>q:Hp,q(X) -+ HP,-q(X) is an isomorphism. Every tp E Hp,q(X) can be writ-ten uniquely as where tpk E HP,q + 2k( X) is primitive. §9. PURE SPINORS 335 One might imagine that there is a direct relationship between Clifford cohomology and the standard Dolbeault cohomology of X. This is indeed true. There is an explicit element in the Hodge automorphism group which relates the two. This is computed in Michelsohn [1]. It shows in particular that Hr-s• n-r-s(x;w) W(X;QS(W)), and so the Clifford cohomology is independent of the Kahler metric chosen on X. The role played by Clifford multiplication in H*'*(X) does however depend on the metric. §9. Pure Spinors, Complex Structures, and Twistors Thus far we have said relatively little about the role of spinors in local riemannian geometry although there remains much to be discovered in this area. There are, however, several places where spinors enter naturally. For example there is a notion due to E. Cartan of spinors of "pure" type which are related to almost complex structures. These spinors are also related to the calibrations introduced in Harvey-Lawson [3]. The interesting fact is that the simplest spinors give rise, under "squaring," to the most complicated differential forms. Whenever there is a parallel spinor on a manifold, there is a reduc-tion of the holonomy group. Via Bochner's method, spinors play a central role in constructing and understanding manifolds with reduced holonomy. This is particularly true of the exceptional cases of G2 and Spin7 holonomy. This section and the next one are devoted to a discussion of the topics just mentioned. We begin with the notion of a pure spinor. Fix [Rn with its standard inner product <".) and extend this metric C-linearly to en = [Rn ®Iffi C. Let Cfn = Cfn ®Iffi C be the associated Clifford algebra, and let $(; be a fundamental Cfn-module, i.e., an irreducible complex spinor space. For each spinor (1 E $(;, we consider the C-linear map given by jAv) == v . (1. (9.1) Generically, this map is injective. However, there are interesting spinors for which dim(ker j,,) > 0, and it is these that we shall now examine. We begin with the following definition. DEFINITION 9.1. A complex subs pace Vc cn is said to be isotropic (with respect to the bilinear form <".») if <v,w) = 0 for all V,W E V. We define a hermitian inner product ( ... ) on en by setting (v,w) = <v,w). Clearly, if V c en is an isotropic subspace, then V 1. V in this 336 IV. APPLICATIONS hermitian mner product. in partIcular, therefore, we have 2 dimICV n. (9.2) Lemma 9.2. For any non-zero spinor (J, the subspace ker j" is isotropic. Proof. If v . (J = W· (J = 0, then (v· W + W· v) . (J = -2< v, w)(J = 0. • DEFINITION 9.3. A spinor (J is pure if ker j" is a maximal isotropic sub-space, i.e., if dim(ker j,,) = [nI2]. Denote by P $ the subset of pure spinors in $IC' and denote by n the set of maximal isotropic subspaces of en. Both P$ and n are naturally acted upon by the group Pin., and the assignment (J ker j" gives a Pinn-equivariant map K: P$ ----+ (9.3) To see that K is equivariant note that for v E en and (J E SIC, we have g(v)·g·(J=g·v·g-1.g.(J=g.v"(J for all gEPinncC£nx. Hence, ker jg" = g(ker j,,) as claimed. REMARK 9.4. Note that in fact we can define a Pinn-invariant "filtra-tion" of the spinor spaces $0 c $1 C $2 C ... c $[n/2] = $IC (9.4) where $k = {(J : dim(ker j,,) [nI2] -k} for each k. The subset P$ = $0 -{O} consists exactly of the pure spinors. The subsets $k are not linear subspaces. As we shall see, there are orthonormal bases of $IC which con-sist entirely of pure spinors. At this point our discussion divides as usual into the two cases, n even and n odd. We shall discuss in detail only the even case. (The reader can easily carry over the arguments and constructions to the odd case.) From this point on we shall assume that n = 2m is an even integer, and further-more that [R2m is oriented. DEFINITION 9.5. An orthogonal almost complex structure on [R2m is an orthogonal transformation J: [R2m --> [R2m which satisfies J2 = -Id. For any such J, an associated unitary basis of [R2m is an ordered orthonormal basis of the form {e1,Je1, ... ,em,Jem}. Any two unitary bases for a given J determine the same orientation. This is called the canonical orientation associated to J. Let <Cm denote the set of all orthogonal almost complex structures on [R2m. It is easily seen that <Cm is a homogeneous space for the group 02m. It falls into two connected components <c;:; and <c;;; where <c;:; S02m/Um consists of those almost complex structures whose canonical orientation is positive (i.e., agrees with the given one on [R2m). §9. PURE SPINORS Associated to any J E <Cm there is a decomposition C2m = V(J) EB V(J), where 337 V(J) == {v E C2m: Jv = -iv} = {vo + iJvo: Vo E 1R2m}. (9.5) For any Vo E 1R2m we have that <vo + iJvo, Vo + iJvo> = <vo, vo> -<Jvo,1vo> + 2i<vo,1vo> = 0, and so the space V(J) is isotropic. Con-versely, given any m-dimensional isotropic subspace V C C2m, there is a unique J E <Cm so that V = V(J). To see this note that if one writes C2m = R2m EB iR2m, then V(J) is just the graph of J. Hence, there is an 02m-equi-variant bijection (9.6) which associates to J the isotropic subspace V(J). Let denote the component corresponding to Recall now that with respect to the complex volume element = ime1· .. e2m we have a decomposition = $;' EB into + 1 and -1 eigenspaces respectively. Lemma 9.6. If a E is pure, then either (J E $;' or (J E Proof. Let Vj = (1/J2)(ej + if), 1 j m, be a hermitian orthonormal basis of V = ker j". Since Vi 1. Vj for i =P j and Vi 1. Vj for all i,j, we see that e1Jl' ... ,emJm is an orthonormal basis of 1R2m. Note that (ej + ifj)a = ° => ieJj(J = a for each j. Hence, ime1fl··· emfma = = (J, and a E $;' or a E as claimed. • This lemma gives a decomposition P$ = pr II P$-of the pure spinor space into positive and negative types. Let IP(P$+) denote the projectiviza-tion of the positive pure spinor space, i.e., IP(P$+) = P$+ / where we say that a (J' if a = ta' for some t E C. Each of the spaces IP(P$± ), <c;; and are acted upon by Spin2m, in fact by S02m. Proposition 9.7. The maps a H K(a) and J H V(J) induce S02m-equi-variant diffeomorphisms and Proof. The second line follows immediately from the first by a change of orientation. The map V has already been shown to be an equivariant bijection between homogeneous spaces, and is therefore a diffeomor-phism. It remains only to show that the equivariant map K is a bijection. We construct K-1 as follows. Fix V E and let J E be the associated complex structure. Choose a unitary basis {e1,Je1, ... ,em,Jem} of 1R2m and set 338 IV. APPLICATIONS as in §S. Define and (9.7) and note that by (S.3): All of the elements wj' Wk for 1 j,k n, commute in Cizm. (9.S) Furthermore, by (S.3) these elements satisfy the identities and WjWj = WjWj = 0 ejwj = wjej for all cp,1jJ E $<r:-(9.9) (9.10) (9.11 ) (9.12) (9.13) for each j, where (v,w) denotes any Spinzm-invariant hermitian inner product in $<r:-' These identities easily imply the following. Fix any j and let W be a linear subspace invariant under multiplication by ej and Jej. Then there is a hermitian orthogonal direct sum decomposition W= Wj where and and where PCj: W W is defined by Pe/w) = Rj·W. By (9.12) we see that Pej maps to Wj isomorphically and so dim = dim Wj. We first apply this process with W = $<r:-and j = 1 to obtain a decom-position $<r:-= $1 EB $'1 where $1 = ker Pe, is invariant under ez,Jez, ... ,em' Jem• We next set W = $1 and j = 2 to obtain a decomposition $1 = $z EB where $z = ker(pc,) n ker(p.,) and dim<r:-$z = 2m-Z. Proceeding inductively we eventually construct (9.14) The complex volume form W<r:-= imeJe1 •.• emJem has the value + 1 on $rn because: ep = 0 = -iejJep = (1. Therefore, $m c $:; We clearly have that V(J) = ker j" for (1 E $rn. Hence, $m is independent of the choice of unitary basis and the map V H [$m] gives the desired map K-1 •• §9. PURE SPINORS 339 Globalizing this result to manifolds gives the following: Proposition 9.8. Let X be an oriented riemannian manifold of dimension 2m. Then the orthogonal almost complex structures on X, whose canonical orientation is positive, are in natural one-to-one correspondence with cross-sections of the projectivized bundle IP(P$+) of positive pure spinors on X. In particular, X is Kiihler if and only if there is a parallel cross section of IP(P$+). Note. IP(P$+) is an SOzm-bundle and is globally defined whether or not X is a spin manifold. DEFINITION 9.9. The bundle r(X) == IP(P$+) is called the twistor space of X. The total space of r(X) carries a canonical almost complex structure defined as follows. Note to begin that the fibres ofthe projection n: r(X) X are naturally homogeneous complex manifolds SOZm/Um)' The rie-mannian connection of X determines a field of horizontal planes :Ye for n, that is, it determines a canonical decomposition T( r(X)) = 1/ EB :Ye where 1/ is the field of tangent planes to the fibres. As noted, 1/ has an almost complex structure integrable on the fibres. The bundle :Ye has a "tautological" almost complex structure defined, via the identification n*::Ye J -> T *X, to be the structure J itself. (Here we consider r(X) as the bundle of positively oriented almost complex structures on the tangent spaces of X.) In spinor terms the almost complex structure on r(X) can be written as follows. Note that :Ye = n*TX and so $a!(:Ye) = n*$a!(X). Now whenever, a vector bundle is pulled over its own projectivization, a line bundle splits off tautologically. Let A,>\" denote the tautological line bundle in n*$a!(X) and let Ail' denote the line bundle in $a!(1/) corresponding to the complex structure on the fibres. Then the line bundle A = Ail' ® A,>\" C $a!(1/) ® $a!(:Ye) C $a!(1/ EB :Ye) = $a!(r(X)) gives the projective pure spinor field defining the almost complex structure on r(X). We now consider the question of integrability. Let J be an almost com-plex structure defined over an oriented riemannian 2m-manifold X, and let Vc TX @ C be the associated field of totally isotropic m-planes. The structure J is said to be integrable if it comes from an honest complex structure on X, i.e., if there is a holomorphically related system of local complex coordinate charts so that in each coordinate system z l' ... 'Zm' the field V is given by V = spanc {a/az1, ••• ,a/azm}' 340 IV. APPLICATIONS The fundamental theorem of Newlander and Nirenberg states that J is integrable if and only if J qV) (9.15) i.e., if and only if for any pair of complex vector fields v,w with values in V, the Lie bracket [v,w J again has values in V. Suppose now that a is a (locally defined) pure spinor field whose pro-jective class defines J. Then since V == ker j" we see that locally qV) = {v E qTX @ q: v'a = a}. Proposition 9.10. The almost complex structure defined by a pure spinor field a is integrable if and only if a satisfies the equation (9.16) for all v,w E qV) where V = ker j". Proof. Fix v,w E qV) and differentiate the condition W· a = a with respect to v. This gives the equation a = Vv(w'a) = (Vvw)'a + w·Vva. Interchanging v and wand then subtracting shows that a = (V vW -V wV) . a + w . V va -v . V wa. Since Vvw -Vwv = [v,wJ, we conclude that v'Vwa = w'Vva<=> [v,w la = a-<=> [v,w J E qV). • Note that the equations (9.16) depend only on the projective class of a, since for any smooth function f, we have (v.Vw -w'Vv)(fa) = f(v' V w -W' V v)a. The equations (9.16) also depend only on the conformal class of the underlying riemannian metric. Proposition 9.10 has an interesting reformulation in terms of the twistor space r(X). As mentioned above, r(X) carries a canonical almost complex structure. Now a Cl-map between almost complex manifolds f: (X,J x) -+ (Y,J y) will be called holomorphic ifits differential f * is everywhere J-linear, i.e., if f * 0 J x = J y 0 f *. Theorem 9.11. (Michelsohn [3J) Let X be an oriented (even-dimensionaQ riemannian manifold with an almost complex structure determined by a pro-jective spinor field s E q r(X)). Then this almost complex structure is inte-grable if and only if s is holomorphic. Note. More succinctly one could say that cross-sections of r(X) induce almost complex structures, and holomorphic cross-sections induce integrable ones. However, the condition that a cross-section s be holomorphic is not §9. PURE SPINORS 341 linear since the complex structure on X depends itself on s. It is rather of the form: ass = 0, where as denotes the Cauchy-Riemann operator associated to s. In this form the equation is reminiscent of many basic equations in geometry, such as the minimal surface equation fo,/ f = 0, the Yang-Mills equation AV RV = ° (cf, Lawson [1 J, [2J), and the condition for balanced metrics (d*)"'w = ° (cf. Michelsohn [2J). Proof of Theorem 9.11. It suffices to work locally, so we may choose a positive spinor field (J E r(p$+) with [(J J = s. Let V = ker j" be the field of (0,1 )-subspaces as above. The condition that s = [(J J be holomorphic is equivalent to the condition that V v(J = Av(J for all v E qV) (9.17) where )'v is a function depending on v and where V is the riemannian connection on P$+. We must show that conditions (9.16) and (9.17) are equivalent. Let {el" .. ,en} be a local hermitian frame field for V, and recall from (9.14) that the complex line field s is given by s = "i ker(Il,,). Now equation (9.16) and the fact that e] = ° show that -elFtp = elS,p = eijVei(J = ° for all I,j. Therefore we have ejViip E " ker(llii.) = s c $+, and since ejViip E $-, we conclude that ejVep = ° for allj. (9.18) Condition (9.16) says that the CCO(X)-bilinear form on qV) given by f3(v,w) = wVv(J is symmetric. Hence, (9.18) f3 = 0, and we conclude that V v(J E n ker(lle) = [(J J for all v E q V). Therefore, (9.16) (9.18) (9.17). On the other hand if (9.17) holds, then wVv(J = ° for all v,w E qV) and so (9.16) holds. • REMARK. Theorem 9.11 could be modified by changing the almost com-plex structure on r(X) to be the one induced by s itself, i.e., by lifting Js uniformly to all horizontal spaces above each point (instead of twisting along the fibre). At the point SE r(X) these two complex structures agree. In low dimensions these constructions are particularly interesting for the following reason: REMARK 9.12. In dimensions 2m :s;; 6 every non-zero positive (or negative) spinor is pure, i.e., P$± = $i -{a}. This is simply because the group Spinzm acts transitively on the unit sphere in $i in these dimensions. (Transitivity also holds for 2m = 8 if one restricts to the real spinors $±, but it does not hold for the complex ones.) As a consequence we have the following fact: Proposition 9.13. On an oriented manifold of dimension four or six, every nowhere-vanishing spinor field uniquely determines an almost complex struc-ture (via (9.3)). 342 IV. APPLICATIONS Let X be an oriented riemannian 4-manifold. A 2-form cp on X is said to be self-dual (of anti-self-dual) if *cp = cp (*cp = -cp respectively). Since the Hodge *-operator satisfies (*)2 = 1, there is an orthogonal decom-position A2(X) = A+ EB A_ where A± = {cp E A2(X): *cp = ±c p } . Now the twistor space r(X) = IP(P$+) = 1P($t) can be identified with the bundle of unit self-duaI2-forms by associating to a complex structure J its Kiihler form WJ E A+ (given by .j2wAU,W) = <JU,W»). It is a 2-sphere bundle over X with a canon-ical almost complex structure. The following theorem has played an important role in understanding the structure of the self-dual Yang-Mills equations over riemannian 4-manifolds. The seminal ideas for these applications were due to R. Penrose and R. Ward (see Ward and Wells [1]). A riemannian 4-manifold is called self-dual if its Weyl conformal tensor W is self-dual, i.e., if * W = Wwhere W is considered as a 2-form with values in Hom(TX,TX). Theorem 9.14 (Atiyah-Hitchin-Singer [1 ]). The canonical almost complex structure on the twistor space of a riemannian 4-manifold X is integrable if and only if X is self-dual. We refer the reader to the original paper for details. The arguments in-volve the so-called "twistor equation" which is roughly the complement of the Dirac equation. One class of manifolds whose twistor space always has an integrable complex structure is the set of manifolds of constant sectional curvature. For the case of the sphere s2m, one has r(S2m) J 2m+ 1 S02m+ dU2m S02m+2/Um+ l' The map n: J2m+ 1 s2m associates to an isotropic m-plane V C c2m+ 1 the unit vector e E 1R2m+ 1 such that C2m+ 1 = V EB V EB Ce. (This determines e up to sign. Since 1R2m + 1 = Re{ V EB V} EB IRe, this sign is determined by fixing an orientation on 1R2m+ 1.) In two beautiful papers, E. Calabi [2], [3] used r(S2m) to classify the harmonic maps from S2 to sn. He first showed that a harmonic map cp: S2 sn must lie in a geodesic subsphere of even dimension. He then showed that any harmonic map cp: S2 s2m lifts to a holomorphic map <1>: S2 1 which is horizontal for the projection n. We now return to the question of almost complex structures. Suppose that (J E q$t) is a field of pure spinors over a 2m-manifold X. Then as we have seen, the projective class [(J] E qlP(P$t)) determines an almost complex structure on X. However, the field itself carries more information. It determines a trivialization of the canonical bundle AcV associated to the structure. To see this we need the following lemma. Let $e be the ir-reducible complex representation for C£2m' For any (J E $e we define the §9. PURE SPINORS 343 isotropy group of a by G" == {g E Spinzm: ga = a}. Lemma 9.15. For any pure spinor a E p$+, one has that G" = SUm where SUm c Spinzm is conjugate to the lifting of the standard embedding SUm c SOZm' Proof. Set V = ker j" c eZm as above, and note that if ga = a, then g(V) = V, where by definition g(v) = g' V· g-l = Ady(v). Now V = {x + iJ x: x E [RZm} where J is the complex structure determined by a. The action of Spinzm on eZm = [RZm EB i[RZm given by Ad, preserves real and imaginary parts. Therefore, if g E G", then g(x + iJx) = g(x) + ig(Jx) E V for all x, and so g(Jx) = J(g(x)) for all x E [RZm. Hence, the image of G" under the covering map Ad: Spinzm ---+ SO Zm is contained in the subgroup Um = {y E SOZm: yJ = Jy}. We now show that Ad(G,,) S SUm. Given g E G", there exists a unitary basis {e1,Jel"" ,em,Jem} for [RZm which "diagonalizes" g, i.e., so that g(ek + iJek) = eiZ8k(ek + iJek) for real numbers 0 2 Ok < nand 1 2 k 2 m. This means that g = ±TI (cos Ok -sin OkekJed· k Now, (ek + iJ eda = 0 = eka = -iJ eka = eJ eka = -ia for each k. Hence, ga = ±TI (cos Ok + isin 0k)a = ±eiI:8ka. k Since ga = a, we have 2 L Ok == 0 (mod 2n) and so detdg) = ei2I:8k = 1 as claimed. The above argument shows that for any g E Ad -l(SUm) we have that ga = ± a. Since SUm is simply-connected, Ad -l(SUm) has two connected components. The identity component fixes a, the other component con-tains - 1 and hence cannot fix a. • Alternatively we could consider the Lie algebra g" = {<p E AZ[RZm: <pa = O} of G". For any <p E g", there is a unitary basis of [RZm so that <p = I AkekJek• Since ekJeka = -ia, we have that <pa = -i I Aka and so I Ak = O. This condition characterizes SUm in Um• 344 IV. APPLICATIONS It is incidentally not true that: G" SUm = (1 is pure. For example, in dimension eight we could take (1 = (10 + (10 where (10 and (10 are pure and determine almost complex structures J 0 and -J 0 respectively. For any 2m-dimensional spin manifold X, Lemma 9.15 implies Proposition 9.16. Each globally defined pure spinor field (J on X determines a unique reduction of the structure group of X to SUm. Proposition 9.17. X admits a pure spinor field which is parallel if and only if X is Kithler and Ricci flat. Proof. If (1 is. parallel, then the holonomy group of X is contained in Ad(G,,) = SUm. This means that the almost complex structure is parallel and the canonical bundle is flat. Conversely, if X is Kiihler and Ricci-flat, then $e = and both Ag and Ac provide parallel pure spinors. _ Corollary 9.18. Suppose dim(X) = 4 or 6. Then X admits a non-zero paral-lel spinor field if and only if X is Kithler and Ricci-flat. Proof. If (J E r($d is parallel, then each component (J± E r($t) is parallel, and as noted in 9.12 each component is pure. _ Using 1I.8.10, we now conclude the following: Theorem 9.19. Let X be a compact spin 4-manifold with A(X) #-0. Then any riemannian metric on X with scalar curvature /( ° is Kithler and Ricei-flat. In fact, any such metric admits a parallel quaternion structure, i.e., parallel orthogonal complex structures I and J with IJ = -JI. Proof. If /( 0, then by 11.8.8 and 11.8.10 and the Index Theorem, we see that X carries a parallel spinor field (J. Each component (1+ and (J-is also parallel. Assume that (J+ #-0. Since $::' is an IHl-line bundle (and the lHl-structure is preserved by the connection), we see that (1+, i(1+, j(1+ and k(J+ give a complete parallelization of $::', that is, if one positive spinor is parallel, then all positive spinors are parallel. We conclude that X car-ries a family 1P($.n S2 of parallel complex structures, each of which makes X into a Kiihler manifold. _ It should be pointed out that by Yau's solution of the Calabi Conjec-ture [2J, [3J, we know that Ricci-flat Kiihler manifolds exist in all dimensions. REMARK 9.20. A theorem similar to 9.19 was asserted in Hitchin [1] for all dimensions. It was based on an incorrect calculation that G" SUm for all (J. Indeed, one finds counterexamples already in dimension eight. This will be discussed in the next section. §1O. REDUCED HOLONOMY AND CALIBRATIONS 345 We conclude this section with a remark about what happens in odd dimensions. Let $e be one of the two irreducible complex spinor spaces for C£Zm + 1 and let (J E $e be a pure spinoL The totally isotropic subspace V = ker j" c CZm + 1 has dimension m and we get a hermitian orthogonal decomposition cZm + 1 = V EB V EB Ce where e E IRZm+ 1. This gives an orthogonal decomposition IRZm+1 = VIffi EB IRe where VIffi @ C = V EB V (i.e., VIffi has a complex structure). Assuming lIell = 1, we see that e E V is determined up to sign. A choice for e cor-responds to a choice of orientation on IRZm+ 1. In summary then, each pure spinor (J E $e determines a pair (e,J) where e E 1R2m+ 1 is a unit vector and J is an orthogonal almost complex structure on e.L. A global pure spinor field (J on a spin (2m + I)-manifold X determines, as above, a reduction of the structure group to SUm. If V(J = 0, then X splits locally as a rieman-nian product X X 0 x IR where X ° is Kiihler and Ricci-flat. In dimensions three and five every non-zero spinor is pure. §10. Reduced Holonomy and Calibrations The holonomy group of a connected complete riemannian n-manifold X is defined as follows. Fix a point x E X and to each piecewise smooth loop y based at x, let hy: TxX --> TxX be the orthogonal transformation given by parallel translation around y. These transformations form a subgroup :Yex c O(TxX) On whose conjugacy class as a subgroup of On is inde-pendent of the choice of base point x. This conjugacy class :Ye(X) is called the holonomy group of X. Its identity component :Ye(X)O is called the local holonomy group of X (see Kobayashi-Nomizu [1].) If X = yk X zn-k is a riemannian product, then one easily sees that :Ye(X) = :Ye(Y) x :Ye(Z) C Ok X 0n-k C On, and if n1(X) = 0, then by a theorem of de Rham the converse is true. It is therefore sensible to consider manifolds which are irreducible. This means the universal covering is not a riemannian product. In 1955 Marcel Berger [1] classified the possible holonomy groups for irreducible riemannian manifolds. There has been subsequent work refin-ing Berger's list (see Bryant [1], [2] for a brief history). We now know that if X is irreducible and not locally symmetric, then :Ye(X)O must be one of the following: SOn (the generic case) Um (n = 2m; Kiihler) SP1 . SPm (n = 4m; Quaternionic Kiihler) SUm (Kiihler and Ric = 0) Spm (Quaternionic Kiihler and Ric = ° 346 IV. APPLICATIONS or must be one of the two exceptional cases: G2 (n = 7) Spin7 (n = 8) where by definition SPI' SPm = SPI X SP"'/£'2 and £'2 is generated by ( -1, -1). Metrics of each of the above types are known to exist. We have already seen that: 3f(X) c Um <===> there exists a parallel projective pure spinor field on X 3f(X) c SUm <===> there exists a parallel pure spinor field on X. In this and the next section we shall show that parallel spinors are also related to the existence of SPm, G2, and Spin7 holonomy metrics. These results are due to the authors and Reese Harvey (see Michelsohn [3]). The fundamental elementary point is the following: Proposition 10.1. Let X be an n-dimensional (riemannian) spin manifold on which there exists a globally parallel spinor field a. Then at any point x the holonomy group satisfies :Yt'x c Gax where Gax = {g E Spin(TxX): gax = ax}. In other words, :Yt'(X) c Ga (10.1) where Ga denotes the conjugacy class of Gax in Spinn (which is defined independently of x because a is parallel). Conversely, if(10.1) is satisfied for a spinor ax at some point x, then (Ix extends to a globally parallel field a on X. Proof. If Va = 0, then the holonomy transformation hy E Spin(TxX) generated by parallel translation around any loop y must preserve a. Hence, :Yt'x c G ax' Conversely, if :Yt'x c G ax' then parallel translating a x from x to y is path-independent and defines a globally parallel field. • We know that on any spin manifold with A #-0, any metric with zero scalar curvature has parallel spinor fields. Therefore in light of Proposition 10.1, it would be interesting to understand the structure of the isotropy groups Ga c Spinn• For n 6 we have already seen in §9 that Ga c SU[n/2l' so the first interesting case occurs when n = 7. Recall that the irreducible real representation $ of Spin7 has dimension 8. Furthermore, Spin7 acts transitively on the unit sphere giving a diffeomorphism S7 Spin7/G2 (see I.8.2 forward). Hence, all non-zero spinors are essentially equivalent, and we have the following: Proposition 10.2. Let $ be the irreducible real spinor space for Spin7' Then for any non-zero a E $, §1O. REDUCED HOLONOMY AND CALIBRATIONS 347 Corollary 10.3. A 7-dimensional spin manifold has holonomy c Gz if and only if it carries a non-trivial parallel spinor field. The group Spin8 has three irreducible 8-dimensional real representations A + : Spin8 SO($+) A - : Spins SO($-) related by the triality automorphism (see 1.8). Each is transitive on the unit sphere and therefore determines a unique conjugacy class of sub-group. That is, in each representation the isotropy subgroups of any two non-zero elements are conjugate. Proposition 10.4. There are three distinct conjugacy classes of subgroups -:? Spin7 c Spins -0 Spin; defined as the isotropy groups Spin7 = Gv, Spini = Ga± for non-zero elements v E [R8 and O'± E $±. These three conjugacy classes are cyclically permuted by the triality outer automorphism of Spin8' Furthermore, the conjugacy classes Spini are conjugate in the enlarged group Pin8. In fact, = e' Spin; . e-1 (10.2) for any e E S7 c Pin8 (where S7 c [R8 C C£8 are the standard embeddings). The covering homomorphism Ad: Spin8 SOs gives embeddings Ad: Spini SOs (10.3) which are not conjugate in S08 but which are conjugate in Os. Proof. Let Z 7Lz EB 7Lz denote the center of Spins. Then the non-conjugacy of Spin7' and Spin; follows from the fact that they intersect Z in three distinct subgroups (cf. 1.8). Since the representations are permuted by triality, so are these subgroups. Given e E S7 c [Rs, and O'E $+, we have that ea E $-and clearly that Gea = eGae-1. This proves (10.2). To see that (10.3) is an embedding, note that (ker Ad) ("\ Spini' = {1}. To see that Ad(Spinn and Ad(Spin;) are not conjugate in SOs, note that: = Ad(Spin;) for some g E SOs would imply that y = Spin; for yE Spins with Ad(y) = g. On the other hand (10.2) shows that Ad(Spini') are conjugate by elements Ad(e) (= reflection in the hyperplane eJ.) in 08' • The embeddings (10.3) give a single conjugacy class of the subgroup Spin7 c Os and it is exactly this representation which appears in Berger's list. Consequently we have 348 IV. APPLICATIONS Corollary 10.5. An 8-dimensional spin manifold has holonomy c Spin7 if and only if it carries a non-trivial parallel spinor field. Note that if (J is a parallel real spinor field on an 8-manifold, then (J decomposes (J = (J+ + (J-where both (J+ and (J-are parallel. If both (J+ and (J -are non-trivial, we get a further reduction of the holonomy group to Spin; n Spin';-Gz. (To see this, note simply that under triality we have Spin; n Spin';-Spin; n Ad -1(S07) G" for (J as in Proposi-tion 10.2.) Recall that the complex spinor spaces $i for Spins are just the complexifications of the real ones. Write $;' = $+ EB i$+ and consider an element (J <c = (J 0 + i(J 1 EO $;'. Then for g EO Spins, g(J <c = (J <c if and only if g(Jo = (Jo and g(J1 = (J1, and so G"c = G"o n G", If (J 0,(J 1 EO $ + are linearly dependent (over IR), then G "c Spin7. If they are linearly independent, then G"c Spin6 SU4. In this second case G"c remains unchanged if we replace (J <c by 8 <c = 80 + i81 where {8 0,8 d is any orthonormal basis of spanD;l((JO,(J1)' Up to real scalars this orthonor-mality is equivalent to the fact that <8<c,8<c) = 0. One easily checks that in this dimension, the pure spinors are exactly those which are isotropic with respect to the natural real structure on $;'. REMARK. It should be pointed out that R. Bryant [1], [2] has found a number of beautiful examples of manifolds with Gz and Spinrholonomy. We shall now show that the propositions above give a simple neces-sary and sufficient condition for the existence of a topological Gz or Spin7-structure. By this we mean the following. Let X be a differentiable n-manifold and fix a Lie group G c GLn(IR). Then by a topological G-structure on X we mean a topological reduction of the structure group of T X from GLn(lR) to G. This is defined to be a G-equivariant embedding PG c P GdX) of a principal G-bundle into the frame bundle of X. Reducing the structure group to On C GLn(lR) is equivalent to choosing a riemannian metric on X. If G c On is a closed subgroup, then reducing the structure group from On to G is tautologically equivalent to finding a cross-section of the bundle Po(X)/G ---+ X whose fibres are the homogeneous spaces On/G. Theorem 10.6. Let X be a differentiable 7-manifold. Then X carries a topo-logical Gz-structure if and only if it is a spin manifold. Proof. If the structure group of TX can be reduced to Gz, then because n1GZ = 0, X is a spin manifold (see Remark 1.1.10). Conversely, suppose §1O. REDUCED HOLONOMY AND CALlBRA nONS 349 X is spin and let $ be the irreducible real spinor bundle of X. Then since fibre-dimeS) = 8 > dim(X), there exists a nowhere vanishing cross-section a of $ (cf. Milnor-Stasheff [1 J). By Proposition 10.2 we can identify the unit sphere bundle in $ with PSPin(X)jGZ' Therefore the normalized sec-tion a/llall gives us a Gz-reduction. • Note that if X is a spin 7-manifold, then the topological Gz-structures on X are in one-to-one correspondence with global spinor fields a E re$) such that Ilall == 1. The story for Spin7-reductions is quite similar. Theorem 10.7. Let X be a differentiable 8-manifold. Then X carries a to-pological Spin7-structure if and only if X is spin and either X($+) = 0 or X($-) = 0, where $± denote the irreducible real spinor bundles on X. Equiv-alently, X carries a Spin7-structure if and only if wl(X) = wz(X) = 0 and for an appropriate choice of orientation on X we have that (10A) Note. Reversing the orientation of X leaves the Pontrjagin classes un-changed but reverses the sign of X(X) in H8(X). Thus, condition (lOA) could be replaced by the condition that for any orientation on X, one of the two equations holds in H8(X). Note that for manifolds of the type X8 = Si X y7, equation (lOA) is automatically satisfied. Furthermore, X8 is spin iff y7 is spin, and Theo-rem 10.6 applies. There are other special cases of some interest. Corollary 10.8. Let X be a complex manifold of dimension 4. Then X carries a topological Spin; -structure if and only if cl[ci -4c1cZ + 8c3J = 0 orollary 10.9. Let M and N be compact spin 4-manifolds. Then the pro-uct X = M x N carries a topological Spin7-structure if and only if 9sig(M)sig(N) = 4X(M)X(N) n particular, M x M has such a structure if and only if 3sig(M) = ± 2x(M). roof of Theorem 10.7. If X is spin and X($+) = 0, then there is a nowhere anishing section a E re$+) which, via Proposition lOA, reduces the struc-ure group of Spin;' In fact, by lOA we have the identification (10.5) 350 IV. APPLICA nONS where S($ +) denotes the unit sphere bundle in $ +. Hence, the Spin;-reductions are in one-to-one correspondence with spinor fields (J E r($+) such that II(JII == 1. If X($-) = 0, the analogous discussion holds. Suppose conversely that the structure group of TX has a Spin7-reduction, i.e., suppose there is a Spin7-equivariant embedding PSPin7 '-+ P08(X) of a principal Spin7-bundle into the orthogonal frame bundle of X (for some riemannian metric). Since Spin7 is simply-connected, this determines a spin structure: PSPin8(X) -+ PS08(X) c P08(X), in which PSpin7 lifts to a Spin7-equivariant embedding PSpin7 C PSPin8(X), which in turn corresponds by (10.5) to a cross-section of S($+). Hence, X($+) = 0. Note. The reader may find the lack of ambiguity in this last paragraph somewhat unsettling. If so, let us examine the argument more closely. In saying that PSpin7 C P08(X) is "Spin7-equivariant" we must specify an embedding Spin7 c Os. This choice is important only up to conjugation in Os. (To see this, fix 9 E Os and consider the map g: P08(X) -+ P08(X) which converts the given Os-action to the conjugated "g Osg -1" -action.) Now the two subgroups Ad(Spin·n are conjugate in Os but not in SOs. We choose an embedding whose class in SOs is Ad(Spin;). Since Spin7 is connected, the embedding PSPin7 <=-+ P08(X) clearly fixes an orientation on X, i.e., it selects a sub-bundle PS08(X) c Po.(X). With this choice of orientation, PSPin7 lifts to PSPint C PSpin8(X). Suppose on the other hand that we had chosen an embedding Spin7 c SOs of Ad(Spin';-)-type, which differed from the first choice by an element 9 E Os (with det 9 = -1). Our given reduction/: PSPin7 <=-+ po.(X) must now be conjugated to go/ 0 g-l: PSpin7 <=-+ P08(X). This new reduction chooses the opposite orientation on X and lifts to a Spin';--subbundle of PSPin8(X). Theproofthatx($±) = P1(X)Z -4pz(X) ± 8X(X) is a good exercise in the algebra of characteristic classes, which we leave to the reader. • It is an interesting fact that spinors in dimensions seven and eight are also related to certain exceptional geometries of subvarieties introduced in Harvey-Lawson [1], [2], [3]. The fundamental idea is this: Let <p be a differential p-form on a riemannian manifold X normalized so that (10.6) for every oriented tangent p-plane P on X. This is equivalent to the con-dition that fz <p ;£ vol(Z) (10.7) for every oriented p-dimensional C1-submanifold Z of finite volume in X. If such a manifold Z has the property that fz <p = vol(Z) (10.8) §1O. REDUCED HOLONOMY AND CALIBRA nONS 351 then Z is called a tp-submanifold. These creatures are interesting for the following reason. Let Z c X be a compact oriented p-dimensional sub-manifold, possibly with boundary. We say that Z is homologically volume-minimizing ifvol(Z) vol(Z') for all such Z' with the property that Z -Z' is a boundary (of some singular (p + I)-chain in X). When Z is not com-pact, it is called homologically volume-minimizing if this property holds for every compact subdomain-with-boundary in Z. Proposition 10.10. Suppose cp satisfies (10.6) and dcp = O. Then every cp-submanifold is homologically volume-minimizing in X. Proof. Let Z be a compact cp-submanifold and Z' a competitor with Z -Z' = OC for some singular chain c. Then by (10.7), (10.8) and Stokes' Theorem, we have vol(Z) = fz cp = fz. cp vol(Z'). • This proposition shows that every cp-submanifold is in particular a minimal submanifold, i.e., its mean curvature vector field is identically zero. It is therefore roughly as regular as X itself. For example, if X is real analytic, so is any cp-submanifold. The above definition extends im-mediately from submanifolds to integral currents (see Harvey-Lawson [3]). The "cp-subvarieties" can then have singularities, although they remain homologically volume-minimizing. DEFINITION 10.11. If cp satisfies (10.6) and dcp = 0, then cp is called a calibration. The pair (X,cp) is a calibrated manifold and the family of cp-submanifolds is the associated calibrated geometry. Calibrations are interesting for the special cases, not the general one. We know that few topological reductions of the structure group of a mani-fold are "integrable" in the sense that they are holonomy reductions. Simi-larly, few calibrations are "integrable" in the sense that the associated geometry of submanifolds is large. However, many interesting examples do exist. EXAMPLE 10.12 (Kiihler geometry; Un-case). Let X be a Kiihler mani-fold of complex dimension n. Fix p, 1 p n, and consider the 2p-form 1 cp = -(}JP P p! where (}J is the Kiihler form of X. Then Cpp is a calibration and the asso-ciated family of cpp-submanifolds (cpp-subvarieties) consists exactly of the complex analytic submanifolds (subvarieties) of dimension p in X. Propo-sition 10.8 says that every complex analytic subvariety is homologically volume-minimizing in X (a result due to H. Federer [1]). 352 IV. APPLICA nONS EXAMPLE 10.13 (special lagrangian geometry; SUn-case). Let X be a Ricci-flat Kiihler manifold of complex dimension n, and assume that the canonical line bundle x == T* X is trivial. Consider the n-form A = Re(Q) where Q is a parallel holomorphic n-form (i.e., section of K). Then A is a calibration and the associated geometry is called special lagrangian. It is shown in Harvey-Lawson [3] that the speciallagrangian subvarieties are as plentiful as the complex sub varieties. There are three rich exceptional geometries which appear in dimensions seven and eight. EXAMPLE 10.14 (associative and coassociative geometry; Gz-case). Con-sider euclidean space = Im q), where CO denotes the Cayley numbers, and define a parallel 3-form qJ on by setting qJ(x,y,z) = <x' y, z) Then qJ is a calibration with a rich geometry of submanifolds. It is called the associative geometry because the submanifolds are also defined by the vanishing of the associator [x,y,z] = (xy)z -x(yz). The dual form IjJ = *qJ gives a geometry of 4-folds called the coassociative geometry. These geometries exist (and are rich) on any riemannian 7-manifold with Gz-holonomy. Note that qJ and IjJ are clearly Gz-invariant since they are defined using Cayley multiplication. In fact qJ and IjJ are the only non-trivial exterior p-forms, 0 < p < 7, which are Gz-invariant. Any riemannian 7-manifold with Gz-holonomy carries parallel forms qJ and IjJ which at each point are equivalent (over S07) to those given above. Furthermore, in any such manifold the geometry of associative 3-folds and coassociative 4-folds is vast. In particular, every analytic submanifold of dimension two is contained in an associative submanifold (whose germ along the sur-face is uniquely determined). EXAMPLE 10.15 (Cayley geometry; Spin7-case). Consider euclidean space = CO and define a parallel 4-form <I> on by setting <I>(x,y,z,w) = <x x y x z, w) where the triple cross product is defined by x x y x z = !{x(yz) -z(yx)}. Then <I> is a calibration which, as shown in Harvey-Lawson [3], has a rich geometry of submanifolds, called the Cayley geometry. It includes as special cases all complex submanifolds and all special langrangian sub-manifolds for a 6-dimensional family of complex structures on If we consider c as Im CO = l-L C CO, then <I>11I'!7 = IjJ and 1 L <I> qJ. §1O. REDUCED HOLONOMY AND CALIBRA nONS 353 The form <I> is Spin7-invariant. It is, in fact, the unique Spin7-invariant pjorm on for 0 < p < 8. Any riemannian 8-manifold with Spin7-holo-nomy carries a parallel 4-form of this type and has a large geometry of Cayley submanifolds. The forms <I> and qJ above were first discovered by Bonan [1 J. There is a direct and intimate relationship between spinors and cali-brations. It comes about by what can be roughly termed the process of "squaring." We shall examine this process to the point that meets our needs here. For further discussion the reader is referred to the book of Reese Harvey [1 J. Let n = 2m be an even integer and recall the fundamental algebra isomorphism (10.8) We introduce on $e a Pinzm-invariant hermitian metric (.,.) by which we can identify $t with $e. ($e is the same real vector space as $e but with the opposite complex structure, i.e., with scalar multiplication by t in $e replaced by t.) The identification $e ® $e Homd$e,$d is then determined by associating to a pair of spinors (J b(J Z EO $e, the elementary endomorphism L(Tl ® (Tlr) == (r, (J z)(J 1 Inverting (10.8) now gives the following: Proposition 10.16. There is an isomorphism for rEO $e. $e ® $<[; C£Zm which associates to a pair of spinors (J b(J Z EO $<[;, the unique element, qJ EO C£zm such that qJ. r = (r, (J z)(J 1 for all r EO $<[;. Under this isomorphism we have for each IjJ EO C£zm, an identijication of endomorphisms and where LrjJ and RrjJ denote respectively left and right multiplication by IjJ on C£Zm, and where 1jJ* == rJ.(ljJt). Via 1.1.3 the isomorphism above determines a canonical identijication (10.9) which is Pinzm-equivariant, where 9 EO Pinzm acts on $e ® $<[; by I1g ® I1g, and on A*CZm by the complexijication of the standard representation ofOzm on forms. Proof. All assertions but the last are easily checked. For the last, one need only recall that rJ.(gt) = 9 -1 for 9 EO Pinzm and therefore I1g ® I1g = I1g ® 1)* Adg the standard representation of 9 on A *Czm. 354 IV. APPLICA nONS There is a corresponding real version of this result. For simplicity we shall consider only the cases where C£n is a real matrix algebra. Suppose n == 6 or 8 (mod 8) and let $ denote an irreducible real C£n-module. Then Clifford multiplication gives an isomorphism J1.: C£n --+ HomD;l($, $). When n == 7 (mod 8), there are two distinct isomorphisms J1.± :C£! --+ HomD;l($±,$±) where $+ (and $-) are the irreducible real C£n-modules on which the volume form w acts by Id (and -Id respectively), and where C£! = (1 ± w) . C£n' Inverses to these maps are given explicitly as follows: Proposition 10.17. Let $ be an irreducible real C£n-module where n == 6 or 8 (mod 8), and assume $ is provided with a Pinn-invariant inner product < .,. ). Then assigning to a pair O';r EO $ the endomorph ism Le; ® k) = <', r)O' deter-mines an isomorphism (10.10) under which J1.", ® J1.rjJ* L", ® RrjJ where 1jJ* = rx(ljJt). Via I.1.3 this gives a canonical Pinn-equivariant isomorphism (10.11) where g EO Pinn acts on $ ® $ by J1.g ® J1.g• The isomorphism (10.11) decomposes into two isomorphisms: 02$ Efl AP[Rn P ;: 3 or 4(mod 4) A2$ Efl AP[Rn p;: 1 or 2 (mod 4) (10.12) (10.13) under the canonical decomposition $ ® $ = (02$) EB (A2$) into symmetric and skew-symmetric 2-forms respectively. When n == 7 (mod 8), all analogous statements hold with $ replaced by $+ (or $-) and with C£n replaced by C£: (or C£; respectively). Proof. The initial statements are similar to those in Proposition 10.16 and are proved analogously. For statements (10.12) and (10.13), we note first that under the isomorphism (10.10), symmetric tensors go to self-adjoint transformations and skew-symmetric tensors go to skew-adjoint trans-formations. On the other hand Clifford multiplication by a p-form cp has the property that (J1.",)* = 1),11",. This proves (10.12) and (10.13). Details of the case where n == 7 (mod 8) are left to the reader. • §1O. REDUCED HOLONOMY AND CALIBRA nONS 355 For any spinor (J in $ (or $+ if n == 7 (mod 8)), this proposition gives a decomposition of the "square": (J ® (J = CfJo + CfJ3 + CfJ4 + CfJ7 + CfJs + ... + CfJ4t-1 + CfJ4t +... (10.14) with CfJp EO for each p. Under special assumptions about (J, certain of these components will vanish or be related to one another by the Hodge *-isomorphism. We shall first consider the case where n = 8. Theorem 10.18. Let $ denote the irreducible real and let $ = $+ EB $-be its standard decomposition into ± 1-eigenspaces for the volume form w. Then for any unit-length spinor (J EO $ +, the decomposition (10.14) becomes (J®(J=l+<I>+w where <I> EO A is the Cayley 410rm defined in 10.13. Proof. To begin consider (J ® (J EO Since W(J = (J, we have by Proposi-tion 10.17 that (J ® (J = (w(J) ® (J = w((J ® (J) and similarly that (J ® (J = (J ® (w(J) = ((J ® (J) . rx(wt) = ((J ® (J)w. Now left and right multiplication are related to the Hodge *-operator by (5.35) of Chapter H. This shows that if we write (J ® (J = I CfJp as in (10.14), then p(p-1) p(p+ 1) I (_l)-Z-* CfJp = I CfJp = L (-1)-z-* CfJp· Hence CfJp = 0 for p"# 0,4,8, and furthermore *CfJ4 = CfJ4' *CfJo = CfJs· The identity element 1 EO can be written as 1 = I (Ja ® (Ja where {(J 1> .•• ,(J 16} is any orthonormal basis of $. It follows by choosing (J 1 = (J, that (1, (J ® (J) = 1. Therefore (J ® (J = 1 + <I> + W for some <I> EO (We are using here the inner product on for which 1 and eh ... eip' 1 i1 < ... < ip, form an orthonormal basis. This differs from the stan-dard one on by a factor of l6.) Observe now that since the isotropy subgroup of any nonzero element (J EO $ + is Spin;, the imagc Ad(Spinn c: SOs must leave invariant each component of (J ® (J. In particular <I> is a Spin; -invariant 4-form. This identifies it as a multiple of the Cayley 4-form. To determine which multiple, we proceed as follows. Fix orthonormal vectors el' ... ,e4. Then <(J ® (J, e1 ... e4) = <(J ® (J, e1 ... e4·1) = «e4··· e1(J) ® (J, 1) = <e4 ... e1(J, (J) 11(Jllz = 1. (10.15) This proves that the A 4 -component of (J ® (J is a calibration, i.e., it satisfies (10.6). Note that we have equality in (10.15) if and only if e4··· e1(J = (J. Set w = e1 ••. e4. Since WZ = 1 and w$+ = $+, we can decompose $+ = $! EB into the ± 1 eigenspaces of w. Multiplying by e1eS maps $! isomorphically onto $:, so dim $! = 4. The group Spin7 is transitive 356 IV. APPLICA nONS on the unit sphere in $+ so we may assume (J E $!. Hence equality does occur in (10.15) for some choices of eh' .. ,e4. This proves that the A 4 -component of (J ® (J is exactly the Cayley 4-form. • Entirely analogous arguments prove the following: Theorem 10.19. Let $ be the irreducible real C£7-module on which the volumeform w acts by + 1. Thenfor any unit spinor (J E $, the decomposition (10.14) becomes (J®(J=l+q;+IjJ+w where q; E A is the associative calibration and IjJ = *q; E A is the coassociative calibration. It is clear that squaring spinors is a natural way to produce interesting calibrations in all dimensions. (This interesting fact was first noticed and proved by Dadok and Harvey [1].) The method is efficient both locally and globally on a manifold. It is also efficient for analysing the condition dq; = 0 as we shall now see. Let <I> be a Cayley 4-form and consider the orbit GLs . <I> c A This orbit is independent of the choice of <I> and of the underlying inner prod-uct. It consists precisely of all 4-forms whose isotropy group in GLs is isomorphic to Spin7' We call the forms in Ca == GLs' <1>, the 4-forms of Cayley type. Note that any such form <1>' determines uniquely an inner product on which makes <1>' a Cayley calibration. Theorem 10.20. Let X be a smooth oriented 8-manifold, and suppose <I> is a smooth 4-form on X which is of Cayley type at each point. Then <I> is closed if and only if <I> is parallel in the riemannian metric it determines. Therefore, to find a closed 4-form of Cayley type is to find a metric with holonomy c Spin7' Proof. Introduce the metric for which <I> is a Cayley 4-form. Let (J be a local spinor field so that (J ® (J = 1 + <I> + w. Since *<1> = <I> we see that d<l> = 0 -= d*<I> = 0 -= D<I> = 0 -= D((J ® (J) = 0 where D d + d* is the Dirac operator on C£(X) discussed in Chapter n, §5. Let I/J denote the Atiyah-Singer operator on the real spinor bundle $. Theorem 10.19 will be a direct consequence of the following lemma: Lemma 10.21. For any (J E re$) with II(JII == 1, we have IID((J ® (J)112 = 11I/J(J112 + IIV(JW· Proof. From 10.17 we see that D((J ® (J) = I ejVe/(J ® (J) = (I ejVep) ® (J + I (ep) ® (Vep) = (I/J(J) ® (J + I (ep) ® (Vep)· §11. SPINOR COHOMOLOGY; MANIFOLDS WITH Cl = 0 357 Since IICTllz == 1, we see that <VCT,CT) == O. Furthermore <eiCT,ep) = -<el!iCT,CT) = bij. It follows that IID(CT ® CT)W = IlffiCTllz + I. <eiCT,ep)<Ve,CT, Vep) = IlffiCTllz + I.llve,CTllz = IlffiCTllz + IIVCTW· • In analogy with the above we define a 3-form qJ E A 3[R7 to be of associa-tive type if G", == {g E GL7 : g*qJ = qJ} is isomorphic to Gz. Bryant observed that the set of such 3-forms is open in A3[R7. Each such 3-form determines a unique inner product on [R7 which makes it an associative 3-form. Arguing as in the proof of 10.20 gives the following: Theorem 10.22. Let X be a smooth oriented 7-manifold and suppose qJ is a smooth 3-form on X which is of associative type at each point. Then for the riemannian metric determined by qJ, theform qJ is closed and coclosed if and only if it is parallel. Therefore, to find a metric with holonomy c Gz it suffices to find a closed 3-form qJ, pointwise of associative type, such that d(*",qJ) = 0 where *", is the Hodge *-operator for the metric determined by qJ. Characterizations of this type were formulated and used in the funda-mental work of Bryant [1],[2]. §11. Spinor Cohomology and Complex Manifolds with Vanishing First Chern Class In this section we shall examine the geometry of Dirac bundles over complex manifolds. We shall recapture a number of classical results and prove some new ones. As an application we shall show the existence of compact manifolds with SPn-holonomy. The results here are due to the second author and can be found in Michelsohn [1]. Fix a compact Kiih1er manifold X of dimension n, and let S be a holomorphic bundle of left modules over C£(X) equipped with a hermitian metric and its associated canonical hermitian connection V. (This is the riemannian connection determined uniquely by the condition that CT E reS) is holomorphic ifand only ifVJvCT = iVvCT for all v E TX; (see Wells [1].) We assume that this metric and connection make Sa Dirac bundle over X. We define operators fiJ and on reS) by setting n fiJ = " 8·V-L, 1 EJ j= 1 n = I. ejVEj j=l (11.1 ) 358 IV. APPLICATIONS where {Cl' ... ,cn} is any local hermitian frame field defined as in §§8 and 9. The curvature RS of the canonical hermitian connection is of type (1,1), i.e., RSe e = RSe-e-= 0 for all i,j·. From this and the identities (8.3) it foI-l. J I. J lows that = 0 and =0 (11.2) These operators are easily shown to be formal adjoints of one another. Proposition 11.1. There is a parallel orthogonal direct sum decomposition of S into holomorphic sub-bundles S = SO EB Si EB . . . EB S" with the property that r(Si+l) for all i. and Proof. Let el,Jel, ... ,en,Jen be a local unitary frame field on X and define Cj = -!-(ej -Jej) as above. Consider the family of commuting elements WI>' .. ,Wn, W1, ..• ,wn defined as in (9.7) by setting Wj = -Cij and Wj = -eh. To each (possibly empty) subset 1= {il, ... ,ir} c {l, ... ,n} with complementary subset UI> ... ,jn-r} we associate the element W =W· "'W'W' "'W' I II lr}1 }n-r and we denote III = r. From the relations (9.9)-(9.11) we can write n n 1 = TI (Wj + wj) = L nr j=1 r=O where nr = L WI (11.3) III =r The operators nr are independent of the choice of unitary frame field and are therefore globally defined on X. They have the following basic prop-erties for all r, sand j: if r i= S and for (1, rES (11.4) (11.5) (11.6) (11.7) ( 11.8) Properties (11.5)-(11.8) follow directly from (9.8)-(9.13), as does the fact that nr E c£O.Zr-n(x) for each r. Since 0 = V(l) = L Vnr and V preserves sections of the subbundle C£M(X), we then obtain property (11.4). We §11. SPINOR COHOMOLOGY: MANIFOLDS WITH Cl = 0 359 now define r = 0,1, ... ,n From the properties above we see that the n/s form a parallel family of orthogonal projection operators. Therefore by (11.3) we have the orthog-onal decomposition S = EBr sr. We now show that each sr is holomorphic. Let a E r(S) be a (local) holomorphic section, and write ar = nra. Then for any v E TX we have that VJv(ar) = niVJva) = nr(ia) = iar. Hence, each ar is holomorphic, and one sees immediately that the space of such local cross sections spans the fibre of sr at each point. Therefore sr is a holomorphic sub-bundle. The fact that !:0(r(Sr)) c r(sr+ 1) and c r(sr-l) follows directly from (11.3) and (11.8) .• We now have defined a complex o r(SO) r(Sl) ... r(sn) 0 ( 11.9) with adjoint complex @ iiI iiI o +---r(SO) +---r(Sl) +---... +---r(sn) +---O. (11.1 0) This complex is elliptic, i.e., at any non-zero cotangent vector the symbol sequence is exact. To see this note that the principal symbol of !:0 at E TX is given by Clifford multiplication = == -Sim-ilarly we have = The exactness of (11.9) follows from the iden-tity + = DEFINITION 11.2. The rth cohomology group of the Dirac bundle S is the quotient J('r(x S) = ker(!:0lnsr») . , !:0(r(sr 1)) From the ellipticity of the complex (11.9) it follows that each of the vector spaces J('r(x,S) is finite dimensional. Taking S = C£(X) and using both !:0 and we essentially recover the Clifford cohomology groups of X (see §8). This viewpoint gives a simple unified approach to the fundamental vanishing theorems of Kahler geometry. Let us define elliptic self-adjoint operators V*V and V*V on r(S) by n V*V = -L V;j,tj j= 1 withcJ,'" ,cn as above, They have the property thatJ <V*Va,a) = S IVal2 where V is defined by V va = t(Vv -iVJv)a, Consequently we have that ker(V*V) = {holo, sections of S}, ker(V*V) = {antiholo, sections of S}, 360 IV. APPLICATIONS We now define associated curvature operators n me= L j,k= I Applying the arguments of n.8 and using the fact that = RL.i'k = 0 for all i,j, gives the following (Michelsohn [I]). Theorem 11.3. + = V*V + me = V*V + me. Taking the average of these two formulas gives a formula of the type presented in §8 of Chapter 11. We now consider a holomorphic hermitian line bundle, A, over X. The curvature RA of A is an imaginary valued, i-invariant 2-form (i.e., (I,I)-form) on X. In terms of a hermitian frame field el,' .. ,en as above, we have for all j,k; that is, the matrix ajk = is hermitian symmetric. If this matrix is positive definite at each point, then A is called a positive hermitian line bundle. (If ajk is negative definite at each point, then A is called nega-tive. In this case A * is positive.) It is straightforward to see that any closed imaginary (I,I)-form representing the first Chern class of A is the curvature form of a hermitian metric on A. We now have a vanishing theorem of "Kodaira type." Theorem 11.4. Let X be a compact Kiihler manifold and let S be a holomor-phic Dirac bundle as above. Thenfor any positive line bundle A there exists an integer N such that for m ;:;: N for all r > O. Proof. The curvature of the bundle S ® Am is given by ® t) = ® t + ma ® Since A is positive we may choose local hermitian frames so that = -A/Jjk where 0 < Al A2 ... )0", It follows that ® t) = (I elkRLka) ® t + m I (elka) ® hk hk = ® t -m( )Ojqp) ® t = + mra] ® t §11. SPINOR COHOMOLOGY: MANIFOLDS WITH Cl = 0 361 r = -L )OjCij = L AjWj. Note that for any a E S we have (ra, a) = L Aj(Wp, a) = L AiwJa, a) = L AjllWpW Al L IIwpl12 = Al L (wp, a) = AI(wa, a) where w = L wj• This element has the fundamental property that which follows from the elementary fact that WiW, = w, if i E I and WiW, = 0 otherwise. Consequently, if a E sr, i.e., if nra = a, then wa = ra. It follows that on nSr ® Am) + mrA) ® 1 where A> 0 is the minimum of Al on X. For r 1 and m sufficiently large we have ® > 0, and it follows from 11.3 that + > 0 on nSr ® Am) .• There is also a vanishing theorem of "Nakano type" for the Clifford bundle. We refer to Michelsohn [1] for the proof. Theorem 11.5. If A is a negative line bundle over a compact Kahler manifold X, then for all q < O. Let us now suppose that X is a Kahler spin manifold, i.e., that cl(X) == 0 (mod 2), and let Src be the bundle of complex spinors on X with its canonical riemannian connection. This bundle is holomorphic; in fact, it is equiva-lent as a complex vector bundle to At T* X ® b 1/2 where b -I = T* X is the canonical bundle (see App. D.). Since X is Kahler, the canonical con-nection on Src is also the canonical hermitian one. We can therefore apply Theorem 11.3. To compute the terms mrc and mrc which appear there, we need to know the curvature tensor of Src. From (4.37) of Chapter 11 we see that this can be expressed in terms of the Riemann curvature tensor R of X by the formula 2n = t L <RY.W1'/i' 1'/j)1'/i . 1'/{ i.j= I where 1'/1"" ,1'/2n is any real orthonormal basis of the tangent space. Choosing a unitary basis el,Jel, ... ,en,Jen and writing {cJ as above, we can reexpress this as n = L {<Ry.WCj,Ek)ej• Ck' + <RY.Wej,Ck)Cj· ek'} j,k= I = 2 L <Ry,WCj,Ek)ej • Ck' + L <Ry,wcj' Sj) j,k j 362 IV. APPLICA nONS where we use the fact that eik + Skej = -bjk. It follows that n mIC = L e/kRlj.ek j,k = 1 (11.11) = 2 L R Jklm e j • Sk • el • em' + I, R Jklle j • ek' j,k,l,m j,k,l where RJklm denotes <Rej,ekel.em)' Now the Bianchi identity states that R]klm = RJlkm + RlkJm = RJ1km· That is, R]klm is symmetric in k and I. But Skel = -SISk' and so equation (11.11) becomes This relates to the Ricci form of X, which for tangent vectors V and W is given by Ric(V,W) = -I, {<Rej,yej' W) + <RJej,yJej, W)} j = -2i I, <Rej,e/V, W). j Thus, we are able to write mIC as 1 mIC = --I, Ric(Sjh)ej' Sk·· 2 j,k Since Ric is hermitian symmetric we can choose our basis so that Ric(Sjh) = -!-)'jbjk where Aj = Ric(e;,ej) = Ric(Jej,Jej) for j = 1, ... ,n are the eigenvalues. As before, we write -eij = Wj and carry out a similar analysis for mIC' Thus we obtain the following: Theorem 11.6. Let X be a KCihler manifold equipped with a spin structure. Then on the bundle of spinors $IC --lI, --1I, !!fi!!fi + !!fi!!fi = V*V + -A ill· = V*V + -)..w· 4 j J J 4' J' where Ab ... ,An are the eigenvalues of the Ricci tensor. Taking the average of these two formulas gives the Lichnerowicz Theorem (11.8.8). To see this we observe that V*V + V*V = V*V 2 and 1 AiWj + wj) = Aj = "2 K J J (11.12) where K = trace[J;!(Ric) is the scalar curvature of X. Theorem 11.6 easily gives the following: §11. SPINOR COHOMOLOGY: MANIFOLDS WITH c, = 0 363 Corollary 11.7. Let A be a hermitian line bundle over a K(ihler manifold X with curvature form RA, and assume that X is spin. If Ric > 2RA, then .Y{"(X, $1[; ® A) = 0 (11.13) for all r > O. Similarly, if Ric > -2RA, then (I I ./3) holds for all r < n. Proof. Straightforward computation shows that the terms ml[; and ml[; in Theorem 11.3 for the Dirac bundle $1[; ® A are exactly given by Clifford multiplication by the elements (11.14) Choosing a basis which diagonalizes the hermitian form tRic -R with eigenvalues /11' ... ,/1n we find that ml[; = L /1jWj. This proves the first state-ment. The second is proved similarly. _ Taking the average of the two formulas of Theorem 11.3 in this case and applying (11.12) gives the formula (cf. Theorem D.12) + = V*V + tK + QA (11.15) where QA is the Clifford element corresponding to the curvature 2-form of A. Theorem 11.8. Let X be a compact K(ihler manifold equipped with a spin structure, and let A be a hermitian line bundle over X. If the scalar curvature K satisfies the inequalitytK > IA11 + ... + IAnl at each point, where Al' ... ,An are the eigenvalues of the curvature form of A, then the cohomolog y groups .Y{"(X, $1[; @ A) = 0 for all r. In particular, by the Atiyah-Singer Index Theorem, {ch A· A(X)} [X] = o. It is interesting to specialize our discussion to the case where A = ll/s where b = M:TX is the anticanonical line bundle of X (the line bundle generated by the global section el ... en in C£(X)). An easy adaptation of the arguments given for D.2 shows the following (see Michelsohn [1]): Lemma 11.9. Suppose cl(X) = kIX where k is odd and IX E H2(X;Z) is indivi-sible (i.e., not a non-trivial integral multiple of any other class in H2(X;Z)). Thenfor any odd integer m, there is a bundle oftheform $1[; ® bm/2k globally 364 IV. APPLICATIONS defined on X where locally $e is the bundle of spinors with its canonical connection. For any rational number t, the curvature of (5t with its natural metric is just tRd where for any unit vector e, = Ric(e,e) = Ric(Je,Je). If we choose a unitary basis e1,Jel"" ,en,Jen so that Ric(ej,ek) = Aj(5jk' then = tRic(sj, cd = (t/2)AAb and applying (1l.l4) gives the following: Lemma 11.10. On r($e ® (5t/2) + = V*V + (l + t) f }OjWj 4 j= I --1 f = V*V + -(1 -t) }OjWj 4 j= 1 where AI, ... ,An are the eigenvalues of the Ricci tensor on X. REMARK 11.11. The operators which appear here can be diagonalized as follows. Suppose that (]' E ® (5t/2, i.e., that nr(1 = (1. From the fact that nr = LIII=r WI and that if i E I if i If: I we see that on ® (5,/2 we have AjWj = )' AIWI J IIl=r where and and if i El if i If: I, when I is the multi index {i I, ... , iJ It follows from (9.9)-(9.13) that multi-plication by the elements WI constitutes a family of orthogonal projections onto mutually perpendicular subspaces of $e ® (5'/2. These formulas give a delicate and precise analysis of what conditions are necessary for the positivity of L A/Vj and L AjWj. Note in particular that the numbers {AI: III = r} are precisely the eigenvalues of Ric acting on A'TX as a derivation. We see from Lemma 11.9 that ® (51/2 always exists globally. It cor-responds naturally to the bundle A o'*T* X (see App. D), and there is an isomorphism $e ® (51/2) H'(X,(!J), where (!J is the structure sheaf of X. Corollary 11.12. On ® (51/2), --1 --+ = V*V + -L }oIWI = V*V 2 III =r §11. SPINOR COHOMOLOGY: MANIFOLDS WITH Cl = 0 365 In particular, if Ric > 0, then .Y{'r(x, $1[; ® 151/2) = 0 for all r > 0 and {ch 151/2. A(X)}[X] = Td(X) = 1. Arguing as above, we can conclude the following: Theorem 11.13. Let A be a hermitian line bundle over X and suppose that (1 + t)Ric + 2RA > O. If $1[; ® 151/2 ® A exists globally on X, then .Y{'r(x, $1[; ® 151/2 ® A) = 0 (11.17) for all r> O. Similarly, if (1 -t)Ric -2RA > 0, then (11.17) holds for all r < n. We now average the formulas of Lemma 11.10 and obtain Theorem 11.14. Let X be a compact Kahler manifold with Cl(X) = ka, k E 71+ where a E H2(X;7l.) is indivisible. Let $1[; ® bP/2k be the twisted bundle of spinors where b1/2k is a local holomorphic 2kth root of the anti-canonical bundle and where p + k == 0 (mod 2). Then on ® bP/2k) + = V*V + -21 L [(k + p)AI + (k -p)ArJWI· kill =r In particular, if Ric > 0, then .Y{'r(x, $e ® bP/2k) = 0 for all r whenever Ipl < k and p + k is even. We apply the Atiyah-Singer Theorem to the complex ® bP/2k to get Theorem 11.15. Let X be a compact complex manifold such that Cl(X) = ka, k E 7l.+, where a is indivisible. If X admits a Ricci-positive Kahler metric, then the Hilbert polynomial Pa(t) = {eita. A(X)}[X] vanishes for all integers t such that It I < k and t + k is even. From Yau [2], [3] we know that X carries a Ricci-positive Kahler metric if and only if cl(b) = c1(X) is represented by a closed positive (1,1)-form. To illustrate Theorem 11.15 we consider complex projective n-space IP"(C) and let WE H2(1P"(C); Z) = 7l. denote the standard generator. Then Cl = (n + l)w and a direct calculation shows that 1 " P w(t) = 2" I n (t -n + 2j -1). n. J= 1 This is predicted by Theorem 11.15 since IP"(C) carries a Kahler metric of positive curvature. 366 IV. APPLICATIONS From the work ofYau [2], [3] we know that the hypersurface V"(d) c [p"+ l(e) carries a Kahler metric of positive Ricci curvature for d n + 1. Furthermore, for n 2, cl(V"(d)) = (n + 2 -d)w where w is a generator of H2(V"(d); &'). It follows that the polynomial P wet) on V"(d) has zeros at n -d -2j for j = 0, ... ,n -d. Theorem ILlS has the following consequence: Corollary 11.16. Let X be a compact complex manifold of dimension n. If X admits a Kiihler metric of positive Ricci curvature, then k ,r CI(X) for k>n+1. Proof. The Hilbert polynomial of IUS is of degree n and can have therefore no more than n zeros. • One case of Theorem 11.l4 is of particular interest since it involves only the scalar curvature K = 2(AI + ... + A"). Corollary 11.17. Let X, band k be as in Theorem 11.l4. If the scalar curvature K of X satisfies K > 0, then .Ye"(X, $1[; ® bP/2k) = 0 .Ye°(X, $1[; ® bP/2k) = 0 for all p > -k for all p < k where p + k is even. Similarly, if K < 0, then .Ye"(X, $1[; ® bP/2k) = 0 for all p < -k for all p > k where p + k is even. In particular if K > 0, then all the plurigenera of X are zero (cf. Yau [I]). This last statement follows by setting p = k -2kg for g E &' +. The gth plurigenus of X is exactly the dimension of .Ye°(X, $1[; ® bi-9) HO(X, (!)(b-9». Note that the formula in Theorem 11.14 is particularly simple when Ric == O. From Yau's proof of the profound conjectures of Calabi we know exactly when such metrics exist. Theorem 11.18 (Yau [2]). Let X be a compact complex manifold which admits at least one Kiihler metric. Then X carries a Ricci-fiat Kiihler metric if and only if cl(X) = O. We can now apply the Splitting Theorem of Cheeger and Gromoll [2] to conclude that any compact Ricci flat Kahler manifold X has a finite covering X which decomposes into a Kahler product X = T x Xo §11. SPINOR COHOMOLOGY: MANIFOLDS WITH Cl = 0 367 where T is a flat complex torus and where X 0 is a compact Ricci-flat manifold which is simply-connected. When Ric == 0, Theorem 11.14 gives us the following simple identity between operators (11.18) on $e. In this case there is a canonical connection-preserving isometry = A o'*(X) = of graded bundles, which carries !!fi to a and therefore identifies ,#'r(X,$d with H'(X,(!)) (cf. App. D). Equation (11.18) therefore implies the following: Proposition 11.19. On a compact Ricci-flat Kahler manifold, any harmonic spinor field is parallel. Equivalently, any harmonic (O,r)-jorm is parallel. Suppose now that X is a compact simply-connected Ricci-flat Kahler manifold of dimension n. Then the line bundles $g A 0.0 and $'J:; A O.n are trivial (they have parallel cross-sections). The index of the elliptic complex (11.9) in this case is just A(X) = L (-l}'dim ,#'r(X,$d = L (-lydim Hr(x,(!)) = Td(X) and dim ,#'0 = dim ,#'1 = 1. Moreover, by the generalized Hodge Decom-position Theorem (cf. 111.5.5) we have ,#'r(X,$d ker(!!fi.@ + .@!!fi)lm() = ker(V*VIr($(J Consequently, if Td(X) of-1 + (-1)", then there must exist non-trivial parallel sections of (i.e., parallel (O,r)-forms) for some r of-O,n. This implies that the local holonomy group G of X is properly contained in SUn' If G is a product of two non-trivial groups, then X is a product manifold. Otherwise G belongs to the list of Berger discussed in §1O, and we conclude that either X is locally symmetric or G = SPnI2, for n even and > 2. It is elementary that locally symmetric, Ricci-flat manifolds are flat (and therefore not simply-connected in the compact case). Combining the three paragraphs above proves the following general structure theorem. Theorem 11.20 (Michelsohn [1 ]). Let X be a compact simply-connected Kiihler manifold with Cl(X) = 0. Then there is a finite covering manifold X of X which is biholomorphically equivalent to a product of compact Kahler manifolds with vanishing first Chern class X=TxX1x"'XXkxY where T is a complex torus and each of X 10 .•• ,X ko Y is simply-connected, where Td(X) = if dim Xj is odd if dim X j is even and where Y admits a metric with Spm-holonomy. 368 IV. APPLICATIONS This decomposition theorem is a wonderfully effective device for de-tecting manifolds with Spm-holonomy. Corollary 11.21. Let X be a compact simply-connected Kiihler manifold of dimension 2m with Cl (X) = O. If Td(X) -# 0 or 2k for some k, then X carries a metric with Spm-holonomy. Alternatively, if X is not a product and Td(X) -# 2, then the same conclusion holds. Theorem 11.20 with its corollary first appeared in Michelsohn [1] in pre-publication form. Before printing, it was modified due to the publication of an announcement of the non-existence of compact manifolds with Spm-holonomy for m 2. This announcement was subsequently found to be in error. The first counterexample was produced by A. Fujiki using the results above. To construct this example we first take the symmetric square of a Kummer surface. This is the compact analytic space X 0 = V2(4) X V2(4)/1'2 where 1'2 is generated by the interchange of factors in the product. Resolving the singularities along the diagonal gives a compact complex 4-manifold X. Alternatively, we could construct X by first blow-ing up the diagonal (i.e., by replacing the diagonal by its projectivized normal bundle) and then dividing by the natural extension of the "flip" to this space. The manifold X is a certain component of the Hilbert scheme ofO-cycles on V2(4). It is straightforward to see that X is simply-connected and algebraic (hence Kahler) and that cl(X) = O. However, further com-putation shows that Td(X) = 3. Consequently the Calabi-Yau metric on X, given by Theorem 11.18, has Sp2-holonomy. (See Beauville [1] for further discussion and examples.) §12. The Positive Mass Conjecture in General Relativity The classical theory of gravity, as enunciated by Einstein, is a subject formulated in the language of differential geometry. In many ways it stands apart from other theories of modern physics. One of its curious features is that there seems to be no satisfactory way of defining an energy density for a gravitational field. Nevertheless, there is a concept of the total energy of a gravitating system, which is defined in terms of the asymptotic behavior of the field at large distances. It is in many ways essential to the theory that this energy be 0 (and = 0 only on flat Minkowski space). The proof of this "positive mass conjecture" has a long history during which many special cases were established. The key case of space-times admitting a maximal space-like hypersurface was first proved by Schoen and Yau [3] using minimal surface techniques. Using Jang's equation they subsequently adapted their techniques to prove the general result [5]. §12. POSITIVE MASS CONJECTURE 369 Of interest here is the fact that an alternative proof of this theorem can be given using the Dirac operator on spinors. This proof was found by E. Witten [1]. Very roughly the idea is as follows: Consider a space-time X (i.e., a 4-dimensional Lorentzian manifold), with a properly embedded space-like hypersurface M c X. We assume that X satisfies Einstein's equations Ric -!gK = T where the energy-momentum tensor Tis positive on time-like vectors. It is assumed that as one goes to infinity in M, the geometry of M c X is, in a very explicit way, asymptotic to the standard geometry 1R3 c 1R3,1 in Minkowski space. In this context Witten considers the following problem. Let S denote the spinor bundle of X with its canonical connection V, and consider the restriction of (S,V) to the hypersurface M. On this bundle one has a Dirac operator defined by 3 = L ej' Ve 1 J where e1,e2,e3 is any local orthonormal frame field on M. One now applies Bochner's method and writes = V*V +,3 (12.1 ) where, since V is the metric connection of X (and not of M), the term 3 involves the second fundamental form of the hypersurface M in X. Com-putation shows that the positivity of the tensor T on the normal vectors to M implies the positivity of ,3. This in turn shows that on suitable sub-spaces of qS) the operator is invertible. Since S is asymptotically fiat, one can consider solutions of the equation = 0 which are asymptotically constant. Witten shows that there are unique solutions for each constant value at infinity. Let a be such a solution and consider a region n c M with smooth boundary an. From (12.1) we have 0= In {<V*Va,a) + <,3(a),a)} = -En(a) + In {IIVa112 + <3(a), a)} where En(a) is an integral over an which satisfies (12.2) since 3 O. Witten shows that the inequality (12.2) viewed asymptoti-cally, establishes the positivity of the energy of the system. For the complete story the reader is referred to the original paper of Witten [1] and also to the rigorous account of Parker and Taubes [1]. APPENDIX A Principal G-Bundles Let X be a paracompact Hausdorff space and G a topological group. A principal G-bundle over X is essentially a bundle of "affine G-spaces" over X. To be precise, it is a fibre bundle n : P -+ X together with a continuous, right action of G on P which preserves the fibres and acts simply and transitively on them. Thus, the fibres are exactly the orbits of G. More-over, every point in X has a neighborhood U and a homeomorphism hu:n-1(U) =. U x G of the form hu(p) = (n(p),y(p)) and with the prop-erty that h(pg) = (n(p), y(p)g) for all g E G. Thus, the bundle is locally of the form UxG 1 U where G acts by multiplication on the right. Two principal G-bundles n: P -+ X and n': P' -+ X are said to be equivalent if there is a homeomorphism H: P -+ P' so that \) X commutes and so that H(pg) = H(p)g for all g E G. The equivalence classes of principal G-bundles over X will be denoted by PrinG(X). EXAMPLE. Let n: P -+ X be a 2-sheeted covering space of X, and let G = 7l.2• The group 7l.2 acts on P by interchanging the sheets. This is clearly a principal 7l.2-bundle. In fact it is not difficult to see that Prinz2(X) COV2(X) for any manifold X. More generally, any normal covering of a manifold is a principal r-bundle where r is the group of deck transformations of the covering (with the discrete topology). EXAMPLE. Let E be a real n-dimensional vector bundle over X, and let PGL(E) be the bundle of bases in E, i.e., the bundle whose fibre at x EX is the set of all bases for the vector space Ex. This is a principal GLn-bundle where GLn is the group of invertible n x n real matrices. The action 370 PRINCIPAL G-BUNDLES 371 of GLn on PGL(E) is defined as follows. Fix a matrix g = ((aij)) in GLn. Then given a basis p = (VI' ... ,vn) of Ex at a point x, we set pg =' (V'I' ... where j = 1, ... ,n This action is clearly continuous and is simple and transitive on the fibres. If E is oriented, we may consider the bundle P GU (E) of oriented bases in E. The construction above makes this a principal GLn+ -bundle where GLn+ =' {g E GLn : det(g) > O}. If E is riemannian, we can consider the bundle P o(E) of orthonormal bases. This is a principal On-bundle where On is the orthogonal group. If X is oriented we get the bundle Pso(E) of oriented orthonormal bases, which is a principal SOn-bundle, where SOn =' {g E On : det(g) = I}. There are natural complex and quaternionic analogues of these constructions. Recall that a general fibre bundle B 4 X with fibre F and structure group G <:; Homeo(F) is given by the following data. There is an open cover I1lt = {Ua}aEA of X, and over each Ua there is a local trivialization n-I(Ua) Ua x F so that pr 0 ha = n (where pr is pro-jection onto U a). The change of trivialization over U n U fl is of the form h 0 h-' (U a n Up) x F (U a n Up) X F where ha 0 hp I(x,f) = (x, gap(x)f) and gap: U a n Up -+ G are continuous functions called the transition func-tions of the bundle. They satisfy the cocycle condition (A.I) in U an Up n U y' The bundle can be reassembled from this data by pasting together the local products {U a X F} ad with these homeomorphisms. Any fibre bundle B with structure group G as above has an associated principal G-bundle P G(B). It is obtained by simply replacing F by G in the local products and then pasting together the {Ua X G}aEA by the same transition functions, where gaP(x) acts on G by multiplication on the left. Note that we are free to multiply on the right by elements of G. Since right and left multiplication commute, this multiplication by elements of G on the right makes sense when we assemble the bundle P G(B). Note, however, that while each fibre looks like G, there is no preferred element, i.e., no "identity." The original fibre bundle B can be recaptured from P G(B) as an asso-ciated bundle, that is, B P G(F) x <p F where <p: G -+ Homeo(F) (see II.2). We now observe that every principal G-bundle over X can be presented by transition functions gap: Ua n Up -+ G multiplying G on the left as above. Of course a principal G-bundle is a fibre bundle and therefore is always given by a family of transition functions Gap: Ua nU p -+ Homeo(G) over 372 APPENDIX A some open cover {U a} of X. Since the bundle is principal, these functions must commute pointwise with right multiplication by G. Let gaP(x) = Then GaP(x)(g) = Gap(x)(l)g = gap(x)g and we are reduced to the simpler case, as claimed. Thus, we have seen that every principal G-bundle on X is given by a pair (0Zt, {gap}) where 0Zt = {Ua}ad is an open cover of X and where gap: U a (\ Up -+ G are continuous functions sa!isfying the cocycle condi-tion (A. 1 ). Such a pair can be thought of as a Cech l-cocycle with coeffi-cients in G (or, more precisely, in the sheaf of germs of continuous maps to G). One can easily check that two such bundles, constructed from cocycles {gap} and on 0Zt are equivalent if and only if there exist continuous maps ga: U a -+ G for each IX such that (A. 2) in U a (\ Up for all IX, {3. (This can be considered a "Cech-coboundary" condition.) Therefore, we define two l-cocycles {gap} and on 0Zt to be equiva-lent iff there is a "Cech O-cochain" {ga} ad such that (A.2) holds. The set of equivalence classes will be denoted by Hl(OZt; G). This set naturally represents the equivalence classes of principal G-bundles on X which can be trivialized over the open sets of 0Zt. Suppose now that ("Y,j) is a refinement of 0Zt, i.e., "1/ is an open cover of X and j: "1/ -+ 0Zt is a map such that V £ j(V) for all V E "1/. Then by restriction we get a map r 1I""IL: Hl(OZt; G) -+ Hl("I/; G) which can be shown to be independent of the refinement function j. These maps satisfy the relation r 1I""IL = r 11"11" 0 r 1I""IL for successive refinements "If' -+ "1/ -+ 0Zt. Thus we can take the direct limit This limit naturally represents the equivalence classes of principal G-bundles on X. That is, PrinG(X) Hl(X; G). If G is abelian, Hl(X; G) is simply the first Cech cohomology group of X with coefficients in G. EXAMPLE. Let G = 7lz. Then 7lz-bundles over X are precisely the two-fold coverings of X, and the correspondence Covz(X) Hl(X; 7lz) is just the isomorphism given in Lemma 1.1. REMARK A.1. If X is a Coo-manifold and G is a Lie group, we can require all the maps gap, ga' etc. in the discussion above to be differen-tiable. The resulting set, denoted Hl(X; G)oo, represents classes of smooth principal G-bundles over X. The natural map Hl(X; G)oo -+ Hl(X; G) can PRINCIPAL G-BUNDLES 373 be shown to be a bijection so we shall in general drop the reference to smoothness. An analogous remark can be made concerning complex manifolds X and complex Lie groups G where the maps gap and ga are required to be holomorphic. In this case, however, the corresponding map to Hl(X;G) is far from a bijection in general. (An exception is the case where X is Stein.) Note that if G is not abelian, Hl(X; G) is not a group. It is merely a set with a distinguished element given by the trivial G-bundle. Nonetheless, if (A.3) is an exact sequence of topological groups, the standard arguments in Cech cohomology theory (cf. Hirzebruch [1, Ch.I, §2] show that for para-compact spaces X, there is an exact sequence: {*} _ HO(X;K) HO(X;G) HO(X;G/) _ Hl(X;G/) of pointed sets. HO(X; G) is the set of O-cocycles and is easily identified with the space of continuous maps from X to G. The maps i* andj* are given in the obvious way by coefficient homomorphisms. Note that for a principal G-bundle P, the corresponding principal G'-bundle j*(P) can be obtained by dividing P by the action of K on the right. REMARK A.2. If the group K in (A.3) is abelian, then H2(X;K) is defined and the exact sequence (A.4) can be extended to ... -Hl(X; K) -Hl(X; G) -Hl(X; G/) _ H2(X; K) (A.5) EXAMPLE A.3. For the sequence 0-SOn -On -7l.2 -0, the induced map W1 :H1(X;On) -Hl(X;71.2) is just the first Stiefel-Whitney class. Clearly w1(P) = 0 if and only if P comes from an SOn-bundle, i.e., if and only if P is orientable. EXAMPLE A.4. For the sequence o -71.2 -Spinn SOn -0, the induced cobundary map W2: Hl(X; SOn) ----. H2(X; 7l.2) is the second Stiefel-Whitney class. Clearly w2(P) = 0 if and only if P comes from a Spinn-bundle, i.e., if and only if P carries a spin structure. 374 APPENDIX A There is a caution, however. As shown in §1, distinct spin structures may give abstractly equivalent principal Spinn-bundles. Note that the definition of a Cech coboundary leads to a nice com-binatorial definition of W2(P), Let (Oft, {g,.p}) be a cocycle representing P, where each set U,. (\ Up is simply-connected. Lift each map g,.p to a map g,.p: U,. (\ U P Spinn, and define w,.py = g,.pgpygy,. (A.6) in U,. (\ Up (\ Uy. Since == 1, we see that w,.py: U,. n Up (\ Uy 7l.2• This 7l.2-cocycle represents W2(P), EXAMPLE A.5. Consider the sequence o ---+ 7l. ---+ ---+ SI ---+ O. Since all groups here are abelian, the exact sequence in cohomology con-tinues indefinitely. It is an elementary exercise to show that = 0 for all i > 0 and for any manifold X. (Here carries the standard, not the discrete, topology.) We thereby get an isomorphism Cl :HI(X;SI) H2(x;71.) (A.7) called the first Chern class. This shows that the equivalence classes of principal SI-bundles are in natural 1-to-1 correspondence with elements of H2(X;71.) Note. For a full and clear discussion of the basic subject of fibre bundles the reader is referred to the classic book of Steenrod [1]. We conclude this section with some remarks about transferring bundles from one space to another. Let P X be a principal G-bundle over a space X and let f: Y X be a continuous map. Consider the set of points f*P == {(y,p) E Y x P J(Y) = n(p)}, (A.S) in the product Y x P, and note that projection of Y x P onto its factors induces maps on f* P which make the following diagram commute: f*P P ;r 1 1" (A.9) We now observe that if:f* P Y is a principal G-bundle over Y. It follows directly from definitions that G acts on f*P and is simple and transitive PRINCIPAL G-BUNDLES 375 on the fibres of ii. The map j commutes with this action. Any set of local trivializations of P over a family of open sets I1lt = {Ua}aEA in X naturally determine local trivializations of f* P over the family of open sets f*11lt = {I-I Ua}aEA in Y. The transition functions {gap} for this cover lift back to transition functions (A. 10) for f*11lt. DEFINITION A.6. The principal G-bundle f* P -+ Y is called the bundle induced from P -+ X by the mapping f. It is clear that if two bundles P and P' are equivalent on X, then f* P and f* P' are equivalent on Y. (Lift the equivalence.) Thus we have a map (A. 11) From equation (A.l 0) we see that the co cycle for f* P is just pull-back via f of the cocycle for P. Thus, (A.tt) corresponds to the usual induced mapping on Cech cohomology groups: f* : Hl(X; G) ---+ Hl(y; G). This "pull-back" mapping on principal bundles has two important properties: Proposition A.7. For continuous maps X Y Z, the induced maps PrinG(Z) PrinF( Y) -!: PrinG(X) satisfy (g 0 f)* = f* 0 g* (A.t2) Proof. That f*(g* P) = (g 0 f)* P is obvious from the definition. • Proposition A.S. Let Y be a compact Hausdorffspace. Iftwomapsfo: Y -+ X and fl : Y -+ X are homotopic, then = fT-We leave the proof as an exercise for the reader. APPENDIX B Classifyinn Spaces and Characteristic Classes The point of this appendix is to briefly summarize the basic facts from the theory of classifying spaces and characteristic classes for principal G-bundles. Details can be found in Milnor-Stasheff [1] and Husemoller [1]. For simplicity we shall assume throughout this appendix that G is a Lie group. Furthermore all spaces will be assumed to have base points, and all maps will be base-point-preserving. All spaces will also be assumed to have the homotopy type of a countable CW-complex. DEFINITION B.I. A classifying space for the group G is a connected topo-logical space BG, together with a principal G-bundle EG -+ BG, such that the following is true. For any compact Hausdorff space X, there is a one-to-one correspondence between the equivalence classes of principal G-bundles on X and the homotopy classes of maps from X to BG, given by associating to each map I: X -+ BG, the induced bundle 1* EG over X (see the end of App. A). Thus the induced bundle construction gives a natural bijection PrinG(X) [X, BGJ. (B.1) Note that as a special case we have PrinG(Sn) nn(BG). (B.2) Given two different classifying spaces for G, say BoG and B 1 G, there must exist mappings 10: BoG -+ B 1 G and /1: B1 G -+ BoG such that IrEi+1G EiG for i E 7L2• This implies that (/;+10 11)*EiG E;G, and therefore /;+ la/; is homotopic to the identity on BiG for i E 7L2• Thus BoG and B 1 G are homotopy equivalent, i.e., BG is well defined up to homotopy type. The bundle EG -+ BG is called the universal principal G-bundle. This bundle has the following homotopy characterization (see Steenrod [1] for example). Theorem B.2. Let E -+ B be a principal G-bundle with the property that the total space 01 E is contractible. Then (B, E) is a classifying space lor G. The theory has definite interest due to the following: Theorem B.3. For any Lie group there exists a classifying space. 376 CHARACTERISTIC CLASSES 377 The general construction of classifying spaces for principal bundles is due to Milnor [1,2] and proceeds as follows. Given a group G, we consider the n-fold join G * ... * G. (The join of two spaces X and Y is obtained from X x Yx [0,1] by collapsing {x} x Yx {O} to a point for each x and also collapsing X x {y} x {1} to a point for each y.) [0,1] y x x y x [0,1] X x Yx [0,1] ---+ X* Y. The space G * .,. * G is (n -I)-connected. It also has a free G-action ob-tained by mUltiplying simultaneously on the right in each of the factors. We pass to the limit as n -+ 00, and define EG == G * G * G * .. '. This is infinitely-connected and has a free right G-action. We set BG = EG/G with n: EG -+ BG the quotient map. From Theorem B.2, this bundle is classifying. Of course this construction of Milnor can (and has) been carried out in enormous generality. It should be noted that since EG is conllactible, the long exact sequence of homotopy groups for a fib ration implies that (B.3) for all n 1. Thus, the homotopy groups of BG present nothing new. However, the homology of the space is quite interesting. We begin the discussion with the following. Consider the singular cohomology H*(BG; A) with coefficients in a ring A. DEFINITION B.4. Each non-zero class in H*(BG; A) is a universal charac-teristic class for principal G-bundles. Fix a class C E Hk(BG; A). Then for any principal G-bundle P -+ X there is a map Jp: X -+ BG so that P JtEG. We define the c-characteristic 378 APPENDIX B class of P to be the class (BA) Since Jp is well defined up to homotopy, the class c(P) is uniquely defined. Observe that any such characteristic class transforms naturally in the following sense. Given a principal G-bundle P -4 X and a continuous map F: Y -4 X, the c-characteristic classes satisfy: c(F* P) = F*c(P) (B.5) To see this we simply note that JF*P = Jp 0 F, and so c(F* P) = J;"p(c) = F*J;(c) = F*c(P). Given a continuous homomorphism cp: H -4 G of Lie groups, there is a corresponding continuous map Bcp:BH--BG (B.6) which classifies the principal G-bundle over BH associated to the repre-sentation cp. That is, (Bcp)*(EG) = EH x'" G where EH x'" G denotes the quotient of EH x G by the action of H given by setting cl\(p, g) == (ph -1, hg) for p E EH, g E G and hE H. This associa-tion also transforms naturally. That is, for any composite homomorphism H 54 K .!4 G, we have B(tfJ 0 cp) = B(tfJ) 0 B(cp). If the homomorphism cp: H -4 G is a homotopy equivalence, then so is the mapping Bcp: BH -4 BG. This is easily seen by applying the 5-lemma to the induced map on the long exact homotopy sequences of the fibrations H--EH--BH and G--EG--BG. (Recall that BH and BG are assumed to be homotopy equivalent to countable CW-complexes.) This gives the following result: Proposition B.S. Let G be a connected Lie group and K G a maximal compact subgroup. Then BG BK, that is, the classifying spaces oJ G and K are homotopy equivalent. Proof It follows from the Iwasawa decomposition of G (cf. Helgason [1]) that the inclusion K 4 G is a homotopy equivalence. _ It follows, for example, that BOn BGLn(lR), BSOn BGL:(IR), BUn BGLn(C), BSUn BSLn(C), BSpn BSPn(C), etc. CHARACTERISTIC CLASSES 379 Note that, in general, each mapping of the type Bcp: BH -4 BG induces a transformation (Bcp)*: H*(BG; A) -4 H*(BH; A) of universal characteris-tic classes. Any non-zero element which goes to zero under this transforma-tion is a universal obstruction to the reduction of the structure group from G to H. That is, if c is such a class and if P -4 X is a principal G-bundle with c(P) #-0, then P cannot be written as an associated bundle P = P' x'" G (as above) for any principal H-bundle P' -4 X. The cases of basic interest here are those of the compact classical groups. For these cases there are useful alternative constructions of the classifying spaces. For K = IR, C or 1Hl, let GK(n, N) denote the Grassmann manifold of all n-dimensional K-linear subspaces of KN. There are natural inclusions GK(n, N) c GK(n, N + N') induced by the "first-coordinate" inclusions KN c KN+NI. Taking the direct limit, we get the Grassmannian of K-planes in KOO. There is a canonical K-vector bundle !En -4 GK(n, (0) whose fibre at a plane PE GK(n, (0) consists of all the vectors in that plane. We shall show that BOn GIffi(n, (0) BUn GC(n, (0) BSPn GIHl(n, (0). (B.7) The principal fibrations in each case are obtained by taking the appro-priate bundle of bases (orthonormal, unitary, or symplectic) in the ca-nonical bundle !En. Thereby, !En becomes the universal (real, complex or quaternionic) n-plane bundle over the classifying space (BOn, BUn' or BSPn respectively). We begin our proof of these claims with the real case. For this we note that for any N, we have GIffi(n, N) 0N/(ON-n X On). The bundle of ortho-normal frames for the canonical n-plane bundle !En -4 GIffi(n, N) is just the Stiefel manifold 0N/ON-n. Hence, the bundle of orthonormal frames for !En -4 GIffi(n, (0) is just the direct limit over N of the principal On-bundles 0N/ON-n --0N/(ON-n X On) It is an elementary exercise, using the fibrations Ok -4 Ok + 1 -4 sk, to see that 0N/ON-n is (N -n -2)-connected. Hence the infinite Stiefel mani-fold limN 0N/ON-n is contractible, and the limiting fibration must repre-sent the universal bundle EOn -4 BOn by Theorem B.2. 380 APPENDIX B The arguments for BUn and BSPn are entirely analogous. Here one uses the facts that GC(n,N);:;:UN/(UN_nxUn) and GIH1(n,N);:;: SPN/(SPN-n X SPn). We omit the details. The reader has undoubtedly noticed that BOn can be considered as the classifying space for real n-dimensional vector bundles. That is, for a com-pact Hausdorff space X, the equivalence classes of such bundles corre-spond one-to-one with elements of [X, BOn] by associating to f: X BOn the vector bundle f*fEn. This follows easily from the discussion above by passing to the bundles of orthonormal bases. The corresponding remarks hold for BUn and BSPn. It should be noted that the fact that GK(n, (0) is a classifying space for n-dimensional vector bundles (over K) can be proved by a direct geometric construction (see Milnor-Stasheff [1]). We now examine the cohomology of the basic classifying spaces. These spaces are infinite-dimensional and have non-zero classes in arbitrarily high dimensions. Nevertheless, as rings under the cup-product multiplica-tion, they are quite understandable. Basic references for all the following results are Milnor-Stasheff [1], Steenrod-Epstein [1], and Borel [2]. We begin with £:z-cohomology. Theorem B.7. The co homology ring H*(BOn; £:z) is a £:z-polynomial ring on canonical generators wk E Hk(BOn; £:z) for k = 1, ... ,no Note. The classes W1, ... 'Wn are characterized inductively by the following fact. The kernel of the homomorphism H*(BOn;£:z) H*(BOn-1;£:Z) induced by the standard inclusion On -1 On, is the principal ideal <wn). Analogous statements characterize the canonical generators in the subsequent theorems. The class Wk is called the universal kth Stiefel-Whitney class. Thus to any n-dimensional real vector bundle E X classified by a map fE: X BOn, we have the associated kth Stiefel-Whitney class of E, wk(E) == f1(wk). An important concept is that of the total Stiefel-Whitney class W == 1 + W1 + ... + Wn. It satisfies the Whitney product formula for the sum of two bundles w(E EEl E') = w(E) u w(E') which translates in longhand to the equations k wk(E EEl E') = L wi(E) U Wk-i(E') i=O (B.8) CHARACTERISTIC CLASSES 381 for k = 1,2, .... This can be considered as a statement about the induced map on cohomology induced by B(On x On') -4 BOn+n, under the usual "diagonal block" inclusion On X On' C On+n" Given a smooth manifold X, we define the kth Stiefel-Whitney class of X to be wk(TX). It is an important fact that the Stiefel-Whitney classes of a compact smooth manifold are invariants of the homotopy type of the manifold. That is, given a homotopy equivalence f: Y -4 X between com-pact smooth manifolds, we have that f*wiX) = Wk(Y) (see Milnor-Stasheff [1], p. 131.) This will not be true of many other characteristic classes of manifolds. Consider now the space BSOn. This can be realized as above as the Grassmannian of oriented n-planes in /Roo. There are natural maps BSOn -4 BOn -4 BZz = K(Zz, 1). This sequence is a fibration, and we have the following: Theorem B.8. The cohomology ring H*(BSOn; Zz) is a Zz-polynomial ring Zz[ wZ' •• , ,wn] where Wk denotes the lift of the universal kth Stiefel-Whitney class by the map BSOn -4 BOn. Finally we consider the space BSpinn which sits in a fibration BSpinn -4 BSOn -+ K(Zz, 2). REMARK B.9. The cohomology ring H*(BSpinn; Z2) is quite complex and since its precise structure is not used here, we refer the interested reader to the basic paper of D. Quillen [1]. (The "stable" groups H*(BSpin; Z2) were first computed by E, Thomas [1].) The Zz-cohomology of the spaces BUn and BSPn is merely the mod 2 reduction of the integral cohomology, and we have the following: Theorem B.IO. The cohomology ring H*(BUn; Z) is a Z-polynomial ring on canonical generators ck E HZk(BUn; Z) for k = 1, ' .. ,no The class Ck is called the universal kth Chern class. Thus to any n-dimensional complex vector bundle E -4 X classified by a map fE: X -4 BUn, we have the associated kth Chern class ck(E) = !1(ck). There is the concept of the total Chern class c = 1 + Cl + Cz + .. , + cn, and again there is a product formula c(E EEl E') = c(E) u c(E'). (B.9) Given any complex manifold X, the tangent bundle is a complex vector bundle. We call ciX) == ck(TX) the kth Chern class of X. 382 APPENDIX B There is now a ring isomorphism H*(BUn; £:z) £:z[cI, ... ,cn] where ck is of degree 2k for each k. The class ck is the mod 2 reduction of ck. The natural inclusion Un C 0Zn induces a map BUn -+ BOzn. It can be shown that under this map WZk f--+ ck and, of course, WZk+ I -+ O. Hence, for any complex vector bundle E, and WZk+I(E) = 0 for all k. The picture is much the same for quaternion bundles. Theorem B.ll. The co homology ring H*(BSPn; £:) is a £:-polynomial ring £:[ 0" I, ... ,O"n] on canonical generators O"k E H4k(BSPn; £:) for k = 1, ... ,no Under the natural map BSPn -+ BUZn we have that CZk f--+ O"k and CZk + I f--+ O. The integral co homology of BOn is a more complicated story due to the presence of 2-torsion. However, away from the prime 2, things become nice. Theorem B.12. Let A be any integral domain containing t (e.g., £:[t] or Q). Then H*(BOn; A) is the polynomial algebra A[pI' ... ,P[nm] on canonical generators Pk E H4\BOn; A) for k = 1, ... ,[ n12]. Choose A = 0. Then the class Pk is called the universal kth rational Pontrjagin class. Given a real vector bundle E -+ X classified by fE : X -> BOn, we define the kth rational Pontrjagin class of E to be piE) = f1(Pk). There is a total rational Pontrjagin class P = 1 + PI + ... + P[n/Zl and a product formula p(E EEl E') = p(E) u p(E') (B.I0) in H*(X; 0). There are rational Pontrjagin classes of a smooth manifold which, in the compact case, are homeomorphism invariants. To understand the integral case we must examine the Bockstein homo-morphism. The coefficient sequence 0 -+ £: .3:..£: -+ £:z -+ 0 gives rise to a long exact sequence where f3 is called the Bockstein. Note that the kernel of f3 is the set of classes in H*(X; £:2) which are mod 2 reductions of integral classes. CHARACTERISTIC CLASSES 383 Theorem B.13. The integral cohomology H*(BOn; £:) is an additive direct sum £:[Pl' ... ,P[n(2]] EEl Image(f») where Pk E H4k(BOn; £:) becomes the element Pk of Theorem B.12 under tensor product with A, and where f) is the Bockstein homomorphism. The class Pk is called the universal kth integral Pontrjagin class. These classes have the following property. Consider the homomorphism On -4 Un induced by complexification. The Chern classes lift back over the in-duced map BOn -4 BUn. The classes CZk+ 1 go to zero, and (B.lt ) It is conventional to define the Pontrjagin classes of a real vector bundle E by this formula, i.e., by setting (B.12) Observe now that the subset f)(H*(BOn; £:z)) c H*(BOn; £:) consists en-tirely of 2-torsion elements. Important among these are the classes (B.13) which measure whether the (k -l)st Stiefel-Whitney class is the mod 2 reduction of an integral class. Of course all the Wk's vanish when pulled back to BUn or BSPn under the maps BUn -4 BOzn and BSPn -4 B04n. We now examine BSOn. Theorem B.14. Let A be an integral domain containing 1. Then for each n, there are ring homomorphisms H*(BSOzn+ 1; A) = A[Pl' ... ,Pn] H*(BSOzn; A) = A[Pl' ... ,Pn,X]/(XZ -Pn) where Pk E H4k(BSO *; A) and where X E Hn(BSO Zn; A). The elements Pk are the images of the A-Pontrjagin classes (of Theorem B.l2) under the map BSOn -4 BOn. The class X is called the universal Euler class. It can be defined as usual for any oriented vector bundle E -4 X by setting X(E) = n(X). (For odd-dimensional bundles it is defined to be zero.) There is a product formula X(E EEl E') = X(E) u X(E'). (B.l4) There is a natural definition of X as an integral co homology class. (see Milnor-Stasheff [1 ]). Under the map BUn -4 BSOzn, the class X pulls back to Cn. Under mod 2 reduction, X becomes Wn in Hn(BSOzn; £:z). The analogue of Theorem B.l3 holds for H*(BSOn; £:). APPENDIX C Orientation Classes and Thom Isomorphisms in K-Theory The point of this appendix is to examine the notion of an orientation class for K-theory, KO-theory, or KR-theory on a vector bundle. Such classes exist only on bundles with an appropriate structure, and when they exist they determine a "Thorn isomorphism" for the given theory. These can be compared with the standard Thorn isomorphism for co homology via the Chern character. For more elaboration of the matters discussed here the reader is referred to the book of Karoubi [2]. Let n: E -4 X denote a vector bundle over a locally compact space X. This bundle may be real or complex, or even Real (when X is provided with an involution). Let k-* denote any of the theories K;pt, KO;pt or KR;pt. (This last theory is defined only for spaces with an involution. We shall let the reader make the obvious adaptations of the exposition to fit this case.) Proposition c.l. Via the projection n: E -4 X, k-*(E) is canonically a k-*(X)-module. Proof. Given any two locally compact spaces A and B, the outer tensor product induces a graded, bi-additive map defined as follows. Fix elements [VO,vl; a] E k(A x [Rm) = k-m(A) and [Wo, W1; r] E k(B x [Rn) = k -neE). Choose extensions of the isomorphisms (J: Vo -4 V1 and r: Wo -4 Wb which are defined outside compact subsets, to all ofAx [Rm and B x [Rn respectively. Fix a metric in each of the bundles, and let (J* : V1 -4 Vo and r* : W1 -4 Wo be the adjoint morphisms to (J and r. Let V; [gJ J-lj denote the outer tensor product, i.e., V; [gJ J-lj = (prjV;) ® (pr!J-lj) where prl and prz are the projections ofAx B x [Rn+m onto A x [Rm and B x [Rn respectively. Then the product of [VO,vl; (J] and [WO,W1;r] is defined to be [UO,U1;p] where 384 and where THOM ISOMORPHISMS IN K-THEORY _ ((J [8] 1 p-l[8]r -1 r*). (J* 1 From the fact that pp* = ((J(J* 1 + 1 r*r 0 ) o (J*(J [8] 1 + 1 [8] rr* 385 we see that p is an isomorphism at any point where either (J or r is an isomorphism. Hence, p is an isomorphism outside a compact subset of A x B x IRn+m. All choices involved in this definition are unique up to homotopy, so the product is well defined. The desired module multiplication is now given by the composition k-*(X) x k-*(E) k-*(X x E) k-*(E) where the second homomorphism is induced by the proper map (n x Id): E X x E. Verification that this multiplication is associative and dis-tributive is left to the reader. • From this point on we shall assume that X is compact. DEFINITION C.2. A class U E k(E) = kO(E) is said to be a k-theory orien-tation for the bundle E if k-*(E) is a free k-*(X)-module with generator u. EXAMPLE C.3. Let E = X x cm X be a trivialized hermitian vector bundle and define (C.l) where !Cm n* E denotes the trivial m-plane bundle on E and where (Jx,v(cp) = v A cP -V* L cP for (x,v) E X X cm and qJ E Age0!Cm. By using the identification 1R2m cm and taking the canonical orientation, this element can be rewritten as u = ,$e;,£l ] (C.2) where $e = Eel Se n*$dE) is the irreducible complex graded Cezm-module (extended trivially over E) and where ,£lx,v(cp) = V· qJ is given by Clifford multiplication. The fundamental assertion of the Bott Periodicity Theorem is that u gives a K-theory orientation on E (see 9.20, (9.8) and 9.28 of Chapter I). EXAMPLE C.4. Let E = X X 1R8m X be a trivialized riemannian vec-tor bundle, and define (C.3) 386 APPENDIX C where $ = $+ EEl $-= n*$(E) is the irreducible real graded Ctsm-module (extended trivially over E) and where J.lx,v(cp) = V'cp is given by Clifford multiplication. The Bott Periodicity Theorem (see 9.22, (9.8) and 9.28 of Chapter I) states that u is a KO-theory orien-tation for E. EXAMPLE C.5. Let E = X x cm = X x ([Rm EEl mm) -.':. X be a trivial-ized Real vector bundle over a compact space X with trivial involution. Set u = R + iL] EKRcpt(E) (C.4) where ctm = EEl is extended trivially over E and where (R + iL)x,u+iv(CP) = cP' u + iv' cP at any point (x, u + iv) E X x ([Rm EEl i[Rm). From the (l,l)-Periodicity Theorem of 1.10 we see that u defines a KR-theory orientation on E. We now return to the general case of a vector bundle over a compact space X. DEFINITION C.6. A class u E k(E) is said to have the Bott periodicity property if u determines a k-theory orientation in any local trivialization of E over a closed subset C c X, i.e., if k -*(Elcl is a free k -*( C)-module generated by u, whenever Elc is trivial. Theorem C.7 Let n: E -4 X be a vector bundle over a compact space X. Then any class u E k(E) with the Bott periodicity property is a k-theory orientation for E. Proof. Let 1 be a covering of X by closed subsets such that Elc; is trivial for each j. We proceed by induction on N. For N = 1 the state-ment is obvious. Suppose we have proved the assertion for E restricted to A = Cl U '" U CN-l (or to any closed subset of A). Set B = CN' For each theory k-* there exists a Mayer-Vietoris sequence with connecting homomorphisms of degree 1 (see Karoubi [2]). One easily checks that multiplication by the appropriate restrictions of u gives a morphism of exact sequences: .•. -4 k-i(A u B) -4 k-i(A) EEl k-i(B) -4 k-i(A (\ B) -4 k-i+l(A u B) -4'" 1 1 1 1 .•• -4 k-i(EIAuB) -4 k-i(EIA) EEl k-i(EIB) -4 k-i(EIAnB) -4 k-i+ l(EIAUB) -4'" By inductioin the vertical arrows are isomorphisms from k-*(A), k-*(B) and k-*(A (\ B) (since A (\ B is a closed subset of A). By the 5-lemma we THOM ISOMORPHISMS IN K-THEORY 387 conclude that the vertical arrows are isomorphisms also from k-*(A u B). The same argument clearly also applies to any closed subset of A u B. This completes the proof. _ Theorem C.7 has the following immediate consequences: Theorem C.S (The Thorn isomorphism in K-theory for Um-bundles). Let n: E -4 X be a complex hermitian vector bundle over a compact space X. Then the class A (E) = [n*AevenE n*AOddE·(J] E K (E) - 1 IC' IC, cpt where (Je(<P) = e A cp -e* L cp, is a K-theory orientation for E. In particular, the map i! : K(X) -4 Kcpt(E) given by i!(a) = (n*a) . A-l(E) is an isomorphism. Proof. By Example C.3, A_1(E) has the Bott periodicity property. _ Theorem C.9 (The Thorn isomorphism in KO-theory for Spinsm-bundles). Let n: E -4 X be a real 8m-dimensional bundle with a spin structure, over a compact space X. Consider the class S(E) = [n*$+,n*$-;u] E KOcpt(E) where $ = $+ EEl $-is the irreducible graded real spinor bundle of E and where fle(CP) = e· cp is given by Clifford multiplication. Then S(E) is a KO-theory orientation for E. In particular, the map i!: KO(X) -4 KOcpt(E) given by i!(a) = (n*a) . S(E) is an isomorphism. Proof. By Example C.4, S(E) has the Bott periodicity property. _ Theorem C.lO (The Thorn isomorphism in KR-theory for Real bundles). Let n: Eo -4 X be a real vector bundle over a compact space, and let E = Eo ® C = Eo EEl iEo be the associated Real bundle (with trivial invo-lution on X). Then the class U(E) = [n*C£O(Eo), n*C£l(Eo); R + iL] E KRcpt(E) is a KR-theory orientationfor E (where (R + iL)(cp) = cpe -ie' cp at e + ie' E Eo EEl iEo above a point in X). In particular, the map i,: KR(X) -4 KRcpt(E) given by i!(a) = (n*a) . U(E) is an isomorphism. 388 APPENDIX C Proof. By Example C.S, U(E) has the Bott periodicity property. _ REMARK C.ll. For a general Real bundle n: E ...... X over a compact space X with involution, the class A-l(E) given in C.S defines an element in KRcpt(E) which is always a KR-theory orientation for E (see Atiyah [2]). Theorem Cl2 (The Thorn isomorphisms in K-theory for S02m-' Spin2m-and Let n: E ...... X be an oriented real vector bundle of dimension 2m over a compact space X. Then the class o(E) = [n*Ct+(E),n*Ct-(E);Jl] E Kcpt(E) ® Q defined in (Il/.12.12) is a (K ® Q)-theory orientation for E. If E has a spin structure, then the class s(E) = E Kcpt(E) defined by (///.12.13) is a K-theory orientation for E. This remains true for any Spine-structure on E where $dE) = EB $i(E) is the associated irreducible spinor bundle (see App. D). The associated maps i!: K(X) ® Q --+ Kcpt(E) ® Q and given by i!(a) = (n*a) . o(E) given by i!(a) = (n*a) . s(E) (in the Spine-case) are isomorphisms. Proof. Consider first the spin (or Spine) case. In any local trivialization of E, say Elc ..::. C X 1R2m, the class s(E) becomes the element which is the pull-back to the product of the canonical generator of Kcpt(1R2m) given by the Atiyah-Bott-Shapiro Isomorphism. Hence, s(E) has the Bott periodicity property for K -theory. Suppose now that E is not necessarily spin. From the isomorphism: ct2m = Homd$IC' $d = $IC ® we see that in any local trivialization Elc ..::. C X 1R2m, the class o(E) becomes o(Eld = ® $i ® Jl] = = Si; Jl] = 2ms(E). Hence, after tensoring with Q, o(E) has the Bott periodicity property, and the proof is complete. _ THOM ISOMORPHISMS IN K-THEORY 389 The Thorn isomorphisms given above can be compared to the standard Thorn isomorphism in cohomology via the Chern character. Formulas for this are given in Chapter Ill, §12 (see (1II.12.14) and (III.12.IS». REMARK C.13. Each of the Thorn isomorphisms given above can be extended to bundles defined over a locally compact space X. The orienta-tion classes given explicitly in the Theorems do not define classes in k(E) when X is not compact. Nevertheless, one can easily show that they do pair with elements in k(X) to give elements in k(E). Basically, these orientation classes have compact support in "vertical slices" and elements of k(X) lift to classes with compact support in the "horizontal directions," and so the product has compact support on E. The reader has probably noted that if X is a manifold and E = T* X is its cotangent bundle, then the orientation classes given in the theorems above are the principal symbols of certain fundamental differential opera-tors. In particular, Theorem C.S shows that on any Srn-dimensional spin manifold, the principal symbol of the Atiyah-Singer operator is an orien-tation class for the KO-theory of the cotangent bundle. Similarly, if X is any even-dimensional spin (or Spine) manifold, then by Theorem C.12 the principal symbol of the complex Atiyah-Singer operator gives an orienta-tion class for the K-theory of the cotangent bundle. Viewed in this way, it is clear that the Atiyah-Singer operator is a fundamental one. Twisting this operator with a general coefficient bundle generates the KO-theory (or K-theory) of T*X freely as a module over KO(X) (or K(X) respec-tively). Otherwise stated, at the level of principal symbols, every elliptic operator on X is an Atiyah-Singer operator with coefficients in some bundle. The proof of Theorem C.7 carries over directly to give the following useful case of the Leray-Hirsch Theorem. Suppose E -+ X is a complex vector bundle over a compact space X, and let p: iP>(E) -+ X be the asso-ciated projective bundle, i.e., the bundle whose fibre at each point x is the projective space of all complex lines through the origin in Ex. Note that H*(iP>(E); Z) becomes an H*(X; Z)-module under the homomorphism p*. Consider now the "tautological" complex line bundle t -+ iP>(E) whose fibre at to E iP>(EJ c iP>(E) consists of all vectors v E to c Ex. Set u = c1(t) E H2(iP>(E); Z). Theorem C.14. Suppose E -+ X is a complex vector bundle of rank k. Then H*(iP>(E);Z) is a free H*(X;Z)-module with basis l,u,u2, ... ,Uk-1. Proof. When E is the trivial bundle, this follows directly from the Kunneth formula. In general one chooses a covering 1 of X and proceeds by induction using the Mayer-Vietoris sequence exactly as in the proof of Theorem C. 7. • APPENDIX D Spine.Manifolds The notion of a Spine-manifold is an important one, but it was not treated in the main body of the text to avoid too much congestion in the expo-sition. It is essentially the "complex analogue" of the notion of a spin manifold, and much of the discussion of this case can be carried out in parallel with the spin case. We begin with the complex version of a question mentioned in the introduction. Let X be an oriented riemannian manifold, and ask whether there exists a bundle of irreducible complex modules for the bundle ct(X). If X is spin, the answer is certainly yes. (One merely takes the associated bundle $1[; == PSPin(X) V where V an irreducible complex ctn-module.) However, the existence of a spin-structure is not necessary for the construction of such bundles. To see this we examine the complex representations of Spinn more closely. Suppose dl[;: Spinn -+ UN is a complex spinor representation, i.e., one coming from an irreducible Ctn-module. Let z: U I 4 UN denote the cen-ter, i.e., the scalar multiples of the identity. Then we get a homomorphism dl[; x z: Spinn x U I -+ UN which clearly has the element ( -1, -1) in its kernel. Dividing by this element gives the group == Spinn x 4'2 U I' Note that there is a short exact sequence (D.1) o --+ Z2 --+ SOn X U I --+ 1 (D.2) where the subgroup Z2 c is generated by the element [( -1,1)] = [(1,-1)]. Now we have c ctn ® C as a multiplicative subgroup of the group of units. (In fact it is obtained by "tensoring" Spinn with the unit complex numbers.) To construct the bundle of irreducible complex modules over X we proceed as in the spin case. With the sequence (D.2) in mind, we ask: Does there exist a principal over X which admits a bundle mapping: (DJ) for some principal U I-bundle PU1 over X? By "equivariant" we mean that SPIN'-MANIFOLDS 391 = for all pE PSpin' and all g E We analyse this in the spirit of Appendix A. Consider the exact sequence Hl(X;SOn) EEl Hl(X;Ul) H2(X;/f2) (DA) determined by the coefficient sequence (0.2). The coboundary map asso-ciates to a pair (Pso,puJ the element w2(pso) + c\(PuJ where Cl is the mod 2 reduction of the first Ch ern class of PUt (see Example A.5). Under the isomorphism (A.7) this coboundary map can be rewritten as Hl(X; SOn) EEl H2(X; 1') H2(X; 1'2) (D.5) where p: H2(X; 1') -+ H2(x; 1'2) is mod 2 reduction. Consequently, given the frame bundle Pso(X), we can find the bundle (D.3) provided that w2(Pso(X» = p(u) for some u E H2(X; 1'), i.e., provided that w2(X) is the mod 2 reduction of an integral class. Of course this argument carries over to any principal SOn-bundle. DEFINITION D.l. Let PSOn be a principal SOn-bundle over X. A Spinc-structure on Pso consists of a principal U cbundle PUt and also a principal with a bundle map The class c E H2(X; 1') corresponding to PUt under the isomorphism H2(X; 1') Prinu,(X) (cf. A.5), is called the canonical class of the Spine-structure. The argument given above proves the following: Theorem D.2. A principal SOn-bundle P carries a if and only ifw2(P) is the mod 2 reduction of an integral class, that is, if and only if W3(P) = O. (See B.13 for a discussion of the classes It should be noted that Theorem D.2 could be established in the spirit of Theorem l.4ff by considering the appropriate 2-fold coverings of the spaces PSOn X PUt. DEFINITION D.3. An oriented riemannian manifold with a Spine-struc-ture on its tangent (frame) bundle is called a Spinc-manifold. The theorem above has the following immediate corollary: COROLLARY D.4. An orientable manifold X carries a Spine-structure (i.e., X can be made into a Spine-manifold), if and only if the second Stiefel-lthitney class wz(X) is the mod 2 reduction of an integral class, i.e., if and only if W3(X) = O. 392 APPENDIX D If a manifold carries one Spine-structure, it often carries many. They are parameterized by the elements in 2H2(X; if) EEl HI(X; if2)' There are two important cases where a Spine-structure exists and is canonically determined: EXAMPLE D.S. Any bundle with a spin structure carries a canonically determined Spine-structure. The Spine-bundle is obtained as PSpinc == PSPin x.l:2 V I where if2 acts diagonally by (-1, -1) and where V I of course denotes the trivial circle bundle. We see then that any spin manifold is canonically a Spine-manifold. EXAMPLE D.6. Any complex vector bundle E carries a canonically deter-mined Spine-structure. To see this, note first that w2(E) == cl(E) (mod 2), and so Theorem D.1 implies that E carries Spine-structures. To see that there is a canonical one we proceed as follows. Introduce a hermitian metric in E and let PujE) be the principal Un-bundle of unitary frames. There is a canonical homomorphism j: Un L----. (D.6) which we shall construct explicitly below. It is a lifting of the homomor-phism Un -+ S02n X Ul given by g I---> (i(g), det(g)) where i:Un 4 S02n is the standard inclusion. Thus, we have a commutative diagram The canonical principal Spine bundle for E is now constructed as the associated bundle. Pspinc(E) == PujE) x j Note, incidentally, that the associated U I-bundle in this case is PuJE) == PujE) xdet VI' (D.S) (D.9) This is the principal bundle of the complex line bundle A"E whose first Chern class is cl(E), i.e., cl(A"E) = cl(E). One consequence of this discussion is that every complex manifold, in fact, every almost complex manifold, is canonically a Spine-manifold. Before moving on we shall provide the details of the homomorphism (D.6). It can be shown to exist by proving the existence of the lifting (D.7) using covering space theory. However, it can be given explicitly as follows. Let g E Un and choose a unitary basis {el' ... ,en} of C" in which g has the form: g diag{eiB" ..• ,eiBn}. Let {el,Jeb •.• ,en,Jen} be the canonically SPIN'-MANIFOLDS associated orthonormal basis of [R2n = en. Then . nn ( Ok . Ok ) j(g) == k= 1 cos 2 + Sill 2 ekJek x e in Spin'2n = Spin2n XU:2 SI. 393 (D.lO) We have seen that spin manifolds and complex manifolds are all Spinc-manifolds. In fact, it requires some searching about to find an orientable manifold which is not Spinc. Note, for example, that [p>4n+I([R), although neither spin nor complex, is Spinc. It was observed by Landweber and Stong that perhaps the simplest manifold which is not Spinc is the orientable 5-manifold SU3/S03 whose only non-zero mod 2 cohomology classes are 1, W2, W3 and W2 . W3. Since SqI(W2) = W3 of-0, one concludes that W3 of-O. EXAMPLE D.7. (Universal non-Spinc-manifolds) We shall construct here a family of open simply-connected n-manifolds Un(p) which are not Spinc for each n 9. These examples are universal in the sense that every simply connected non-Spinc-manifold of dimension n 9, carries some Un(p) as an open submanifold. We fix a positive integer p and consider the cell complex L == S2 U", D3 where the map oD3 = S2 S2 is of degree 2P• We can embed this com-plex into [R6. In fact there is a natural PL-embedding given as follows. Consider S2 od3, where d3 denotes the standard 3-simplex, and embed the mapping cone C", = S2 U'" S2 -+ S2 * S2 S5 in the obvious way. Here "*" denotes the join. This clearly extends to an embedding S2 U'" D3 4 S2 * D3 D6. Let U1: be a regular neighborhood of L in [R6. This is an open parallel-izable manifold. We now claim that there is a 3-dimensional real vector bundle E -+ U 1: which restricts to be the non-trivial bundle on S2 C L. Since U1: retracts onto L, we need only find a map F: L -+ BS03, which when restricted to S2 represents the non-zero element in 7r2(BS03). How-ever, since 7r2(BS03) 7rl(S03) "Z2, any map f: S2 -+ BS03 extends to L. Therefore this bundle exists. We define the manifold Un(p) as follows. Consider L embedded in the total space of E as a subset of the zero section; i.e., L C U 1: C E. For each n 9, let Un(p) be a regular neighborhood of L x {O} C E x [Rn-9. Theorem D.S. The manifolds Un(p), p 1, are not Spinc. Furthermore, any simply-connected manifold xn of dimension n 9 is not Spinc if and only if for some p 1, there exists an embedding Un(p) 4 X" as an open submanifold. 394 APPENDIX D Proof. Fix p 1 and n 9 and set U = Un(p). Let S2 cUbe the non-trivial 2-sphere. Then TUls2 Els2 EEl (trivial), and so Wz(U)[S2] = w2(E)[S2] =1= 0 by construction. Since H2(U;1') = 0 and 7r1(U) = 0, we see that w2(U) is not the mod 2 reduction of an integral class. (Note that U is homotopy equivalent to L.) Hence, U is not Spine. Of course any open submanifold of a Spine-manifold is Spine. Hence, we need only show that a simply-connected n-manifold xn which is not Spine must contain some un(p) as an open submanifold. Since 7rIX = 0, we have an isomorphism 7r2X -='> H z(X; 1'). Since X is not Spine, the homo-morphism W2 E H2(X; 1'2) Hom(7r2X, 1'2) is not the mod 2 reduction of an element in H2(X; 1') Hom(7r2X,1'). Hence, W2 is non-zero on some torsion class rx of order 2P in 7r2X. Now rx is represented by a map f: S2 ...... X which extends to a map F: un(p) ...... X (since 2Prx = 0 in 7r2X). The induced bundle F*TX is clearly equivalent to E EEl (trivial) Tun(p). Hence, by Smale-Hirsch immersion theory, F is homotopic to an immersion un(p) c..... xn. The immersion can be made an embedding by putting L into general position and shrinking Un(p) to a small neighborhood of L. This completes the proof. _ We did not need to invoke Smale-Hirsch Theory here. The embedding un(p) 4 xn which is homotopic to F can be built explicitly. A tubular neighborhood of S2 C Un(p) is diffeomorphic to a tubular neighborhood of S2 C Xn, since the normal bundles are equivalent. Completing the em-bedding to a neighborhood of D3 C Un(p) is straightforward. For compact examples we add a boundary to Un(p) and consider the double xn(p) == Un(p) Uo Un(p) where Un(p) == Un(p) U aUn(p). Let us return our attention to manifolds which are Spine. There is an alternative approach to this concept which comes from considering vector bundles. DEFINITION D.9. Let X be a Spine-manifold of dimension n. By a com-plex spinor bundle for X we mean a vector bundle S associated to a repre-sentation of Spine by Clifford multiplication, i.e., S == PSpinc(X) V where V is a complex Ctn-module and d: ...... GL(V) is given by restriction of the Ctn-representation to c ctn ® c. If the represen-tation of ctn is irreducible, we say that S is fundamental. These spinor bundles are bundles of complex modules over ct(X). When n is even there exists only one fundamental spinor bundle for X, denoted S(X). It splits into a direct sum (D.11 ) SPIN'-MANIFOLDS 395 where S± (X) = (1 ± w,dS(X) and where = in/2el .•. en is, as usual, the volume form. Multiplication by a non-zero tangent vector maps S± (X) to S+(X) and is invertible. When n is odd, there are two irreducible complex representations of ctn• However they are equivalent when restricted to Hence, there is only one fundamental spinor bundle S(X) for any Spinc-manifold X. For a spin manifold X, the bundle S(X) is just the usual complex spinor bundle. For a complex manifold X with its canonical SpinC-structure, we have that S(X) = AO.* that is, S(X) is simply the direct sum of the complex exterior powers of the tangent bundle, considered as a complex bundle (which, in turn, is isomorphic to the direct sum of the bundles of (O,q)-forms, ° ;::; q ;::; n). This follows directly from the interpretation of the complex spinor repre-sentation given in 1.5 (see 1.5.25 and the attending discussion). In partic-ular, from this we see that the Clifford multiplication Ct(X) Q9[J;! S(X) ---> S(X) is generated as follows. For each tangent vector v, consider the linear map At;TxX !:.+ given by Jlv(cp) = v 1\ cP -v* L cP where v* is the l-form corresponding to v under the hermitian metric on X. Repeating this operation gives Jlv(Jlvcp) = -iivWcp, and so by the universal property of Clifford algebras, the map Jl extends to a representation of Ct(X). Since each Jlv is complex linear the repre-sentation is complex. It is instructive to examine these fundamental spinor bundles as one varies the Spine-structure on a given manifold. To do this it is good to have clearly in mind the elementary isomorphisms: (D.12) where Vectl(X) denotes the set of equivalence classes of complex line bundles on X and where the map to H2 is given by the first Chern class see Example A.5). EXAMPLE D.10. Let X be a spin manifold with its canonical Spine-tructure. We can change this structure by changing the U I-bundle as ollows. Pick an element rx E H2(X; Z), and let Pu,(rx) E Prinsl(X) and IX E Vectl(X) be the corresponding elements under (D. 12). Then we de-ne the rxth Spine-structure by (D.l3) 396 APPENDIX D Note that this has a map (D.14) where Pupa) = PU,(a)/£:2 is the "square" of Pu,(a). (In particular A2a = A;.) We now let Sa(X) be the fundamental spinor bundle for the ath Spinc-structure on X. Then it is easy to see that Sa(X) = S(X) ® Aa' (D.1S) Thus, changing the Spine-structure on X by an element a E H2(X; £:) amounts to twisting the fundamental spinor bundle by the complex line bundle Aa' This is a general fact. The group H2(X; £:) acts on the set of Spinc-structures on a Spine-manifold X, by ten so ring the fundamental spinor bundle with complex line bundles (modulo this action, the Spine-structures correspond one-to-one with elements of Hl(X; £:2)') Notice that tensoring with line bundles preserves the fibre-dimension and thus takes one bundle of irreducible modules over ct(X) to another. A good example of this phenomenon is the following. Suppose X is a complex manifold which is also spin. This means that w2(X) == cl(X) == 0 (mod 2). Now the class -Cl (X) E H2(X; £:) corresponds to the complex line bundle K == T* X, called the canonical bundle. Since -Cl (X) = Cl (K) is even, there exist square roots Kl/2 for K. The different square roots are parameterized by elements of Hl(X; £:2), corresponding to choices of spin structure. Now the canonical spinor bundle S(X) = for X as a Spinc-manifold, and the canonical spinor bundle $dX) for X as a spin manifold are related by the equation (D.16) Much of this business of Spine-manifolds becomes transparent when viewed in terms of the fundamental spinor bundles. For example, let us return to the original problem of trying to construct a fundamental spinor bundle (i.e., a bundle of irreducible complex modules for ct(X)) over a manifold X. Locally of course we can always do it. Let {Ua}aEA be a cov-ering of X so that U a, n ... n U ak is contractible for all ab ... ,an" On each U a' bundles can be trivialized and we can find a fundamental com-plex spinor bundle of the form Ua x V where V is an irreducible ctn ® ([>module. Now in passing from Ua to Up we look for transition functions gap: Ua n Up -+ Spinn so that 0 gap = gap: Ua n Up -+ SOn are the cor-responding transition functions for Pso(X). Now the existence of a com-patible set of choices is equivalent to the vanishing of the Cech cocycle Wapy == gapgpygya: Ua n Up n Uy £:2 = in H2(X; £:2)' (D.17) SPIN'-MANIFOLDS 397 Suppose now that the class [W ] E H2(X; if2) is the mod 2 reduction of an integral class 1I/'E H2(X; if), and let A be the complex line bundle cor-responding to 11/'. Consider the problem of finding a square root of A, i.e., a line bundle A 1/2 with (A 1/2)2 = A. Let Yap: Van V p --+ Sl be the transition functions for X Since Van V p is contractible we can extract a square root Yap == yW: Va n V p --+ Sl. However, the compatibility is just the Cech co cycle (D.18) where 0---+ if2 ---+ Sl Sl ---+ 0 and (J(z) == Z2. The class [w'J E H2(X; if2) is just the co boundary of A E H1(X; Sl) under the associated long exact sequence in cohomology. In fact, consider the following commutative diagram: H1(X;Sl) H1(X;Sl) H2(X;if2) 111 111 H2(X;if) H2(X;if) H2(X;if2) It is clear that our obstructions agree, i.e., [w'J = p(c1(A)) = p(1I/') = [wJ. And so [w ] + [w'J = 0 in the wonderful world of if 2. This means essentially that while we cannot construct the spinor bundle and we cannot construct A 1/2, we can construct their product. Adjusting by coboundaries we can choose gap and Yap so that Wapy == Thus the tran-sition functions Gap: Va n Vp ---+ Spinn XU:2 Sl defined by GaP = gaP X Yap have Wapy == GapGpyGya == 0, and thus determine a global bundle we may think of as S(X) = So(X) ® A 1/2 (D.19) where So(X) is the fundamental spinor bundle for the possibly non-existent spin structure on X, and where A 1/2 is the possibly non-existent square root of A with C1(A) == W2(X) (mod 2). Since S(X) is a bundle of modules over ct(X) we need only introduce an adapted metric and connection to make it a Dirac bundle. We do this as follows. Fix any line bundle A as above and choose a unitary connection on A. This induces a connection on the bundle A 1/2 when it exists. Locally both So(X) and A 1/2 exist and So(X) carries a canonical (riemannian) con-nection. We give the bundle S(X) = So(X) ® A 1/2 the tensor product con-nection. It is not difficult to see that this connection is well defined globally. 398 APPENDIX D It is easy to check, using the arguments of §4, that this connection makes S(X) into a Dirac bundle. Notice that the line bundle A corresponds to the principal U l-bundle PU1 associated to the Spine-structure, i.e., Pu,(A) = PSpinc(X) X p U 1 where p : -+ U 1 is induced from the projection Spin. x U 1 -+ U 1. Let us summarIze: Proposition 0.11. Let X be a Spinc-manifold with associated complex line bundle A. Then to each U l-connection w on A there is a canonically asso-ciated connection on P Spinc. It is just the lift of the product of the canonical riemannian connection with w via the covering map PSpinc -+ Pso x Pu,(A). This connection makes any complex spinor bundle for X into a Dirac bundle. Associated to any unitary connection on A there is the curvature 2-form O. This is a closed 2-form, and the class [(2n)-1 a] E H2(X; Ill) represents C1(A) or more precisely the image of Cl(A) under the map H2(X;Z)-+ H2(X; Ill). We call this the real Chern class of l. The curvature form is defined as follows. For a local trivialization of Pu,(A), the connection 1-form becomes simply a real-valued 1-form w. Then O=dw. If we change trivializations by a transition function g: V n V' -+ U 1, then the new connection 1-form is w' = w + 9 -1 dg. Locally we have 9 = ei9 for some function e. The equation above becomes w' = w + i de. It follows that dw' = dw, that is, the curvature 2-form a is well defined globally. Suppose now that Wl and W2 are two different connections on A with as-sociated curvature forms 01 and O2. Then it is easy to see that W1 -W2 = a for a globally defined I-form 0(. Hence, 01 -O2 = dO(, and we see that the de Rham class [a] is independent of choice of connection. We are now prepared to derive the Bochner-type identity for these spinor bundles. Theorem 0.12. Let X be a Spinc-manifold with associated line bundle A, and fix a connection on A with curvature 2jorm O. Let S be a bundle of complex spinors on X with the canonical connection, and let D be the Dirae operator on S. Then I D2 = V*V + t K + t a I where K denotes the scalar curvature of X and where a denotes Clifford multiplication by the 2jorm o. Proof. The computation is local, so we can assume S(X) = So(X) ® A 1/2 where So(X) is a spinor bundle for the local spin structure. We may now SPIN'-MANIFOLDS 399 apply Theorem 11.8.17, and it is sufficient to compute the term Recall that ® z) = I (ehG') ® R;;::k(Z) j<k = I (ehG') ® (!n(ej,ek)iz) j<k = .I n(ej,ek)ehG') ® z J<k = n(ejh)ej /\ ek)G' ] ® z = G') ® z • We used here the fact that the curvature of the induced connection on Al/2 is !n. We also use the fact that = n(v,w)iz. Note that Clifford multiplication by in is a hermitian symmetric operation. Given this theorem, the following lemma is relevant: Lemma 0.13. Let A be a complex line bundle on a manifold X. Then any closed 2jorm n which represents the real Chern class of A can be realized as the curvature form of a U l-connection on A. NOTE D.14. This means, in particular, that if Cl(A) is a torsion class in H2(X; Z), then A admits a flat connection. Proof. Choose any U l-connection Wo on 1J and let no be its curvature form. Since n and no represent the same de Rham class, there is a I-form 0( so that dO( = n -no. Let w = Wo + 0(. Then the curvature of w is no + dO( = n .• Suppose now that X has even dimension and consider the Dirac operator D+: rs+(X) --rS-(X) for the splitting (D.1I). Using (D.19) and the Atiyah-Singer Formula, we obtain the following: Theorem 0.15. Let X be a compact Spine-manifold of even dimension, and let D denote the Dirac operator of the fundamental complex spinor bundle. Then (D.20) where c is the canonical class of the Spine-structure, i.e., c = Cl(A) for the associated complex line bundle A. 400 APPENDIX D Combining this with D.13 and D.14 gives the following: Corollary 0.16. Let X be a compact Spine-manifold such that w2(X) is the mod 2 reduction of a torsion class. If X carries a metric of positive scalar curvature, then A(X) = o. Corollary 0.17. Let X be a compact Spine-manifold with canonical class c E H2(X; £'). Let n be any 2-form representing the de Rham class of c in H2(X; £') ® lIt Then there is a riemannian metric on X whose scalar curva-ture satisfies only if {eYe. A(X) }eX] = O. Here the norm 11'11 is taken in the same metric. It is given by Ilnll = I IAil where n = I Ahi-l /\ e2i for the diagonalizing orthonormal frame ebe2, ... at the point. It is a nice exercise to apply this theorem to IPn(C) with the standard Fubini-Study metric, and then to verify directly the vanishing of the "Hilbert polynomial" at certain integers. It is interesting to examine the Dirac operator on a complex hermitian manifold X, considered as a Spine-manifold. Recall that S(X) = A 0,*. The splitting S(X) = S+(X) EB S-(X) corresponds to the even-odd decomposi-tion A 0,* = A O,even EB A O.odd. The Dirac operator can be identified with 8 + 8* : A O,even -----+ A O,odd where 8* is the hermitian adjoint of the operator 8. The index of this operator is the Todd genus of X. If c l(X) == 0 (mod 2), we can choose a spin structure on X, i.e., we choose a square root K1/2 of the line bundle K = An,O. 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A-genus, 231, 283, 284 A-degree of J, 309 A-enlargeable, 309 A-genus, 138, 161, 233, 256, 280, 292 higher --,310 of h ypersurfaces, 138 A-operator, 137 si-c1ass, 92, 100, 152, 162, 163,276,292, 298, 313 Adams operation, 236 Adams Theorem, 46 Adams-Milnor Theorem, 93 adjoint -operator, 187, 194 -representation, 12 algebra associated graded -, 10 Clifford -, 7, 27, 29 exterior -, 10, 24, 38 filtered -, 10 matrix -isomorphisms, 26 E,-graded -, 9 almost complex structure, 336 integrable ---, 339 analytic index, 135, 141, 142, 192, 194, 201, 254 -Clifford -, 141, 142, 216, 217 -G--,212 -mod 2-,147 -of a family, 208 -of Mishchenko-Fomenko, 312 anti-self-dual, 342 associated -bundle, 93, 371 -graded algebra, 10 associative -calibration, 352 -geometry, 352 associator, 352 asymptotic principal symbol, 245 Atiyah-Bott-Shapiro isomorphisms Index - for K-theory, 69 -for KR-theory, 75 Atiyah-Milnor-Singer invariant, 144, 276 Atiyah-Segal-Singer Fixed Point Formula, 261 Atiyah-Singer Index Theorem, 244 cohomological form, 254, 255, 258 -for families, 269 -for real families, 272 - for Cf,-linear operators, 273 Atiyah-Singer operator, 121, 137,256, 267,274 Cfn-linear, 140 conformal invariance of -, 133 twisted -, 139 automorphism a--, 9, 95, 127 triality -,55 bad end, 324 Bianchi identity, 112, 362 big covering spaces, 302 Bochner -method, 153, 335 -formulas, 155-157 -Theorem, 157 Bochner-Lichnerowicz formulas, 160, 164, 398 Bockstein homomorphism, 382 Bott periodicity unitary case, 62, 63, 67, 225 orthogonal case, 63, 67, 222 (1,1) case, 72, 73 -property, 386 bundle associated -,93,371 induced -,375 -of bases, 94, 370 -of orthonormal bases, 111, 371 principal-, 370 virtual -, 59 418 INDEX C*-algebra of a discrete group, 311 calibration, calibrated manifold, calibrated geometry, 351 associative, 352 Cayley, 352 coassociative, 352 speciallagrangian, 352 canonical -bundle of complex manifold, 361, 396 -bundle over Grasssmannians, 379 -class of a Spin'-structure, 391 -orientation of a complex structure, 336 -riemannian connection, 112 Cartan-Dieudonne Theorem, 17 Cayley numbers, 51 character, 212 -ring, 212 characteristic class Chern, 227, 281, 374, 381, 398 Pontrjagin (integral), 228, 282, 283, 382 Stiefel-Whitney, 79, 81, 83, 84, 373, 380 universal, 377 Chern class divisibility of --,281 first --, 374 kth --,381 total--,277,381 vanishing first --, 367 Chern character, 234, 235, 237, 238 reduced --,316 --commutativity defect, 241 Cik-Dirac bundle, 140 Cik-bundle, 140,214 Dirac -,140 riemannian -,215 d"2-graded -,215 Cik-linear operator Atiyah-Singer, 140, 144-153 differential, 215 Fredholm, 217 d"2-graded,215 classical pseudodifferential operator, 244 classification of --Clifford algebras, 27, 29 --irreducible representations of Clifford algebras, 32, 33 classifying space, 376 Clifford algebra 7 --classification, 27, 29 --periodicity, 27 --representations, 30, 37 Clifford modules, 30 Grothendieck group of --, 33,40, 73 Real--,73 d"z-graded --,39,73 Clifford -bundle, 95 -bundle of a manifold, 111 -co homology, 332 -diamond, 331 -difference element, 236 -group, 36 -index, 216, 217 -multiplication, 30 coassociative geometry, 352 cobordism, 90,92, 100 cocycle condition, 371 cohomological formulas for the index, 254, 255, 258, 399 cohomology Clifford, 332 with compact supports, 239 Dolbeault, 335 spin or, 357, 359 cokernel, 192 compact operator, 192 compact support bundle with--, 313 cochain with --, 239 cohomology with --,238 K-theory with--, 66 compactly enlargeable, 303 complex -spinor bundle, 96, 394 -spin or representation, 36 -volume element, 34, 99,135 conformal invariance of the Dirac operator, 133 conjugate bundle, 227 connected sum, 91 connection, 101 canonical riemannian -, 112 -I-form, 101, 103 -laplacian, 154 -on spinor bundles, 110 projected -, 282 tensor product -, 122 constant at infinity, 303 contracting 6 -maps, 302, 317 contraction, 24 convolution product, 173 covariant derivative, 102 --as derivation, 107-8 --on spinor bundles, 110 riemannian --, 103 second --, 154 curvature, 101 -2-form, 101, 398 -identities, 112, 126 -of sphere, 162 -on spinor bundles, 110 -operator, 158 positive scalar-, 160-165,297-326 Ricci -,156 scalar -, 160 -transformation, 106 de Rham -complex, 125 -cohomology, 125 degree -of a map (constant at infinity), 303 pure-,8 derivative co variant -, 102 exterior -, 123 invariant second -, 154 diffeomorphism spin structure-preserving -, 86 difference element, 100 Clifford --,236 spinor --,238 differential operator 167 Ci.-antilinear --,216 Ci.-linear --,215 Ci.-linear Z2-graded --, 215 elliptic--, 113, 167, 188 G-invariant --,211 See also: operator and elliptic operator dimension geometric -of a bundle, 47 -of some classical groups, 56 Dirac bundle, 114 Ci.---, 140 rth cohomology group of --,359 Dirac laplacian, 113 Dirac operator, 112 Atiyah-Singer --, 121, 137,256, 267,274 Ci.-linear --, 140 conformal invariance of --, 133 essential self-adjointness of --, 117 kernel of --, 116, 117 --on Clifford bundle, 123 INDEX principal symbol of --, 169 self-adjointness of --, 114 eigenspace, 196 eigenvalues, 196 growth of -, 196 Einstein's equations, 369 elliptic complex, 332 elliptic estimates, 194 elliptic operators, 113, 168, 188, 191 continuous family of --,204,206 homotopy of --, 204 product family of --,205 -pseudodifferential-, 191 enlargeable, 302ff . .4--,309 compactly -, 303 weakly-, 318 a dense, 196 a-con tracting, 302 --on 2-forms, 317 e-hyperspherical, 317 a-local operator, 182 419 essential self-adjointness of Dirac operator, 117 essentially positive self-adjoint operator, 219 Euller characteristic operator, 136 Euler class, 227, 383 even part of Clifford algebra, 9 exceptional isomorphisms of Lie groups, 50, 56, 58 excision property, 248 exotic spheres, 93, 162, 292 exponential mapping, 13 exterior -algebra, 10, 24, 38 -derivative, 123 family of --differential operators, 205 --elliptic operators, 206, 208 --real operators, 210, 270, 272 --riemannian manifolds, 269 fibre integration over the -, 239 filtered algebra, 10 fixed-point -formula, 267 -set, 261 formal adjoint, 187, 194 formal development, 179 420 Fourier transform 171 frame bundle, 111 Fredholm operators, 192,201 and K-theory, 208, 211, 222 Cik-antilinear, 217 Cik-linear, 217 families of --, 208 G-invariant -212 G2-structures, 347, 348, 357 G-signature theorem, 264 G-bundle, 259 G-index, 212, 259 -theorem, 260 G-manifold, 259 G-spin theorem, 267 G-structure topological -, 348 Gallot-Meyer Theorem, 158 geometric dimension, 47 good presentation of bundles, 171 graded Hilbert module, 217 Grothendieck group of Clifford modules, 33,40, 73 group -actions, 29 Clifford -,36 isotropy -,43 K--, 59 -of units in a Clifford algebra, 12 orthogonal-, 13 Pin-,14 special orthogonal-, 17 Spin-,14 Gysin homomorphism, 239 harmonic -p-forms, 125, 195 --space, 332 -spinor, 160, 161,367 heat kernel, 198, 277 trace of -, 199 hermitian line bundle positive -, 360 higher A-genus associated to u, 310 Hilbert polynomial, 365 Hilbert module graded --, 217 Hirzebruch -L-sequence, 232 -Signature Theorem, 233, 256 -Riemann-Roch Theorem, 258 INDEX Hodge Decomposition Theorem, 195, 332 Hodge laplacian, 123 holonomy group, 345 Spn--, 357, 367, 368 homotopic elliptic operators, 204, 214 homotopy sphere, 93, 162,292 hyperspherical, 317 hypersurfaces V"(d) 88, 90 A-genus of --, 138 immersions into Il\I:n, 283, 286 incompressible surface, 326 index (analytic), 141, 142,208,312 Clifford -, 216, 217 G--,212 mod 2-,147 -of a Cik-linear operator, 141, 142 -of an elliptic operator, 135, 194,254 -of a family, 208 -of a Fredholm operator, 192, 201 -on an odd-dimensional manifold, 257 index (topological), 244 G--, 259 -of a family of operators, 269 -of a real family of operators, 272 topological invariance of -,201 induced bundle, 375 infinitely smoothing operator, 179, 186 integrable almost complex structure, 339 integrality theorems, 280, 400 integration over the fibre, 239 interior product, 24 interpolation of Sobolev norms, 175 invariant second derivative, 154 invariant Atiyah-Milnor-Singer -, 144 irreducible -bundle of Ci(E)-modules, 97 -manifold, 345 -representation 31 isotropic subspace, 335 isotropy group, 43 -of a spin or, 343 join, 377 kth power operation, 235 K-theory,59 equivariant -, 259 higher groups, 61 multiplication in -, 59,66 -of a point, 62, 63,149 real-,60 Real-,70 reduced -, 60 relative -, 61 -with compact supports, 66 K3 surface, 90 Kiihler manifold, 330 -with Cl = 0,367 kernel, 192 Kervaire semicharacteristic, 143, 290 Kervaire sphere, 162 Kodaira-type vanishing theorem, 360 Kuiper's Theorem, 208 Kummer surface, 90 U-section, 116 L-genus, 233 L-sequence, 232 L-operator, 24, 95, 128 lagrangian special-, 352 Laplace-Beltrami operator, 168 laplacian Hodge-, 123 connection -, 154 Lefschetz number, 212 Leray-Hirsch Theorem, 389 Lichnerowicz Theorem, 160, 362 Lie algebra of Spin., 40 Lie groups (exceptional isomorphisms), 50, 56,58 line bundle tautological--,47 local trivialization, 371 local holonomy group, 345 loop space, 222 Minkowski space, 369 mod-2 index, 147 modules Clifford -,30 decompositions of --, 35 Z,-graded -,39 Grothendieck group of Clifford -, 33, 40,73 multiplication Clifford -, 30 multiplicative property, 249 for sphere bundles, 252 multiplicative sequence, 229 .4-,231 Hirzebruch L-, 232 Todd,230 INDEX Nakano type vanishing theorem, 360 negative hermitian line bundle, 360 non-negative curvature operator, 158 norm, 15 uniform Ck __ , 172 Novikov Conjecture, 312 odd part of Clifford algebra, 9 operators adjoint -, 187, 194 Atiyah-Singer -, 121, 137, 274 bounded -, 192 Cf.-linear Atiyah-Singer -, 140 compact -, 192 differential-, 167 elliptic -, 113, 168, 188 elliptic pseudodifferential -, 191 essentially positive -, 219 Euler characteristic -, 136 families of -, 204, 206 Fredholm -, 192,201,217 G--, 211 heat -,198 Laplace-Beltrami -, 168 positive self-adjoint -, 198 pseudodifferential-, 178, 179, 186 signature -, 136 smoothing -, 179, 186 orientation k-theory -, 385 oriented -bundle, 78 -volume element, 21 orthogonal -almost complex structure, 336 -group, 13 parallel, 139 parametrix, 117, 190, 191 Pauli matrices, 120 periodici ty -in Clifford algebras, 27 Bott -,62,63,67, 72, 73, 222, 225 Pin group, 14 plurigenus, 366 Poincare duality, 239 Pontrjagin classes, 228, 382, 383 positive -curvature operator, 158 -hermitian line bundle, 360 -self-adjoint operator, 198 421 422 INDEX positive scalar curvature, 160-165,297-326 -and higher A-genera, 311 obstructions to -, 301 -on bad ends, 324 -on compact manifolds, 299 -on complete manifolds, 306, 310, 319, 322, 324 -enlargeable manifolds, 306, 310 -3-manifolds, 324 -4-manifolds, 344 -weakly enlargeable manifolds, 319 -under surgery, 299 the space of metrics with -, 326-330 power operation, 235 primitive cohomology class, 334 principal bundle, 370 --of tangent frames, 94, 111 --of bases, 370 --of orthonormal frames, 78, 371 -spin --,80 universal--,376 principal symbol, 113, 168,243 asymptotic --,245 --ofa pseudodifferential operator, 186 regularly G-homotopic --,214 regularly homotopy of --,204 product convolution -, 173 interior -, 24 -in K-theory, 59 smash-,61 warped -, 321 projectivization of a bundle, 225 pseudodifferential operators, 178, 179, 186 classical --, 244 elliptic --, 191 formal development of --, 179 principal symbol of --, 186, 188 symbol of--, 177, 187 pure spinor, 336 Radon-Hurwitz Theorem, 45 rank of a spinor, 43 real -spinor bundle, 96 -spinor representation, 35 -K-ring, 60 Real -K-group, 70 -module, 73 -space, 70 -vector bundle, 70 -Z2-graded module, 73 reduced Chern character, 316 reduced K-ring, 60 reducible representation, 31 refinement of a covering, 372 reflection, 17 regularly homotopic principal symbols, 204, 214 relative K-groups, 61 Relative Index Theorem, 315 Rellich Lemma, 173 representations (See also "modules".) adjoint -, 12 complex spinor -,36 ± decomposition of -, 35 irreducible -,31 -of a Clifford algebra, 30, 37 quaternionic -, 30 real spinor -, 35 -ring, 211 twisted adjoint -, 14 unitarity of -of Clifford algebras, 35 weights of a -,44 Ricci -curvature, 156, 308, 367 -form, 362 Riemann-Roch-Hirzebruch formula, 258 riemannian canonical-connection, 112 -covariant derivative, 103 -curvature identities, 112, 126 -curvature of the sphere, 162 Fundamental Theorem of -Geometry, 112 -structure, 78 ring of virtual representations, 211 Rochlin's Theorem, 89 SI-actions, 291, 292, 295 S3-actions, 291, 296 scalar curvature, 160 positive --, 160-165,297-326 (See also "positive scalar curvature".) Schwartz space, 171 second fundamental form, 282 second covariant derivative, 154 self-adjoint operator, 195 essentially positive --,219 positive --, 199 self-dual 2-form, 342 INDEX 423 Serre duality, 333 signature G-Theorem, 264 -of a manifold, 137, 266, 284, 288, 290, 295, 349 -of a spin manifold, 89, 280 -operator, 136 -Theorem, 233, 256 twisted -operator, 139 smash product, 61 smoothing operator, 179, 186 Sobolev basic -k-norm, 170 -s-norm, 171 -space, 170, 171 -Embedding Theorem, 172, 176 speciallagrangian geometry, 352 special orthogonal group, 17 Spin'-(manifold), (structure), 391 spin -cobordism, 90, 92, 100 -field, 18 -group, 14 -group as a covering group, 19,20 -manifold, 85, 88, 89,90 -structure, 80, 81, 82, 84 -structure preserving, 86, 266 spinor bundle, 96 canonical complex --of a spin manifold, 121, 137 canonical complex --of a Spin' manifold, 394 complex and quaternionic structure on--,98 --decompositions, 99 ;E2-graded --,96,98 spin or -cohomology, 357, 359 -difference element, 238 harmonic -, 160 pure -,336 -representation, 35, 36 Splitting Principal, 226, 237 Stiefel-Whitney class, 380 first--, 79, 83, 84, 373 second --, 81, 83, 84, 373 kth --,380 total--, 380 structure riemannian -,78 spin -,80 support of a pseudodifferential operator, 183 surgery, 299 suspension, 61 symbol class, 169, 243 Real--,271 symbol asymptotic principal-, 245 -of a pseudodifferential operator, 177, 187 principal -, 113, 168, 243 principal -of a pseudodifferential operator, 186, 188 tangent frame bundle, 111 tautological line bundle, 47 tensor product --connection, 122 ;E2-graded --, 11 Thorn class, 152 Thorn isomorphism, 239, 240, 243, 271, 387-89 --in K-theory, 243 Todd sequence and genus, 230, 367-8 topological G-structure, 348 topological index, 244, 269 -G--,259 --of a family of Cik-linear operators, 273 --of a family of complex operators, 269 --of a family of real operators, 272 torsion tensor, 112 total characteristic class, 230 A-,231 ,4-, 231, 237, 238, 242, 254 Chern, 227, 381 Hirzebruch L-, 232, 237, 242 Pontrjagin, 228, 382 Todd, 230, 242 trace of the heat kernel, 199 transition functions, 371 transpose antiautomorphism, 12 triality automorphism, 55 tubular neighborhood, 299 twisted -adjoint representation, 14, 15 -Atiyah-Singer operator, 139 -signature operator, 139 twistor space, 339, 342 424 INDEX uniform Cl-norm, 314 uniform Ck-norm, 172 unitarity of Clifford modules, 37 unitary basis, 336 units multiplicative group of -, 12 universal characteristic classes, 377ff. n-plane bundle, 379 principal G-bundle, 376 vanishing theorem, 155, 157, 158, 160, 164, 360, 361, 363, 365, 366, 400 vector fields --on the sphere, 45, 46 --on manifolds, 288, 289 virtual bundles, 59 volume element, 21, 33, 128 complex --, 34, 99, 135 warped product, 321 weakly enlargeable, 318 wedge product of spaces, 61 weights of a representation, 44 Workhorse Theorem, 180 Z2-graded -algebra, 9 -bundle, 96 Grothendieck group of -modules, 40, 73 -module, 39, 73 -tensor product, 11 Notation Index 11., Ab A, Ak 231 d, d* 123 Ao 267 D (Dirac operator) 112,113 .si. 92, 100, 152,276 DO, D' 135, 136 A(X), A(X), A(X) 233, 138 D+, 136, 137 '!f,U(X) 310 fj 123 A(X;E) 139 I/J 121 .si(X), .si(X,E) 276 I/Jo, I/J', I/J+, 138 A-deg(f) 309 11 140 ad, Ad 12 331, 357 Ad 12 333 ex 9, 95, 127 Da 167 bi(X) 144 '!)X (Poincare duality) 239 deg(f) 303 BG 376 Diff(X), Diff(E; X), Diff(E,F; X) 205 BO., BSO., BU., etc. 378 div 115 C (the complex numbers) i5 (anticanonical bundle) 363 IC(n) 26 i5(E) 242, 388 <e; 336 d (Hodge Laplacian, c,(E) 82, 374 La place-BeItrami) 123, 168 c(E), ck(E) 227, 381 d., d.+, d;; 35 ch, chk 234 36 chG,chg 260 ch, 285 cl; 316 clX(V,q) 12 Cf(V,q) 7 [E,F;a] 64,67 [Eo,.·· ,E.;a" ... ,a.] 64,67 EG 376 ' oj,6j 331 Cf°(V,q), Cf'(V,q) 9 Fg 261 Ct x(V,q) 12 3', 3'(H ,,H 2) 201 Cf., Cf:, Cf", 20 3' x 210 Cf. 27 3'.. 210 Ct;, Ct; 34 Cf(E), Cfo(E), cr '(E) 95 3'G 212 3'k, 3'k 217 Cf±(E) 107 3't,tyk 219 Cf(X) 111 Cf(X) 330 Cf±(X) 128, 136 Cfp,q(X) 331 coker( ) 192 G., g. 343 Yu 118 [(E) 102 fcpt 115 Cov2(X) 78 IHl (the quaternions) 25 X(E) 227, 383 lHl(n) 26 42{) NOTATION INDEX H, HP 125 Hept 239 Hl(X;G) 372 :Yt', P, if 331 :Yt'(X) (the holonomy group of X) )f'p.q(X), Hp,q(X), )f'p,q(X; W) Hp,q(X; W) 332 :Yt"(X,S) (spin or cohomology) i! 387, 388 i(go,gd, i(g,y) 329 Jm, J;, 336, 337 ind 135, 194,208 indk 141,206 indG, ind. 212 J 330 $ 332 kX 14 k-*(E) 384 ker 116, 192 K(X) 59 K(X,Y) 61 K(X) 60 K-'(X), K-i(X), K-'(X,Y) 61 K-*(pt) 62 345 359 Kept(X), K;p:(X), K;p:(X,Y) 66,67 KO(X) 60 'KO-'(X), KO-'(X), KO-'(X,y) KOept(X), KOe-p:(X), KO;p;(X,Y) KR(X), KR-'(X), KR-'(XY), KR"'(X,y) 70,71 KRept(X), 72, 73 KG(X) 259 K 160 L, L 24,95, 128 L, Lk, I., Lk 232 L(X), L(X), L(X), L(X) 233 Le 263 Le 265 e 116 Li(E) 170 L.(X), L.(X,Y) 64 L.(X)ePt' L.(X,Y)ept 67 p. (in K-theory) 64 P, .P(H;.H 2) 201 P' 203 p. (in operator theory) 223 A,(E) 235 A_I 240, 387 NV, A*V 10 A*(X) 123 61,62 66,67 9Jl., 33 WI., 40 9JlR"s, 9JlR"s, 9Jl"s' IDl"s 73 MSpink 91 Jl 67, 69, 74, 75, 96, 97 (]I (the octonions) 51 On,O"s 19 O(V,q) 9 Opm(E,F), 205 W (connection) 101 W (volume) 21,128 Wc 34, 135 OJj,65j 338, 358 W/ 358 n (curvature) 101 90 92 n3'k 222 PIE), pj(E) 228, 382 PGdE) 370 PGL+(E) 82,371 porE) 78,371 Pso(E), PSPin(E) 80,371 PGdX) 94 Pso(X), PSPin(X) 111 PSp,nc(X) 370 P(V,q), P(V,q), Pin(V,q) 13,14 Pin., Pin", 19 PrinG(X) 370 Pt, p$±, P(P$±) 336, 337 9I'(X) 326 PIE) 225, 389 P'(Il\I:), P'(iC), P'(K) 47 n, 358 "'k(E) 236 'f'COm(E,F) 244 'f'DOm(E,F) 186 'f'DOm 178 'f'DOK,m 183 Q (the rational numbers) Q[[x ]]-228 q", 19 Il\I: (the real numbers) ll\I:(n) 26 Rv w 105,106 110 R 158 9\ 155 9\E 164 NOTATION INDEX 9\,> 9\e 360 R(G) 211 Ric 156 Pn 39; d(Pn), N Pn, @Pn 94, 95 p, 16 f:I' 171 Iti 139 $, $e 121 $t, $e 137 [S+,S-;JlJ, [st,se;JlJ 69,70 S ® E 122 s(E) 238 s(E) 242, 388 S(E), SdE) 96 SO(E), S'(E) 99, 135 S±(E), Sf(E) 98,99, 129, 136 sig(X) 137 sig(X; E) 139 sig(X,g) 263 sig(X; E,g) 265 SL.(IR), SLn(iC), SLn(lHl) SLn(lR) 56 55 SL:(K) 58 SO(V,q), SP(V,q) 17 SO., SO", 19 SOn, spinn 41 SPn> SUn 49 Spin(V, q) 14 Spin., Spin" 19 Spin?" 5'6 Spin' 390 sym (symbol of an operator) Symm 177 tr( ) 186 tr( ) 243 tr,( ) 113, 168 8,( ) 245 trj (elementary symmetric functions) 227 tr. (Pauli matrices) 120 Tn 306 TX, T*X 94 ff(V) 8 '1:(E) 241 Tde, Td(X), Tdk 230 top-ind 244, 269, 272 top-indG 259 '!"(X) 339 e 93 Un 49 VX 16 vn(d) 88 V(X) 58 w,(E) 79,373 w2(E) 81, 373 wk(E), wk(X) 85, 380 Wk(E) 383 20 So 41 @ 11,40 om 168 x p 371 # (connected sum) 91 * 333 384 L 24 ( )! 239, 241, 387, 388 ( (in theory of multiplicative seq uences) 229 C) (Fourier transform) 171 (v 1\ w)(x) 41 [v,w J 40 u*v 173 II Ilk Illb II Ilu. II Iln,L2 170,171 172 194 194 427 ERRATA 1. In Proposition 1.3, assume k has characteristic zero. 2. On page 11, line 24, assume that A and .i9are filtered. 3. On page 43, line 10, the Lie group is also connected. 4. On page 192, line 22, change "Sobolev Embedding" to "Rellich." 5. On page 257, lines 2 and 3 of Remark 13.11, add the following omitted line: "Any elliptic operator P can be converted to a pseudodifferential operator of degree zero." 6. On page 278, line 33, " ... compact connected spin manifolds." 7. In Definition 3.2 of Chapter IV, the action should be effective on each connected component of X.