fiber Cohen advanced

The Topology of Fiber Bundles Lecture Notes Ralph L. Cohen Dept. of Mathematics Stanford University Contents Introduction v Chapter 1. Locally Trival Fibrations 1 1. Definitions and examples 1 1.1. Vector Bundles 3 1.2. Lie Groups and Principal Bundles 7 1.3. Clutching Functions and Structure Groups 15 2. Pull Backs and Bundle Algebra 21 2.1. Pull Backs 21 2.2. The tangent bundle of Projective Space 24 2.3. K - theory 25 2.4. Differential Forms 30 2.5. Connections and Curvature 33 2.6. The Levi - Civita Connection 39 Chapter 2. Classification of Bundles 45 1. The homotopy invariance of fiber bundles 45 2. Universal bundles and classifying spaces 50 3. Classifying Gauge Groups 60 4. Existence of universal bundles: the Milnor join construction and the simplicial classifying space 63 4.1. The join construction 63 4.2. Simplicial spaces and classifying spaces 66 5. Some Applications 72 5.1. Line bundles over projective spaces 73 5.2. Structures on bundles and homotopy liftings 74 5.3. Embedded bundles and K-theory 77 5.4. Representations and flat connections 78 Chapter 3. Characteristic Classes 81 1. Preliminaries 81 2. Chern Classes and Stiefel - Whitney Classes 85 iii iv CONTENTS 2.1. The Thom Isomorphism Theorem 88 2.2. The Gysin sequence 94 2.3. Proof of theorem 3.5 95 3. The product formula and the splitting principle 97 4. Applications 102 4.1. Characteristic classes of manifolds 102 4.2. Normal bundles and immersions 105 5. Pontrjagin Classes 108 5.1. Orientations and Complex Conjugates 109 5.2. Pontrjagin classes 111 5.3. Oriented characteristic classes 114 6. Connections, Curvature, and Characteristic Classes 115 Chapter 4. Homotopy Theory of Fibrations 125 1. Homotopy Groups 125 2. Fibrations 130 3. Obstruction Theory 135 4. Eilenberg - MacLane Spaces 140 4.1. Obstruction theory and the existence of Eilenberg - MacLane spaces 140 4.2. The Hopf - Whitney theorem and the classification theorem for Eilenberg - MacLane spaces 144 5. Spectral Sequences 152 5.1. The spectral sequence of a filtration 152 5.2. The Leray - Serre spectral sequence for a fibration 157 5.3. Applications I: The Hurewicz theorem 160 5.4. Applications II: H∗(ΩSn) andH∗(U(n)) 162 5.5. Applications III: Spin andSpin Cstructures 165 Bibliography 171 Introduction Fiber Bundles and more general fibrations are basic objects of study in many areas of mathe- matics. A fiber bundle with base space Band fiberFcan be viewed as a parameterized family of objects, each “isomorphic” to F, where the family is parameterized by points in B. For example a vector bundle over a space Bis a parameterized family of vector spaces Vx, one for each point x∈B. Given a Lie group G, a principal G- bundle over a space Bcan be viewed as a parameterized family of spacesFx, each with a free, transitive action of G(so in particular each Fxis homeomorphic to G). A covering space is also an example of a fiber bundle where the fibers are discrete sets. Sheaves and “fibrations” are generalizations of the notion of fiber bundles and are fundamental objects in Algebraic Geometry and Algebraic Topology, respectively. Fiber bundles and fibrations encode topological and geometric information about the spaces over which they are defined. Here are but a few observations on their impact in mathematics. •A structure such as an orientation, a framing, an almost complex structure, a spin structure, and a Riemannian metric are all constructions on the the tangent bundle of a manifold. •The exact sequence in homotopy groups, and the Leray - Serre spectral sequence for ho- mology groups of a fibration have been basic tools in Algebraic Topology for nearly half a century. •Understanding algebraic sections of algebraic bundles over a projective variety is a basic goal in algebraic geometry. •K- theory, a type of classification of vector bundles over a topological space is at the same time an important homotopy invariant of the space, and a quantity for encoding index information about elliptic differential operators. •The Yang - Mills partial differential equations are defined on the space of connections on a principal bundle over a Riemannian two dimensional or four dimensional manifold. The properties of the solution space have had a great impact on our understanding of four dimensional Differential Topology in the last fifteen years. In these notes we will study basic topological properties of fiber bundles and fibrations. We will study their definitions, and constructions, while considering many examples. A main goal of these notes is to develop the topology needed to classify principal bundles, and to discuss various models of their classifying spaces. We will compute the cohomology of the classifying spaces of O(n) andU(n), and use them to study K- theory. These calculations will also allow us to describe characteristic v vi INTRODUCTION classes for these bundles which we use in a variety of applications including the study of framings, orientations, almost complex structures, Spin andSpin C- structures, vector fields, and immersions of manifolds. We will also discuss the Chern-Weil method of defining characteristic classes (via the curvature of a connection), and show how all U(n) - characteristic classes can be defined this way. We also use these techniques to consider the topological implications when a bundle admits a flat connection. Finally, we study the algebraic topology of fibrations. This includes the exact sequence in homotopy groups of a fibration, general obstruction theory, including the interpretation of characteristic classes as obstructions to the existence of cross sections, and the construction and properties of Eilenberg - MacLane spaces. We then study the spectral sequence of a filtration and the Leray - Serre spectral sequence for a fibration. A variety of applications are given, including the Hurewicz theorem in homotopy theory, and a calculation of the homology of certain loop spaces and Lie groups. None of the results in these notes are new. This is meant to be an exposition of classical topics that are of importance to students in any area of topology and geometry. There are several good references for many of the topics covered here, in particular the classical text of Steenrod on fiber bundles (describing the classification theorem) [ 39], the book of Milnor and Stasheff on characteristic classes of vector bundles [ 31], the texts of Whitehead [ 42] and Mosher and Tangora [ 32] for the results in homotopy theory. When good references are available we may not include the details of all the proofs. These notes grew out of a graduate topology course I gave at Stanford University during the Spring term, 1998. I am very grateful to the students in that course for comments on earlier versions of these notes. R.L. Cohen Stanford University August, 1998 CHAPTER 1 Locally Trival Fibrations In this chapter we define our basic object of study: locally trivial fibrations, or “fiber bundles”. We discuss many examples, including covering spaces, vector bundles, and principal bundles. We also describe various constructions on bundles, including pull-backs, sums, and products. We then study the homotopy invariance of bundles, and use it in several applications. Throughout all that follows, all spaces will be Hausdorff and paracompact. 1. Definitions and examples LetBbe connected space with a basepoint b0∈B, andp:E→Bbe a continuous map. Definition 1.1.The mapp:E→Bis alocally trivial fibration, or fiber bundle, with fiber F if it satisfies the following properties: (1)p−1(b0) =F (2)p:E→Bis surjective (3) For every point x∈Bthere is an open neighborhood Ux⊂Band a “fiber preserving homeomorphism” Ψ Ux:p−1(Ux)→Ux×F, that is a homeomorphism making the following diagram commute: p−1(Ux)ΨUx−−−−→∼=Ux×F p/arrowbt/arrowbtproj Ux =Ux Some examples: •The projection map X×F−→Xis the trivial fibration over Xwith fiberF. •LetS1⊂Cbe the unit circle with basepoint 1 ∈S1. Consider the map fn:S1→S1given byfn(z) =zn.Thenfn:S1→S1is a locally trivial fibration with fiber a set of ndistinct points (the nthroots of unity in S1). •Letexp:R→S1be given by exp(t) =e2πit∈S1. Thenexpis a locally trivial fibration with fiber the integers Z. 1 2 1. LOCALLY TRIVAL FIBRATIONS •Recall that the n- dimensional real projective space RPnis defined by RPn=Sn/∼ wherex∼−x, forx∈Sn⊂Rn+1. Letp:Sn→RPnbe the projection map. This is a locally trivial fibration with fiber the two point set. •Here is the complex analogue of the last example. Let S2n+1be the unit sphere in Cn+1. Recall that the complex projective space CPnis defined by CPn=S2n+1/∼ wherex∼ux, wherex∈S2n+1⊂Cn, andu∈S1⊂C.Then the projection p:S2n+1→ CPnis a locally trivial fibration with fiber S1. •Consider the Moebeus band M= [0,1]×[0,1]/∼where (t,0)∼(1−t,1). LetCbe the “center circle” C={(1/2,s)∈M}and consider the projection p:M→C (t,s)→(1/2,s). This map is a locally trivial fibration with fiber [0 ,1]. Given a fiber bundle p:E→Bwith fiberF, the space Bis called the base space and the space Eis called the total space. We will denote this data by a triple ( F,E,B ). Definition 1.2.A map (or “morphism”) of fiber bundles Φ : ( F1,E1,B1)→(F2,E2,B2) is a pair of basepoint preserving continuous maps ¯φ:E1→E2andφ:B1→B2making the following diagram commute: E1¯φ−−−−→E2 p1/arrowbt/arrowbtp2 B1−−−−→ φB2 Notice that such a map of fibrations determines a continuous map of the fibers, φ0:F1→F2. A map of fiber bundles Φ : ( F1,E1,B1)→(F2,E2,B2) is an isomorphism if there is an inverse map of fibrations Φ−1: (F2,E2,B2)→(F1,E1,B1) so that Φ◦Φ−1= Φ−1◦Φ = 1. Finally we say that a fibration ( F,E,B ) istrivial if it isomorphic to the trivial fibration B×F→ B. Exercise. Verify that all of the above examples of fiber bundles are all nontrivial except for the first one. 1. DEFINITIONS AND EXAMPLES 3 The notion of a locally trivial fibration is quite general and includes examples of many types. For example you may have already noticed that covering spaces are examples of locally trivial fibrations. In fact one may simply define a covering space to be a locally trivial fibration with discrete fiber. Two other very important classes of examples of locally trivial fiber bundles are vector bundles andprincipal bundles. We now describe these notions in some detail. 1.1. Vector Bundles. Definition 1.3.Ann- dimensional vector bundle over a field kis a locally trivial fibration p:E→Bwith fiber an n- dimensional k- vector space Vsatisfying the additional requirement that the local trivializations ψ:p−1(U)→U×V induce k- linear transformations on each fiber. That is, restricted to each x∈U,ψdefines a k- linear transformation (and thus isomorphism) ψ:p−1(x)∼=−−−−→ {x}×V. It is common to denote the data ( V,E,B ) defining an n- dimensional vector bundle by a Greek letter, e.gζ. A “map” or “morphism” of vector bundles Φ : ζ→ξis a map of fiber bundles as defined above, with the added requirement that when restricted to each fiber, ¯φis ak- linear transformation. Examples •Given ann- dimensional kvector space V, thenB×V→Bis the corresponding trivial bundle over the base space B. Notice that since all n- dimensional trivial bundles over B are isomorphic, we denote it (or more precisely, its isomorphism class) by /epsilon1n. •Consider the “ Moebeus line bundle” µdefined to the the one dimensional real vector bundle (“line bundle”) over the circle given as follows. Let E= [0,1]×R/∼where (0,t)∼(1,−t). LetCbe the “middle” circle C={(s,0)∈E}. Thenµis the line bundle defined by the projection p:E→C (s,t)→(s,0). •Define the real line bundle γ1over the projective space RPnas follows. Let x∈Sn. Let [x]∈RPn=Sn/∼be the class represented by x. Then [x] determines (and is determined by) the line through the origin in Rn+1going through x. It is well defined since both representatives of [ x] (xand−x) determine the same line. Thus RPncan be thought of as 4 1. LOCALLY TRIVAL FIBRATIONS the space of lines through the origin in Rn+1. LetE={([x],v) : [x]∈RPn,v∈[x]}. Then γ1is the line bundle defined by the projection p:E→RPn ([x],v)→[x]. Exercise. Verify that the RP1is a homeomorphic to a circle, and the line bundle γ1over RP1is isomorphic to the Moebeus line bundle µ. •By abuse of notation we let γ1also denote the complex line bundle over CPndefined analogously to the realline bundle γ1overRPnabove. •LetGrk(Rn) (respectively Grk(Cn)) be the space whose points are k- dimensional subvec- tor spaces of Rn(respectively Cn). These spaces are called “Grassmannian” manifolds, and are topologized as follows. Let Vk(Rn) denote the space of injective linear transformations fromRktoRn. LetVk(Cn) denote the analogous space of injective linear transformations Ck/arrowhookleft→Cn. These spaces are called “Stiefel manifolds”, and can be thought of as spaces of n×kmatrices of rank k. These spaces are given topologies as subspaces of the appropriate vector space of matrices. To define Grk(Rn) andGrk(Cn), we put an equivalence relation onVk(Rn) andVk(Cn) by saying that two transformations AandBare equivalent if they have the same image in Rn(orCn). If viewed as matrices, then A∼Bif and only if there is an element C∈GL(k,R) (orGL(k,C)) so thatA=BC. Then the equivalence classes of these matrices are completely determined by their image in Rn(orCn), i.e the equivalence class is determined completely by a k- dimensional subspace of Rn(orCn). Thus we define Grk(Rn) =Vk(Rn)/∼andGrk(Cn) =Vk(Cn)/∼ with the corresponding quotient topologies. Consider the vector bundle γkoverGrk(Rn) whose total space Eis the subspace of Grk(Rn)×Cndefined by E={(W,ω) :W∈Grk(Rn) andω∈W⊂Rn}. Thenγkis the vector bundle given by the natural projection E→Grk(Rn) (W,ω)→W For reasons that will become more apparent later in these notes, the bundles γkare called the “universal” or “canonical” k- dimensional bundles over the Grassmannians. Exercises 1. Verify that γkis ak-dimensional real vector bundle over Grk(Rn). 1. DEFINITIONS AND EXAMPLES 5 2. Define the analogous bundle (which by abuse of notation we also call γk) over Grk(Cn). Verify that it is a k-dimensional complex vector bundle over Grk(Cn). 3. Verify that RPn−1=Gr1(Rn) and that the line bundle γ1defined above is the universal bundle. Do the analogous exercise with CPn−1andGr1(Cn). •Notice that the universal bundle γkover the Grassmanians Grk(Rn) andGrk(Cn) come equipped with embeddings (i.e injective vector bundle maps) in the trivial bundles Grk(Rn)× RnandGrk(Cn)×Cnrespectively. We can define the orthogonal complement bundles γ⊥ k to be then−kdimensional bundles whose total spaces are given by E⊥ k={(W,ν)∈Grk(Rn)×Rn:ν⊥W} and similarly over Grk(Cn). Observe that the natural projection to the Grassmannian definesn−kdimensional vector bundles (over RandCrespectively). •Perhaps the most important class of vector bundles are tangent bundles over differentiable manifolds. To define the tangent bundle, for now we will assume we have a k- dimensional closed manifold Membedded in Euclidean space Rn. (We will define the tangent bundle independently of such an embedding later in this chapter.) The tangent bundle τMhas total space TM defined to be the subspace of M×Rngiven as follows. TM={(x,v)∈M×Rn:vis tangent to Matx.} wherevbeing tangent to Matxmeans there is a smooth curve α: (−/epsilon1,/epsilon1)→M⊂Rn for some/epsilon1>0, satisfying the properties α(0) =xanddα dt(0) =v. The natural projection p:TM→M (x,v)→x defines the tangent bundle τM. The fiber of the tangent bundle at x∈M,p−1(x) is called the tangent space ofMat x, and is denoted by TxM. We say that a manifold is parallelizable if its tangent bundle is trivial. Parallelizable manifolds form an important class of manifolds, but as we will see below, not all manifolds are parallelizable. Exercise. Verify that TM is indeed a k- dimensional vector bundle embedded in M×Rn. 6 1. LOCALLY TRIVAL FIBRATIONS •LetM⊂Rnbe ak- dimensional closed manifold embedded in Rnas above. We define then−kdimensional “normal” bundleνn−k(M) to be the orthogonal complement of the tangent bundle. That is, νn−k(M) has total space Nn−k(M) defined to be the subspace ofM×Rngiven by Nn−k(M) ={(x,u)∈M×Rn:u⊥TxM⊂Rn}. The natural projection p:Nn−k(M)→M (x,u)→x defines the normal bundle νn−k(M). Givenx∈M, the fiber at x,p−1(x) is called the normal space at x, and is denoted Nx(M). An important notion associated to vector bundles (and in fact all fibrations) is the notion of a (cross) section. Definition 1.4.Given a fiber bundle p:E→B a sectionsis a continuous map s:B→Esuch thatp◦s=identity :B→B. Notice that every vector bundle has a section, namely the zero section z:B→E x→0x where 0xis the origin in the vector space p−1(x). However most geometrically interesting sections have few zero’s. Indeed as we will see later, an appropriate count of the number of zero’s of a section of an n- dimensional bundle over an n- dimensional manifold is an important topological invariant of that bundle (called the “Euler number”). In particular an interesting geometric question is to determine when a vector bundle has a nowhere zero section, and if it does, how many linearly independent sections it has. (Sections {s1,···,sm}are said to be linearly independant if the vectors {s1(x),···,sm(x)}are linearly independent for every x∈B.) These questions are classical in the case where the vector bundle is the tangent bundle. A section of the tangent bundle is called a vector field. The question of how many linearly independent vector fields exist on the sphere Sn was answered by J.F. Adams [ 2] in the early 1960’s using sophisticated techniques of homotopy theory. Exercises (from [31]) 1. DEFINITIONS AND EXAMPLES 7 1. Letx∈Sn, and [x]∈RPnbe the corresponding element. Consider the functions fi,j: RPn→Rdefined by fi,j([x]) =xixj. Show that these functions define a diffeomorphism between RPnand the submanifold of R(n+1)2consisting of all symmetric ( n+1)×(n+1) matrices Aof trace 1 satisfying AA=A. 2. Use exercise 1 to show that RPnis compact. 3. Prove that an n-dimensional vector bundle ζhasn- linearly independent sections if and only ifζis trivial. 4. Show that the unit sphere Snadmits a nowhere zero vector field if nis odd. Show that the normal bundle of Sn⊂Rn+1is a trivial line bundle for all n. 5. IfSnadmits a nowhere zero vector field show that the identity map of Snis homotopic to the antipodal map. For neven show that the antipodal map of Snis homotopic to the reflection r(x1,···,xn+1) = (−x1,x2,···,xn+1); and therefore has degree −1. Combining these facts, show that Snis not parallelizable for neven, n≥2. 1.2. Lie Groups and Principal Bundles. Lie groups play a central role in bundle theory. In this section we give a basic description of Lie groups, their actions on manifolds (and other spaces), and one of the main objects of study in these notes, their principal bundles. Definition 1.5.A Lie group is a topological group Gwhich has the structure of a differentiable manifold. Moreover the multiplication map G×G→G and the inverse map G→G g→g−1 are required to be differentiable maps. The following is an important basic property of the differential topology of Lie groups. Theorem 1.1.LetGbe a Lie group. Then Gis parallelizable. That is, its tangent bundle τG is trivial. 8 1. LOCALLY TRIVAL FIBRATIONS Proof. Let 1∈Gdenote the identity element, and T1Gthe tangent space of Gat 1. IfGis an n- dimensional manifold, T1Gis ann- dimensional vector space. We define a bundle isomorphism of the tangent bundle τGwith the trivial bundle G×T1(G), which, on the total space level is given by a map φ:G×T1G−→TG defined as follows. Let g∈G. Then multiplication by gon the right is a diffeomorphism ×g:G→G x→xg Since×gis a diffeomorphism, its derivative is a linear isomorphism at every point: Dg(x) :TxG∼=−−−−→TxgG. We can now define φ:G×T1G→TG by φ(g,v) =Dg(1)(v)∈TgG. Clearlyφis a bundle isomorphism. /square Principal bundles are basically parameterized families of topological groups, and often Lie groups. In order to define the notion carefully we first review some basic properties of group actions. Recall that a right action of topological group Gon a space Xis a map µ:X×G→X (x,g)→xg satisfying the basic properties (1)x·1 =xfor allx∈X (2)x(g1g2) = (xg1)g2for allx∈Xandg1, g2∈G. Notice that given such an action, every element gacts as a homeomorphism, since action by g−1is its inverse. Thus the group action µdefines a map µ:G→Homeo (X) whereHomeo (X) denotes the group of homeomorphisms of X. The two conditions listed above are equivalent to the requirement that µ:G→Homeo (X) be a group homomorphism. IfGis a Lie group and Mis a smooth manifold with a right G- action. We say that the action issmooth if the homomorphism µdefined above factors through a homomorphism µ:G→Diffeo (M) whereDiffeo (M) is the group of diffeomorphisms of M. 1. DEFINITIONS AND EXAMPLES 9 LetXbe a space with a right G- action. Given x∈X, letxG={xg:g∈G}⊂X. This is called the orbit ofxunder theG- action. The isotropy subgroup of x,Iso(x), is defined by Iso(x) ={g∈G:xg=x}Notice that the map G→xG defined by sending gtoxgdefines a homeomorphism from the coset space to the orbit G/Iso (x)∼=−−−−→xG⊂X. A group action on a space Xis said to be transitive if the space Xis the orbit of a single point, X=xG. Notice that if X=x0Gfor somex0∈X, thenX=xGforanyx∈X. Notice furthermore that the transitivity condition is equivalent to saying that for any two points x1,x2∈X, there is an element g∈Gsuch thatx1=x2g. Finally notice that if Xhas a transitive G- action, then the above discussion about isotropy subgroups implies that there exists a subgroup H < G and a homeomorphism G/H∼=−−−−→X. Of course if Xis smooth,Gis a Lie group, and the action is smooth, then the above map would be a diffeomorphism. A group action is said to be (fixed point) free if the isotropy groups of every point xare trivial, Iso(x) ={1} for allx∈X. Said another way, the action is free if and only if the only time there is an equation of the form xg=xis ifg= 1∈G. That is, if for g∈G, the fixed point set Fix(g)⊂Xis the set Fix(g) ={x∈X:xg=x}, then the action is free if and only if Fix(g) =∅for allg/negationslash= 1∈G. We are now able to define principal bundles. Definition 1.6.LetGbe a topological group. A principalGbundle is a fiber bundle p:E→B with fiberF=Gsatisfying the following properties. (1) The total space Ehas a free, fiberwise right Gaction. That is, it has a free group action making the following diagram commute: E×Gµ−−−−→E p×/epsilon1/arrowbt/arrowbtp B×{1} =B where/epsilon1is the constant map. 10 1. LOCALLY TRIVAL FIBRATIONS (2) The induced action on fibers µ:p−1(x)×G→p−1(x) is free and transitive. (3) There exist local trivializations ψ:p−1(U)∼=−−−−→U×G that are equivariant. That is, the following diagrams commute: p−1(U)×Gψ×1−−−−→∼=U×G×G µ/arrowbt/arrowbt1×mult. p−1(U)∼=−−−−→ ψU×G. Notice that in a principal G- bundle, the group Gacts freely on the total space E. It is natural to ask if a free group action suffices to induce a principal G- bundle. That is, suppose Eis a space with a free, right Gaction, and define Bto be the orbit space B=E/G =E/∼ wherey1∼y2if and only if there exists a g∈Gwithy1=y2g(i.e if and only if their orbits are equal:y1G=y2G). Definep:E→Bto be the natural projection, E→E/G. Then the fibers are the orbits, p−1([y]) =yG. So forp:E→Bto be a principal bundle we must check the local triviality condition. In general for this to hold we need the following extra condition. Definition 1.7.A group action E×G→Ehas slices if projection onto the orbit space p:E→E/G has local sections. That is, around every x∈E/G there is a neighborhood Uand a continuous map s:U→Esuch thatp◦s=id:U→U. Proposition 1.2.IfEhas a freeGaction with slices, then the projection map p:E→E/G is a principal G- bundle. Proof. We need to verify the local triviality condition. Let x∈E/G. LetUbe an open set aroundxadmitting a section s:U→E. Define a local trivialization ψ:U×G→p−1(U) byψ(y,g) =s(y)·g. Clearlyψis a local trivialization. /square 1. DEFINITIONS AND EXAMPLES 11 The following result is originally due to A. Gleason [ 15], and its proof can be found in Steenrod’s book [ 39]. It is quite helpful in studying free group actions. Theorem 1.3.LetEbe a smooth manifold, having a free, smooth G- action, where Gis a compact Lie group. Then the action has slices. In particular, the projection map p:E→E/G defines a principal G- bundle. The following was one of the early theorems in fiber bundle theory, appearing originally in H. Samelson’s thesis. [ 34] Corollary 1.4.LetGbe a Lie group, and let H < G be a compact subgroup. Then the projection onto the orbit space p:G→G/H is a principal H- bundle. Examples. •The projection map p:S2n+1→CPnis a principal S1- bundle. •LetVk(Rn) be the Stiefel manifold of rank k n×kmatrices described above. Then the projection map p:Vk(Rn)→Grk(Rn) is a principal GL(k,R) - bundle. Similarly the projection map p:Vk(Cn)→Grk(Cn) is a principal GL(k,C) - bundle. •LetVk(Rn)O⊂Rdenote those n×kmatrices whose k- columns are orthonormal n- dimensional vectors. This is the Stiefel manifold of orthonormal k- frames in Rn. Then the induced projection map p:Vk(Rn)O→Grk(Rn) is a principal O(k) - bundle. Similarly, if Vk(Cn)Uis the space of orthonormal k- frames inCn(with respect to the standard Hermitian inner product), then the projection map p:Vk(Cn)U→Grk(Cn) is a principal U(n) - bundle. 12 1. LOCALLY TRIVAL FIBRATIONS •There is a diffeomorphism ρ:U(n)/U(n−1)∼=−−−−→S2n−1 and the projection map U(n)→S2n−1is a principal U(n−1) - bundle. To see this, notice that U(n) acts transitively on the unit sphere in Cn(i.eS2n−1). Moreover the isotropy subgroup of the point e1= (1,0,···,0)∈S2n−1are those elements A∈U(n) which have first column equal to e1= (1,0,···,0). Such matrices also have first row = (1,0,···,0). That is, Ais of the form A=/parenleftBigg 1 0 0A/prime/parenrightBigg whereA/primeis an element of U(n−1). Thus the isotropy subgroup Iso(e1)∼=U(n−1) and the result follows. Notice that a similar argument gives a diffeomorphism SO(n)/SO(n−1)∼=Sn−1. •There is a diffeomorphism ρ:U(n)/U(n−k)∼=−−−−→Vk(Cn)U. The argument here is similar to the above, noticing that U(n) acts transitively on Vk(Cn)U, and the isotopy subgroup of the n×kmatrix e= 1 0 0··· 0 0 1 0··· 0 0 0 1··· 0 ......... 0 0··· 0 1 0 0··· 0 0 ......... 0 0··· 0 0  consist of matrices in U(n) of them form  1 0 0··· 0 0 0··· 0 0 0 1 0··· 0......... 0 0 1··· 0......... 0 0··· 0 1 0 0 ··· 0 0 (0) ( B)  whereBis an (n−k)×(n−k) dimensional unitary matrix. •A similar argument shows that there are diffeomorphisms ρ:U(n)/(U(k)×U(n−k))∼=−−−−→Grk(Cn) 1. DEFINITIONS AND EXAMPLES 13 and ρ:O(n)/(O(k)×O(n−k))∼=−−−−→Grk(Rn) Principal bundles define other fiber bundles in the presence of group actions. Namely, suppose p:E→Bbe a principal G- bundle and Fis a space with a cellular right group action. Then the product space E×Fhas the “diagonal” group action ( e,f)g= (eg,fg ). Consider the orbit space, E×GF= (E×F)/G. Then the induced projection map p:E×GF→B is a locally trivial fibration with fiber F. For example we have the following important class of fiber bundles. Proposition 1.5.LetGbe a compact Lie group and K <H <G closed subgroups. Then the projection map of coset spaces p:G/K→G/H is a locally trivial fibration with fiber H/K . Proof. Observe that G/K∼=G×HH/K whereHacts onH/K in the natural way. Moreover the projection map p:G/K→G/H is the projection can be viewed as the projection G/K =G×HH/K→G/H and so is the H/K - fiber bundle induced by the H- principal bundle G→G/H via the action of Hon the coset space H/K . /square Example We know by the above examples, that U(2)/U(1)∼=S3, and thatU(2)/U(1)×U(1)∼=Gr1(C2) = CP1∼=S2. Therefore there is a principal U(1) - fibration p:U(2)/U(1)→U(2)/U(1)×U(1), or equivalently, a principal U(1) =S1fibration p:S3→S2. This fibration is the well known “Hopf fibration”, and is of central importance in both geometry and algebraic topology. In particular, as we will see later, the map from S3toS2gives an nontrivial element in the homotopy group π3(S2), which from the naive point of view is quite surprising. It says, that, in a sense that can be made precise, there is a “three dimensional hole” in S2that cannot be filled. Many people (eg. Whitehead, see [ 43]) refer to this discovery as the beginning of modern homotopy theory. 14 1. LOCALLY TRIVAL FIBRATIONS The fact that the Hopf fibration is a locally trivial fibration also leads to an interesting geometric observation. First, it is not difficult to see directly (and we will prove this later) that one can take the upper and lower hemispheres of S2to be a cover of S2over which the Hopf fibration is trivial. That is, there are local trivializations, ψ+:D2 +×S1→p−1(D2 +) and ψ−:D2 −×S1→p−1(D2 −) whereD2 +andD2 −are the upper and lower hemispheres of S2, respectively. Putting these two local trivializations together yields the following classical result: Theorem 1.6.The sphere S3is homeomorphic to the union of two solid tori D2×S1whose intersection is their common torus boundary, S1×S1. As another example of fiber bundles induced by principal bundles, suppose that ρ:G→GL(n,R) is a representation of a topological group G, andp:E→Bis a principal Gbundle. Then let Rn(ρ) denote the space Rnwith the action of Ggiven by the representation ρ. Then the projection E×GRn(ρ)→B is a vector bundle. Exercise. Letp:Vk(Rn)→Grk(Rn) be the principal bundle described above. Let Rnhave the standard GL(n,R) representation. Proved that the induced vector bundle p:Vk(Rn)×GL(n,R)Rn is isomorphic to the universal bundle γkdescribed in the last section. In the last section we discussed sections of vector bundles and in particular vector fields. For principal bundles, the existence of a section (or lack thereof) completely determines the triviality of the bundle. Theorem 1.7.A principal G- bundlep:E→Bis trivial if and only if it has a section. 1. DEFINITIONS AND EXAMPLES 15 Proof. Ifp:E→Bis isomorphic to the trivial bundle B×G→B, then clearly it has a section. So we therefore only need to prove the converse. Supposes:B→Eis a section of the principal bundle p:E→B. Define the map ψ:B×G→E byψ(b,g) =s(b)gwhere multiplication on the right by gis given by the right G- action of Gon E. It is straightforward to check that ψis an isomorphism of principal G- bundles, and hence a trivialization of E. /square 1.3. Clutching Functions and Structure Groups. Letp:E→Bbe a locally trivial fibration with fiber F. Cover the basespace Bby a collection of open sets {Uα}equipped with local trivializations ψα:Uα×F∼=−−−−→p−1(Uα). Let us compare the local trivializations on the intersection: Uα∩Uβ: Uα∩Uβ×Fψβ−−−−→∼=p−1(Uα∩Uβ)ψ−1 α−−−−→∼=Uα∩Uβ×F. For everyx∈Uα∩Uβ,ψ−1 α◦ψβdetermines a homeomorphism of the fiber F. That is, this composition determines a map φα,β:Uα∩Uβ→Homeo (F). These maps are called the clutching functions of the fiber bundle. When the bundle is a real n- dimensional vector bundle then the clutching functions are of the form φα,β:Uα∩Uβ→GL(n,R). Similarly, complex vector bundles have clutching functions that take values in GL(n,C). Ifp:E→Bis aG- principal - bundle, then the clutching functions take values in G: φα,β:Uα∩Uβ→G. In general for a bundle p:E→Bwith fiberF, the group in which the clutching values take values is called the structure group of the bundle. If no group is specified, then the structure group is the homeomorphism group Homeo (F). For example if the bundle is smooth, then we are requiring the structure group to be the subgroup of diffeomorphisms, Diffeo (F). The clutching functions and the associated structure group completely determine the isomor- phism type of the bundle. Namely, given an open covering of a space B, and a compatible family of clutching functions φα,β:Uα∩Uβ→G, and a space Fupon which the group acts, we can form the space E=/uniondisplay αUα×F/∼ where ifx∈Uα∩Uβ, then (x,f)∈Uα×Fis identified with ( x,fφα,β(x))∈Uβ×F.Eis the total space of a locally trivial fibration over Bwith fiberFand structure group G. If the original data of clutching functions came from locally trivializations of a bundle, then notice that the construction 16 1. LOCALLY TRIVAL FIBRATIONS ofEabove yields a description of the total space of the bundle. Thus we have a description of the total space of a fiber bundle completely in terms of the family of clutching functions. Supposeζis ann- dimensional vector bundle with projection map p:E→Band local trivializations ψα:Uα×Rn→p−1(Uα). Then the clutching functions take values in the general linear group φα,β:Uα∩Uβ→GL(n,R). So the total space Ehas the form E=/uniontext αUα×Rn/∼as above. We can then form the corresponding principalGL(n,R) bundle with total space EGL=/uniondisplay αUα×GL(n,R) with the same clutching functions. That is, for x∈Uα∩Uβ, (x,g)∈Uα×GL(n,R) is identified with (x,g·φα,β(x))∈Uβ×GL(n,R). The principal bundle p:EGL→B is called the associated principal bundle to the vector bundle ζ, or sometimes is referred to as the associated frame bundle. Observe also that this process is reversable. Namely if p:P→Xis a principal GL(n,R) - bundle with clutching functions θα,β:Vα∩Vβ→GL(n,R), then there is an associated vector bundle p:PRn→Xwhere PRn=/uniondisplay αVα×Rn where ifx∈Vα∩Vβ, then (x,v)∈Vα×Rnis identified with ( x,v·θα,β(x))∈Vβ×Rn. This correspondence between vector bundles and principal bundles proves the following result: Theorem 1.8.LetVectR n(X)andVectC n(X)denote the set of isomorphism classes of real and complexn- dimensional vector bundles ovr Xrespectively. For a Lie group GletPrinG(X)denote the set of isomorphism classes of principal G- bundles. Then there are bijective correspondences VectR n(X)∼=−−−−→PrinGL(n,R)(X) VectC n(X)∼=−−−−→PrinGL(n,C)(X). This correspondence and theorem 1.6 allows for the following method of determining whether a vector bundle is trivial: Corollary 1.9.A vector bundle ζ:p:E→Bis trivial if and only if its associated principal GL(n)- bundlep:EGL→Badmits a section. 1. DEFINITIONS AND EXAMPLES 17 The use of clutching functions also allows us to describe the tangent bundle of a manifold, τM, independently of any embedding into Euclidean space. This is done as follows. Recall that a smooth, n- dimensional manifold Madmits a differentiable atlas, which consists of an open cover {Uα}ofMand homeomorphisms to open sets in Rn: fα:Uα∼=−−−−→Vα⊂Rn. For intersections, let Uα,β=fα(Uα∩Uβ)⊂Rn. The smooth structure on Mcomes from the requirement that on the intersections, the compositions Uα,βf−1 α−−−−→Uα∩Uβfβ−−−−→Uβ,α are required to be diffeomorphisms of open sets in Rn. Using a fixed trivialization of the tangent bundle of Rn,TRn∼=Rn×Rn, (and hence of the tangent bundle of any open set in Rn), then we can define, for x∈Uα∩Uβ, the linear isomorphism φα,β(x) :Rn→Rnto be the derivative of the above composition φα,β(x) :Rn=Tfα(x)Uα,βD(fβ◦f−1 α)(x)−−−−−−−−−→ Tfβ(x)Uα,β=Rn. This defines clutching functions φα,β=D(fβ◦f−1 α) :Uα∩Uβ→GL(n,R) from which we can form the tangent bundle TM=/uniondisplay αUα×Rn/∼ with the identifications as described above. Exercise. Verify that this definition of the tangent bundle is isomorphic to the one given in section 1.1 when the manifold is embedded in Euclidean space. Clutching functions and structure groups are also useful in studying structures on principal bundles and their associated vector bundles. Definition 1.8.Letp:P→Bbe a principal G- bundle, and let H <G be a subgroup. Pis said to have a reduction of its structure group to Hif and only if Pis isomorphic to a bundle whose clutching functions take values in H: φα,β:Uα∩Uβ→H <G. Exercise. 18 1. LOCALLY TRIVAL FIBRATIONS LetP→Xbe a principal G- bundle. Then Phas a reduction of its structure group to H <G if and only if there is a principal H- bundle ˜P→Xand an isomorphism of Gbundles, ˜P×HG∼=−−−−→P /arrowbt/arrowbt X =X Definition 1.9.LetH <GL (n,R). Then an H- structure on an n- dimensional vector bundle ζis a reduction of the structure group of its associated GL(n,R) - principal bundle to H. Examples. •A{1}<GL (n,R) - structure on a vector bundle ( or its associated principal bundle) is a trivialization or framing of the bundle. A framed manifold is a manifold with a framing of its tangent bundle. •Given a 2n- dimensional real vector bundle ζ, an almost complex structure onζis a GL(n,C)< GL (2n,R) structure on its associated principal bundle. An almost complex structure on a manifold is an almost complex structure on its tangent bundle. We now study two examples of vector bundle structures in some detail: Euclidean structures, and orientations. Example 1: O (n)- structures and Euclidean structures on vector bundles. Recall that a Euclidean vector space is a real vector space Vtogether with a positive definite quadratic function µ:V→R. Specifically, the statement that µis quadratic means that it can written in the form µ(v) =/summationdisplay iαi(v)βi(v) where each αiandβi:V→Ris linear. The statement that µis positive definite means that µ(v)>0 forv/negationslash= 0. Positive definite quadratic functions arise from, and give rise to inner products (i.e symmetric bilinear pairings ( v,w)→v·w) defined by v·w=1 2(µ(v+w)−µ(v)−µ(w)). Notice that if we write |v|=√v·vthen|v|2=µ(v). So in particular there is a metric on V. This notion generalizes to vector bundles in the following way. 1. DEFINITIONS AND EXAMPLES 19 Definition 1.10.A Euclidean vector bundle is a real vector bundle ζ:p:E→Btogether with a map µ:E→R which when restricted to each fiber is a positive definite quadratic function. That is, µinduces a Euclidean structure on each fiber. Exercise. Show that an O(n) - structure on a vector bundle ζgives rise to a Euclidean structure on ζ. Conversely, a Euclidean structure on ζgives rise to an O(n) - structure. Hint. Make the constructions directly in terms of the clutching functions. Definition 1.11.A smooth Euclidean structure on the tangent bundle µ:TM→Ris called aRiemannian structure on M. Exercises. 1.Existence theorem for Euclidean metrics. Using a partition of unity, show that any vector bundle over a paracompact space can be given a Euclidean metric. 2.Isometry theorem. Letµandµ/primebe two different Euclidean metrics on the same vector bundle ζ:p:E→B. Prove that there exists a homeomorphism f:E→Ewhich carries each fiber isomorphically onto itself, so that the composition µ◦f:E→Ris equal toµ/prime. (Hint. Use the fact that every positive definite matrix Acan be xpressd uniquely as the square of a positive definite matrix√ A. The power series expansion /radicalbig (tI+X) =√ t(I+1 2tX−1 8t2X2+−··· ), is valid providing that the characteristic roots of tI+X=Alie between 0 and 2 t. This shows that the function A→√ Ais smooth.) Example 2: S L(n,R)- structures and orientations. Recall that an orientation of a real n- dimensional vector space Vis an equivalence class of basis forV, where two bases {v1,···,vn}and{w1,···,wn}are equivalent (i.e determine the same orientation) if and only if the change of basis matrix A= (ai,j), wherewi=/summationtext jai,jvjhas positive determinant, det(A)>0. LetOr(V) be the set of orientations of V. Notice that Or(V) is a two point set. 20 1. LOCALLY TRIVAL FIBRATIONS For a vector bundle ζ:p:E→B, an orientation is a continuous choice of orientations of each fiber. Said more precisely, we may define the “orientation double cover” Or(ζ) to be the two - fold covering space Or(ζ) =EGL×GL(n,R)Or(Rn) whereEGLis the associated principal bundle, and where GL(n,R) acts onOr(Rn) by matrix multiplication on a basis representing the orientation. Definition 1.12.ζisorientable if the orientation double cover Or(ζ) admits a section. A choice of section is an orientation of ζ. This definition is reasonable, in that a continuous section of Or(ζ) is a continuous choice of orientations of the fibers of ζ. Recall that SL(n,R)< GL (n,R) andSO(n)< O(n) are the subgroups consisting of matrices with positive determinants. The following is now straightforward. Theorem 1.10.Ann- dimensional vector bundle ζhas an orientation if and only if it has a SL(n,R)- structure. Similarly a Euclidean vector bundle is orientable if and only if it has a SO(n) - structure. Choices of these structures are equivalent to choices of orientations. Finally, a manifold is said to be orientable if its tangent bundle τMis orientable. 2. PULL BACKS AND BUNDLE ALGEBRA 21 2. Pull Backs and Bundle Algebra In this section we describe the notion of the pull back of a bundle along a continuous map. We then use it to describe constructions on bundles such as direct sums, tensor products, symmetric and exterior products, and homomorphisms. We use direct sums to give a description of the tangent bundle of projective space in terms of line bundles. We then study group completions, and define the notion of K- theory. Finally we use exterior products of bundles to study differential forms, and introduce the notions of connections, and curvature. 2.1. Pull Backs. Letp:E→Bbe a fiber bundle with fiber F. LetA⊂Bbe a subspace. The restriction of EtoA, writtenE|Ais simply given by E|A=p−1(A). The restriction of the projection p:E|A→Ais clearly still a locally trivial fibration with fiber F. This notion generalizes from inclusions of subsets A⊂Bto general maps f:X→Bin the form of the pull back bundle over X,f∗(E). This bundle is defined by f∗(E) ={(x,u)∈X×E:f(x) =p(u)}. Proposition 1.11.The map pf:f∗(E)→X (x,u)→x is a locally trivial fibration with fiber F. Furthermore if ι:A /arrowhookleft→Bis an inclusion of a subspace, then the pull-back ι∗(E)is equal to the restriction E|A. Proof. Let{Uα}be a collection of open sets in Bandψα:Uα×F→p−1(Uα) local trivial- izations of the bundle p:E→B. Then{f−1(Uα)}is an open cover of X, and the maps ψα(f) :f−1(Uα)×F→p−1 f(f−1(Uα)) defined by ( x,y)→(x,ψα(f(x),y)) are clearly local trivializations. This proves the first statement in the proposition. The second statement is obvious. /square We now use the pull back construction to define certain algebraic constructions on bundles. Letp1:E1→B1andp2:E2→B2be fiber bundles with fibers F1andF2respectively. Then the cartesian product p1×p2:E1×E2→B1×B2 is clearly a fiber bundle with fiber F1×F2. In the case when B1=B2=B,we can consider the pull back (or restriction) of this cartesian product bundle via the diagonal map 22 1. LOCALLY TRIVAL FIBRATIONS ∆ :B /arrowhookleft→B×B x→(x,x). Then the pull-back ∆∗(E1×E2)→Bis a fiber bundle with fiber F1×F2, is defined to be the internal product, or Whitney sum of the fiber bundles E1andE2. It is written E1⊕E2= ∆∗(E1×E2). Notice that if E1andE2areG1andG2principal bundles respectively, then E1⊕E2is a principal G1×G2- bundle. Similarly, if E1andE2arenandmdimensional vector bundles respectively, then E1⊕E2is ann+m- dimensional vector bundle. E1⊕E2is called the Whitney sum of the vector bundles. Notice that the clutching functions of E1⊕E2naturally lie in GL(n,R)×GL(m,R) which is thought of as a subgroup of GL(n+m,R) consisting of ( n+m)×(n+m) - dimensional matrices of the form /parenleftBigg A0 0B/parenrightBigg whereA∈GL(n,R) andB∈GL(m,R). We now describe other algebraic constructions on vector bundles. The first is a generalization of the fact that a given a subspace of a vector space, the ambient vector space splits as a direct sum of the subspace and the quotient space. Letη:Eη→Bbe ak- dimensional vector bundle and ζ:Eζ→Bann- dimensional bundle. Letι:η /arrowhookleft→ζbe a linear embedding of vector bundles. So on each fiber ιis a linear embedding of ak- dimensional vector space into an n- dimensional vector space. Define ζ/ηto be the vector bundle whose fiber at xisEζ x/Eη x. Exercise. Verify that ζ/ηis ann−k- dimensional vector bundle over B. Theorem 1.12.There is a splitting of vector bundles ζ∼=η⊕ζ/η. Proof. Giveζa Euclidean structure. Define η⊥⊂ζto be the subbundle whose fiber at xis the orthogonal complement Eη⊥ x={v∈Eζ x:v·w= 0 for allw∈Eη x} Then clearly there is an isomorphism of bundles η⊕η⊥∼=ζ. 2. PULL BACKS AND BUNDLE ALGEBRA 23 Moreover the composition η⊥⊂ζ→ζ/η is also an isomorphism. The theorem follows. /square Corollary 1.13.Letζbe a Euclidean n- dimensional vector bundle. Then ζhas aO(k)× O(n−k)- structure if and only if ζadmits ak- dimensional subbundle η⊂ζ. We now describe the dual of a vector bundle. So let ζ:Eζ→Bbe ann- dimensional bundle. Its dual,ζ∗:Eζ∗→Bis the bundle whose fiber at x∈Bis the dual vector space Eζ∗ x=Hom (Eζ,R). If {φα,β:Uα∩Uβ→GL(n,R)} are clutching functions for ζ, then {φ∗ α,β:Uα∩Uβ→GL(n,R)} form the clutching functions for ζ∗, whereφ∗ α,β(x) is the adjoint (transpose) of φα,β(x). The dual of a complex bundle is defined similarly. Exercise. Prove that ζandζ∗are isomorophic vector bundles. HInt. Giveζa Euclidean structure. Now letη:Eη→Bbe ak- dimensional, and as above, ζ:Eζ→Bann- dimensional bundle. We define the tensor product bundle η⊗ζto be the bundle whose fiber at x∈Bis the tensor product of vector spaces, Eη x⊗Eζ x. The clutching fucntions can be thought of as compositions of the form φη⊗ζ α,β:Uα∩Uβφη α,β×φζ α,β−−−−−−−→ GL(k,R)×GL(n,R)⊗−−−−→GL(kn,R) where the tensor product of two linear transformations A:V1→V2andB:W1→W2is the induced linear transformation A⊗B:V1⊗W1→V2⊗W2. With these two constructions we are now able to define the “homomorphism bundle”, Hom (η,ζ). This will be the bundle whose fiber at x∈Bis thek·m- dimensional vector space of linear transformations Hom (Eη x,Eζ x)∼=(Eη x)∗⊗Eζ x. So as bundles we can define Hom (η,ζ) =η∗⊗ζ. Observation. A bundle homomorphsim θ:η→ζassigns to every x∈Ba linear transforma- tion of the fibers, θx:Eη x→Eζ x. Thus a bundle homomorphism can be thought of as a section of the bundle Hom (η,ζ). That is, there is a bijection between the space of sections, Γ( Hom (η,ζ)) and the space of bundle homomorphisms, {θ:η→ζ}. 24 1. LOCALLY TRIVAL FIBRATIONS 2.2. The tangent bundle of Projective Space. We now use these constructions to identify the tangent bundle of projective spaces, τRPnandτCPn. We study the real case first. Recall the canonical line bundle, γ1:Eγ1→RPn. If [x]∈RPnis viewed as a line in Rn+1, then the fiber Eγ1 [x]is the one dimensional space of vectors in the line [ x]. Thusγ1has a natural embedding into the trivial n+ 1 - dimensional bundle /epsilon1:RPn×Rn+1→RPnvia Eγ1={([x],u)∈RPn×Rn+1:u∈[x]}/arrowhookleft→RPn×Rn+1. Letγ⊥ 1be then- dimensional orthogonal complement bundle of this embedding. Theorem 1.14.There is an isomorphism of the tangent bundle with the homomorphism bundle τRPn∼=Hom (γ1,γ⊥ 1) Proof. Letp:Sn→RPnbe the natural projection. For x∈Sn, recall that the tangent space ofSncan be described as TxSn={(x,v)∈Sn×Rn+1:x·v= 0}. Notice that ( x,v)∈TxSnand (−x,−v)∈T−xSnhave the same image in T[x]RPnunder the derivativeDp:TSn→TRPn. Sincepis a local diffeomorphism, Dp(x) :TxSn→T[x]RPnis an isomorphism for every x∈Sn. ThusT[x]RPncan be identified with the space of pairs T[x]RPn={(x,v),(−x,−v) :x,v∈Rn+1,|x|= 1, x·v= 0}. Ifx∈Sn, letLx= [x] denote the line through ±xinRn+1. Then a pair ( x,v),(−x,−v)∈ T[x]RPnis uniquely determined by a linear transformation /lscript:Lx→L⊥ /lscript(tx) =tv. ThusT[x]RPnis canonically isomorphic to Hom (Eγ1x,Eγ⊥ 1x), and so τRPn∼=Hom (γ1,γ⊥ 1), as claimed. /square The following description of the τRPn⊕/epsilon11will be quite helpful to us in future calculations of characteristic classes. Theorem 1.15.The Whiney sum of the tangent bundle and a trivial line bundle, τRPn⊕/epsilon11is isomorphic to the Whitney sum of n+ 1copies of the canonical line bundle γ1, τRPn⊕/epsilon11∼=⊕n+1γ1. 2. PULL BACKS AND BUNDLE ALGEBRA 25 Proof. Consider the line bundle Hom (γ1,γ1) over RPn. This line bundle is trivial since it has a canonical nowhere zero section ι(x) = 1 :Eγ1 [x]→Eγ1 [x]. We therefore have τRPn⊕/epsilon11∼=τRPn⊕Hom (γ1,γ1) ∼=Hom (γ1,γ⊥ 1)⊕Hom (γ1,γ1) ∼=Hom (γ1,γ⊥ 1⊕γ1) ∼=Hom (γ1,/epsilon1n+1) ∼=⊕n+1γ∗ 1 ∼=⊕n+1γ1 as claimed. /square The following are complex analogues of the above theorems and are proved in the same way. Theorem 1.16. τCPn∼=CHom C(γ1,γ⊥ 1) and τCPn⊕/epsilon11∼=⊕n+1γ∗ 1, where∼=CandHom Cdenote isomorphisms and homomorphisms of complex bundles, respectively. Note.γ∗is not isomorphic as complex vector bundles to γ1. It is isomorphic to γ1with the conjugate complex structure. We will discuss this phenomenon more later. 2.3. K - theory. LetVect∗(X) =⊕n≥0Vectn(X) where, as above, Vectn(X) denotes the set of isomorphism classes of n- dimensional complex bundles over X.Vect∗ R(X) denotes the analogous set of real vector bundles. In both these cases Vect0(X) denotes, by convention, the one point set, representing the unique zero dimensional vector bundle. Now the Whitney sum operation induces pairings Vectn(X)×Vectm(X)⊕−−−−→Vectn+m(X) which in turn give Vect∗(X) the structure of an abelian monoid. Notice that it is indeed abelian because given vector bundles ηandζwe have an obvious isomorphism η⊕ζ∼=ζ⊕η. The “zero” in this monoid structure is the unique element of Vect0(X). 26 1. LOCALLY TRIVAL FIBRATIONS Given an abelian monoid, A, there is a construction due to Grothendieck of its group completion K(A).Formally,K(A) is the smallest abelian group equipped with a homomorphism of monoids, ι: A→K(A). It is smallest in the sense if Gis any abelian group and φ:A→Gis any homomorphism of monoids, then there is a unique extension of φto a map of abelian groups ¯φ:K(A)→Gmaking the diagram commute: Aι−−−−→K(A) φ/arrowbt/arrowbt¯φ G =G This formal property, called the universal property, characterizes K(A), and can be taken to be the definition. However there is a much more explicit description. Basically the group completion K(A) is obtained by formally adjoining inverses to the elements of A. That is, an element of K(A) can be thought of as a formal difference α−β, whereα,β∈A. Strictly speaking we have the following definition. Definition 1.13.LetF(A) be the free abelian group generated by the elements of A, and let R(A) denote the subgroup of F(A) generated by elements of the form a⊕b−(a+b) wherea,b∈A. Here “⊕” is the group operation in the free abelian group and “+” is the addition in the monoid structure of A. We then define the Grothendieck group completion K(A) to be the quotient group K(A) =F(A)/R(A). Notice that an element of K(A) is of the form θ=/summationdisplay iniai−/summationdisplay jmjbj where theni’s andmj’s are positive integers, and each aiandbj∈A. That is, by the relations in R(A), we may write θ=α−β whereα=/summationtext iniai∈A, andβ=/summationtext jmjbj∈A. Notice also that the composition ι:A⊂F(A)→F(A)/R(A) =K(A) is a homomorphism of monoids, and clearly has the universal property described above. We can now make the following definition. Definition 1.14.Given a space X, its complex and real (or orthogonal) K- theories are defined to be the Grothendieck group completions of the abelian monoids of isomorphism classes of vector bundles: K(X) =K(Vect∗(X)) KO(X) =K(Vect∗ R(X) 2. PULL BACKS AND BUNDLE ALGEBRA 27 An element α=ζ−η∈K(X) is often referred to as a “virtual vector bundle” over X. Notice that the discusion of the tangent bundles of projective spaces above (section 2.2) can be interpreted in K-theoretic language as follows: Proposition 1.17.As elements of K(CPn), we have the equation [τCPn] = (n+ 1)[γ∗ 1]−[1] where [m]∈K(X)refers to the class represented by the trivial bundle of dimension m. Similarly, in the orthogonal K- theoryKO(RPn)we have the equation [τRPn] = (n+ 1)[γ1]−[1]. Notice that for a point, Vect∗(pt) =Z+, the nonnegative integers, since there is precisely one vector bundle over a point (i.e vector space) of each dimension. Thus K(pt)∼=KO(pt)∼=Z. Notice furthermore that by taking tensor products there are pairings Vectm(X)×Vectn(X)⊗−−−−→Vectmn(X). The following is verified by a simple check of definitions. Proposition 1.18.The tensor product pairing of vector bundles gives K(X)andKO(X)the structure of commutative rings. Now given a bundle ζoverY, and a map f:X→Y, we saw in the previous section how to define the pull-back, f∗(ζ) overX. This defines a homomorphism of abelian monoids f∗:Vect∗(Y)→Vect∗(X). After group completing we have the following: Proposition 1.19.A continuous map f:X→Yinduces ring homomorphisms, f∗:K(Y)→K(X) and f∗:KO(Y)→KO(X). 28 1. LOCALLY TRIVAL FIBRATIONS In particular, consider the inclusion of a basepoint x0/arrowhookleft→X. This induces a map of rings, called the augmentation, /epsilon1:K(X)→K(x0)∼=Z. This map is a split surjection of rings, because the constant map c:X→x0induces a right inverse of/epsilon1,c∗:Z=K(x0)→K(X). Notice that the augmentation can be viewed as the “dimension” map in that when restricted to the monoid Vect∗(X), then/epsilon1:Vectm(X)→{m}⊂Z. That is, on an element ζ−η∈K(X),/epsilon1(ζ−η) =dim(ζ)−dim(η). We then define the reduced K-theory as follows. Definition 1.15.The reduced K- theory of X, denoted ˜K(X) is defined to be the kernel of the augmentation map ˜K(X) =ker{/epsilon1:K(X)→Z} and so consists of classes ζ−η∈K(X) such that dim(ζ) =dim(η). The reduced orthogonal K- theory, ˜KO(X) is defined similarly. The following is an immediate consequence of the above observations: Proposition 1.20.There are natural splittings of rings K(X)∼=˜K(X)⊕Z KO(X)∼=˜KO(X)⊕Z. Clearly then the reduced K- theory is the interesting part of K- theory. Notice that a bundle ζ∈Vectn(X) determines the element [ ζ]−[n]∈˜K(X), where [n] is theK- theory class of the trivialn- dimensional bundle. The definitions of K- theory are somewhat abstract. The following discussion makes it clear precisely what K- theory measures in the case of compact spaces. Definition 1.16.Letζandηbe vector bundles over a space X.ζandηare said to be stably isomorphic if for some mandn, there is an isomorphism ζ⊕/epsilon1n∼=η⊕/epsilon1m where, as above, /epsilon1kdenotes the trivial bundle of dimension k. We letSVect(X) denote the set of stable isomorphism classes of vector bundles over X. Notice thatSVect(X) is also an abelian monoid under Whitney sum, and that since any two trivial bundles are stably isomorphic, and that adding a trivial bundle to a bundle does not change the stable isomorphic class, then any trivial bundle represents the zero element of SVect(X). 2. PULL BACKS AND BUNDLE ALGEBRA 29 Theorem 1.21.LetXbe a compact space, then SVect(X)is an abelian group and is isomorphic to the reduced K-theory, SVect(X)∼=˜K(X). Proof. A main component of the proof is the following result, which we will prove in the next chapter when we study the classification of vector bundles. Theorem 1.22.Every vector bundle over a compact space can be embedded in a trivial bundle. That is, ifζis a bundle over a compact space X, then for sufficiently large N > 0, there is bundle embedding ζ /arrowhookleft→/epsilon1N. We use this result in the following way in order to prove the above theorem. Let ζbe a bundle over a compact space X. Then by this result we can find an embedding ζ /arrowhookleft→/epsilon1N. Letζ⊥be the orthogonal complement bundle to this embedding. So that ζ⊕ζ⊥=/epsilon1N. Since/epsilon1Nrepresents the zero element in SVect(X), then as an equation in SVect(X) this becomes [ζ] + [ζ⊥] = 0. Thus every element in SVect(X) is invertible in the monoid structure, and hence SVect(X) is an abelian group. To prove thatSVect(X) is isomorphic to ˜K(X), notice that the natural surjection of Vect∗(X) ontoSVect(X) is a morphism of abelian monoids, and since SVect(X) is an abelian group, this surjection extends linearly to a surjective homomorphism of abelian groups, ρ:K(X)→SVect(X). Since [/epsilon1n] = [n]∈K(X) maps to zero in SVect(X) underρ, this map factors through a surjective homomorphism from reduced K- theory, which by abuse of notation we also call ρ, ρ:˜K(X)→SVect(X). To prove that ρis a injective (and hence an isomorphism), we will construct a left inverse to ρ. This is done by considering the composition Vect∗(X)ι−−−−→K(X)→˜K(X) which is given by mapping an n- dimensional bundle ζto [ζ]−[n]. This map clearly sends two bundles which are stably isomorphic to the same class in ˜K(X), and hence factors through a homomorphism j:SVect(X)→˜K(X). 30 1. LOCALLY TRIVAL FIBRATIONS By checking its values on bundles, it becomes clear that the composition j◦ρ:˜K(X)→SVect(X)→ ˜K(X) is the identity map. This proves the theorem. /square We end this section with the following observation. As we said above, in the next chapter we will study the classification of bundles. In the process we will show that homotopic maps induce isomorphic pull - back bundles, and therefore homotopy equivalences induce bijections, via pulling back, on the sets of isomorphism classes of bundles. This tells us that K-theory is a “homotopy invariant” of topological spaces and continuous maps between them. More precisely, the results of the next chapter will imply the following important properties of K- theory. Theorem 1.23.Letf:X→Yandg:X→Ybe homotopic maps. then the pull back homomorphisms are equal f∗=g∗:K(Y)→K(X) and f∗=g∗:KO(Y)→KO(X). This can be expressed in categorical language as follows: (Notice the similarity of role K- theory plays in the following theorem to cohomology theory.) Theorem 1.24.The assignments X→K(X)andX→KO∗(X)are contravariant functors from the category of topological spaces and homotopy classes of continuous maps to the category of rings and ring homomorphisms. 2.4. Differential Forms. In the next two sections we describe certain differentiable construc- tions on bundles over smooth manifolds that are basic in geometric analysis. We begin by recalling some “multilinear algebra”. LetVbe a vector space over a field k. LetT(V) be the associated tensor algebra T(V) =⊕n≥0V⊗n whereV0=k. The algebra structure is comes from the natural pairings V⊗n⊗V⊗m =−−−−→V⊗(n+m). Recall that the exterior algebra Λ(V) =T(V)/A whereA⊂T(V) is the two sided ideal generated by {a⊗b+b⊗a:a, b∈V}. The algebra Λ( V) inherits the grading from the tensor algebra, Λ( V) =⊕n≥0Λk(V),and the induced multiplication is called the “wedge product”, u∧v. Recall that if Vis ann- dimensional vector space, Λk(V) is an/parenleftbign k/parenrightbig - dimensional vector space. 2. PULL BACKS AND BUNDLE ALGEBRA 31 Assume now that Vis a real vector space. An element of the dual space, ( V⊗n)∗=Hom (V⊗n,R) is a multilinear form V×···×V→R. An element of the dual space (Λk(V))∗is an alternating form, i.e a multilinear function θso that θ(vσ(1),···,vσ(k)) =sgn(σ)θ(v1,···,vk) whereσ∈Σkis any permutation. LetAk(V) = (Λk(V))∗be the space of alternating k- forms. Let U⊂Rnbe an open set. Recall the following definition. Definition 1.17.A differential k- form on the open set U⊂Rnis a smooth function ω:U→Ak(Rn). By convention, 0 -forms are just smooth functions, f:U→R. Notice that given such a smooth function, its differential, dfassigns to a point x∈U⊂Rna linear map on tangent spaces, df(x) :Rn=TxRn→Tf(x)R=R. That is,df:U→(Rn)∗, and hence is a one form on U. Let Ωk(U) denote the space of k- forms on the open set U. Recall that any k-formω∈Ωk(U) can be written in the form (2.1) ω(x) =/summationdisplay IfI(x)dxI where the sum is taken over all sequences of length kof integers from 1 to n,I= (i1,···,ik), fI:U→Ris a smooth function, and where dxI=dxi1∧···∧dxik. Heredxidenotes the differential of the function xi:U⊂Rn→Rwhich is the projection onto the ith- coordinate. Recall also that there is an exterior derivative, d: Ωk(U)→Ωk+1(U) defined by d(fdxI) =df∧dxI==k/summationdisplay j=1∂f ∂xjdxj∧dxI A simple calculation shows that d2(ω) =d(dω) = 0, using the symmetry of second order partial derivatives. These constructions can be extended to arbitrary manifolds in the following way. Given an n- dimensional smooth manifold M, let Λk(τ(M)) be the/parenleftbign k/parenrightbig - dimensional vector bundle whose fiber atx∈Mis thek- fold exterior product, of the tangent space, Λk(TxM). 32 1. LOCALLY TRIVAL FIBRATIONS Exercise. Define clutching functions of Λk(τ(M)) in terms of clutching functions of the tangent bundle, τ(M) Definition 1.18.A differential k-form onMis a section of the dual bundle, Λk(τ(M))∗∼=Λk(τ∗(M))∼=Hom (Λk(τ(M)),/epsilon11). That is, the space of k-forms is given by the space of sections, Ωk(M) = Γ(Λk(τ∗(M))). So ak-formω∈Ωk(M) assigns to x∈Man alternating kform on its tangent space, ω(x) :TxM×···×TxM→R. and hence given a local chart with a local coordinate system, then locally ωcan be written in the form (2.1). Since differentiation is a local operation, we may extend the definition of the exterior derivative of forms on open sets in Rnto alln- manifolds, d: Ωk(M)→Ωk+1(M). In particular, the zero forms are the space of functions, Ω0(M) =C∞(M;R),and forf∈Ω0(M), thendf∈Ω1(M) = Γ(τ(M)∗) is the 1 -form defined by the differential, df(x) :TxM→Tf(x)R=R. Now as above, d2(ω) = 0 for any form ω. Thus we have a cochain complex, called the deRham complex, (2.2)Ω0(M)d−−−−→ Ω1(M)d−−−−→ ···d−−−−→ Ωk−1(M)d−−−−→ Ωk(M)d−−−−→ Ωk+1(M) d−−−−→ ···d−−−−→ Ωn(M)d−−−−→ 0. Recall that a k- formωwithdω= 0 is called a closed form. Ak- formωin the image of d, i.e ω=dηfor someη∈Ωk−1(M) is called an exact form. The quotient vector space of closed forms modulo exact forms defined the “deRham cohomology” group: Definition 1.19. Hk deRham (M) ={closedk- forms}/{exactk- forms}. 2. PULL BACKS AND BUNDLE ALGEBRA 33 The famous de Rham theorem asserts that these cohomology groups are isomorphic to singular cohomology with R- coefficients. To see the relationship, let Ck(M) be the space of k- dimensional singular chains on M, (i.e the free abelian group generated by smooth singular simplices σ: ∆k→ M), and let Ck(M;R) =Hom (Ck(M),R) be the space of real valued singular cochains. Notice that a k-formωgives rise to a k- dimensional singular cochain in that it acts on a singular simplex σ: ∆k→Mby /angbracketleftω,σ/angbracketright=/integraldisplay σω. This defines a homomorphism γ: Ωk(M)→Ck(M;R) for eachk. Exercise. Prove that γis a map of cochain complexes. That is, γ(dω) =δγ(ω) whereδ:Ck(M;R)→Ck+1(M;R) is the singular coboundary operator. Hint. Use Stokes’ theorem. We refer the reader to [ 5] for a proof of the deRham Theorem: Theorem 1.25.The map of cochain complexes, γ: Ω∗(M)→C∗(M;R) is a chain homotopy equivalence. Therefore it induces an isomorphism in cohomology H∗ deRham (M)∼=−−−−→H∗(M;R). 2.5. Connections and Curvature. In modern geometry, differential topology, and geometric analysis, one often needs to study not only smooth functions on a manifold, but more generally, spaces of smooth sections of a vector bundle Γ( ζ). (Notice that sections of bundles are indeed a generalization of smooth functions in that the space of sections of the n- dimensional trivial bundle over a manifold M, Γ(/epsilon1n) =C∞(M;Rn) =⊕nC∞(M;R).) Similarly, one needs to study differential forms that take values in vector bundles. These are defined as follows. Definition 1.20.Letζbe a smooth bundle over a manifold M. A differential k- form with values inζis defined to be a smooth section of the bundle of homomorphisms, Hom (Λk(τ(M)),ζ) = Λk(τ(M)∗)⊗ζ. 34 1. LOCALLY TRIVAL FIBRATIONS We write the space of k-forms with values in ζas Ωk(M;ζ) = Γ(Λk(τ(M)∗⊗ζ). The zero forms are simply the space of sections, Ω0(M;ζ) = Γ(ζ). Notice that if ζis the trivial bundleζ=/epsilon1n, then one gets standard forms, Ωk(M;/epsilon1n) = Ωk(M)⊗Rn=⊕nΩk(M). Even though spaces of forms with values in a bundle are easy to define, there is no canonical analogue of the exterior derivative. There do however exist differential operators D: Ωk(M;ζ)→Ωk+1(M;ζ) that satisfy familiar product formulas. These operators are called covariant derivatives (orconnec- tions ) and are related to the notion of a connection on a principal bundle, which we now define and study. LetGbe a compact Lie group. Recall that the tangent bundle τGhas a canonical trivialization ψ:G×T1G→TG (g,v)→D(/lscriptg)(v) where for any g∈G,/lscriptg:G→Gis the map given by left multiplication by g, andD(/lscriptg) : ThG→TghGis its derivative. rgandD(rg) will denote the analogous maps corresponding to right multiplication. The differential of right multiplication on Gdefines a right action of Gon the tangent bundle τG. We claim that the trivialization ψis equivariant with respect to this action, if we take as the right action of GonT1Gto be the adjoint action: T1G×G→T1G (v,g)→D(/lscriptg−1)(v)D(rg). Exercise. Verify this claim. As is standard, we identify T1Gwith the Lie algebra g. This action is referred to as the adjoint representation of the Lie group Gon its Lie algebra g. Now let p:P→M be a smooth principal G-bundle over a manifold M. This adjoint representation induces a vector bundlead(P), (2.3) ad(P) :P×Gg→M. 2. PULL BACKS AND BUNDLE ALGEBRA 35 This bundle has the following relevance. Let p∗(τM) :p∗(TM)→Pbe the pull - back over the total space Pof the tangent bundle of M. We have a surjective map of bundles τP→p∗(τM). DefineTFPto be the kernel bundle of this map. So the fiber of TFPat a pointy∈Pis the kernel of the surjective linear transformation Dp(y) :TyP→Tp(y)M. Notice that the right action of Gon the total space of the principal bundle Pdefines an action of Gon the tangent bundle τP, which restricts to an action of GonTFP. Furthermore, by recognizing that the fibers are equivariantly homeomorphic to the Lie group G, the following is a direct consequence of the above considerations: Proposition 1.26.TFPis naturally isomorphic to the pull - back of the adjoint bundle, TFP∼=p∗(ad(P)). Thus we have an exact sequence of G- equivariant vector bundles over P: (2.4) 0→p∗(ad(P))→τPDp−−−−→p∗(τM)→0. Recall that short exact sequences of bundles split as Whitney sums. A connection is a G- equivariant splitting of this sequence: Definition 1.21.Aconnection on the principal bundle Pis aG- equivariant splitting ωA:τP→p∗(ad(P)) of the above sequence of vector bundles. That is, ωAdefines aG- equivariant isomorphism ωA⊕Dp:τP→p∗(ad(P))⊕p∗(τM). The following is an important description of the space of connections on P,A(P). Proposition 1.27.The space of connections on the principal bundle P,A(P), is an affine space modeled on the infinite dimensional vector space of one forms on Mwith values in the bundle ad(P),Ω1(M;ad(P)). Proof. Consider two connections ωAandωB, ωA,ωB:τP→p∗(ad(P). Since these are splittings of the exact sequence 2.4, they are both the identity when restricted to p∗(ad(P))/arrowhookleft→τP. Thus their difference, ωA−ωBis zero when restricted to p∗(ad(P)). By the exact sequence it therefore factors as a composition ωA−ωB:τP→p∗(τM)α−−−−→p∗(ad(P)) 36 1. LOCALLY TRIVAL FIBRATIONS for some bundle homomorphism α:p∗(τM)→p∗(ad(P)). That is, for every y∈P,αdefines a linear transformation αy:p∗(TM)y→p∗(ad(P))y. Hence for every y∈P,αdefines (and is defined by) a linear transformation αy:Tp(y)M→ad(P)p(y). Furthermore, the fact that both ωAandωBareequivariant splittings says that ωA−ωBis equivariant, which translates to the fact that αyonly depends on the orbit of yunder theG- action. That is, αy=αyg:Tp(y)M→ad(P)p(y) for everyg∈G. Thusαyonly depends on p(y)∈M. Hence for every x∈M,αdefnes, and is defined by, a linear transformation αx:TxM→ad(P)x. Thusαmay be viewed as a section of the bundle of homormorphisms, Hom (τM,ad (P)), and hence is a one form, α∈Ω1(M;ad(P)). Thus any two connections on Pdiffer by an element in Ω1(M;ad(P)) in this sense. Now reversing the procedure, an element β∈Ω1(M;ad(P)) defines an equivariant homomor- phism of bundles over P, β:p∗(τM)→p∗(ad(P)). By adding the composition τPDp−−−−→p∗(τM)β−−−−→p∗(ad(P)) to any connection (equivariant splitting) ωA:τP→p∗(ad(P)) one produces a new equivariant splitting of τP, and hence a new connection. The proposition follows. /square Remark. Even though the space of connections A(P) is affine, it is not, in general a vector space. There is no “zero” in A(P) since there is no pre-chosen, canonical connection. The one exception to this, of course, is when Pis the trivial G- bundle, P=M×G→M. In this case there is an obvious equvariant splitting of τP, which serves as the “zero” in A(P). Moreover in this case the adjoint bundle ad(P) is also trivial, ad(P) =M×g→M. 2. PULL BACKS AND BUNDLE ALGEBRA 37 Hence there is a canonical identification of the space of connections on the trivial bundle with Ω1(M;g) = Ω1(M)⊗g. Letp:P→Mbe a principal G- bundle and let ωA∈A(P) be a connection. The curvature FAofωAis a two form FA∈Ω2(M;ad(P)) which measures to what extent the splitting ωAcommutes with the braket operation on vector fields. More precisely, let XandYbe vector fields on M. The connection ωAdefines an equivariant splitting of τPand hence defines a “horizontal” lifting of these vector fields, which we denote by ˜X and˜Yrespectively. Definition 1.22.The curvature FA∈Ω2(M;ad(P)) is defined by FA(X,Y) =ωA[˜X,˜Y]. For those unfamiliar with the bracket operation on vector fields, we refer you to [ 38] Another important construction with connections is the associated covariant derivative which is defined as follows. Definition 1.23.The covariant derivative induced by the connection ωA DA: Ω0(M;ad(P))→Ω1(M;ad(P)) is defined by DA(σ)(X) = [ ˜X,σ]. whereXis a vector field on M. The notion of covariant derivative, and hence connection, extends to vector bundles as well. Let ζ:p:Eζ→Mbe a finite dimensional vector bundle over M. Definition 1.24.A connection on ζ(or a covariant derivative) is a linear transformation DA: Ω0(M;ζ)→Ω1(M;ζ) that satisfies the Leibnitz rule (2.5) DA(fφ) =df⊗φ+fDA(φ) for anyf∈C∞(M;R)and anyφ∈Ω0(M;ζ). 38 1. LOCALLY TRIVAL FIBRATIONS Now we can model the space of connections on a vector bundle, A(ζ) similarly to how we modeled the space of connections on a principal bundle A(P). Namely, given any two connections DAandDBonζand a function f∈C∞(M;R), one can take the convex combination f·DA+ (1−f)·DB and obtain a new connection. From this it is not difficult to see the following. We leave the proof as an exercise to the reader. Proposition 1.28.The space of connections on the vector bundle ζ,A(ζ)is an affine space modeled on the vector space of one forms Ω1(M;End(ζ)), whereEnd(ζ)is the bundle of endomor- phisms ofζ. LetXbe a vector field on MandDAa connection on the vector bundle ζ. The covariant derivative in the direction of X, which we denote by ( DA)Xis an operator on the space of sections ofζ, (DA)X: Ω0(M;ζ)→Ω0(M;ζ) defined by (DA)X(σ) =/angbracketleftDA(φ);X/angbracketright. One can then define the curvature FA∈Ω2(M;End(ζ)) by defining its action on a pair of vector fieldsXandYto be (2.6) FA(X,Y) = (DA)X(DA)Y−(DA)Y(DA)X−(DA)[X,Y]. To interpret this formula notice that a - priori FA(X,Y) is a second order differential operator on the space of sections of ζ. However a direct calculation shows that for f∈C∞(M;R) andσ∈Ω0(M;ζ), then FA(X,Y)(fσ) =fFA(X,Y)(σ) and henceFA(X,Y) is in fact a zero - order operator on Ω0(M;ζ). But a zero order operator on the space of sections of ζis a section of the endomorphism bundle End(ζ). ThusFAassigns to any pair of vector fields XandYa section of End(ζ). Moreover it is straightforward to check that this assignment is tensorial in XandY(i.eFA(fX,Y ) =FA(X,fY ) =fFA(X,Y)). ThusFAis an element of Ω2(M;End(ζ)). The curvature measures the lack of commutativity in second order partial covariant derivatives. Given a connection on a bundle ζthe linear mapping DA: Ω0(M;ζ)→Ω1(M;ζ) extends to a deRham type sequence, Ω0(M;ζ)DA−−−−→ Ω1(M;ζ)DA−−−−→ Ω2(M;ζ)DA−−−−→ ··· 2. PULL BACKS AND BUNDLE ALGEBRA 39 where forσ∈Ωp(M;ζ),DA(σ) is thep+ 1 -form defined by the formula DA(σ)(X0,···,Xp) =p/summationdisplay j=0(−1)j(DA)Xj(σ(X0,···,ˆXj,···,Xp)) (2.7) +/summationdisplay i<j(−1)i+jσ([Xi,Xj],X0,···,ˆXi,···,ˆXj,···,Xp). We observe that unlike with the standard deRham exterior derivative (which can be viewed as a connection on the trivial line bundle), it is not generally true that DA◦DA= 0. In fact we have the following, whose proof is a direct calculation that we leave to the reader. Proposition 1.29. DA◦DA=FA: Ω0(M;ζ)→Ω2(M;ζ) where in this context the curvature FAis interpreted as a assigning to a section σ∈Ω0(M;ζ)the2 - formFA(σ)which associates to vector fields XandYthe section FA(X,Y)(σ)as defined in (2.6). Thus the curvature of a connection FAcan also be viewed as measuring the extent to which the covariant derivative DAfails to form a cochain complex on the space of differential forms with values in the bundle ζ. However it is always true that the covariant derivative of the curvature tensor is zero. This is the well known Bianchi identity (see [38] for a complete discussion). Theorem 1.30.Let A be a connection on a vector bundle ζ. Then DAFA= 0. We end this section by observing that if Pis a principal G- bundle with a connection ωA, then any representation of Gon a finite dimensional vector space Vinduces a connection on the corresponding vector bundle P×GV→M. We refer the reader to [ 16] and [ 38] for thorough discussions of the various ways of viewing connec- tions. [ 3] has a nice, brief discussion of connections on principal bundles, and [ 14] and [ 23] have similarly concise discussions of connections on vector bundles. 2.6. The Levi - Civita Connection. LetMbe a manifold equipped with a Riemannian structure. Recall that this is a Euclidean structure on its tangent bundle. In this section we will show how this structure induces a connection, or covariant derivative, on the tangent bundle. This connection is called the Levi - Civita connection associated to the Riemannian structure. Our treatment of this topic follows that of Milnor and Stasheff [ 31] 40 1. LOCALLY TRIVAL FIBRATIONS LetDA: Ω0(M;ζ)→Ω1(M;ζ) be a connection (or covariant derivative) on an n- dimensional vector bundle ζ. Its curvature is a two- form with values in the endomorphism bundle FA∈Ω2(M;End(ζ)) The endomorphism bundle can be described alternatively as follows. Let Eζbe the principal GL(n,R) bundle associated to ζ. Then of course ζ=Eζ⊗GL(n,R)Rn.The endomorphism bun- dle can then be described as follows. The proof is an easy exercise that we leave to the reader. Proposition 1.31. End(ζ)∼=ad(ζ) =Eζ×GL(n,R)Mn(R) whereGL(n,R)acts onMn(R)by conjugation, A·B=ABA−1. Letωbe a differential p- form onMwith values in End(ζ), ω∈Ωp(M;End(ζ))∼=Ωp(M;ad(ζ)) = Ωp(M;Eζ×GL(n,R)Mn(R)). Then on a coordinate chart U⊂Mwith local trivialization ψ:ζ|U∼=U×Cnforζ, (and hence the induced coordinate chart and local trivialization for ad(ζ)),ωcan be viewed as an n×nmatrix of p-forms onM. We write ω= (ωi,j). Of course this description depends on the coordinate chart and local trivialization chosen, but at anyx∈U, then by the above proposition, two trivializations yield conjugate matrices. That is, if (ωi,j(x)) and (ω/prime i,j(x)) are two matrix descriptions of ω(x) defined by two different local trivializations ofζ|U, then there exists an A∈GL(n,C) with A(ωi,j(x))A−1= (ω/prime i,j(x)). Now suppose the bundle ζis equipped with a Euclidean structure. As seen earlier in this chapter this is equivalent to its associated principal GL(n,R) - bundleEζhaving a reduction to the structure groupO(n). We letEO(n)→Mdenote this principal O(n) - bundle. Now the Lie algebra o(n) ofO(n) (i.e the tangent space T1(O(n))) is a subspace of the Lie algebra ofGL(n,R), i.e o(n)⊂Mn(R). The following is well known (see, for example[ 35]) 2. PULL BACKS AND BUNDLE ALGEBRA 41 Proposition 1.32.The Lie algebra o(n)⊂Mn(R)is the subspace consisting of skew symmetric n×n- matrices. That is, A∈o(n)if and only if At=−A whereAtis the transpose. So ifζhas a Euclidean structure, we can form the adjoint bundle adO(ζ) =EO(n)×O(n)o(n)⊂Eζ×GL(n,R)Mn(R) =ad(ζ) where, again O(n) acts on o(n) by conjugation. Now suppose DAis an orthogonal connection on ζ. That is, it is induced by a connection on the principal O(n) - bundleEO(n)→M. The following is fairly clear, and we leave its proof as an exercise. Corollary 1.33.IfDAis an orthogonal connection on a Euclidean bundle ζ, then the curvature FAlies in the space of o(n)valued two forms FA∈Ω2(M;adO(ζ))⊂Ω2(M;ad(ζ)) = Ω2(M;End(ζ)). Furthermore, on a coordinate chart U⊂Mwith local trivialization ψ:ζ|U∼=U×Cnthat preserves the Euclidean structure, we may write the form FAas a skew - symmetric matrix of two forms, FA|U= (ωi,j)i,j= 1,···,n where each ωi,j∈Ω2(M)andωi,j=−ωj,i. In fact the connection DAitself can be written as skew symmetric matrix of one forms DA|U= (αi,j) where each αi,j∈Ω1(M). We now describe the notion of a “symmetric” connection on the cotangent bundle of a manifold, and then show that if the manifold is equipped with a Riemannian structure (i.e there is a Euclidean structure on the (co) - tangent bundle), then there is a unique symmetric, orthogonal connection on the cotangent bundle. Definition 1.25.A connection DAon the cotangent bundle τ∗Missymmetric (ortorsion free )if the composition Γ(τ∗) = Ω0(M;τ∗)DA−−−−→ Ω1(M;τ∗) = Γ(τ∗⊗τ∗)∧−−−−→ Γ(Λ2τ∗) is equal to the exterior derivative d. 42 1. LOCALLY TRIVAL FIBRATIONS In terms of local coordinates x1,···,xn, if we write (2.8) DA(dxk) =/summationdisplay i,jΓk i,jdxi⊗dxj (the functions Γk i,jare called the “Christoffel symbols”), then the requirement that DAis symmetric is that the image/summationtext i,jΓk i,jdxi⊗dxjbe equal to the exterior derivative d(dxk) = 0. This implies that the Christoffel symbols Γk i,jmust be symmetric in iandj. The following is straightforward to verify. Lemma 1.34.A connection DAonτ∗is symmetric if and only if the covariant derivative of the differential of any smooth function DA(df)∈Γ(τ∗⊗τ∗) is a symmetric tensor. That is, if ψ1,···,ψnform a local basis of sections of τ∗, and we write the corresponding local expression DA(df) =/summationdisplay i,jai,jψi⊗ψj thenai,j=aj,i. We now show that the (co)-tangent bundle of a Riemannian metric has a preferred connection. Theorem 1.35.The cotangent bundle τ∗Mof a Riemannian manifold has a unique orthogo- nal, symmetric connection. (It is orthogonal with respect to the Euclidean structure defined by the Riemannian metric.) Proof. LetUbe an open neighborhood in Mwith a trivialization ψ:U×Rn:→τ∗ |U which preserves the Euclidean structure. ψdefinesnorthonormal sections of τ∗ |U,ψ1,···,ψn. The ψj’s constitute an orthonormal basis of one forms on M. We will show that there is one and only one skew-symmetric matrix ( αi,j) of one forms such that dψk=/summationdisplay αk,j∧ψj. We can then define a connection DAonτ∗ |Uby requiring that DA(ψk) =/summationdisplay αk,j⊗ψj. It is then clear that DAis the unique symmetric connection which is compatible with the metric. Since the local connections are unique, they glue together to yield a unique global connection with this property. 2. PULL BACKS AND BUNDLE ALGEBRA 43 In order to prove the existence and uniqueness of the skew symmetric matrix of one forms ( αi,j) we need the following combinatorial observation. Anyn×n×narray of real valued functions Ai,j,kcan be written uniquely as the sum of an arrayBi,j,kwhich is symmetric in i,j, and an array Ci,j,kwhich is skew symmetric in j,k. To see this, consider the formulas Bi,j,k=1 2(Ai,j,k+Aj,i,k−Ak,i,j−Ak,j,i+Aj,k,i+Ai,k,j) Ci,j,k=1 2(Ai,j,k−Aj,i,k+Ak,i,j+Ak,j,i−Aj,k,i−Ai,k,j) Uniqueness would follow since if an array Di,j,kwere both symmetric in i,jand skew symmetric in j,k, then one would have Di,j,k=Dj,i,k=−Dj,k,i=−Dk,j,i=Dk,i,j=Di,k,j=−Di,j,k and hence all the entries are zero. Now choose functions Ai,j,ksuch that dψk=/summationdisplay Ai,j,kψi∧ψj and setAi,j,k=Bi,j,k+Ci,j,kas above. It then follows that dψk=/summationdisplay Ci,j,kψi∧ψj by the symmetry of the Bi,j,k’s. Then we define the one forms αk,j=/summationdisplay Ci,j,kψi. They clearly form the unique skew symmetric matrix of one forms with dψk=/summationtextαk,j∧ψj.This proves the lemma. /square This preferred connection on the (co)tangent bundle of a Riemannian metric is called the Levi - Civita connection. Statements about the curvature of a metric on a manifold are actually statements about the curvature form of the Levi - Civita connection associated to the Riemannian metric. For example, a “flat metric” on a manifold is a Riemannian structure whose corresponding Levi-Civita connection has zero curvature form. As is fairly clear, these connections form a central object of study in Riemannian geometry. CHAPTER 2 Classification of Bundles In this chapter we prove Steenrod’s classification theorem of principal G- bundles, and the corresponding classification theorem of vector bundles. This theorem states that for every group G, there is a “classifying space” BGwith a well defined homotopy type so that the homotopy classes of maps from a space X, [X,BG ],is in bijective correspondence with the set of isomorphism classes of principal G- bundles,PrinG(X). We then describe various examples and constructions of these classifying spaces, and use them to study structures on principal bundles, vector bundles, and manifolds. 1. The homotopy invariance of fiber bundles The goal of this section is to prove the following theorem, and to examine certain applications such as the classification of principal bundles over spheres in terms of the homotopy groups of Lie groups. Theorem 2.1.Letp:E→Bbe a fiber bundle with fiber F, and letf0:X→Bandf1:X→B be homotopic maps.Then the pull - back bundles are isomorphic, f∗ 0(E)∼=f∗ 1(E). The main step in the proof of this theorem is the basic Covering Homotopy Theorem for fiber bundles which we now state and prove. Theorem 2.2.Covering Homotopy theorem. Letp0:E→Bandq:Z→Ybe fiber bundles with the same fiber, F, whereBis normal and locally compact. Let h0be a bundle map E˜h0−−−−→Z p/arrowbt/arrowbtq B−−−−→ h0Y 45 46 2. CLASSIFICATION OF BUNDLES LetH:B×I→Ybe a homotopy of h0(i.eh0=H|B×{0}.) Then there exists a covering of the homotopyHby a bundle map E×I˜H−−−−→Z p×1/arrowbt/arrowbtq B×I−−−−→ HY. Proof. We prove the theorem here when the base space Bis compact. The natural extension is to whenBhas the homotopy type of a CW - complex. The proof in full generality can be found in Steenrod’s book [ 39]. The idea of the proof is to decompose the homotopy Hinto homotopies that take place in local neighborhoods where the bundle is trivial. The theorem is obviously true for trivial bundles, and so the homotopy Hcan be covered on each local neighborhood. One then must be careful to patch the coverings together so as to obtain a global covering of the homotopy H. Since the space Xis compact, we may assume that the pull - back bundle H∗(Z)→B×Ihas locally trivial neighborhoods of the form {Uα×Ij}, where{Uα}is a locally trivial covering of B(i.e there are local trivializations φα,β:Uα×F→p−1(Uα)), andI1,···,Iris a finite sequence of open intervals covering I= [0,1], so that each Ijmeets only Ij−1andIj+1nontrivially. Choose numbers 0 =t0<t1<···<tr= 1 so thattj∈Ij∩Ij+1. We assume inductively that the covering homotopy ˜H(x,t) has been defined E×[0,tj] so as to satisfy the theorem over this part. For eachx∈B, there is a pair of neighborhoods ( W,W/prime) such that for x∈W,¯W⊂W/primeand ¯W/prime⊂Uαfor someUα. Choose a finite number of such pairs ( Wi,W/prime i), (i= 1,···,s) coveringB. Then the Urysohn lemma implies there is a map ui:B→[tj,tj+1] such that ui(¯Wi) =tj+1and uj(B−W/prime i) =tj. Defineτ0(x) =tjforx∈B, and τi(x) =max(u1(x),···,ui(x)), x∈B, i = 1,···,s. Then tj=τ0(x)≤τ1(x)≤···≤ts(x) =tj+1. DefineBito be the set of pairs ( x,t) such that tj≤t≤τi(x). LetEibe the part of E×Ilying overBi. Then we have a sequence of total spaces of bundles E×tj=E0⊂E1⊂···⊂Es=E×[tj,tj+1]. We suppose inductively that ˜Hhas been defined on Ei−1and we now define its extension over Ei. By the definition of the τ’s, the setBi−Bi−1is contained in W/prime i×[tj,tj+1]; and by the definition of theW’s,¯W/primei×[tj,tj+1]⊂Uα×Ijwhich maps via Hto a locally trivial neighborhood, say Vk, 1. THE HOMOTOPY INVARIANCE OF FIBER BUNDLES 47 forq:Z→Y. Sayφk:Vk×F→q−1(Vk) is a local trivialization. In particular we can define ρk:q−1(Vk)→Fto be the inverse of φkfollowed by the projection onto F. We now define ˜H(e,t) =φk(H(x,t),ρ(˜H(e,τi−1(x))) where (e,t)∈Ei−Ei−1andx=p(e)∈B. It is now a straightforward verification that this extension of ˜His indeed a bundle map on Ei. This then completes the inductive step. /square We now prove theorem 2.1 using the covering homotopy theorem. Proof. Letp:E→B, andf0;X→Bandf1:X→Bbe as in the statement of the theorem. LetH:X×I→Bbe a homotopy with H0=f0andG1=f1. Now by the covering homotopy theorem there is a covering homotopy ˜H:f∗ 0(E)×I→Ethat covers H:X×I→B. By definition this defines a map of bundles over X×I, that by abuse of notation we also call ˜H, f∗ 0(E)×I˜H−−−−→H∗(E) /arrowbt/arrowbt X×I−−−−→ =X×I. This is clearly a bundle isomorphism since it induces the identity map on both the base space and on the fibers. Restricting this isomorphism to X×{1}, and noting that since H1=f1, we get a bundle isomorphism f∗ 0(E)˜H−−−−→∼=f∗ 1(E) /arrowbt/arrowbt X×{1} −−−−→ =X×{1}. This proves theorem 2.1 /square We now derive certain consequences of this theorem. Corollary 2.3.Letp:E→Bbe a principal G- bundle over a connected space B. Then for any spaceXthe pull back construction gives a well defined map from the set of homotopy classes of maps from XtoBto the set of isomorphism classes of principal G- bundles, ρE: [X,B]→PrinG(X). 48 2. CLASSIFICATION OF BUNDLES Definition 2.1.A principal G- bundlep:EG→BGis called universal if the pull back construction ρEG: [X,BG ]→PrinG(X) is a bijection for every space X. In this case the base space of the universal bundle BGis called a classifying space forG(or for principal G- bundles). The main goal of this chapter is to prove that universal bundles exist for every group G, and that the classifying spaces are unique up to homotopy type. Applying theorem 2.1 to vector bundles gives the following, which was claimed at the end of chapter 1. Corollary 2.4.Iff0:X→Yandf1:X→Yare homotopic, they induce the same homomorphism of abelian monoids, f∗ 0=f∗ 1:Vect∗(Y)→Vect∗(X) Vect∗ R(Y)→Vect∗ R(X) and hence of Ktheories f∗ 0=f∗ 1:K(Y)→K(X) KO(Y)→KO(X) Corollary 2.5.Iff:X→Yis a homotopy equivalence, then it induces isomorphisms f∗:PrinG(Y)∼=−−−−→PrinG(X) Vect∗(Y)∼=−−−−→Vect∗(X) K(Y)∼=−−−−→K(X) Corollary 2.6.Any fiber bundle over a contractible space is trivial. Proof. IfXis contractible, it is homotopy equivalent to a point. Apply the above corollary. /square The following result is a classification theorem for bundles over spheres. It begins to describe why understanding the homotopy type of Lie groups is so important in Topology. Theorem 2.7.There is a bijective correspondence between principal bundles and homotopy groups PrinG(Sn)∼=πn−1(G) 1. THE HOMOTOPY INVARIANCE OF FIBER BUNDLES 49 where as a set πn−1G= [Sn−1,x0;G,{1}], which refers to (based) homotopy classes of basepoint preserving maps from the sphere Sn−1with basepoint x0∈Sn−1, to the group Gwith basepoint the identity 1∈G. Proof. Letp:E→Snbe aG- bundle. Write Snas the union of its upper and lower hemispheres, Sn=Dn +∪Sn−1Dn −. SinceDn +andDn −are both contractible, the above corollary says that Erestricted to each of these hemispheres is trivial. Morever if we fix a trivialization of the fiber of Eat the basepoint x0∈Sn−1⊂Sn, then we can extend this trivialization to both the upper and lower hemispheres. We may therefore write E= (Dn +×G)∪θ(Dn −×G) whereθis a clutching function defined on the equator, θ:Sn−1→G. That is, Econsists of the two trivial components, ( Dn +×G) and (Dn −×G) where ifx∈Sn−1, then (x,g)∈(Dn +×G) is identified with ( x,θ(x)g)∈(Dn −×G). Notice that since our original trivializations extended a common trivialization on the basepoint x0∈Sn−1, then the trivialization θ:Sn−1→Gmaps the basepointx0to the identity 1 ∈G. The assignment of a bundle its clutching function, will define our correspondence Θ :PrinG(Sn)→πn−1G. To see that this correspondence is well defined we need to check that if E1is isomorphic to E2, then the corresponding clutching functions θ1andθ2are homotopic. Let Ψ : E1→E2be an isomorphism. We may assume this isomorphism respects the given trivializations of these fibers of these bundles over the basepoint x0∈Sn−1⊂Sn. Then the isomorphism Ψ determines an isomorphism (Dn +×G)∪θ1(Dn −×G)Ψ−−−−→∼=(Dn +×G)∪θ2(Dn −×G). By restricting to the hemispheres, the isomorphism Ψ defines maps Ψ+:Dn +→G and Ψ−:Dn −→G which both map the basepoint x0∈Sn−1to the identity 1∈G, and furthermore have the property that forx∈Sn−1, Ψ+(x)θ1(x) =θ2(x)Ψ−(x), or, Ψ +(x)θ1(x)Ψ−(x)−1=θ2(x)∈G. Now by considering the linear homotopy Ψ +(tx)θ1(x)Ψ−(tx)−1 fort∈[0,1], we see that θ2(x) is homotopic to Ψ +(0)θ1(x)Ψ−(0)−1, where the two zeros in this description refer to the origins of Dn +andDn −respectively, i.e the north and south poles of the sphereSn. Now since Ψ +and Ψ −are defined on connected spaces, their images lie in a connected component of the group G. Since their image on the basepoint x0∈Sn−1are both the identity, 50 2. CLASSIFICATION OF BUNDLES there exist paths α+(t) andα−(t) inSnthat start when t= 0 at Ψ +(0) and Ψ −(0) respectively, and both end at t= 1 at the identity 1 ∈G. Then the homotopy α+(t)θ1(x)α−(t)−1is a homotopy from the map Ψ +(0)θ1(x)Ψ−(0)−1to the map θ1(x). Since the first of these maps is homotopic to θ2(x), we have that θ1is homotopic to θ2, as claimed. This implies that the map Θ : PrinG(Sn)→πn−1G is well defined. The fact that Θ is surjective comes from the fact that every map Sn−1→Gcan be viewed as the clutching function of the bundle E= (Dn +×G)∪θ(Dn −×G) as seen in our discussion of clutching functions in chapter 1. We now show that Θ is injective. That is, suppose E1andE2have homotopic clutching functions, θ1/similarequalθ2:Sn−1→G. We need to show that E1is isomorphic to E2As above we write E1= (Dn +×G)∪θ1(Dn −×G) and E2= (Dn +×G)∪θ2(Dn −×G). LetH:Sn−1×[−1,1]→Gbe a homotopy so that H1=θ1andH1=θ2. Identify the closure of an open neighborhood Nof the equator Sn−1inSnwithSn−1×[−1,1] WriteD+=D2 +∪¯Nand D−=D2 −∪¯NThenD+andD−are topologically closed disks and hence contractible, with D+∩D−=¯N∼=Sn−1×[−1,1]. Thus we may form the principal G- bundle E=D+×G∪HD−×G where by abuse of notation, Hrefers to the composition ¯N∼=Sn−1×[−1,1]H−−−−→G. We leave it to the interested reader to verify that Eis isomorphic to both E1andE2. This completes the proof of the theorem. /square 2. Universal bundles and classifying spaces The goal of this section is to study universal principal G- bundles, the resulting classification theorem, and the corresponding classifying spaces. We will discuss several examples including the universal bundle for any subgroup of the general linear group. We postpone the proof of the existence of universal bundles for all groups until the next section. In order to identify universal bundles, we need to recall the following definition from homotopy theory. 2. UNIVERSAL BUNDLES AND CLASSIFYING SPACES 51 Definition 2.2.A spaceXis said to be aspherical if all of its homotopy groups are trivial, πn(X) = 0 for all n≥0. Equivalently, a space Xis aspherical if every map from a sphere Sn→Xcan be extended to a map of its bounding disk, Dn+1→X. Note. A famous theorem of J.H.C. Whitehead states that if Xhas the homotopy type of a CW - complex, then Xbeing aspherical is equivalent to Xbeing contractible (see [ 44]). The following is the main result of this section. It identifies when a principal bundle is universal. Theorem 2.8.Letp:E→Bbe a principal G- bundle, where the total space Eis aspherical. Then this bundle is universal in the sense that if Xis any space, the induced pull-back map ψ: [X,B]→PrinG(X) f→f∗(E) is a bijective correspondence. For the purposes of these notes we will prove the theorem in the setting where the action of Gon the total space Eiscellular. That is, there is a CW - decomposition of the space Ewhich, in an appropriate sense, is respected by the group action. There is very little loss of generality in these assumptions, since the actions of compact Lie groups on manifolds, and algebraic actions on projective varieties satisfy this property. For the proof of the theorem in its full generality we refer the reader to Steenrod’s book [ 39], and for a full reference on equivariant CW - complexes and how they approximate a wide range of group actions, we refer the reader to [ 24] In order to make the notion of cellular action precise, we need to define the notion of an equivariant CW - complex, or aG-CW -complex. The idea is the following. Recall that a CW - complex is a space that is made up out of disks of various dimensions whose interiors are disjoint. In particular it can be built up skeleton by skeleton, and the ( k+ 1)stskeletonX(k+1)is constructed out of the kthskeletonX(k)by attaching ( k+ 1) - dimensional disks via “attaching maps”,Sk→X(k). A “G-CW - complex” is one that has a group action so that the orbits of the points on the interior of a cell are uniform in the sense that each point in a cell Dkhas the same isotropy subgroup, sayH, and the orbit of a cell itself is of the form G/H×Dk. This leads to the following definition. Definition 2.3.AG-CW - complex is a space with G-actionXwhich is topologically the direct limit of G- invariant subspaces {X(k)}called the equivariant skeleta, X(0)⊂X(1)⊂···⊂X(k−1)⊂X(k)⊂···X 52 2. CLASSIFICATION OF BUNDLES where for each k≥0 there is a countable collection of kdimensional disks, subgroups of G, and maps of boundary spheres {Dk j, Hj<G, φj:∂Dk j×G/Hj=Sk−1 j×G/Hj→X(k−1)j∈Ik} so that (1) Each “attaching map” φj:Sk−1 j×G/Hj→X(k−1)isG-equivariant, and (2) X(k)=Xk−1)/uniondisplay φjj∈Ij(Dk j×G/Hj). This notation means that each “ disk orbit ” Dk j×G/Hjis attached to X(k−1)via the map φj: Sk−1 j×G/Hj→X(k−1). We leave the following as an exercise to the reader. Exercise. Prove that when Xis aG-CW complex the orbit space X/G has the an induced structure of a (non-equivariant) CW - complex. Note. Observe that in a G-CW complexXwith a freeGaction, all disk orbits are of the form Dk×G, since all isotropy subgroups are trivial. We now prove the above theorem under the assumption that the principal bundle p:E→B has the property that with respect to group action of GonE, thenEhas the structure of a G-CW - complex. The basespace is then given the induced CW - structure. The spaces Xin the statement of the theorem are assumed to be of the homotopy type of CW - complexes. Proof. We first prove that the pull - back map ψ: [X,B]→PrinG(X) is surjective. So let q:P→Xbe a principal G- bundle, with PaG-CW - complex. We prove there is aG- equivariant map h:P→Ethat maps each orbit pGhomeomorphically onto its image, h(y)G. We prove this by induction on the equivariant skeleta of P. So assume inductively that the maphhas been constructed on the ( k−1) - skeleton, hk−1:P(k−1)→E. Since the action of GonPis free, all the k- dimensional disk orbits are of the form Dk×G. LetDk j×Gbe a disk orbit in the G-CW - structure of the k- skeletonP(k). Consider the disk Dk j×{1}⊂Dk j×G. Then the map hk−1extends toDk j×{1}if and only if the composition Sk−1 j×{1}⊂Sk−1 j×Gφj−−−−→P(k−1)hk−1−−−−→E 2. UNIVERSAL BUNDLES AND CLASSIFYING SPACES 53 is null homotopic. But since Eis aspherical, any such map is null homotopic and extends to a map of the disk, γ:Dk j×{1}→E. Now extend γequivariantly to a map hk,j:Dk j×G→E. By construction hk,jmaps the orbit of each point x∈Dk jequivariantly to the orbit of γ(x) inE. Since both orbits are isomorphic to G(because the action of Gon bothPandEare free), this map is a homeomorphism on orbits. Taking the collection of the extensions hk,jtogether then gives an extension hk:P(k)→E with the required properties. This completes the inductive step. Thus we may conclude we have a G- equivariant map h:P→Ethat is a homeomorphism on the orbits. Hence it induces a map on the orbit space f:P/G =X→E/G =Bmaking the following diagram commute Ph−−−−→E q/arrowbt/arrowbtp X−−−−→ fB Sincehinduces a homeomorphism on each orbit, the maps handfdetermine a homeomorphism of principalG- bundles which induces an equivariant isomorphism on each fiber. This implies that h induces an isomorphism of principal bundles to the pull - back Ph−−−−→∼=f∗(E) q/arrowbt/arrowbtp X−−−−→ =X. Thus the isomorphism class [ P]∈PrinG(X) is given by f∗(E). That is, [ P] =ψ(f), and hence ψ: [X,B]→PrinG(X) is surjective. We now prove ψis injective. To do this, assume f0:X→Bandf1:X→Bare maps so that there is an isomorphism Φ :f∗ 0(E)∼=−−−−→f∗ 1(E). We need to prove that f0andf1are homotopic maps. Now by the cellular approximation theorem (see [ 37]) we can find cellular maps homotopic to f0andf1respectively. We therefore assume without loss of generality that f0andf1are cellular. This, together with the assumption that Eis a G-CW complex, gives the pull back bundles f∗ 0(E) andf∗ 1(E) the structure of G-CW complexes. Define a principal G- bundleE→X×Iby E=f∗ 0(E)×[0,1/2]∪Φf∗ 1(E)×[1/2,1] wherev∈f∗ 0(E)×{1/2}is identified with Φ( v)∈f∗ 1(E)×{1/2}.Ealso has the structure of a G- CW - complex. 54 2. CLASSIFICATION OF BUNDLES Now by the same kind of inductive argument that was used in the surjectivity argument above, we can find an equivariant map H:E→Ethat induces a homeomorphism on each orbit, and that extends the obvious maps f∗ 0(E)×{0}→Eandf∗ 1(E)×{1}→E. The induced map on orbit spaces F:E/G=X×I→E/G =B is a homotopy between f0andf1. This proves the correspondence Ψ is injective, and completes the proof of the theorem. /square The following result establishes the homotopy uniqueness of universal bundles. Theorem 2.9.LetE1→B1andE2→B2be universal principal G- bundles. Then there is a bundle map E1˜h−−−−→E2/arrowbt/arrowbt B1−−−−→ hB2 so thathis a homotopy equivalence. Proof. The fact that E2→B2is a universal bundle means, by 2.8 that there is a “classifying map”h:B1→B2and an isomorphism ˜h:E1→h∗(E2). Equivalently, ˜hcan be thought of as a bundle map ˜h:E!→E2lying overh:B1→B2. Similarly, using the universal property of E1→B1, we get a classifying map g:B2→B1and an isomorphism ˜ g:E2→g∗(E1), or equivalently, a bundle map ˜g:E2→E1. Notice that the composition g◦f:B1→B2→B1 is a map whose pull back, (g◦f)∗(E1) =g∗(f∗(E1)) ∼=g∗(E2) ∼=E1. That is, (g◦f)∗(E1)∼=id∗(E1), and hence by 2.8 we have g◦f/similarequalid:B1→B1. Similarly, f◦g/similarequalid:B2→B2. Thusfandgare homotopy inverses of each other. /square Because of this theorem, the basespace of a universal principal G- bundle has a well defined homotopy type. We denote this homotopy type by BG, and refer to it as the classifying space of the groupG. We also use the notation EGto denote the total space of a universal G- bundle. We have the following immediate result about the homotopy groups of the classifying space BG. 2. UNIVERSAL BUNDLES AND CLASSIFYING SPACES 55 Corollary 2.10.For any group G, there is an isomorphism of homotopy groups, πn−1G∼=πn(BG). Proof. By considering 2.7 and 2.8 we see that both of these homotopy groups are in bijective correspondence with the set of principal bundles PrinG(Sn). To realize this bijection by a group homomorphism, consider the “suspension” of the group G, ΣGobtained by attaching two cones on Galong the equator. That is, ΣG=G×[−1,1]/∼ where all points of the form ( g,1), (h,−1), or (1,t) are identified to a single point. Notice that this suspension construction can be applied to any space with a basepoint, and in particular Σ Sn−1∼=Sn. Consider the principal GbundleEover ΣGdefined to be trivial on both cones with clutching functionid:G×{0}=−−−−→Gon the equator. That is, if C+=G×[0,1]/∼⊂ΣGandC−= G×[−1,0]⊂ΣEare the upper and lower cones, respectively, then E= (C+×G)∪id(C−×G) where ((g,0),h)∈C+×Gis identified with (( g,0)gh∈C−×G. Then by 2.8 there is a classifying map f: ΣG→BG such thatf∗(EG)∼=E. Now for any space X, let ΩXbe the loop space ofX, ΩX={γ: [−1,1]→Xsuch thatγ(−1) =γ(1) =x0∈X} wherex0∈Xis a fixed basepoint. Then the map f: ΣG→BGdetermines a map (its adjoint) ¯f:G→ΩBG defined by ¯f(g)(t) =f(g,t). But now the loop space Ω Xof any connected space Xhas the property thatπn−1(ΩX) =πn(X) (see the exercise below). We then have the induced group homomorphism πn−1(G)¯f∗−−−−→πn−1(ΩBG)∼=−−−−→πn(BG) which induces the bijective correspondence described above. /square Exercises. 1. Prove that for any connected space X, there is an isomorphism πn−1(ΩX)∼=πn(X). 2. Prove that the composition πn−1(G)¯f∗−−−−→πn−1(ΩBG)∼=−−−−→πn(BG) 56 2. CLASSIFICATION OF BUNDLES in the above proof yields the bijection associated with identifying both πn−1(G) andπn(BG) with PrinG(Sn). We recall the following definition from homotopy theory. Definition 2.4.An Eilenberg - MacLane space of type ( G,n) is a space Xsuch that πk(X) =  G if k = n 0 otherwise We writeK(G,n) for an Eilenberg - MacLane space of type ( G,n). Recall that for n≥2, the homotopy groups πn(X) are abelian groups, so in this K(G,n) only exists Corollary 2.11.Letπbe a discrete group. Then the classifying space Bπis an Eilenberg - MacLane space K(π,1). Examples. •Rhas a free, cellular action of the integers Zby (t,n)→t+n t∈R, n∈Z. SinceRis contractible, R/Z=S1=BZ=K(Z,1). •The inclusion Sn⊂Sn+1as the equator is clearly null homotopic since the inclusion obviously extends to a map of the disk. Hence the direct limit space lim−→nSn=∪nSn=S∞ is aspherical. Now Z2acts freely on Snby the antipodal map, as described in chapter one. The inclusions Sn⊂Sn+1are equivariant and hence there is an induced free action ofZ2onS∞. Thus the projection map S∞→S∞/Z2=RP∞ is a universal principal Z2=O(1) - bundle, and so RP∞=BO(1) =BZ2=K(Z2,1) . •Similarly, the inclusion of the unit sphere in Cninto the unit sphere in Cn+1gives an the inclusionS2n−1⊂S2n+1which is null homotopic. It is also equivariant with respect to the freeS1=U(1) - action given by (complex) scalar multiplication. Then the limit S∞=∪nS2n+1is aspherical with a free S1action. We therefore have that the projection S∞→S∞/S1=CP∞ 2. UNIVERSAL BUNDLES AND CLASSIFYING SPACES 57 is a principal S1=U(1) bundle. Hence we have CP∞=BS1=BU(1). Moreover since S1is aK(Z,1), then we have that CP∞=K(Z,2). •The cyclic groups Znare subgroups of U(1) and so they act freely on S∞as well. Thus the projection maps S∞→S∞/Zn is a universal principal Znbundle. The quotient space S∞/Znis denotedL∞(n) and is referred to as the infinite Zn- lens space. These examples allow us to give the following description of line bundles and their relation to cohomology. We first recall a well known theorem in homotopy theory. This theorem will be discussed further in chapter 4. We refer the reader to [ 42] for details. Theorem 2.12.LetGbe an abelian group. Then there is a natural isomorphism φ:Hn(K(G,n);G)∼=−−−−→Hom (G,G). Letι∈Hn(K(G,n);G) beφ−1(id). This is called the fundamental class. Then if Xhas the homotopy type of a CW - complex, the mapping [X,K (G,n)]→Hn(X;G) f→f∗(ι) is a bijective correspondence. With this we can now prove the following: Theorem 2.13.There are bijective correspondences which allow us to classify complex line bundles, Vect1(X)∼=PrinU(1)(X)∼=[X,BU (1)] = [X,CP∞]∼=[X,K (Z,2)]∼=H2(X;Z) where the last correspondence takes a map f:X→CP∞to the class c1=f∗(c)∈H2(X), wherec∈H2(CP∞)is the generator. In the composition of these correspondences, the class c1∈ H2(X)corresponding to a line bundle ζ∈Vect1(X)is called the first Chern class of ζ(or of the corresponding principal U(1)- bundle). 58 2. CLASSIFICATION OF BUNDLES Proof. These correspondences follow directly from the above considerations, once we recall thatVect1(X)∼=PrinGL(1,C)(X)∼=[X,BGL (1,C)], and that CP∞is a model for BGL (1,C) as well asBU(1). This is because, we can express CP∞in its homogeneous form as CP∞= lim−→n(Cn+1−{0})/GL(1,C), and that lim−→n(Cn+1−{0}) is an aspherical space with a free action of GL(1,C) =C∗./square There is a similar theorem classifying real line bundles: Theorem 2.14.There are bijective correspondences Vect1 R(X)∼=PrinO(1)(X)∼=[X,BO (1)] = [X,RP∞]∼=[X,K (Z2,1)]∼=H1(X;Z2) where the last correspondence takes a map f:X→RP∞to the class w1=f∗(w)∈H1(X;Z2), wherew∈H1(RP∞;Z2)is the generator. In the composition of these correspondences, the class w1∈H1(X;Z2)corresponding to a line bundle ζ∈Vect1 R(X)is called the first Stiefel - Whitney class ofζ(or of the corresponding principal O(1)- bundle). More Examples. •LetVn(CN) be the Stieflel - manifold studied in the last chapter. We claim that the inclusion of vector spaces CN⊂C2Nas the first N- coordinates induces an inclusion Vn(CN)/arrowhookleft→Vn(C2N) which is null homotopic. To see this, let ι:Cn→C2Nbe a fixed linear embedding, whose image lies in the last N- coordinates in C2N. Then given any ρ∈Vn(CN)⊂Vn(C2N), thent·ι+ (1−t)·ρfort∈[0,1] defines a one parameter family of linear embeddings of CninC2N, and hence a contraction of the image of Vn(CN) onto the elementι. Hence the limiting space Vn(C∞) is aspherical with a free GL(n,C) - action. Therefore the projection Vn(C∞)→Vn(C∞)/GL(n,C) =Grn(C∞) is a universal GL(n,C) - bundle. Hence the infinite Grassmannian is the classifying space Grn(C∞) =BGL (n,C) and so we have a classification Vectn(X)∼=PrinGL(n,C)(X)∼=[X,BGL (n,C)]∼=[X,Grn(C∞)]. 2. UNIVERSAL BUNDLES AND CLASSIFYING SPACES 59 •A simlar argument shows that the infinite unitary Stiefel manifold, VU n(C∞) is aspherical with a free U(n) - action. Thus the projection VU n(C∞)→Vn(C∞)/U(n) =Grn(C∞) is a universal principal U(n) - bundle. Hence the infinite Grassmanian Grn(C∞) is the classifying space for U(n) bundles as well, Grn(C∞) =BU(n). The fact that this Grassmannian is both BGL (n,C) andBU(n) reflects the fact that every n- dimensional complex vector bundle has a U(n) - structure. •We have similar universal GL(n,R) andO(n) - bundles: Vn(R∞)→Vn(R∞)/GL(n,R) =Grn(R∞) and VO n(R∞)→VO n(R∞)/O(n) =Grn(R∞). Thus we have Grn(R∞) =BGL (n,R) =BO(n) and so this infinite dimensional Grassmannian classifies real n- dimensional vector bundles as well as principal O(n) - bundles. Now suppose p:EG→EG/G =BGis a universal G- bundle. Suppose further that H <G is a subgroup. Then Hacts freely on EGas well, and hence the projection EG→EG/H is a universal H- bundle. Hence EG/H =BH. Using the infinite dimensional Stiefel manifolds described above, this observation gives us models for the classifying spaces for any subgroup of a general linear group. So for example if we have a subgroup (i.e a faithful representation) H⊂ GL(n,C), then BH=Vn(C∞)/H. This observation also leads to the following useful fact. Proposition 2.15.. Letp:EG→BGbe a universal principal G- bundle, and let H < G . Then there is a fiber bundle BH→BG with fiber the orbit space G/H . 60 2. CLASSIFICATION OF BUNDLES Proof. This bundle is given by G/H→EG×GG/H→EG/G =BG together with the observation that EG×GG/H =EG/H =BH. /square 3. Classifying Gauge Groups In this section we describe the classifying space of the group of automorphisms of a principal G - bundle, or the gauge group of the bundle. We describe the classifying space in two different ways: in terms of the space of connections on the bundle, and in terms of the mapping space of the base manifold to the classifying space BG. These constructions are important in Yang - Mills theory, and we refer the reader to [ 3] and [ 11] for more details. LetAbe a connection on a principal bundle P−→MwhereMis a closed manifold equipped with a Riemannian metric. The Yang - Mills functional applied to A,YM(A) is the square of the L2norm of the curvature, YM(A) =1 2/integraldisplay M/bardblFA/bardbl2d(vol). We viewYM as a mappingYM :A(P)−→R. The relevance of the gauge group in Yang - Mills theory is that this is the group of symmetries of AthatYM preserves. Definition 2.5.The gauge group G(P) of the principal bundle Pis the group of bundle au- tomorphisms of P−→M. That is, an element φ∈G(P) is a bundle isomorphism of Pwith itself lying over the identity: Pφ−−−−→∼=P /arrowbt/arrowbt M=−−−−→M. Equivalently,G(P) is the groupG(P) =AutG(P) ofG- equivariant diffeomorphisms of the space P. The gauge group G(P) can be thought of in several equivalent ways. The following one is particularly useful. Consider the conjugation action of the Lie group Gon itself, G×G−→G (g,h)−→ghg−1. 3. CLASSIFYING GAUGE GROUPS 61 This left action defines a fiber bundle Ad(P) =P×GG−→P/G =M with fiberG. We leave the following as an exercise for the reader. Proposition 2.16.The gauge group of a principal bundle P−→Mis naturally isomorphic (as topological groups) the the group of sections of Ad(P),C∞(M;Ad(P)). The gauge group G(P) acts on the space of connections A(P) by the pull - back construction. More generally, if f:P→Qis any smooth map of principal G- bundles and Ais a connection on Q, then there is a natural pull back connection f∗(A) onQ, defined by pulling back the equivariant splitting of τQto an equivariant splitting of τPin the obvious way. The pull - back construction for automorphisms φ:P−→Pdefines an action of G(P) onA(P). We leave the proof of the following is an exercise for the reader. Proposition 2.17.LetPbe the trivial bundle M×G→M. Then the gauge group G(P)is given by the function space from MtoG, G(P)∼=C∞(M;G). Furthermore if φ:M→Gis identified with an element of G(P), andA∈Ω1(M;g)is identified with an element of A(G), then the induced action of φonGis given by φ∗(A) =φ−1Aφ+φ−1dφ. It is not difficult to see that in general the gauge group G(P) does not act freely on the space of connectionsA(P). However there is an important subgroup G0(P)<G(P) that does. This is the group of based gauge transformations. To define this group, let x0∈Mbe a fixed basepoint, and letPx0be the fiber of Patx0. Definition 2.6.The based gauge group G0(P) is a subgroup of the group of bundle automor- phismsG(P) which pointwise fix the fiber Px). That is, G0(P) ={φ∈G(P) : ifv∈Px0thenφ(v) =v}. Theorem 2.18.The based gauge group G0(P)acts freely on the space of connections A(P). 62 2. CLASSIFICATION OF BUNDLES Proof. Suppose that A∈A(P) is a fixed point of φ∈G0(P). That is, φ∗(A) =A. We need to show that φ= 1. The equivariant splitting ωAgiven by a connection Adefines a notion of parallel transport in P along curves in M(see [16]) . It is not difficult to see that the statement φ∗(A) =Aimplies that application of the automorphism φcommutes with parallel transport. Now let w∈Pxbe a point in the fiber of an element x∈M. Given curve γinMbetween the basepoint x0andxone sees that φ(w) =Tγ(φ(Tγ−1(w)) whereTγis parallel transport along γ. But since Tγ−1(w)∈Px0andφ∈G0(P), φ(Tγ−1(w)) =w. Henceφ(w) =w, that is,φ= 1. /square Remark. Notice that this argument actually says that if A∈A(P) is the fixed point of anygauge transformation φ∈G(P), thenφis determined by its action on a single fiber. LetB(P) andB0(P) be the orbit spaces of connections on Pup to guage and based gauge equivalence respectively, B(P) =A(P)/G(P)B0(P) =A(P)/G0(P). Now it is straightforward to check directly that the Yang - Mills functional in invariant under gauge transformations. Thus it yields maps YM :B(P)→RandYM :B0(P)→R. It is therefore important to understand the homotopy types of these orbit spaces. Because of the freeness of the action of G0(P), the homotopy type of the orbit space G0(P) is easier to understand. We end this section with a discussion of its homotopy type. Since the space of connections A(P) is affine, it is contractible. Moreover it is possible to show that the free action of the based gauge groupG0(P) has local slices (see [ 11]). Thus we have B0(P) =A(P)/G0(P) is the classifying space of the based gauge group, B0(P) =BG0(P). But the classifying spaces of the gauge groups are relatively easy to understand. (see [ 3].) Theorem 2.19.LetG−→EG−→BGbe a universal principal bundle for the Lie group G(so thatEGis aspherical). Let y0∈BGbe a fixed basepoint. Then there are homotopy equivalences BG(P)/similarequalMapP(M,BG )andB0(P)/similarequalBG0(P)/similarequalMapP 0(M,BG ) 4. EXISTENCE OF UNIVERSAL BUNDLES: THE MILNOR JOIN CONSTRUCTION AND THE SIMPLICIAL CLASSIFYING SPACE 63 whereMap (M,BG )is the space of all continuous maps from MtoBGandMap 0(M,BG )is the space of those maps that preserve the basepoints. The superscript Pdenotes the path component of these mapping spaces consisting of the homotopy class of maps that classify the principal G- bundle P. Proof. Consider the space of all G- equivariant maps from PtoEG,MapG(P,EG ). The gauge groupG(P)∼=AutG(P) acts freely on the left of this space by composition. It is easy to see thatMapG(P,EG ) is aspherical, and its orbit space is given by the space of maps from the G- orbit space ofP(=M) to theG- orbit space of EG(= BG), MapG(P,EG )/G(P)∼=MapP(M,BG ). This proves that Map (M,BG ) =BG(P). Similarly MapG 0(P,EG ), the space of G- equivariant maps that send the fiber Px0to the fiber EGy0, is an aspherical space with a free G0(P) action, whose orbit space is MapP 0(M,BG ). HenceMapP 0(M,BG ) =BG0(P). /square 4. Existence of universal bundles: the Milnor join construction and the simplicial classifying space In the last section we proved a “recognition principle” for universal principal Gbundles. Namely, if the total space of a principal G- bundlep:E→Bis aspherical, then it is universal. We also proved a homotopy uniqueness theorem, stating among other things that the homotopy type of the base space of a universal bundle, i.e the classifying space BG, is well defined. We also described many examples of universal bundles, and particular have a model for the classifying space BG, using Stiefel manifolds, for every subgroup of a general linear group. The goal of this section is to prove the general existence theorem. Namely, for every group G, there is a universal principal G- bundlep:EG→BG. We will give two constructions of the universal bundle and the corresponding classifying space. One, due to Milnor [ 30] involves taking the “infinite join” of a group with itself. The other is an example of a simplicial space, called the simplicial bar construction. It is originally due to Eilenberg and MacLane [ 12]. These constructions are essentially equivalent and both yield G-CW - complexes. Since they are so useful in algebraic topology and combinatorics, we will also take this opportunity to introduce the notion of a general simplicial space and show how these classifying spaces are important examples. 4.1. The join construction. The “join” between two spaces XandY, writtenX∗Yis the space of all lines connecting points in Xto points in Y. The following is a more precise definition: 64 2. CLASSIFICATION OF BUNDLES Definition 2.7.The joinX∗Yis defined by X∗Y=X×I×Y/∼ whereI= [0,1] is the unit interval and the equivalence relation is given by ( x,0,y1)∼(x,0,y2) for any two points y1, y2∈Y, and similarly ( x1,1,y)∼(x2,1,y) for any two points x1, x2∈X. A point (x,t,y )∈X∗Yshould be viewed as a point on the line connecting the points xandy. Here are some examples. Examples. •Letybe a single point. Then X∗yis the cone CX=X×I/X×{1}. •LetY={y1,y2}be the space consisting of two distinct points. Then X∗Yis the suspension Σ Xdiscussed earlier. Notice that the suspension can be viewed as the union of two cones, with vertices y1andy2respectively, attached along the equator. •Exercise. Prove that the join of two spheres, is another sphere, Sn∗Sm∼=Sn+m+1. •Let{x0,···,xk}be a collection of k+ 1 - distinct points. Then the k- fold join x0∗x1∗···∗xkis the convex hull of these points and hence is by the k- dimensional simplex ∆kwith vertices{x0,···,xk}. Observe that the space Xsits naturally as a subspace of the join X∗Yas endpoints of line segments, ι:X /arrowhookleft→X∗Y x→(x,0,y). Notice that this formula for the inclusion makes sense and does not depend on the choice of y∈Y. There is a similar embedding j:Y /arrowhookleft→X∗Y y→(x,1,y). Lemma 2.20.The inclusions ι:X /arrowhookleft→X∗Yandj:Y /arrowhookleft→X∗Yare null homotopic. Proof. Pick a point y0∈Y. By definition, the embedding ι:X→X∗Yfactors as the composition ι:X /arrowhookleft→X∗y0⊂X∗Y x→(x,0,y0). 4. EXISTENCE OF UNIVERSAL BUNDLES: THE MILNOR JOIN CONSTRUCTION AND THE SIMPLICIAL CLASSIFYING SPACE 65 But as observed above, the join X∗y0is the cone on Xand hence contractible. This means that ι is null homotopic, as claimed. The fact that j:Y /arrowhookleft→X∗Yis null homotopic is proved in the same way. /square Now letGbe a group and consider the iterated join G∗(k+1)=G∗G∗···∗G where there are k+ 1 copies of the group element. This space has a free Gaction given by the diagonal action g·(g0,t1,g1,···,tk,gk) = (gg0,t1,gg1,···,tk,ggk). Exercise. 1. Prove that there is a natural G- equivariant map ∆k×Gk+1→G∗(k+1) which is a homeomorphism when restricted to ˜∆k×Gk+1where ˜∆k⊂∆kis the interior. Here G acts on ∆k×Gk+1trivially on the simplex ∆kand diagonally on Gk+1. 2.Use exercise 1 to prove that the iterated join G∗(k+1)has the structure of a G-CW - complex. DefineJ(G) to be the infinite join J(G) = lim k→∞G∗(k+1) where the limit is taken over the embeddings ι:G∗(k+1)/arrowhookleft→G∗(k+2)Since these embedding maps areG-equivariant, we have an induced G- action onJ(G). Theorem 2.21.The projection map p:J(G)→J (G)/G is a universal principal G- bundle. Proof. By the above exercise the space J(G) has the structure of a G-CW - complex with a freeG- action. Therefore by the results of the last section the projection p:J(G)→J (G)/Gis a principal G- bundle. To see that J(G) is aspherical, notice that since Snis compact, any map α:Sn→J (G) is homotopic to one that factors through a finite join (that by abuse of notation we still callα),α:Sn→G∗(n+1)/arrowhookleft→J (G). But by the above lemma the inclusion G∗(n+1)⊂J(G) is null homotopic, and hence so is α. ThusJ(G) is aspherical. By the results of last section, this means that the projection J(G)→J (G)/Gis a universal G- bundle. /square 66 2. CLASSIFICATION OF BUNDLES 4.2. Simplicial spaces and classifying spaces. We therefore now have a universal bundle for every topological group G. We actually know a fair amount about the geometry of the total spaceEG=J(G) which, by the above exercise can be described as the union of simplices, where thek- simplices are parameterized by k+ 1 -tuples of elements of G, EG=J(G) =/uniondisplay k∆k×Gk+1/∼ and so the classifying space can be described by BG=J(G)/G∼=/uniondisplay k∆k×Gk/∼ It turns out that in these constructions, the simplices are glued together along faces, and these gluings are parameterized by the k+ 1 - product maps ∂i:Gk+2→Gk+1given by multiplying the ithand (i+ 1)stcoordinates. Having this type of data (parameterizing spaces of simplices as well as gluing maps) is an example of an object known as a “simplicial set” which is an important combinatorial object in topology. We now describe this notion in more detail and show how these universal G- bundles and classifying spaces can be viewed in these terms. Good references for this theory are [ 9], [26]. The idea of simplicial sets is to provide a combinatorial technique to study cell complexes built out of simplices; i.e simplicial complexes. A simplicial complex Xis built out of a union of simplices, glued along faces. Thus if Xndenotes the indexing set for the n- dimensional simplices of X, then we can write X=/uniondisplay n≥0∆n×Xn/∼ where ∆nis the standard n- simplex in Rn; ∆n={(t1,···,tn)∈Rn: 0≤tj≤1,andn/summationdisplay i=1ti≤1}. The gluing relation in this union can be encoded by set maps among the Xn’s that would tell us for example how to identify an n−1 simplex indexed by an element of Xn−1with a particular face of ann- simplex indexed by an element of Xn. Thus in principal simplicial complexes can be studied purely combinatorially in terms of the sets Xnand set maps between them. The notion of asimplicial set makes this idea precise. Definition 2.8.Asimplicial set X∗is a collection of sets Xn, n≥0 4. EXISTENCE OF UNIVERSAL BUNDLES: THE MILNOR JOIN CONSTRUCTION AND THE SIMPLICIAL CLASSIFYING SPACE 67 together with set maps ∂i:Xn−→Xn−1andsj:Xn−→Xn+1 for 0≤i,j≤ncalled face anddegeneracy maps respectively. These maps are required to satisfy the following compatibility conditions ∂i∂j=∂j−1∂ifori<j sisj=sj+1sifori<j and ∂isj=  sj−1∂ifori<j 1 for i=j, j+ 1 sj∂i−1fori>j + 1 As mentioned above, the maps ∂iandsjencode the combinatorial information necessary for gluing the simplices together. To say precisely how this works, consider the following maps between the standard simplices: δi: ∆n−1−→∆nandσj: ∆n+1−→∆n for 0≤i,j≤ndefined by the formulae δi(t1,···,tn−1) =  (t1,···,ti−1,0,ti,···,tn−1) fori≥1 (1−/summationtextn−1 q=1tq,t1,···,tn−1) fori= 0 and σj(t1,···,tn+1) =  (t1,···,ti−1,ti+ti+1,ti+2,···,tn+1) fori≥1 (t2,···,tn+1) for i= 0. δiincludes ∆n−1in ∆nas theithface, andσjprojects, in a linear fashion, ∆n+1onto itsjth face. We can now define the space associated to the simplicial set X∗as follows. Definition 2.9.The geometric realization of a simplicial set X∗is the space /bardblX∗/bardbl=/uniondisplay n≥0∆n×Xn/∼ where ift∈∆n−1andx∈Xn, then (t,∂i(x))∼(δi(t),x) 68 2. CLASSIFICATION OF BUNDLES and ift∈∆n+1andx∈Xnthen (t,sj(x))∼(σj(t),x). In the topology of /bardblX∗/bardbl, eachXnis assumed to have the discrete topology, so that ∆n×Xnis a discrete set of n- simplices. Thus/bardblX∗/bardblhas one n - simplex for every element of Xn, glued together in a way determined by the face and degeneracy maps. Example. Consider the simplicial set S∗defined as follows. The set of n- simplices is given by Sn=Z/(n+ 1),generated by an element τn. The face maps are given by ∂i(τr n) =  τr n−1ifr≤i≤n τr−1 n−1if 0≤i≤r−1. The degeneracies are given by si(τr n) =  τr n+1ifr≤i≤n τr+1 n+1if 0≤i≤r−1. Notice that there is one zero simplex, two one simplices, one of them the image of the degeneracy s0:S0−→S1, and the other nondegenerate (i.e not in the image of a degeneracy map). Notice also that all simplices in dimensions larger than one are in the image of a degeneracy map. Hence we have that the geometric realization /bardblS∗/bardbl= ∆1/0∼1 =S1. LetX∗be any simplicial set. There is a particularly nice and explicit way for computing the homology of the geometric realization, H∗(/bardblX∗/bardbl). Consider the following chain complex. Define Cn(X∗) to be the free abelian group generated by the set ofn- simplices Xn. Define the homomorphism dn:Cn(X∗)−→Cn−1(X∗) by the formula dn([x]) =n/summationdisplay i=0(−1)i∂i([x]) wherex∈Xn. Proposition 2.22.The homology of the geometric realization H∗(/bardblX∗/bardbl)is the homology of the chain complex −→···dn+1−−−−→Cn(X∗)dn−−−−→Cn−1(X∗)dn−1−−−−→ ···d0−−−−→C0(X∗). 4. EXISTENCE OF UNIVERSAL BUNDLES: THE MILNOR JOIN CONSTRUCTION AND THE SIMPLICIAL CLASSIFYING SPACE 69 Proof. It is straightforward to check that the geometric realization /bardblX∗/bardblis aCW - complex and that this is the associated cellular chain complex. /square Besides being useful computationally, the following result establishes the fact that all CW complexes can be studied simplicially. Theorem 2.23.EveryCW complex has the homotopy type of the geometric realization of a simplicial set. Proof. LetXbe aCW complex. Define the singular simplicial set of X,S(X)∗as follows. ThensimplicesS(X)nis the set of singular n- simplices, S(X)n={c: ∆n−→X}. The face and degeneracy maps are defined by ∂i(c) =c◦δi: ∆n−1−→∆n−→X and sj(c) =c◦σi: ∆n+1−→∆n−→X. Notice that the associated chain complex to S(X)∗as in 2.22 is the singular chain complex of the spaceX. Hence by 2.22 we have that H∗(/bardblS(X)/bardbl)∼=H∗(X). This isomorphism is actually realized by a map of spaces E:/bardblS(X)∗/bardbl−→X defined by the natural evaluation maps ∆n×S(X)n−→X given by (t,c)−→c(t). It is straightforward to check that the map Edoes induce an isomorphism in homology. In fact it induces an isomorphism in homotopy groups. We will not prove this here; it is more technical and we refer the reader to [M] for details. Note that it follows from the homological isomorphism by the Hurewicz theorem if we knew that Xwas simply connected. A map between spaces that induces an isomorphism in homotopy groups is called a weak homotopy equivalence. Thus anyspace is weakly homotopy equivalent to a CW - complex (i.e the geometric realization of its singular simplicial set). But by the Whitehead theorem, two CW complexes that are weakly homotopy equivalent are homotopy equivalent. Hence Xand/bardblS(X)∗/bardblare homotopy equivalent. /square 70 2. CLASSIFICATION OF BUNDLES We next observe that the notion of simplicial set can be generalized as follows. We say that X∗ is asimplicial space if it is a simplicial set (i.e it satisfies definition 2.8) where the sets Xnare topological spaces and the face and degeneracy maps ∂i:Xn−→Xn−1andsj:Xn−→Xn+1 are continuous maps. The definition of the geometric realization of a simplicial space X∗,/bardblX∗/bardbl, is the same as in 2.9 with the proviso that the topology of each ∆n×Xnis the product topology. Notice that since the “set of n- simplices” Xnis actually a space, it is not necessarily true that /bardblX∗/bardbl is aCW complex. However if in fact each Xnis aCW complex and the face and degeneracy maps are cellular, then /bardblX∗/bardbldoes have a natural CW structure induced by the product CW - structures on ∆n×Xn. Notice that this simplicial notion generalizes even further. For example a simplicial group would be defined similarly, where each Xnwould be a group and the face and degeneracy maps are group homomorphisms. Simplicial vector spaces, modules, etc. are defined similarly. The categorical nature of these definitions should by now be coming clear. Indeed most generally one can define a simplicial object in a category Cusing the above definition where now the Xn’s are assumed to be objects in the category and the face and degenarcies are assumed to be morphisms. If the categoryCis a subcategory of the category of sets then geometric realizations can be defined as in 2.9 For example the geometric realization of a simplicial (abelian) group turns out to be a topological (abelian) group.(Try to verify this for yourself!) We now use this simplicial theory to construct universal principal G- bundles and classifying spaces. LetGbe a topological group and let EG∗be the simplicial space defined as follows. The space ofn- simplices is given by the n+ 1 - fold cartesian product EGn=Gn+1. The face maps ∂i:Gn+1−→Gnare given by the formula ∂i(g0,···,gn) = (g0,···,ˆgi,···,gn). The degeneracy maps sj:Gn+1−→Gn+2are given by the formula sj(g0,···,gn) = (g0,···,gj,gj,···,gn). Exercise. Show that the geometric realization /bardblEG∗/bardblis aspherical. Hint. Let /bardblEG∗/bardbl(n)be thenth - skeleton, /bardblEG∗/bardbl(n)=n/uniondisplay p=0∆p×Gp+1. 4. EXISTENCE OF UNIVERSAL BUNDLES: THE MILNOR JOIN CONSTRUCTION AND THE SIMPLICIAL CLASSIFYING SPACE 71 Then show that the inclusion of one skeleton in the next /bardblEG∗/bardbl(n)/arrowhookleft→/bardblEG∗/bardbl(n+1)is null - homotopic. One way of doing this is to establish a homeomorphism between /bardblEG∗/bardbl(n)andn- fold joinG∗···∗G. See [M] for details. Notice that the group Gacts freely on the right of /bardblEG∗/bardblby the rule /bardblEG∗/bardbl×G= /uniondisplay p≥0∆p×Gp+1 ×G−→/bardblEG∗/bardbl (4.1) (t; (g0,···,gp))×g−→(t; (g0g,···,gpg)). Thus we can define EG=/bardblEG∗/bardbl. The projection map p:EG→EG/G =BG is therefore a universal principal G- bundle. This description gives the classfiying space BGan induced simplicial structure described as follows. . LetBG∗be the simplicial space whose n- simplices are the cartesian product BGn=Gn. (4.2) The face and degeneracy maps are given by ∂i(g1,···,gn) =  (g2,···,gn) for i= 0 (g1,···,gigi+1,···gn) for 1≤i≤n−1 (g1,···,gn−1) for i=n. The degeneracy maps are given by sj(g1,···,gn) =  (1,g1,···,gn) for j= 0 (g1,···gj,1,gj+1,···,gn) forj≥1. The simplicial projection map p:EG∗−→BG∗ defined on the level of n- simplicies by p(g0,···,gn) = (g0g−1 1,g1g−1 2,···,gn−1g−1 n) is easily checked to commute with face and degeneracy maps and so induces a map on the level of geometric realizations 72 2. CLASSIFICATION OF BUNDLES p:EG=/bardblEG∗/bardbl−→/bardblBG∗/bardbl which induces a homemorphism BG=EG/G∼=−−−−→ /bardblBG∗/bardbl. Thus for any topological group this construction gives a simplicial space model for its classifying space. This is referred to as the simplicial bar construction. Notice that when Gis discrete the bar construction is a CW complex for the classifying space BG=K(G,1) and 2.22 gives a particularly nice complex for computing its homology. (The homology of a K(G,1) is referred to as the homology of the group G.) Then- chains are the group ring Cn(BG∗) =Z[Gn]∼=Z[G]⊗n and the boundary homomorphisms dn:Z[G]⊗n−→Z[G]⊗n−1 are given by dn(a1⊗···⊗an) = (a2⊗···⊗an)+n−1/summationdisplay i=1(−1)i(a1⊗···⊗aiai+1⊗···⊗an) + (−1)n(a1⊗···⊗an−1). This complex is called the bar complex for computing the homology of a group and was discovered by Eilenberg and MacLane in the mid 1950’s. We end this chapter by observing that the bar construction of the classifying space of a group did not use the full group structure. It only used the existence of an associative multiplication with unit. That is, it did not use the existence of inverse. So in particular one can study the classifying spaceBAof a monoid A. This is an important construction in algebraic - K- theory. 5. Some Applications In a sense much of what we will study in the next chapter are applications of the classification theorem for principal bundles. In this section we describe a few immediate applications. 5. SOME APPLICATIONS 73 5.1. Line bundles over projective spaces. By the classification theorem we know that the set of isomorphism classes of complex line bundles over the projective space CPnis given by Vect1(CPn)∼=PrinGL(1,C)(CPn)∼=PrinU(1)(CPn)∼=[CPn,BU(1)] = [ CPn,CP∞] = [CPn,K(Z,2)]∼=H2(CPn,Z)∼=Z Theorem 2.24.Under the above isomorphism, Vect1(CPn)∼=Z then- fold tensor product of the universal line bundle γ⊗n 1corresponds to the integer n≥0. Proof. The classification theorem says that every line bundle ζoverCPnis the pull back of the universal line bundle via a map fζ:CPn→CP∞. That is, ζ∼=f∗ ζ(γ1). The cohomology class corresponding to ζ,the first chern class c1(ζ),is given by c1(ζ) =f∗ ζ(c)∈H2(CPn)∼=Z wherec∈H2(CP∞)∼=Zis the generator. Clearly ι∗(c)∈H2(CPn) is the generator, where ι:CPn/arrowhookleft→CP∞is natural inclusion. But ι∗(γ1) =γ1∈Vect1(CPn). Thusγ1∈Vect1(CPn)∼=Z corresponds to the generator. To see the effect of taking tensor products, consider the following “tensor product map” BU(1)×···×BU(1)⊗−−−−→BU(1) defined to be the unique map (up to homotopy) that classifies the external tensor product γ1⊗···⊗γ1 overBU(1)×···×BU(1). Using CP∞∼=Gr1(C∞) as our model for BU(1), this tensor product map is given by taking klines/lscript1,···,/lscriptkinC∞and considering the tensor product line /lscript1⊗···⊗/lscriptk⊂C∞⊗···⊗ C∞∼=−−−−→ ψC∞ whereψ:C∞⊗···⊗ C∞∼=C∞is a fixed isomorphism. The induced map τ:CP∞×···× CP∞→CP∞∼=K(Z,2) is determined up to homotopy by what its effect on H2is. Clearly the restriction to each factor is the identity map and so τ∗(c) =c1+···+ck∈H2(CP∞×···× CP∞) =H2(CP∞)⊕···⊕H2(CP∞)∼=Z⊕···⊕ Z wherecidenotes the generator of H2of theithfactor in the product. Therefore the composition tk:CP∞ ∆−−−−→ CP∞×···× CP∞τ−−−−→ CP∞ 74 2. CLASSIFICATION OF BUNDLES has the property that t∗ k(c) =kc∈H2(CP∞). But also we have that on the bundle level, t∗ k(γ1) =γ⊗k 1∈Vect1(CP∞). The theorem now follows. /square We have a similar result for real line vector bundles over real projective spaces. Theorem 2.25.The only nontrivial real line bundle over RPnis the canonical line bundle γ1. Proof. We know that γ1is nontrivial because its restriction to S1=RP1⊂RPnis the Moebeus strip line bundle, which is nonorientable, and hence nontrivial. On the other hand, by the classification theorem, Vect1 R(RPn)∼=[RPn,BGL (1,R)] = [RPn,RP∞] = [RPn,K(Z2,1)]∼=H1(RPn,Z2)∼=Z2. Hence there is only one nontrivial line bundle over RPn. /square . 5.2. Structures on bundles and homotopy liftings. The following theorem is a direct consequence of the classification theorem. We leave its proof as an exercise. Theorem 2.26.. Letp:E→Bbe a principal G- bundle classified by a map f:B→BG. LetH <G be a subgroup. By the naturality of the construction of classifying spaces, this inclusion induces a map (well defined up to homotopy) ι:BH→BG. Then the bundle p:E→Bhas anH - structure (i.e a reduction of its structure group to H) if and only if there is a map ˜f:B→BH so that the composition B˜f−−−−→BHι−−−−→BG is homotopic to f:B→BG. In particular if ˜p:˜E→Bis the principal H- bundle classified by ˜f, then there is an isomorphism of principal Gbundles, ˜E×HG∼=E. The map ˜f:B→BHis called a “lifting” of the classifying map f:B→BG. It is called a lifting because, as we saw at the end of the last section, the map ι:BH→BGcan be viewed as a fiber bundle, by taking our model for BHto beBH=EG/H . Thenιis the projection for the fiber bundle G/H→EG/H =BHι−−−−→EG/G =BG. 5. SOME APPLICATIONS 75 This bundle structure will allow us to analyze in detail what the obstructions are to obtaining a lift ˜fof a classifying map f:B→BG. We will study this is chapter 4. Examples. •An orientation of a bundle classified by a map f:B→BO(k) is a lifting ˜f:B→BSO (k). Notice that the map ι:BSO (k)→BO(k) can be viewed as a two - fold covering map Z2=O(k)/SO(k)→BSO (k)ι−−−−→BO(k). •An almost complex structure of a bundle classified by a map f:B→BO(2n) is a lifting ˜f:B→BU(n). Notice we have a bundle O(2n)/U(n)→BU(n)→BO(2n). The following example will be particularly useful in the next chapter when we define character- istic classes and do calculations with them. Theorem 2.27.A complex bundle vector bundle ζclassified by a map f:B→BU(n)has a nowhere zero section if and only if fhas a lifting ˜f:B→BU(n−1). Similarly a real vector bundleηclassified by a map f:B→BO(n)has a nowhere zero section if and only if fhas a lifting ˜f:B→BO(n−1). Notice we have the following bundles: S2n−1=U(n)/U(n−1)→BU(n−1)→BU(n) and Sn−1=O(n)/O(n−1)→BO(n−1)→BO(n). This theorem says that BU(n−1) forms a sphere bundle ( S2n−1) overBU(n), and similarly, BO(n−1) forms aSn−1- bundle over BO(n). We identify these sphere bundles as follows. Corollary 2.28.The sphere bundles S2n−1→BU(n−1)→BU(n) and Sn−1→BO(n−1)→BO(n) are isomorphic to the unit sphere bundles of the universal vector bundles γnoverBU(n)andBO(n) respectively. 76 2. CLASSIFICATION OF BUNDLES Proof. We consider the complex case. The real case is proved in the same way. Notice that the model for the sphere bundle in the above theorem is the projection map p:BU(n−1) =EU(n)/U(n−1)→EU(n)/U(n) =BU(n). Butγnis the vector bundle EU(n)×U(n)Cn→BU(n) which therefore has unit sphere bundle S(γn) =EU(n)×U(n)S2n−1→BU(n) (5.1) whereS2n−1⊂Cnis the unit sphere with the induced U(n) - action. But S2n−1∼=U(n)/U(n−1) and this diffeomorphism is equivariant with respect to this action. Thus the unit sphere bundle is given by S(γn) =EU(n)×U(n)U(n)/U(n−1)∼=EU(n)/U(n−1) =BU(n−1) as claimed. /square We observe that by using the Grassmannian models for BU(n) andBO(n), then their relation to the sphere bundles can be seen explicitly in the following way. This time we work in the real case. Consider the embedding ι:Grn−1(RN)/arrowhookleft→Grn(RN×R) =Grn(RN+1) defined by (V⊂RN)→(V×R⊂RN×R). Clearly asN→∞ this map becomes a model for the inclusion BO(n−1)/arrowhookleft→BO(n). Now for V∈Grn−1(RN) consider the vector (0 ,1)∈V×R⊂RN×R. This is a unit vector, and so is an element of the fiber of the unit sphere bundle S(γn) overV×R. Hence this association defines a map j:Grn−1(RN)→S(γn) which liftsι:Grn−1(RN)/arrowhookleft→Grn(RN+1). By taking a limit over Nwe get a map j:BO(n−1)→ S(γn). To define a homotopy inverse ρ:S(γn)→BO(n−1), we again work on the finite Grassmannian level. Let (W,w)∈S(γn), the unit sphere bundle over Grn(RK). ThusW⊂RKis ann-dimensional subspace and w∈Wis a unit vector. Let Ww⊂Wdenote the orthogonal complement to the vector winW. ThusWw⊂W⊂RKis ann−1 - dimensional subspace. This association defines a map ρ:S(γn)→Grn−1(RK) and by taking the limit over K, defines a map ρ:S(γn)→BO(n−1). We leave it to the reader to verify that j:BO(n−1)→S(γn) andρ:S(γn)→BO(n−1) are homotopy inverse to each other. 5. SOME APPLICATIONS 77 5.3. Embedded bundles and K-theory. The classification theorem for vector bundles says that for every n- dimensional complex vector bundle ζoverX, there is a classifying map fζ:X→BU(n) so thatζis isomorphic to pull back, f∗(γn) of the universal vector bundle. A similar statement holds for real vector bundles. Using the Grassmannian models for these classifying spaces, we obtain the following as a corollary. Theorem 2.29.Everyn- dimensional complex bundle ζover a space Xcan be embedded in a trivial infinite dimensional bundle, X×C∞. Similarly, every n- dimensional real bundle ηoverX can be embedded in the trivial bundle X×R∞. Proof. Letfζ:X→Grn(C∞) =BU(n) classifyζ. Soζ∼=f∗(γn). But recall that γn={(V,v)∈Grn(C∞)×C∞such thatv∈V.} Henceγnis naturally embedded in the trivial bundle Grn(C∞)×C∞. Thusζ∼=f∗(γn) is naturally embedded in X×C∞. The real case is proved similarly. /square Notice that because of the direct limit topology on Grn(C∞) = lim−→Grn(CN), then ifXis a compact space, any map f:X→Grn(C∞) has image that lies in Grn(CN) for some finite N. But notice that over this finite Grassmannian, γn⊂Grn(CN)×CN. The following is then an immediate corollary. This result was used in chapter one in our discussion about K-theory. Corollary 2.30.IfXis compact, then every n- dimensional complex bundle zeta can be embedded in a trivial bundle X×CNfor someN. The analogous result also holds for real vector bundles. Letf:X→BU(n) classify the n- dimensional complex vector bundle ζ. Then clearly the composition f:X→BU(n)/arrowhookleft→BU(n+ 1) classifies the n+ 1 dimensional vector bundle ζ⊕/epsilon11, where as before, /epsilon11is the one dimensional trivial line bundle. This observation leads to the following. Proposition 2.31.Letζ1andζ2be twon-dimensional vector bundles over Xclassified by f1andf2:X→BU(n)respectively. Then if we add trivial bundles, we get an isomorphism ζ1⊕/epsilon1k∼=ζ2⊕/epsilon1k if and only if the compositions, f1, f2:X→BU(n)/arrowhookleft→BU(n+k) are homotopic. 78 2. CLASSIFICATION OF BUNDLES Now recall from the discussion of K- theory in chapter 1 that the set of stable isomorphism classes of vector bundles SVect(X) is isomorphic to the reduced K- theory, ˜K(X), whenXis compact. This proposition then implies the following important result, which displays how in the case of compact spaces, computing K-theory reduces to a specific homotopy theory calculation. Definition 2.10.LetBUbe the limit of the spaces BU= lim−→nBU(n). Similarly, BO= lim−→nBO(n). Theorem 2.32.ForXcompact there are isomorphisms (bijective correspondences) ˜K(X)∼=SVect(X)∼=[X,BU ] and ˜KO(X)∼=SVect R(X)∼=[X,BO ]. 5.4. Representations and flat connections. Recall the following classification theorem for covering spaces. Theorem 2.33.. LetXbe a connected space. Then the set of isomorphism classes of connected covering spaces, p:E→Xis in bijective correspondence with conjugacy classes of normal subgroups ofπ1(X). This correspondence sends a covering p:E→Bto the image p∗(π1(E))⊂π1(X). Letπ=π1(X) and letp:E→Xbe a connected covering space with π1(E) =N/triangleleftπ. Then the group of deck transformations of Eis the quotient group π/N, and so can be thought of as a principalπ/N - bundle. Viewed this way it is classified by a map fE:X→B(π/N), which on the level of fundamental groups, f∗:π=π1(X)→π1(Bπ/N ) =π/N (5.2) is just the projection on to the quotient space. In particular the universal cover ˜X→Xis the unique simply connected covering space. It is classified by a map γX:X→Bπ which induces an isomorphism on the fundamental group. 5. SOME APPLICATIONS 79 Now letθ:π→Gbe any group homomorphism. By the naturality of classifying spaces this induces a map on classifying spaces, Bθ:Bπ→BG. This induces a principal G- bundle over Xclassified by the composition XγX−−−−→BπBθ−−−−→BG. The bundle this map classifies is given by ˜X×πG→X whereπacts onGvia the homomorphism θ:π→G. This construction defines a map ρ:Hom (π1(X),G)→PrinG(X). Now ifXis a smooth manifold then its universal cover p:˜X→Xinduces an isomorphism on tangent spaces, Dp(x) :Tx˜X→Tp(x)X for everyx∈˜X. Thus, viewed as a principal π- bundle, it has a canonical connection. Notice furthermore that this connection is flat, i.e its curvature is zero. ( Exercise. Check this claim!) Moreover notice that any bundle of the form ˜X×πG→Xhas an induced flat connection. In particular the image of ρ:Hom (π1(X),G)→PrinG(X) consists of principal bundles equipped with flat connections. Notice furthermore that by taking G=GL(n,C) the map ρassigns to an n- dimensional representation an n- dimensional vector bundle with flat connection ρ:Repn(π1(X))→Vectn(X). By taking the sum over all nand passing to the Grothendieck group completion,we get a homomor- phism of rings from the representation ring to K- theory, ρ:R(π1(X))→K(X). An important question is what is the image of this map of rings. Again we know the image is contained in the classes represented by bundles that have flat connections. For X=Bπ, forπa finite group, the following is a famous theorem of Atiyah and Segal: Let /epsilon1:R(π)→Zand/epsilon1:K(Bπ)→Z be the augmentation maps induced by sending a representation or a vector bundle to its dimension. LetI⊂R(π) andI⊂K(Bπ) denote the kernels of these augmentations, i.e the “augmentation 80 2. CLASSIFICATION OF BUNDLES ideals”. Finally let ¯R(π) and ¯K(Bπ) denote the completions of these rings with respect to these ideals. That is, ¯R(π) = lim←−nR(π)/Inand ¯K(Bπ) = lim←−nK(Bπ)/In whereInis the product of the ideal Iwith itselfn- times. Theorem 2.34.(Atiyah and Segal) [4]Forπa finite group, the induced map on the completions of the rings with respect ot the augmentation ideals, ρ:¯R(π)→¯K(Bπ) is an isomorphism. CHAPTER 3 Characteristic Classes In this chapter we define and calculate characteristic classes for principal bundles and vector bundles. Characteristic classes are the basic cohomological invariants of bundles and have a wide variety of applications throughout topology and geometry. Characteristic classes were introduced originally by E. Stiefel in Switzerland and H. Whitney in the United States in the mid 1930’s. Stiefel, who was a student of H. Hopf intoduced in his thesis certain “characteristic homology classes” determined by the tangent bundle of a manifold. At about the same time Whitney studied general sphere bundles, and later introduced the general notion of a characteristic cohomology class coming from a vector bundle, and proved the product formula for their calculation. In the early 1940’s, L. Pontrjagin, in Moscow, introduced new characteristic classes by studying the Grassmannian manifolds, using work of C. Ehresmann from Switzerland. In the mid 1940’s, after just arriving in Princeton from China, S.S Chern defined characteristic classes for complex vector bundles using differential forms and his calculations led a great clarification of the theory. Much of the modern view of characteristic classes has been greatly influenced by the highly influential book of Milnor and Stasheff. This book was originally circulated as lecture notes written in 1957 and finally published in 1974. This book is one of the great textbooks in modern mathematics. These notes follow, in large part, their treatment of the subject. The reader is encouraged to consult their book for further details. 1. Preliminaries Definition 3.1.LetGbe a topological group (possibly with the discrete topology). Then a characteristic class for principal G- bundles is an assignment to each principal G- bundlep:P→B a cohomology class c(P)∈H∗(B) satisfying the following naturality condition. If P1¯f−−−−→P2 p1/arrowbt/arrowbtp2 B1−−−−→ fB2 81 82 3. CHARACTERISTIC CLASSES is a map of principal G- bundles inducing an equivariant homeomorphism on fibers, then f∗(c(P2)) =c(P1)∈H∗(B1). Remarks. 1. In this definition cohomology could be taken with any coefficients, including, for example, DeRham cohomology which has coefficients in the real numbers R. The particular cohomology theory used is referred to as the “values” of the characteristic classes. 2.The same definition of characteristic classes applies to real or complex vector bundles as well as principal bundles. The following is an easy consequence of the definition. Lemma 3.1.Letcbe a characteristic class for principal G- bundles so that ctakes values in Hq(−), forq≥1. Then if/epsilon1is the trivial Gbundle, /epsilon1=X×G→X thenc(/epsilon1) = 0 . Proof. The trivial bundle /epsilon1is the pull - back of the constant map to the one point space e:X→ptof the bundle ν=G→pt. Thusc(/epsilon1) =e∗(c(ν)). Butc(ν)∈Hq(pt) = 0 when q>0./square The following observation is also immediate from the definition. Lemma 3.2.Characteristic classes are invariant under isomorphism. More specifically, Let c be a characteristic class for principal G- bundles. Also let p1:E1→Xandp2:E2→Xbe isomorphic principal G- bundles. Then c(E1) =c(E2)∈H∗(X). Thus for a given space X, a characteristic class ccan be viewed as a map c:PrinG(X)→H∗(X). 3.The naturality property in the definition can be stated in more functorial terms in the following way. Cohomology (with any coefficients) H∗(−) is a contravariant functor from the category hoTop of topological spaces and homotopy classes of maps, to the category Abof abelian groups. By the results of chapter 2, the set of principal G- bundlesPrinG(−) can be viewed as a contravariant functor from the category hoTopto the category of sets Sets. 1. PRELIMINARIES 83 Definition 3.2.(Alternative) A characteristic class is a natural transformation cbetween the functors PrinG(−) andH∗(−): c:PrinG(−)/squigglerightH∗(−) Examples. (1) The first Chern class c(ζ) is a characteristic class on principal U(1) - bundles, or equivalently, complex line bundles. If ζis a line bundle over X, thenc1(ζ)∈H2(X;Z). As we saw in the last chapter, c1is a complete invariant of line bundles. That is to say, the map c1:PrinU(1)(X)→H2(X;Z) is an isomorphism. (2) The first Stiefel - Whitney class w1(η) is a characteristic class of two fold covering spaces (i.e a principal Z2=O(1) - bundles) or of real line bundles. If ηis a real line bundle over a spaceX, thenw1(η)∈H1(X;Z2). Moreover, as we saw in the last chapter, the first Stiefel - Whitney class is a complete invariant of line bundles. That is, the map w1:PrinO(1)(X)→H1(X;Z2) is an isomorphism. We remark that the first Stiefel - Whitney class can be extended to be a characteristic class of real n- dimensional vector bundles (or principal O(n) - bundles) for any n. To see this, consider the subgroupSO(n)< O(n). As we saw in the last chapter, a bundle has an SO(n) structure if and only if it is orientable. Moreover the induced map of classifying spaces gives a 2 - fold covering space or principal O(1) - bundle, Z2=O(1) =O(n)/SO(n)→BSO (n)→BO(n). This covering space defines, via its classifying map w1:BO(n)→BO(1) = RP∞an element w1∈H1(BO(n);Z2) which is the first Stiefel - Whitney class of this covering space. Now letηbe anyn- dimensional real vector bundle over X, and let fη:X→BO(n) be its classifying map. Definition 3.3.The first Stiefel - Whitney class w1(η)∈H1(X;Z2) is defined to be w1(η) =f∗ η(w1)∈H1(X;Z2) The first Chern class c1of ann- dimensional complex vector bundle ζoverXis defined similarly, by pulling back the first Chern class of the principal U(1) - bundle U(1)∼=U(n)/SU(n)→BSU (n)→BU(n) 84 3. CHARACTERISTIC CLASSES via the classifying map fζ:X→BU(n). The following is an immediate consequence of the above lemma and the meaning of SO(n) and SU(n) - structures. Theorem 3.3.Given a complex n- dimensional vector bundle ζoverX, thenc1(ζ)∈H2(X) is zero if and only if ζhas anSU(n)-structure. Furthermore, given a real n- dimensional vector bundle ηoverX, thenw1(η)∈H1(X;Z2)is zero if and only if the bundle ηhas anSO(n)- structure, which is equivalent to ηbeing orientable. We now use the classification theorem for bundles to describe the set of characteristic classes for principal G- bundles. LetRbe a commutative ring and let CharG(R) be the set of all characteristic classes for principal Gbundles that take values in H∗(−;R). Notice that the sum (in cohomology) and the cup product of characteristic classes is again a characteristic class. This gives CharGthe structure of a ring. (Notice that the unit in this ring is the constant characteristic class c(ζ) = 1∈H0(X). Theorem 3.4.There is an isomorphism of rings ρ:CharG(R)∼=−−−−→H∗(BG;R) Proof. Letc∈CharG(R). Define ρ(c) =c(EG)∈H∗(BG;R) whereEG→BGis the universal G- bundle over BG. By definition of the ring structure of CharG(R),ρis a ring homomorphism. Now letγ∈Hq(BG;R). Define the characteristic class cγas follows. Let p:E→Xbe a principalG- bundle classified by a map fE:X→BG. Define cγ(E) =f∗ E(γ)∈Hq(X;R) wheref∗ E:H∗(BG:R)→H∗(X;R) is the cohomology ring homomorphism induced by fE. This association defines a map c:H∗(BG;R)→CharG(R) which immediately seen to be inverse to ρ. /square 2. CHERN CLASSES AND STIEFEL - WHITNEY CLASSES 85 2. Chern Classes and Stiefel - Whitney Classes In this section we compute the rings of unitary characteristic classes CharU(n)(Z) andZ2- valued orthogonal characteristic classes CharO(n)(Z2). These are the characteristic classes of complex and real vector bundles and as such have a great number of applications. By 3.4 computing these rings of characteristic classes reduce to computing the cohomology rings H∗(BU(n);Z) andH∗(BO(n);Z2). The following is the main theorem of this section. Theorem 3.5.a. The ring of U(n)characteristic classes is a polynomial algebra on n- gener- ators, CharU(n)(Z)∼=H∗(BU(n);Z)∼=Z[c1,c2,···,cn] whereci∈H2i(BU(n);Z)is known as the ith- Chern class. b.The ring of Z2- valuedO(n)characteristic classes is a polynomial algebra on n- generators, CharO(n)(Z2)∼=H∗(BO(n);Z2)∼=Z2[w1,w2,···,wn] wherewi∈Hi(BO(n);Z2)is known as the ith- Stiefel - Whitney class. This theorem will be proven by induction on n. Forn= 1BU(1) = CP∞andBO(1) = RP∞ and so the theorem describes the ring structure in the cohomology of these projective spaces. To complete the inductive step we will study the sphere bundles Sn−1→BO(n−1)→BO(n) and S2n−1→BU(n−1)→BU(n) described in the last chapter. In particular recall from 2.28 that in these fibrations, BO(n−1) and BU(n−1) are the unit sphere bundles S(γn) of the universal bundle γnoverBO(n) andBU(n) respectively. Let D(γn) be the unit disk bundles of the universal bundles. That is, in the complex case, D(γn) =EU(n)×U(n)D2n→BU(n) and in the real case, D(γn) =EO(n)×O(n)Dn→BO(n) whereD2n⊂CnandDn⊂Rnare the unit disks, and therefore have the induced unitary and orthogonal group actions. Here is one easy observation about these disk bundles. 86 3. CHARACTERISTIC CLASSES Proposition 3.6.The projection maps p:D(γn) =EU(n)×U(n)D2n→BU(n) and D(γn) =EO(n)×O(n)Dn→BO(n) are homotopy equivalences. Proof. Both of these bundles have zero sections Z:BU(n)→D(γn) andZ:BO(n)→D(γn). In both the complex and real cases, we have p◦Z= 1. To see thatZ◦p/similarequal1 consider the homotopy H:D(γn)×I→D(γn) defined by H(v,t) =tv. /square We will use this result when studying the cohomology exact sequence of the pair ( D(γn),S(γn)): (2.1)···→Hq−1(S(γn))δ−−−−→Hq(D(γn),S(γn))→Hq(D(γn))→Hq(S(γn)) δ−−−−→Hq+1(D(γn),S(γn))→Hq+1(D(γn))→··· Using the above proposition and 2.28 we can substitute H∗(BU(n)) forH∗(D(γn)), andH∗(BU(n− 1)) forH∗(S(γn)) in this sequence to get the following exact sequence (2.2) ···→Hq−1(BU(n−1))δ−−−−→ Hq(D(γn),S(γn))→Hq(BU(n))ι∗ −−−−→Hq(BU(n−1)) δ−−−−→Hq+1(D(γn),S(γn))→Hq+1(BU(n))→··· and we get a similar exact sequence in the real case (2.3) ···→Hq−1(BO(n−1);Z2)δ−−−−→ Hq(D(γn),S(γn);Z2)→Hq(BO(n);Z2)ι∗ −−−−→ Hq(BO(n−1);Z2)δ−−−−→Hq+1(D(γn),S(γn);Z2)→Hq+1(BO(n);Z2)→··· These exact sequences will be quite useful for inductively computing the cohomology of these classifying spaces, but to do so we need a method for computing H∗(D(γn),S(γn)), or more generally, H∗(D(ζ),S(ζ)), whereζis any Euclidean vector bundle and D(ζ) andS(ζ) are the associated unit disk bundles and sphere bundles respectively. The quotient space, (2.4) T(ζ) =D(ζ)/S(ζ) 2. CHERN CLASSES AND STIEFEL - WHITNEY CLASSES 87 is called the Thom space of the bundle ζ. As the name suggests, this construction was first studied by R. Thom [ 41], and has been quite useful in both bundle theory and cobordism theory. Notice that on each fiber (say at x∈X) of then- dimensional disk bundle ζ, the Thom space construction takes the unit n- dimensional disk modulo its boundary n−1 - dimensional sphere which therefore yields ann- dimensional sphere, with marked basepoint, say ∞x∈Sn(ζx) =Dn(ζx)/Sn−1(ζx). The Thom space construction then identifies all the basepoints ∞xto a single point. Notice that for a bundle over a point Rn→pt, the Thom space T(Rn) =Dn/Sn−1=Sn∼=Rn∪∞. More generally, notice that when the basespace Xis compact, then the Thom space is simply the one point compactification of the total space of the vector bundle ζ, (2.5) T(ζ)∼=ζ+=ζ∪∞ where we think of the extra point in this compactification as the common point at infinity assigned to each fiber. In order to compute with the above exact sequences, we will need to study the cohomology of Thom spaces. But before we do we examine the topology of the Thom spaces of product bundles. For this we introduce the “smash product” construction. LetXandYbe spaces with basepoints x0∈Xandy0∈Y. Definition 3.4.The wedgeX∨Yis the “one point union”, X∨Y=X×y0∪x0×Y⊂X×Y. The smash product X∧Yis given by X∧Y=X×Y/X∨Y. Observations. 1. Thekbe a field. Then the Kunneth formula gives ˜H∗(X∧Y;k)∼=˜H∗(X;k)⊗˜H∗(Y;k). 2. LetVandWbe vector spaces, and let V+andW+be their one point compactifications. These are spheres of the same dimension as the respective vector spaces. Then V+∧W+= (V×W)+. So in particular, Sn∧Sm=Sn+m. Proposition 3.7.Letζbe ann- dimensional vector bundle over a space X, and letηbe an m- dimensional bundle over X. Letζ×ηbe the product n+m- dimensional vector bundle over X∧Y. Then the Thom space of ζ×ηis given by T(ζ×η)∼=T(ζ)∧T(η). 88 3. CHARACTERISTIC CLASSES Proof. Notice that the disk bundle is given by D(ζ×η)∼=D(ζ)×D(η) and its boundary sphere bundle is given by S(ζ×η)∼=S(ζ)×D(η)∪D(ζ)×S(η). Thus T(ζ×η) =D(ζ×η)/S(ζ×η)∼=D(ζ)×D(η)/(S(ζ)×D(η)∪D(ζ)×S(η)) ∼=D(ζ)/S(ζ)∧D(η)/S(η) ∼=T(ζ)∧T(η). /square We now proceed to study the cohomology of Thom spaces. 2.1. The Thom Isomorphism Theorem. We begin by describing a cohomological notion of orientability of an vector bundle ζover a space X. Consider the 2 - fold cover over Xdefined as follows. Let Eζbe the principal GL(n,R) bundle associated to ζ. Also letGennbe the set of generators of Hn(Sn)∼=Z. SoGennis a set with two elements. Moreover the general linear group GL(n,R) acts onSn=Rn∪∞ by the usual linear action on Rnextended to have a fixed point at ∞∈Sn. By looking at the induced map on cohomology, there is an action of GL(n,R) onGenn. We can then define the double cover G(ζ) =Eζ×GL(n,R)Genn−→Eζ/GL(n,R) =X. Lemma 3.8.The double covering G(ζ)is isomorphic to the orientation double cover Or(ζ). Proof. Recall from chapter 1 that the orientation double cover Or(ζ) is given by Or(ζ) =Eζ×GL(n,R)Or(Rn) whereOr(Rn) is the two point set consistingof orientations of the vector space Rn. A matrix A∈GL(n,R) acts on this set trivially if and only if the determinant detA is positive. It acts nontrivially (i.e permutes the two elements) if and only if detA is negative. Now the same is true of the action of GL(n,R) onGenn. This is because A∈GL(n,R) induces multiplication by the sign ofdetA onHn(Sn). (Verify this as an exercise!) SinceOr(Rn) andGennare both two point sets with the same action of GL(n,R), the corre- sponding two fold covering spaces Or(ζ) andG(ζ) are isomorphic. /square 2. CHERN CLASSES AND STIEFEL - WHITNEY CLASSES 89 Corollary 3.9.An orientation of an n- dimensional vector bundle ζis equivalent to a section ofG(ζ)and hence defines a continuous family of generators ux∈Hn(Sn(ζx))∼=Z for everyx∈X. HereSn(ζx)is the unit disk of the fiber ζxmodulo its boundary sphere. Sn(ζx)is called the sphere atx. Now recall that given a pair of spaces A⊂Y, there is a relative cup product in cohomology, Hq(Y)⊗Hr(Y,A)∪−−−−→Hq+r(Y,A). So in particular the relative cohomology H∗((Y,A) is a (graded) module over the (graded) ring H∗(Y). In the case of a vector bundle ζover a space X, we then have that H∗(D(ζ),S(ζ)) = ˜H∗(T(ζ)) is a module over H∗(D(ζ))∼=H∗(X).So in particular, given any cohomology class in the Thom space,α∈Hr(T(ζ)) we get an induced homomorphism Hq(X)∪α−−−−→Hq+r(T(ζ)). Our next goal is to prove the famous Thom Isomorphism Theorem which can be stated as follows. Theorem 3.10.Letζbe an oriented n- dimensional real vector bundle over a connected space X. LetRbe any commutative ring. The orientation gives generators ux∈Hn(Sn(ζx);R)∼=R. Then there is a unique class (called the Thom class) in the cohomology of the Thom space u∈Hn(T(ζ);R) so that for every x∈X, if jx:Sn(ζx)/arrowhookleft→D(ζ)/S(ζ) =T(ζ) is the natural inclusion of the sphere at xin the Thom space, then under the induced homomorphism in cohomology, j∗ x:Hn(T(ζ);R)→Hn(Sn(ζx);R)∼=R j∗ x(u) =ux. Furthermore The induced cup product map γ:Hq(X;R)∪u−−−−→ ˜Hq+n(T(ζ);R) is an isomorphism for every q∈Z. So in particular ˜Hr(T(ζ);R) = 0 forr<n . Ifζis not an orientable bundle over X, then the theorem remains true if we take Z2coefficients, R=Z2. 90 3. CHARACTERISTIC CLASSES Proof. We prove the theorem for oriented bundles. We leave the nonorientable case (when R=Z2) to the reader. We also restrict our attention to the case R=Z, since the theorem for general coefficients will follow immediately from this case using the universal coefficient theorem. Case 1:ζis the trivial bundle X×Rn. In this case the Thom space T(ζ) is given by T(ζ) =X×Dn/X×Sn−1. The projection of Xto a point, X→ptdefines a map π:T(ζ) =X×Dn/X×Sn−1→Dn/Sn−1=Sn. Letu∈Hn(T(ζ)) be the image in cohomology of a generator, Z∼=Hn(Sn)π∗ −−−−→Hn(T(ζ)). The fact that taking the cup product with this class Hq(X)∪u−−−−→Hq+n(T(ζ)) =Hq+n(X×Dn,X×Sn−1) =Hq+n(X×Sn,X×pt) is an isomorphism for every q∈Zfollows from the universal coefficient theorem. Case 2:Xis the union of two open sets X=X1∪X2, where we know the Thom isomorphism theorem holds for the restrictions ζi=ζ|Xifori= 1,2 and forζ1,2=ζ|X1∩X2. We prove the theorem for Xusing the Mayer - Vietoris sequence for cohomology. Let X1,2= X1∩X2. →Hq−1(T(ζ1,2))→Hq(T(ζ))→Hq(T(ζ1))⊕Hq(T(ζ2))→Hq((T(ζ1,2))→··· Looking at this sequence when q<n , we see that since Hq(T(ζ1,2)) =Hq(T(ζ1)) =Hq(T(ζ2)) = 0, then by exactness we must have that Hq(T(ζ)) = 0. We now let q=n, and we see that by assumption, Hn(T(ζ1))∼=Hn(T(ζ2))∼=Hn(T(ζ1,2))∼= Z, and that the Thom classes of each of the restriction maps Hn(T(ζ1))→Hn(T(ζ1,2)) and Hn(T(ζ2))→Hn(T(ζ1,2)) correspond. Moreover Hn−1(T(ζ1,2)) = 0. Hence by the exact sequence, Hn(T(ζ))∼=Zand there is a class u∈Hn(T(ζ)) that maps to the direct sum of the Thom classes inHn(T(ζ1))⊕Hn(T(ζ2)). Now forq≥nwe compare the above Mayer - Vietoris sequence with the one of base spaces, →Hq−1(X1,2)→Hq(X)→Hq(X1)⊕Hq(X2)→Hq(X1,2)→··· This sequence maps to the one for Thom spaces by taking the cup product with the Thom classes. By assumption this map is an isomorphism on H∗(Xi),i= 1,2 and onH∗(X1,2). Thus by the Five Lemma it is an isomorphism on H∗(X). This proves the theorem in this case. 2. CHERN CLASSES AND STIEFEL - WHITNEY CLASSES 91 Case 3.Xis covered by finitely many open sets Xi,i= 1,···,kso that the restrictions of the bundle to each Xi,ζiis trivial. The proof in this case is an easy inductive argument (on the number of open sets in the cover), where the inductive step is completed using cases 1 and 2. Notice that this case includes the situation when the basespace Xis compact. Case 4. General Case. We now know the theorem for compact spaces. However it is not necessarily true that the cohomology of a general space (i.e homotopy type of a C.W complex) is determined by the cohomology of its compact subspaces. However it is true that the homology of a spaceXis given by H∗(X)∼=lim−→KH∗(K) where the limit is taken over the partially ordered set of compact subspaces K⊂X.Thus we want to first work in homology and then try to transfer our observations to cohomology. To do this, recall that the construction of the cup product pairing actually comes from a map on the level of cochains, Cq(Y)⊗Cr(Y,A)∪−−−−→Cq+r(Y,A) and therefore has a dual map on the chain level C∗(Y,A)ψ−−−−→C∗(Y)⊗C∗(Y,A). and thus induces a map in homology ψ:Hk(Y,A)→⊕r≥0Hk−r(Y)⊗Hr(Y,A). Hence given α∈Hr(Y,A) we have an induced map in homology (the “slant product”) /α:Hk(Y,A)→Hk−r(Y) defined as follows. If θ∈Hk(Y,A) and ψ(θ) =/summationdisplay jaj⊗bj∈H∗(Y)⊗H∗(Y,A) then /α(θ) =/summationdisplay jα(bj)·aj where by convention, if the degree of a homology class bjis not equal to the degree of α, then α(bj) = 0. Notice that this slant product is dual to the cup product map Hq(Y)∪α−−−−→Hq+r(Y,A). 92 3. CHARACTERISTIC CLASSES Again, by considering the pair ( D(ζ),S(ζ)), and identifying H∗(D(ζ))∼=H∗(X), we can apply the slant product operation to the Thom class, to define a map /u:Hk(T(ζ))→Hk−n(X). which is dual to the Thom map γ:Hq(X)∪u−−−−→Hq+n(T(ζ)).Now since γis an isomorphism in all dimensions when restricted to compact sets, then by the universal coefficient theorem, /u: Hq(T(ζ|K))→Hq−n(K) is an isomorphism for all qand for every compact subset K⊂X. By taking the limit over the partially ordered set of compact subsets of X, we get that /u:Hq(T(ζ))→Hq−n(X) is an isomorphism for all q. Applying the universal coefficient theorem again, we can now conclude that γ:Hk(X)∪u−−−−→Hk+n(T(ζ)) is an isomorphism for all k. This completes the proof of the theorem. /square We now observe that the Thom class of a product of two bundles is the appropriately defined product of the Thom classes. Lemma 3.11.Letζandηbe annandmdimensional oriented vector bundles over XandY respectively. Then the Thom class u(ζ×η)is given by the tensor product: u(ζ×η)∈Hn+m(T(ζ×η)) is equal to u(ζ)⊗u(η)∈Hn(T(ζ))⊗Hm(T(η)) ∼=Hn+m(T(ζ)∧T(η)) =Hn+m(T(ζ×η)). In this description, cohomology is meant to be taken with Z2- coefficients if the bundles are not orientable. Proof.u(ζ)⊗u(η) restricts on each fiber ( x,y)∈X×Yto ux⊗uy∈Hn(Sn(ζx))⊗Hm(Sm(ηy)) ∼=Hn+m(Sn(ζx)∧Sm(ηy)) =Hn+m(Sn+m(ζ×η)(x,y))) which is the generator determined by the product orientation of ζx×ηy. The result follows by the uniqueness of the Thom class. /square We now use the Thom isomorphism theorem to define a characteristic class for oriented vector bundles, called the Euler class. 2. CHERN CLASSES AND STIEFEL - WHITNEY CLASSES 93 Definition 3.5.The Euler class of an oriented, ndimensional bundle ζ, over a connected space X, is then- dimensional cohomology class χ(ζ)∈Hn(X) defined to be the image of the Thom class u(ζ)∈Hn(T(ζ)) under the composition Hn(T(ζ)) =Hn(D(ζ),S(ζ))→Hn(D(ζ))∼=Hn(X). Again, ifζis not orientable, cohomology is taken with Z2- coefficients. Exercise. Verify that the Euler class is a characteristic class according to our definition. The following is then a direct consequence of 3.11. Corollary 3.12.Letζandηbe as in 3.11. Then the Euler class of the product is given by χ(ζ×η) =χ(ζ)⊗χ(η)∈Hn(X)⊗Hm(Y)/arrowhookleft→Hn+m(X×Y). We will also need the following observation. Proposition 3.13.Letηbe an odd dimensional oriented vector bundle over a space X. Say dim(η) = 2n+ 1.Then its Euler class has order two: 2χ(η) = 0∈H2n+1(X). Proof. Consider the bundle map ν:η→η v→−v. Sinceηis odd dimensional, this bundle map is an orientation reversing automorphism of η. This means that ν∗(u) =−u, whereu∈H2n+1(T(η)) is the Thom class. By the definition of the Euler class this in turn implies that ν∗(χ(η)) =−χ(η). But since the Euler class is a characteristic class andνis a bundle map, we must have ν∗(χ(η)) =χ(η). Thusχ(η) =−χ(η). /square 94 3. CHARACTERISTIC CLASSES 2.2. The Gysin sequence. We now input the Thom isomorphism theorem into the cohomol- ogy exact sequence of the pair D(ζ),S(ζ)) in order to obtain an important calculational tool for computing the homology of vector bundles and sphere bundles. Namely, let ζbe an oriented n- dimensional oriented vector bundle over a space X, and consider the exact sequence ···→Hq−1(S(ζ))δ−−−−→Hq(D(ζ),S(ζ))→Hq(D(ζ))→Hq(S(ζ)) δ−−−−→Hq+1(D(ζ),S(ζ))→Hq+1(D(ζ))→··· By identifying H∗(D(ζ),S(ζ)) = ˜H∗(T(ζ)) andH∗(D(ζ))∼=H∗(X), this exact sequence becomes ···→Hq−1(S(ζ))δ−−−−→Hq(T(ζ))→Hq(X)→Hq(S(ζ)) δ−−−−→Hq+1(T(ζ))→Hq+1(X)→··· Finally, by inputting the Thom isomorphism, Hq−n(X)∪u−−−−→∼=Hq(T(ζ))we get the following exact sequence known as the Gysin sequence: (2.6)···→Hq−1(S(ζ))δ−−−−→Hq−n(X)χ−−−−→Hq(X)→Hq(S(ζ)) δ−−−−→Hq−n+1(X)χ−−−−→Hq+1(X)→··· We now make the following observation about the homomorphism χ:Hq(X)→Hq+n(X) in the Gysin sequence. Proposition 3.14.The homomorphism χ:Hq(X)→Hq+n(X)is given by taking the cup product with the Euler class, χ:Hq(X)∪χ−−−−→Hq+n(X). Proof. The theorem is true for q= 0, by definition. Now in general, the map χwas de- fined in terms of the Thom isomorphism γ:Hr(X)∪u−−−−→Hr+n(T(ζ)), which, by definition is a homomorphism of graded H∗(X) - modules. This will then imply that χ:Hq(X)→Hq+n(X) is a homomorphism of graded H∗(X) - modules. Thus χ(α) =χ(1·α) =χ(1)∪αsinceχis anH∗(X) - module homomorphism =χ(ζ)∪α as claimed. /square 2. CHERN CLASSES AND STIEFEL - WHITNEY CLASSES 95 2.3. Proof of theorem 3.5. the goal of this section is to use the Gysin sequence to prove 3.5, which we begin by restating: Theorem 3.15.a. The ring of U(n)characteristic classes is a polynomial algebra on n- generators, CharU(n)(Z)∼=H∗(BU(n);Z)∼=Z[c1,c2,···,cn] whereci∈H2i(BU(n);Z)is known as the ith- Chern class. b.The ring of Z2- valuedO(n)characteristic classes is a polynomial algebra on n- generators, CharO(n)(Z2)∼=H∗(BO(n);Z2)∼=Z2[w1,w2,···,wn] wherewi∈Hi(BO(n);Z2)is known as the ith- Stiefel - Whitney class. Proof. We start by considering the Gysin sequence, applied to the universal bundle γnover BU(n). We input the fact that the sphere bundle S(γn) is given by BU(n−1) see 2.2: (2.7) ···→Hq−1(BU(n−1))δ−−−−→Hq−2n(BU(n))∪χ(γn)−−−−−→ Hq(BU(n))ι∗ −−−−→Hq(BU(n−1)) δ−−−−→Hq−2n+1(BU(n))∪χ(γn)−−−−−→Hq+1(BU(n))→··· and we get a similar exact sequence in the real case (2.8) ···→Hq−1(BO(n−1);Z2)δ−−−−→Hq−n(BO(n);Z2)∪χ(γn)−−−−−→ Hq(BO(n);Z2)ι∗ −−−−→ Hq(BO(n−1);Z2)δ−−−−→Hq−n+1(BO(n));Z2)∪χ(γn)−−−−−→Hq+1(BO(n);Z2)→··· We use these exact sequences to prove the above theorem by induction on n. Forn= 1 then sequence 2.7 reduces to the short exact sequences, 0→Hq−2(BU(1))∪χ(γ1)−−−−→∼=Hq(BU(1))→0 for eachq≥2. We letc1∈H2(BU(1)) =H2(CP∞) be the Euler class χ(γ1). These isomorphisms imply that the ring structure of H∗(BU(1)) is that of a polynomial algebra on this single generator, H∗(BU(1)) =H∗(CP∞) =Z[c1] which is the statement of the theorem in this case. In the real case when n= 1 the Gysin sequence 2.8 reduces to the short exact sequences, 0→Hq−1(BO(1);Z2)∪χ(γ1)−−−−→∼=Hq(BO(1);Z2)→0 96 3. CHARACTERISTIC CLASSES for eachq≥1. We let w1∈H(BO(1);Z2) =H1(RP∞;Z2) be the Euler class χ(γ1). These isomorphisms imply that the ring structure of H∗(BO(1);Z2) is that of a polynomial algebra on this single generator, H∗(BO(1);Z2) =H∗(RP∞;Z2) =Z2[w1] which is the statement of the theorem in this case. We now inductively assume the theorem is true for n−1. That is, H∗(BU(n−1))∼=Z[c1,···,cn−1] andH∗(BO(n−1);Z2)∼=Z2[w1,···,wn−1]. We first consider the Gysin sequence 2.7, and observe that by exactness, for q≤2(n−1), the homomorphism ι∗:Hq(BU(n))→Hq(BU(n−1)) is an isomorphism. That means there are unique classes, c1,···,cn−1∈H∗(BU(n)) that map via ι∗ to the classes of the same name in H∗(BU(n−1)). Furthermore, since ι∗is a ring homomorphism, every polynomial in c1,···,cn−1inH∗(BU(n−1)) is in the image under ι∗of the corresponding polynomial in the these classes in H∗(BU(n)). Hence by our inductive assumption, ι∗:H∗(BU(n))→H∗(BU(n−1)) =Z[c1,···,cn−1] is a split surjection of rings. But by the exactness of the Gysin sequence 2.7 this implies that this long exact splits into short exact sequences, 0→H∗−2n(BU(n))∪χ(γn)−−−−−→H∗(BU(n))ι∗ −−−−→H∗(BU(n−1))∼=Z[c1,···cn−1]→0 Definecn∈H2n(BU(n)) to be the Euler class χ(γn). Then this sequence becomes 0→H∗−2n(BU(n))∪cn−−−−→H∗(BU(n))ι∗ −−−−→ Z[c1,···cn−1]→0 which implies that H∗(BU(n))∼=Z[c1,···,cn]. This completes the inductive step in this case. In the real case now consider the Gysin sequence 2.8, and observe that by exactness, for q<n−1, the homomorphism ι∗:Hq(BO(n);Z2)→Hq(BO(n−1);Z2) is an isomorphism. That means there are unique classes, w1,···,wn−2∈H∗(BO(n);Z2) that map viaι∗to the classes of the same name in H∗(BO(n−1);Z2). In dimension q=n−1, the exactness of the Gysin sequence tells us that the homomorphism ι∗Hn−1(BO(n);Z2)→Hn−1(BO(n−1);Z2) is injective. Also by exactness we see that ι∗is sur- jective if and only if χ(γn)∈Hn(BO(n);Z2) is nonzero. But to see this, by the universal property ofγn, it suffices to prove that there exists some n-dimensional bundle ζwith Euler class χ(ζ)/negationslash= 0. Now by 3.12, the Euler class of the product χ(γk×γn−k) =χ(γk)⊗χ(γn−k)∈Hk(BO(k)×BO(n−k);Z2) =wk⊗wn−k∈H∗(BO(k);Z2)⊗Hn−k(BO(n−k);Z2) 3. THE PRODUCT FORMULA AND THE SPLITTING PRINCIPLE 97 which, by the inductive assumption is nonzero for k≥1. Thusχ(γn)∈Hn(BO(n);Z2) is nonzero, and we define it to be the nthStiefel - Whitney class wn=χ(γn)∈Hn(BO(n);Z2). As observed above, the nontriviality of χ(γn) implies that ι∗Hn−1(BO(n);Z2)→Hn−1(BO(n− 1);Z2) is an isomorphism, and hence there is a unique class wn−1∈Hn−1(BO(n−1);Z2) (as well as w1,···wn−2) restricting to the inductively defined classes of the same names in H∗(BO(n−1);Z2). Furthermore, since ι∗is a ring homomorphism, every polynomial in w1,···,wn−1inH∗(BO(n− 1);Z2) is in the image under ι∗of the corresponding polynomial in the these classes in H∗(BO(n);Z2). Hence by our inductive assumption, ι∗:H∗(BO(n);Z2)→H∗(BO(n−1);Z2) =Z2[w1,···,wn−1] is a split surjection of rings. But by the exactness of the Gysin sequence 2.8 this implies that this long exact splits into short exact sequences, 0→H∗−n(BO(n);Z2)∪wn−−−−→H∗(BO(n);Z2)ι∗ −−−−→H∗(BO(n−1);Z2)∼=Z2[w1,···wn−1]→0 which implies that H∗(BO(n);Z2)∼=Z2[w1,···,wn]. This completes the inductive step and there- fore the proof of the theorem. /square 3. The product formula and the splitting principle Perhaps the most important calculational tool for characteristic classes is the Whitney sum formula, which we now state and prove. Theorem 3.16.a. Letζandηbe vector bundles over a space X. Then the Stiefel - Whitney classes of the Whitney sum bundle ζ⊕ηare given by wk(ζ⊕η) =k/summationdisplay j=0wj(ζ)∪wk−j(η)∈Hk(X;Z2). where by convention, w0= 1∈H0(X;Z2). b. Ifζandηare complex vector bundles, then the Chern classes of the Whitney sum bundle ζ⊕ηare given by ck(ζ⊕η) =k/summationdisplay j=0cj(ζ)∪ck−j(η)∈H2k(X). Again, by convention, c0= 1∈H0(X). 98 3. CHARACTERISTIC CLASSES Proof. We prove the formula in the real case. The complex case is done the same way. Letζbe ann- dimensional vector bundle over X, and letηbe anm- dimensional bundle. LetN=n+m. Since we are computing wk(ζ⊕η), we may assume that k≤N, otherwise this characteristic class is zero. We prove the Whitney sum formula by induction on N≥k. We begin with the case N=k. Sinceζ⊕ηis ak- dimensional bundle, the kthStiefel - Whitney class, wk(ζ⊕η) is equal to the Euler class χ(ζ⊕η). We then have wk(ζ⊕η) =χ(ζ⊕η) =χ(ζ)∪χ(η) by 3.12 =wn(ζ)∪wm(η). This is the Whitney sum formula in this case as one sees by inputting the fact that for a bundle ρ withj >dim (ρ),wj(ρ) = 0. Now inductively assume that the Whitney sum formula holds for computing wkfor any sum of bundles whose sum of dimensions is ≤N−1≥k. Letζhave dimension nandηhave dimension m withn+m=N. To complete the inductive step we need to compute wk(ζ⊕η). Supposeζis classified by a map fζ:X→BO(n), andηis classified by a map fη:X→BO(m). Thenζ⊕ηis classified by the composition fζ⊕η:Xfζ×fη−−−−→BO(n)×BO(m)µ−−−−→BO(n+m) whereµis the map that classifies the product of the universal bundles γn×γmoverBO(n)×BO(m). Equivalently, µis the map on classifying spaces induced by the inclusion of the subgroup O(n)× O(m)/arrowhookleft→O(n+m). Thus to prove the theorem we must show that the map µ:BO(n)×BO(m)→ BO(n+m) has the property that (3.1) µ∗(wk) =k/summationdisplay j=0wj⊗wk−j∈H∗(BO(n);Z2)⊗H∗(BO(m);Z2). For a fixed j≤k, let pj:Hk(BO(n)×BO(m);Z2)→Hj(BO(n);Z2)⊗Hk−j(BO(m);Z2) be the projection onto the summand. So we need to show that pj(µ∗(wk)) =wj⊗wk−j. Now since n+m=N > k , then either j < n ork−j < m (or both). We assume without loss of generality thatj <n . Now by the proof of 3.5 ι∗:Hj(BO(n);Z2)→Hj(BO(j);Z2) is an isomorophism. Moreover we have a commutative diagram: 3. THE PRODUCT FORMULA AND THE SPLITTING PRINCIPLE 99 Hk(BO(N);Z2)µ∗ −−−−→Hk(BO(n)×BO(m);Z2)pj−−−−→Hj(BO(n);Z2)⊗Hk−j(BO(m);Z2) ι∗/arrowbt/arrowbtι∗⊗1 Hk(BO(j+m);Z2)−−−−→ µ∗Hk(BO(j)×BO(m);Z2)−−−−→ pjHj(BO(j);Z2)⊗Hk−j(BO(m);Z2). Sincej < n ,j+m < n +m=Nandι∗(wk) =wk∈Hk(BO(j+m);Z2). This fact and the commutativity of this diagram give, (ι∗⊗1)◦pj◦µ∗(wk) =pj◦µ∗◦ι∗(wk) =pj◦µ∗(wk) =wj⊗wk−jby the inductive assumption. Sinceι∗⊗1 is an isomorphism in this dimension, and since ι∗(wj⊗wk−j) =wj⊗wk−jwe have that pj◦µ∗(wk) =wj⊗wk−j. As remarked above, this suffices to complete the inductive step in the proof of the theorem. /square We can restate the Whitney sum formula in the following convenient way. For an n- dimensional bundleζ, let w(ζ) = 1 +w1(ζ) +w2(ζ) +···+wn(ζ)∈H∗(X;Z2) This is called the total Stiefel - Whitney class. The total Chern class of a complex bundle is defined similarly. The Whitney sum formula can be interpreted as saying these total characteristic classes have the “exponential property” that they take sums to products. That is, we have the following: Corollary 3.17. w(ζ⊕η) =w(ζ)∪w(η) and c(ζ⊕η) =c(ζ)∪c(η). This implies that these characteristic classes are invariants of the stable isomorphism types of bundles: Corollary 3.18.Ifζandηare stably equivalent real vector bundles over a space X, then w(ζ) =w(η)∈H∗(X;Z2), Similarly if they are complex bundles, c(ζ) =c(η)∈H∗(X). 100 3. CHARACTERISTIC CLASSES Proof. Ifζandηare stably equivalent, then ζ⊕/epsilon1m∼=η⊕/epsilon1r for somemandr. So w(ζ⊕/epsilon1m) =w(η⊕/epsilon1r). But by 3.17 w(ζ⊕/epsilon1m) =w(ζ)w(/epsilon1) =w(ζ)·1 =w(ζ). Similarlyw(η⊕/epsilon1r) =w(η). The statement follows. The complex case is proved in the same way. /square By our description of K- theory in chapter 2, we have that these characteristic classes define invariants of K- theory. Theorem 3.19.The Chern classes ciand the Stiefel - Whitney classes widefine natural trans- formations ci:K(X)→H2i(X) and wi:KO(X)→Hi(X;Z2). The total characteristic classes c:K(X)→¯H∗(X) and w:KO(X)→¯H∗(X;Z2) are exponential in the sense that c(α+β) =c(α)c(β)andw(α+β) =w(α)w(β). Here ¯H∗(X)is the direct product ¯H∗(X) =/producttext qHq(X). As an immediate application of these product formulas, we can deduce a “splitting principle” for characteristic classes. We now explain this principle. Recall that an n- dimensional bundle ζoverXsplits as a sum of nline bundles if and only if its associated principal bundle has an O(1)×···×O(1) - structure. That is, the classifying map fζ: X→BO(n) lifts to the n-fold product, BO(1)n. The analogous observation also holds for complex vector bundles. If we have such a lifting, then in cohomology, f∗ ζ:H∗(BO(n);Z2)→H∗(X;Z2) factors through⊗nH∗(BO(1);Z2). The “splitting principle” for characteristic classes says that this cohomological property always happens. 3. THE PRODUCT FORMULA AND THE SPLITTING PRINCIPLE 101 To state this more carefully, recall that H∗(BO(1);Z2) =Z2[w1]. Thus H∗(BO(1)n;Z2)∼=Z2[x1,···,xn] wherexj∈H1is the generator of the cohomology of the jthfactor in this product. Similarly, H∗(BU(1)n)∼=Z[y1,···,yn] whereyj∈H2is the generator of the cohomology of the jthfactor in this product. Notice that the symmetric group Σ nacts on these polynomial algebras by permuting the gen- erators. The subalgebra consisting of polynomials fixed under this symmetric group action is called the algebra of symmetric polynomials, Sym[x1,···,xn] orSym[y1,···,yn]. Theorem 3.20.(Splitting Principle.) The maps µ:BU(1)n→BU(n) andµ:BO(1)n→BO(n) induce injections in cohomology µ∗:H∗(BU(n))→H∗(BU(1)n) andµ∗:H∗(BO(n);Z2)→H∗(BO(1)n;Z2). Furthermore the images of these monomorphisms are the symmetric polynomials H∗(BU(n))∼=Sym[y1,···,yn] andH∗(BO(n);Z2)∼=Sym[x1,···,xn]. Proof. By the Whitney sum formula, µ∗(wj) =/summationdisplay j1+···+jn=jwj1⊗···⊗wjn∈H∗(BO(1);Z2)⊗···⊗H∗(BO(1);Z2). Butwi(γ1) = 0 unless i= 0,1. So µ∗(wj) =/summationdisplay 1≤i1<···<ij≤nxi1···xij∈Z2[x1,···,xn]. This is thejth- elementary symmetric polynomial, σj(x1,···,xn). Thus the image of Z2[w1,···,wn] = H∗(BO(n);Z2) is the subalgebra of Z2[x1,···,xn] generated by the elementary symmetric polyno- mials, Z[σ1,···,σn]. But it is well known that the elementary symmetric polynomials generate Sym[x1,···,xn] (see [ 22]). The complex case is proved similarly. /square This result gives another way of producing characteristic classes which is particularly useful in index theory. Letp(x) be a power series in one variable, which is assumed to have a grading equal to one. Say p(x) =/summationdisplay iaixi. 102 3. CHARACTERISTIC CLASSES Consider the corresponding symmetric power series in n-variables, p(x1,···,xn) =p(x1)···p(xn). Letpj(x1,···,xn) be the homogeneous component of p(x1,···,xn) of grading j. So pj(x1,···,xn) =/summationdisplay i1+···+in=jai1···ainxi1 1···xin n. Sincepjis symmetric, by the splitting principle we can think of pj∈Hj(BO(n);Z2) and hence determines a characteristic class (i.e a polynomial in the Stiefel - Whitney classes). Similarly if we give xgrading 2, we can think of pj∈H2j(BU(n)) and so determines a polynomial in the Chern classes. In particular, given a real valued smooth function y=f(x), its Taylor series pf(x) =/summationtext kf(k)(0) k!xk determines characteristic classes fi∈Hi(BO(n);Z2) orfi∈H2i(BU(n);Z2). Exercise. Consider the examples f(x) =ex,andf(x) =tanh(x). Write the low dimensional characteristic classes fiinH∗(BU(n)) fori= 1,2,3, as explicit polynomials in the Chern classes. 4. Applications In this section all cohomology will be taken with Z2- coefficients, even if not explicitly written. 4.1. Characteristic classes of manifolds. We have seen that the characteristic classes of trivial bundles are trivial. However the converse is not true, as we will now see, by examining the characteristic classes of manifolds. Definition 3.6.The characteristic classes of a manifold M,wj(M),ci(M), are defined to be the characteristic classes of the tangent bundle, τM. Theorem 3.21.wj(Sn) = 0 for allj,n> 0. Proof. As we saw in chapter 1, the normal bundle of the standard embedding Sn/arrowhookleft→Rn+1is a trivial line bundle. Thus τSn⊕/epsilon11∼=/epsilon1n+1 and soτSn+1is stably trivial. The theorem follows. /square 4. APPLICATIONS 103 Of course we know τS2is nontrivial since it has no nowhere zero cross sections. Thus the Stiefel- Whitney classes do not form a complete invariant of the bundle. However they do constitute a very important class of invariants, as we will see below. Writea∈H1(RPn;Z2)∼=Z2as the generator. Then the total Stiefel - Whitney class of the canonical line bundle γ1is w(γ1) = 1 +a∈H∗(RPn). This allows us to compute the Stiefel - Whitney classes of RPn(i.e of the tangent bundle τRPn). Theorem 3.22.w(RPn) = (1 +a)n+1∈H∗(RPn;Z2).Sowj(RPn) =/parenleftbign+1 j/parenrightbig aj∈Hj(RPn). Note: Even though the polynomial (1 + a)n+1has highest degree term an+1, this class is zero in H∗(RPn) sinceHn+1(RPn) = 0. Proof. As seen in chapter 1, τRPn⊕/epsilon11∼=⊕n+1γ1. Thus w(τRPn) =w(τRPn⊕/epsilon11) =w(⊕n+1γ1) =w(γ1)n+1,by the Whitney sum formula = (1 +a)n+1. /square Observation. The same argument shows that the total Chern class of CPnis (4.1) c(CPn) = (1 +a)n+1 wherea∈H2(CPnZ) is the generator. This calculation of the Stiefel - Whitney classes of RPnallows us to rule out the possibility that many of these projective spaces are parallelizable. Corollary 3.23.IfRPnis parallelizable, then nis of the form n= 2k−1for somek. Proof. We show that if n/negationslash= 2k−1 then there is some j >0 such that wj(RPn)/negationslash= 0. But wj(RPn) =/parenleftbign+1 j/parenrightbig aj, so we are reduced to verifying that if mis not a power of 2, then there is aj∈{1,···,m−1}such that/parenleftbigm j/parenrightbig ≡1 mod2. This follows immediately from the following combinatorial lemma, whose proof we leave to the reader. 104 3. CHARACTERISTIC CLASSES Lemma 3.24.Letj∈{1,···,m−1}. Writejandmin their binary representations, m=k/summationdisplay i=0ai2i j=k/summationdisplay i=0bi2i where theai’s andbi’s are either 0or1. Then /parenleftbiggm j/parenrightbigg ≡k/productdisplay i=0/parenleftbiggai bi/parenrightbigg mod2. Note. Here we are adopting the usual conventions that/parenleftbig0 0/parenrightbig = 1 and/parenleftbig0 1/parenrightbig = 0. /square Since we know that Lie groups are parallelizable, this result says that RPncan only have a Lie group structure if nis of the form 2k−1. However a famous theorem of Adams [ 1] says that the onlyRPn’s that are parallelizable are RP1,RP3, andRP7. Now as seen in chapter 2 an n- dimensional vector bundle ζnhask- linearly independent cross sections if and only if ζn∼=ρn−k⊕/epsilon1k for somen−kdimensional bundle ρ. Moreover, having this structure is equivalent to the classifying map fζ:X→BO(n) having a lift (up to homotopy) to a map fρ:X→BO(n−k). Now the Stiefel - Whitney classes give natural obstructions to the existence of such a lift because the mapι:BO(n−k)→BO(n) induces the map of rings ι∗:Z2[w1,···,wn]→Z2[w1,···wn−k] that mapswjtowjforj≤n−k, andwjto 0 forn≥j >n−k. We therefore have the following result. Theorem 3.25.Letζbe ann-dimensional bundle over X. Suppose wk(ζ)is nonzero in Hk(X;Z2). Thenζhas no more than n−klinearly independent cross sections. In particular, if wn(ζ)/negationslash= 0, thenζdoes not have a nowhere zero cross section. This result has applications to the existence of linearly independent vector fields on a manifold. The following is an example. Theorem 3.26.Ifmis even, RPmdoes not have a nowhere zero vector field. 4. APPLICATIONS 105 Proof. By 3.22 wm(RPm) =/parenleftbiggm+ 1 m/parenrightbigg am = (m+ 1)am∈Hm(RPm;Z2). Formeven this is nonzero. Hence wm(RPm)/negationslash= 0. /square 4.2. Normal bundles and immersions. Theorem 3.25 has important applications to the existence of immersions of a manifold Min Euclidean space, which we now discuss. Lete:Mn/curlyrightRn+kbe an immersion. Recall that this means that the derivative at each point, De(x) :TxMn→Te(x)Rn+k=Rn+k is injective. Recall also that the Inverse Function Theorem implies that an immersion is a local embedding. The immersion edefines ak- dimensional normal bundle νk ewhose fiber at x∈Mis the orthogonal complement of the image of TxMninRn+kunderDe(x). In particular we have τMn⊕νk e∼=e∗τRnk∼=/epsilon1n+k. Thus we have the Whitney sum relation among the Stiefel - Whitney classes (4.2) w(Mn)·w(νk e) = 1. So we can compute the Stiefel - Whitney clases of the normal bundle formally as the power series w(νk e) = 1/w(M)∈¯H∗(M;Z2). This proves the following: Proposition 3.27.The Stiefel - Whitney classes of the normal bundle to an immersion e: Mn/curlyrightRn+kare independent of the immersion. They are called the normal Stiefel - Whitney classes, and are written ¯wi(M). These classes are determined by the formula w(M)·¯w(M) = 1. Example. ¯w(RPn) = 1/(1 +a)n+1∈¯H∗(RPn;Z2). So for example, when n= 2k,k >0,w(RP2k) = 1 +a+a2k. This is true since by 3.24/parenleftbig2k+1 r/parenrightbig ≡ 1 mod 2 if and only if r= 0,1,2k. Thus the total normal Stiefel - Whitney class is given by ¯w(RP2k) = 1/(1 +a+a2k) = 1 +a+a2+···+a2k−1. 106 3. CHARACTERISTIC CLASSES Note. The reason this series is truncated a a2k−1is because (1 +a+a2k)(1 +a+a2+···+a2k−1) = 1∈H∗(RPn;Z2) sinceHq(RPn) = 0 forq>n . Corollary 3.28.There is no immersion of RP2kinRNforN≤2k+1−2. Proof. The above calculation shows that ¯ w2k−1(RP2k)/negationslash= 0. Thus it cannot have a normal bundle of dimension less than 2k−1. The result follows. /square In the 1940’s, Whitney proved the following seminal result in the theory of embeddings and immersions [ 45] Theorem 3.29.LetMnbe a closed n- dimensional manifold. Then there is an embedding e:Mn/arrowhookleft→R2n and an immersion ι:Mn/curlyrightR2n−1. Thus combining these results gives the following best immersion dimension for RP2k. Corollary 3.30.RP2khas an immersion in R2k+1−1but not in R2k+1−2. A natural question raised by Whitney’s theorem is to find the best possible immersion dimension for other manifolds, or for some class of manifolds. In general this is a very difficult problem. However by the following important result of Smale and Hirsch [ 17], this is purely a bundle theoretic question, and ultimately a homotopy theoretic question (via classifying maps). Theorem 3.31.LetMnbe a closed n- manifold. Then Mnimmerses in Rn+kif and only if there is ak- dimensional bundle νkoverMnwith τMn⊕νk∼=/epsilon1n+k. Thus questions of immersions boil down to bundle theoretic questions. By classifying space theory they can be viewed as homotopy theoretic questions. More specifically, let ν:M→BO represent the element in ˜KO(X) given by [ν] =−[τM]∈˜KO(X). 4. APPLICATIONS 107 Notice that if viewed with values in BO(N), forNlarge,νclassifies the normal bundle of an embedding of MninRn+N, and in particular ν∗(wi) = ¯wi(M)∈Hi(M;Z2). ν:M→BOis called the “stable normal bundle” map of M. The following is an interpretation of the above theorem of Smale and Hirsch using classifying space theory. Theorem 3.32.Mnadmits an immersion in Rn+kif and only if the stable normal bundle map ν:M→BOhas a homotopy lifting to a map νk:M→BO(k). In the late 1950’s, Wu, in China, computed a formula for how the Steenrod square cohomology operations are affected by Poincare duality in a manifold. W. Massey then used Wu’s formulas to prove the following [ 25]: Write an integer nin its binary expansion n=k/summationdisplay i=0ai·2i where each aiis 0 or 1. Let (4.3) α(n) =k/summationdisplay i=0ai. Soα(n) is the number of ones in the base 2 representation of n. Theorem 3.33.LetMnbe a closed n- dimensional manifold. Then ¯wi(Mn) = 0 fori>n−α(n). Thus Stiefel - Whitney classes give no obstruction to existence of immersions of n- manifolds inR2n−α(n). The conjecture that every n- manifold does indeed immerse in this dimension became known as the “Immersion Conjecture”, and was proved in [ 8]. Theorem 3.34.Every closed manifold Mnimmerses in R2n−α(n) This theorem was proved homotopy theoretically. Namely it was shown that the stable normal bundle map ν:Mn→BOalways has a lift (up to homotopy) to a map Mn→BO(n−α(n)). The theorem then follows from the Hirsch - Smale theorem 3.31. The lifting to BO(n−α(n)) was constructed in two steps. First, by work of Brown and Peterson [ 6] there is a “universal space for 108 3. CHARACTERISTIC CLASSES normal bundles” BO/Inand a map ρ:BO/In→BOwith the property that every stable normal bundle map from an n- manifoldν:Mn→BOlifts to a map ˜ ν:Mn→BO/In. Then the main work in [ 8] was to develop an obstruction theory to analyze the homotopy types of BO/Inand BO(n−α(n)) to show that ρ:BO/In→BOlifts to a map ˜ ρ:BO/In→BO(n−α(n)). The composition Mn ˜ν−−−−→BO/In˜ρ−−−−→BO(n−α(n)) then classifies the normal bundle of an immersion Mn/curlyrightR2n−α(n). This result, and indeed Massey’s theorem 3.33 are best possible, as can be seen by the following example. Letej:RP2j/curlyrightR2j+1−1be an immersion which is guaranteed by Whitney’s theorem. Now writenin its binary expansion n= 2j1+ 2j2+···+ 2jr where the 0≤j1<···<jrandr=α(n). Consider the n- dimensional manifold Mn=RP2j1×···× RP2jr. Consider the product immersion e:Mn=RP2j1×···× RP2jrej1×···×ejr−−−−−−−→ R2j1+1−1×···× R2jr+1−1=R2n−α(n). SinceMn=RP2j1×···× RP2jr, the Whitney sum formula will imply that ¯wn−α(n)(Mn) = ¯w2j1−1(RP2j1)⊗···⊗ ¯w2jr−1(RP2jr) which, by the proof of 3.28 is nonzero. Hence Mndoes not have an immersion in R2n−α(n)−1. Other results along these lines includes a fair amount known about the best immersion dimen- sions of projective spaces (see [ 10]). However the best immersion dimensions of all manifolds with structure, say an orientation or an almost complex structure, is unknown. Also the best embedding dimension for all n- manifolds is unknown. 5. Pontrjagin Classes In this section we define and study Pontrjagin classes. These are integral characteristic classes for real vector bundles and are defined in terms of the Chern classes of the complexification of the bundle. We will then show that polynomials in Pontrjagin classes and the Euler class define all possible characteristic classes for oriented, real vector bundles when the values of the characteristic classes is cohomology with coefficients in an integral domain Rwhich contains 1 /2. By the classifi- cation theorem,to deduce this we must compute H∗(BSO (n);R). For this calculation we follow the treatment given in Milnor and Stasheff [ 31]. 5. PONTRJAGIN CLASSES 109 5.1. Orientations and Complex Conjugates. We begin with a reexamination of certain basic properties of complex vector bundles. LetVbe ann- dimensional C- vector space with basis {v1,···,vn}. By multiplication of these basis vectors by the complex number i, we get a collection of 2 n- vectors{v1,iv1,v2,iv2,···,vn,ivn} which forms a basis for Vas a real 2n- dimensional vector space. This basis then determines an orientation of the underlying real vector space V. Exercise. Show that the orientation of Vthat the basis{v1,iv1,v2,iv2,···,vn,ivn}determines is independent of the choice of the original basis {v1,···,vn} Thus every complex vector space Vhas a canonical orientation. By choosing this orientation for every fiber of a complex vector bundle ζ, we see that every complex vector bundle has a canonical orientation. By the results of section 2 this means that every n- dimensional complex vector bundle ζover a space Xhas a canonical choice of Thom class u∈H2n(T(ζ)) and hence Euler class χ(ζ) =cn(ζ)∈H2n(X). Now given a complex bundle ζthere exists a conjugate bundle ¯ζwhich is equal to ζas a real, 2n- dimensional bundle, but whose complex structure is conjugate. More specifically, recall that a complex structure on a 2 n- dimensional real bundle ζdetermines and is determined by a linear transformation Jζ:ζ→ζ with the property that J2 ζ=Jζ◦Jζ=−id. Ifζhas a complex structure then Jζis just scalar multiplication by the complex number ion each fiber. If we replace Jζby−Jζwe define a new complex structure on ζreferred to as the conjugate complex structure. We write ¯ζto denoteζwith this structure. That is, J¯ζ=−Jη. Notice that the identity map id:ζ→¯ζ is anti-complex linear (or conjugate complex linear) in the sense that id(Jζ·v) =−J¯ζ·id(v). We note that the conjugate bundle ¯ζis often not isomorphic to ζas complex vector bundles. For example, consider the two dimensional sphere as complex projective space S2=CP1=C∪∞. The tangent bundle τCP1has the induced structure as a complex line bundle. Proposition 3.35.The complex line bundles τS2and¯τS2are not isomorphic. 110 3. CHARACTERISTIC CLASSES Proof. Supposeφ:τS2→¯τS2is a isomorphism as complex vector bundles. Then at every tangent space φx:TxS2→TxS2 is a an isomorphism that reverses the complex structure. Any such isomorphism is given by reflection through a line /lscriptxin the tangent plane TxS2. Therefore for every xwe have picked a line /lscriptx⊂TxS2. This defines a (real) one dimensional subbundle /lscriptofτS2, which, by the classification theorem is given by an element of [S2,BO(1)]∼=H1(S2,Z2) = 0. Thus/lscriptis a trivial subbundle of τS2. Hence we can find a nowhere vanishing vector field on S2, which gives us a contradiction. /square Exercise. Let ¯γnbe the conjugate of the universal bundle γnoverBU(n). By the classification theorem, ¯γnis classified by a map q:BU(n)→BU(n) having the property that q∗(γn) = ¯γn. Using the Grassmannian model of BU(n), find an explicit description of a map q:BU(n)→BU(n) with this property. The following describes the effect of conjugating a vector bundle on its Chern classes. Theorem 3.36.ck(¯ζ) = (−1)kck(ζ) Proof. Supposeζis ann- dimensional bundle. By the classification theorem and the functorial property of Chern classes it suffices to prove this theorem when ζis the universal bundle γnover BU(n). Now in our calculations of the cohomology of these classifying spaces, we proved that the inclusionι:BU(k)→BU(n) induces an isomorphism in cohomology in dimension k, ι∗:H2k(BU(n))∼=−−−−→H2k(BU(k)). Hence it suffices to prove this theorem for the universal k- dimensional bundle γkoverBU(k). Nowck(γk) =χ(γk) and similarly, ck(¯γk) =χ(¯γk). So it suffices to prove that χ(γk) = (−1)−kχ(¯γk). But by the observations above, this is equivalent to showing that the canonical orientation of the underlying real 2 k- dimensional bundle from the complex structures of γkand ¯γkare the same if kis even, and opposite if kis odd. To do this we only need to compare the orientations at a single point. Let x∈BU(k) be given by Ck⊂C∞as the first k- coordinates. If{e1,···,ek}forms the standard basis for Ck, then the orientations of γk(x) determined by the complex structures of γk and ¯γkare respectively represented by the real bases {e1,ie1,···,ek,iek}and{e1,−ie1,···,ek,−iek}. 5. PONTRJAGIN CLASSES 111 The change of basis matrix between these two basis has determinant ( −1)k. The theorem follows. /square Now suppose ηis a realn- dimensional vector bundle over a space X, we then let ηCbe its complexification ηC=η⊗RC. ηChas the obvious structure as an n- dimensional complex vector bundle. Proposition 3.37.There is an isomorphism φ:ηC∼=−−−−→ ¯ηC. Proof. Define φ:ηC→¯ηC η×C→η⊗¯C v⊗z→v⊗¯z forv∈ηandz∈C. Clearlyφis an isomorphism of complex vector bundles. /square Corollary 3.38.For a realn- dimensional bundle η, then forkodd, 2ck(ηC) = 0. Proof. By 3.36 and 3.37 ck(ηC) = (−1)kck(ηC). Hence forkoddck(ηC) has order 2. /square 5.2. Pontrjagin classes. We now use these results to define Pontrjagin classes for real vector bundles. Definition 3.7.Letηbe ann- dimensional real vector bundle over a space X. Then define theith- Pontrjagin class pi(η)∈H4i(X;Z) by the formula pi(η) = (−1)ic2i(ηC). Remark. The signs used in this definition are done to make calculations in the next section come out easily. 112 3. CHARACTERISTIC CLASSES As we’ve done with Stiefel - Whitney and Chern classes, define the total Pontrjagin class p(η) = 1 +p1(η) +···+pi(η) +···∈ ¯H∗(X,Z). The following is the Whitney sum formula for Pontrjagin classes, and follows immediately for the Whitney sum formula for Chern classes and 3.38. Theorem 3.39.For real bundles ηandξoverX, we have 2(p(η⊕ξ)−p(η)p(ξ)) = 0∈H∗(X;Z). In particular if Ris a commutative integral domain containing 1/2, then viewed as characteristic classes with values in H∗(X;R), we have p(η⊕ξ) =p(η)p(ξ)∈¯H∗(X:R). Remark. Most often Pontryagin classes are viewed as having values in rational cohomology, and so the formula p(η⊕ξ) =p(η)p(ξ) applies. We now study the Pontrjagin classes of a complex vector bundle. Let ζbe a complex n- dimensional bundle over a space X, and letζC=ζ⊗RCbe the complexification of its underlying real 2n- dimensional bundle. So ζCis a complex 2 n- dimensional bundle. We leave the proof of the following to the reader. Proposition 3.40.As complex 2n- dimensional bundles, ζC∼=ζ⊕¯ζ. This result, together with 3.36 and the definition of Pontrjagin classes imply the following. Corollary 3.41.Letζbe a complex n- dimensional bundle. Then its Pontryagin classes are determined by its Chern classes according to the formula 1−p1+p2−···±pn= (1−c1+c2−···±cn)(1 +c1+c2+···+cn) ∈H∗(X,Z). Example. We will compute the Pontrjagin classes of the tangent bundle of projective space, τCPn. Recall that the total Chern class is given by c(τCPn) = (1 +a)n+1 wherea∈H2(CPn)∼=Zis the generator. Notice that this implies that for the conjugate, ¯ τCPnwe have c(¯τCPn) = (1−a)n+1 5. PONTRJAGIN CLASSES 113 Thus by the above formula we have 1−p1+p2−···±pn= (1 +a)n+1(1−a)n+1 = (1−a2)n+1. We therefore have the formula pk(CPn) =/parenleftbiggn+ 1 k/parenrightbigg a2k∈H4k(CPn). Now letηbe an oriented real n- dimensional vector bundle. Then the complexification ηC= η⊗C=η⊕iηwhich is simply η⊕ηas real vector bundles. Lemma 3.42.The above isomorphism ηC∼=η⊕η of real vector bundles takes the canonical orientation of ηCto(−1)n(n−1) 2times the orientation of η⊕ηinduced from the given orientation of η. Proof. Pick a particular fiber, ηx. Let{v1,···,vn}be aC- basis for V. Then the basis {v1,iv1,···,vnivn}determines the orientation for ηx⊗C. However the basis {v1,···,vn,iv1,···ivn} gives the natural basis for ( η⊕iη)x. The change of basis matrix has determinant ( −1)n(n−1) 2./square Corollary 3.43.Ifηis an oriented 2k- dimensional real vector bundle, then pk(η) =χ(η)2∈H4k(X). Proof. pk(η) = (−1)kc2k(η×C) = (−1)kχ(η⊗C) = (−1)k(−1)k(2k−1)χ(η⊕η) =χ(η⊕η) =χ(η)2. /square 114 3. CHARACTERISTIC CLASSES 5.3. Oriented characteristic classes. We now use the results above to show that Pontrjagin classes and the Euler class yield all possible characteristic classes for oriented vector bundles, if the coefficient ring contains 1 /2. More specifically we prove the following. Theorem 3.44.LetRbe an integral domain containing 1/2. Then H∗(BSO (2n+ 1);R) =R[p1,···,pn] H∗(BSO (2n);R) =R[p1,···,pn−1,χ(γ2n)] Remark. This theorem can be restated by saying that H(BSO (n);R) is generated by {p1,···,p[n/2]}andχ, subject only to the relations χ= 0 ifnis odd χ2=p[n/2]ifnis even. Proof. In this proof all cohomology will be taken with Rcoefficients. We first observe that sinceSO(1) is the trivial group, BSO (1) is contractible, and so H∗(BSO (1)) = 0.This will be the first step in an inductive proof. So we assume the theorem has been proved for BSO (n−1), and we now compute H∗(BSO (n)) using the Gysin sequence: (5.1) ···→Hq−1(BSO (n−1))δ−−−−→Hq−n(BSO (n))∪χ−−−−→ Hq(BSO (n))ι∗ −−−−→ Hq(BSO (n−1))δ−−−−→Hq−n+1(BSO (n))∪χ−−−−→Hq+1(BSO (n))→··· Case 1.nis even. Since the first n/2−1 Pontrjagin classes are defined in H∗(BSO (n)) as well as in H∗(BSO (n−1)), the inductive assumption implies that ι∗:H∗(BSO (n))→H∗(BSO (n−1)) is surjective. Thus the Gysin sequence reduces to short exact sequences 0→Hq(BSO (n))∪χ−−−−→Hq+n(BSO (n))ι∗ −−−−→Hq+n(BSO (n−1))→0. The inductive step then follows. Case 2.nis odd, say n= 2m+ 1. 6. CONNECTIONS, CURVATURE, AND CHARACTERISTIC CLASSES 115 By 3.13 in this case the Euler class χhas order two in integral cohomology. Thus since Rcontains 1/2, in cohomology with Rcoefficients, the Euler class is zero. Thus the Gysin sequence reduces to short exact sequences: 0→Hj(BSO (2m+ 1))ι∗ −−−−→H∗(BSO (2m))→Hj−2m(BSO (2m+ 1))→0. Thus the map ι∗makesH∗(BSO (2m+1)) a subalgebra of H∗(BSO (2m)). This subalgeabra contains the Pontrjagin classes and hence it contains the graded algebra A∗=R[p1,···pm]. By computing ranks we will now show that this is the entire image of ι∗. This will complete the inductive step in this case. So inductively assume that the rank of Aj−1is equal to the rank of Hj(BSO (2m+ 1)). Now we know that every element of Hj(BSO (2m)) can be written uniquely as a sum a+χbwherea∈Aj andb∈Aj−2m. Thus Hj(BSO (2m))∼=Aj⊕Aj−2m which implies that rk(Hj(BSO (2m)) =rk(Aj) +rk(Aj−2m). But by the exactness of the above sequence, rk(Hj(BSO (2m)) =rk(Hj(BSO (2m+ 1)) +rk(Hj−2m(BSO (2m+ 1))). Comparing these two equations, and using our inductive assumption, we conclude that rk(Hj(BSO (2m+ 1)) =rk(Aj). ThusAj=ι∗(Hj(BSO (2m+ 1))), which completes the inductive argument. /square 6. Connections, Curvature, and Characteristic Classes In this section we describe how Chern and Pontrjagin classes can be defined using connections (i.e covariant derivatives) on vector bundles. What we will describe is an introduction to the theory of Chern and Weil that describe the cohomology of a classifying space of a compact Lie group in terms of invariant polynomials on its Lie algebra. The treatment we will follow is from Milnor and Stasheff [ 31]. Definition 3.8.LetMn(C) be the ring of n×nmatrices over C. Then an invariant polynomial onMn(C) is a function P:Mn(C)→C which can be expressed as a complex polynomial in the entries of the matrix, and satisfies, P(ABA−1) =P(B) for everyB∈Mn(C) andA∈GL(n,C). 116 3. CHARACTERISTIC CLASSES Examples. The trace function ( ai,j)→/summationtextn j=1aj,jand the determinant function are examples of invariant polynomials on Mn(C). Now letDA: Ω0(M;ζ)→Ω1(M;ζ) be a connection (or covariant derivative) on a complex n- dimensional vector bundle ζ. Its curvature is a a two- form with values in the endomorphism bundle FA∈Ω2(M;End(ζ)) The endomorphism bundle can be described alternatively as follows. Let Eζbe the principal GL(n,C) bundle associated to ζ. Then of course ζ=Eζ⊗GL(n,C)Cn.The endomorphism bun- dle can then be described as follows. The proof is an easy exercise that we leave to the reader. Proposition 3.45. End(ζ)∼=ad(ζ) =Eζ×GL(n,C)Mn(C) whereGL(n,C)acts onMn(bc)by conjugation, A·B=ABA−1. Letωbe a differential p- form onMwith values in End(ζ), ω∈Ωp(M;End(ζ))∼=Ωp(M;ad(ζ)) = Ωp(M;Eζ×GL(n,C)Mn(C)). Then on a coordinate chart U⊂Mwith local trivialization ψ:ζ|U∼=U×Cnforζ, and hence the induced coordinate chart and local trivialization for ad(ζ),ωcan be viewed as an n×nmatrix ofp -forms onM. We write ω= (ωi,j). Of course this description depends on the coordinate chart and local trivialization chosen, but at anyx∈U, then by the above proposition, two trivializations yield conjugate matrices. That is, if (ωi,j(x)) and (ω/prime i,j(x)) are two matrix descriptions of ω(x) defined by two different local trivializations ofζ|U, then there exists an A∈GL(n,C) with A(ωi,j(x))A−1= (ω/prime i,j(x)). Now letPbe an invariant polynomial on Mn(C) of degree d. Then using the wedge bracket we can applyPto a matrix of pforms, and produce a differential form of top dimension pdonU⊂M: P(ωi,j)∈Ωpd(U).Now since the polynomial Pis invariant under conjugation the form P(ωi,j) is independent of the local trivialization of ζ|U. These forms therefore fit together to give a well defined global form (6.1) P(ω)∈Ω∗(M). IfPis homogeneous of degree d, then (6.2) P(ω)∈Ωpd(M) 6. CONNECTIONS, CURVATURE, AND CHARACTERISTIC CLASSES 117 An important example is when ω=FA∈Ω2(M;End(ζ)) is the curvature form of a connection DAonζ. We have the following fundamental lemma, that will allow us to define characteristic classes in terms of these forms and invariant polynomials. Lemma 3.46.For any connection DAand invariant polynomial (or invariant power series) P, the differential form P(FA)is closed. That is, dP(FA) = 0. Proof. (following Milnor and Stasheff [ 31]) LetPbe an invariant polynomial or power series. We writeP(A) =P(ai,j) where the ai,j’s are the entries of the matrix. We can then consider the matrix of partial derivatives ( ∂P/∂ (xi,j)) where the xi,j’s are indeterminates. Let FA= (ωi,j) be the curvature matrix of two - forms on an open set Uwith a given trivialization. Then the exterior derivative has the following local expression (6.3) dP(FA) =/summationdisplay (∂P/∂ωi,j)dωi,j. In matrix notation this can be written as dP(FA) =trace (P/prime(FA)dFA) Now as seen in chapter 1, on a trivial bundle, and hence on this local coordinate patch, a connection DAcan be viewed as a matrix valued one form, DA= (αi,j) and with respect to which the curvature FAhas the formula ωi,j=dαi,j−/summationdisplay kωi,k∧ωk,j. In matrix notation we write FA=dα−α∧α. Differentiating yields the following form of the Bianchi identity (6.4) dFA=α∧FA−FA∧α. We need the following observation. Claim. The transpose of the matrix of first derivatives of an invariant polynomial (or power series)P/prime(A) commutes with A. 118 3. CHARACTERISTIC CLASSES Proof. LetEj,ibe the matrix with entry 1 in the ( j,i)-th place and zeros in all other coordi- nates. Now differentiate the equation P((I+tEj,i)A) =P(A(I+tEj,i)) with respect to tand then setting t= 0 yields /summationdisplay kAi,k(∂P/∂Aj,k) =/summationdisplay k(∂P/∂Ak,i)Ak,i. Thus the matrix Acommutes with the transpose of ( ∂P/∂Ai,j ) as claimed. /square We now complete the proof of the lemma. Substituting FAfor the matrix of indeterminates in the above claim means we have (6.5) FA∧P/prime(FA) =P/prime(FA)∧FA. Now for notational convenience let X=P/prime(FA)∧α. Then substituting the Bianchi identity 6.4 into 6.3 and using 6.5 we obtain dP(FA) =trace (X∧FA−FA∧X) =/summationdisplay (Xi,j∧ωj,i−ωj,i∧Xi,j). Since eachXi,jcommutes with the 2 - form ωj,i, this sum is zero, which proves the lemma. /square Thus for any connection DAon the complex vector bundle ζoverM, and invariant polynomial P, the formP(FA) represents a deRham cohomology class with complex coefficients. That is, [P(FA)]∈H∗(M:C). Theorem 3.47.The cohomology class [P(FA)]∈H∗(X,C)is independent of the connection DA. Proof. LetDA0andDA1be two connections on ζ. Pull back the bundle ζoverM×Rvia the projection map M×R→M. Call this pull - back bundle ¯ζoverM×R. We get the induced pull back connections ¯DAi,i= 0,1 as well. We can then form the linear combination of connections DA=t¯DA1+ (1−t)DA0. ThenP(FA) is a deRham cocycle on M×R. Now let i= 0 or 1 and consider the inclusions ji:M=M×{i}/arrowhookleft→M×R. The induced connection j∗ i(DA) =DAionζ. But since there is an obvious homotopy between j0andj1and hence the cohomology classes [j∗ 0(P(FA)) =P(FA0)] = [j∗ 1(P(FA)) =P(FA1)]. This proves the theorem. /square 6. CONNECTIONS, CURVATURE, AND CHARACTERISTIC CLASSES 119 Thus the invariant polynomial Pdetermines a cohomology class given any bundle ζover a smooth manifold. It is immediate that these classes are preserved under pull - back, and are hence characteristic classes for U(n) bundles, and hence are given by elements of H∗(BU(n);C)∼=C[c1,···,cn]. In order to see how an invariant polynomial corresponds to a polynomial in the Chern classes we need the following bit of algebra. Recall the elementary symmetric polynomials σ1,···σninn-variables, discussed in section 3. If we view the n- variables as the eigenvalues of an n×nmatrix, we can write (6.6) det(I+tA) = 1 +tσ1(A) +···+tnσn(A). Lemma 3.48.Any invariant polynomial on Mn(C)can be expressed as a polynomial of σ1,···,σn. Proof. GivenA∈Mn(C), chose aBsuch thatBAB−1is in Jordan canonical form. Replacing Bwithdiag(/epsilon1,/epsilon12,···,/epsilon1n)B, we can make the off diagonal entries arbitrarily close to zero. By continuity it follows that P(A) depends only on the diagonal entries of BAB−1, i,e the eigenvalues ofA. SinceP(A) is invariant, it must be a symmetric polynomial of these eigenvalues. Hence it is a polynomial in the elementary symmetric polynomials. /square So we now consider the elementary symmetric polynomials, viewed as invariant polynomials inMn(C). Hence by the above constructions they determine characteristic classes [ σr(FA)]∈ H2r(M;C) whereFAis a connection on a vector bundle ζoverM. Now we’ve seen the elementary symmetric functions before in the context of characteristic classes. Namely we’ve seen that H∗(BU(n)) can be viewed as the subalgebra of symmetric poly- nomials in Z[x1,···xn] =H∗(BU(1)×···×BU(1)),with the Chern class Crcorresponding to the elementary symmetric polynomial σr. This was the phenomenon of the splitting principle. We will now use a splitting principle argument to prove the following. Theorem 3.49.Letζbe a complex n- dimensional vector bundle with connection DA. Then the cohomology class [σr(FA)]∈H2r(X;C)is equal to (2πi)rcr(ζ),forr= 1,···,n. Proof. We first prove this theorem for complex line bundles. That is, n= 1. In this case σ1(FA) =FAwhich is a closed form in Ω2(M;ad(ζ)) = Ω2(M;C) because the adjoint action of GL(1,C) is trivial since it is an abelian group. In particular FAis closed in this case by 3.46. Thus FArepresents a cohomology class in H2(M;C). Moreover as seen above, this cohomology class [ FA] is a characteristic class for line bundles and hence is an element of H2(BU(1);C)∼=Cgenerated by 120 3. CHARACTERISTIC CLASSES the first Chern class c1∈H2(BU(1)). So for this case we need to prove the following generalization of the Gauss - Bonnet theorem. Lemma 3.50.Letζbe a complex line bundle over a manifold Mwith connection DA. Then the curvature form FAis a closed two - form representing the cohomology class [FA] = 2πic1(ζ) = 2πiχ(ζ). Before we prove this lemma we show how this lemma can in fact be interpreted as a generalization of the classical Gauss - Bonnet theorem. So let DAbe a unitary connection on ζ. (That is, DAis induced by a connection on an associated principal U(1) - bundle.) If we view ζas a two dimensional, oriented vector bundle which, to keep notation straight we refer to as ζR, thenDAinduces (and is induced by) a connection DARon the real bundle ζR. Notice that since SO(2)∼=U(1) then orthogonal connections on oriented real two dimensional bundes are equivalent to unitary connections on complex line bundles. SinceSO(2) is abelian, the real adjoint bundle ad(ζR) =EζR×SO(2)M2(R) is trivial. Hence the curvature FARis then a 2×2 matrix valued two - form. FAR∈Ω2(M;M2(R)). Moreover, since the Lie algebra of SO(2) consists of skew symmetric 2 ×2 real matrices, then it is straightforward to check the following relation between the original complex valued connection FA∈Ω2(M;C) and the real curvature form FAR∈Ω1(M;M2(R)). Claim. IfFARis written as the skew symmetric matrix of 2 −forms FAR=/parenleftBigg 0ω −ω0/parenrightBigg ∈Ω2(M;M2(R)) then FA=iω∈Ω2(M;C). When the original connection DARis the Levi - Civita connection associated to a Riemannian metric on the tangent bundle of a Riemann surface, the curvature form ω∈Ω2(M,R) is referred to as the “Gauss - Bonnet”” connection. If dAdenotes the area form with respect to the metric, then we can write ω=κdA 6. CONNECTIONS, CURVATURE, AND CHARACTERISTIC CLASSES 121 thenκis a scalar valued function called the “Gaussian curvature” of the Riemann surface M. In this case, by the claim we have [ FA] = 2πiχ(τ(M)), and since /angbracketleftχ(τ(M)),[M]/angbracketright=χM, WhereχMthe Euler characteristic of M, we have /angbracketleft[FA],[M]/angbracketright=/integraldisplay MFA=i/integraldisplay Mω=i/integraldisplay MκdA. Thus the above lemma applied to this case, which states that /angbracketleft[FA],[M]/angbracketright= 2πiχM is equivalent to the classical Gauss - Bonnet theorem which states that (6.7)/integraldisplay MκdA = 2πχM= 2π(2−2g) wheregis the genus of the Riemann surface M. We now prove the above lemma. Proof. As mentioned above, since [ FA] is a characteristic class for line bundles, and so it is some multiple of the first Chern class, say [ FA] =qc1(ζ). By the naturality, the coefficient q is independent of the bundle. So to evaluate qit is enough to compute it on a specific bundle. We choose the tangent bundle of the unit sphere τS2, equipped with the Levi - Civita connection DAcorresponding to the usual round metric (or equivalently the metric coming from the complex strucureS2=CP1). In this case the Gaussian curvature is constant at one, κ= 1. Moreover since τS2⊕/epsilon11∼=γ1⊕γ1, the Whitney sum formula yields /angbracketleftc1(S2),[S2]/angbracketright= 2/angbracketleftc1(γ1),[S2]/angbracketright= 2. Thus we have /angbracketleft[FA],[S2]/angbracketright=q/angbracketleftc1(S2),[S2]/angbracketright = 2q. 122 3. CHARACTERISTIC CLASSES Putting these facts together yields that 2q=/angbracketleft[FA],[S2]/angbracketright =/integraldisplay S2FA =i/integraldisplay S2κdA =i/integraldisplay S2dA =i·surface area of S2 =i·4π. Henceq= 2πi, as claimed. /square We now proceed with the proof of theorem 3.49 in the case when the bundle is a sum of line bundles. By the splitting principal we will then be able to conclude the theorem is true for all bundles. So letζ=L1⊕···⊕LnwhereL1,···,Lnare complex line bundles over M. LetD1,···,Dn be connections on L!,···,Lnrespectively. Now let DAbe the connection on ζgiven by the sum of these connections DA=D1⊕···⊕Dn. Notice that with respect to any local trivialization, the curvature matrix FAis the diagonal n×n matrix with diagonal entries, the curvatures F1,···,Fnof the connections D1,···Dnrespectively. Thus the invariant polynomial applied to the curvature form σr(FA) is given by the symmetric polynomial in the diagonal entries, σr(FA) =σr(F1,···,Fr). Now since the curvatures Fiare closed 2 - forms on M, we have an equation of cohomology classes [σr(FA)] =σr([F1],···,[Fr]). By the above lemma we therefore have [σr(FA)] =σr([F1],···,[Fn]) =σr((2πi)c1(L1),···,(2πi)c1(Ln)) = (2πi)rσr(c1(L),···,c1(Ln)) sinceσris symmetric = (2πi)rcr(L1⊕···⊕Ln) by the splitting principal 3.20 = (2πi)rcr(ζ) as claimed. This proves the theorem when ζis a sum of line bundles. As observed above, the splitting principal implies that the theorem then must be true for all bundles. /square 6. CONNECTIONS, CURVATURE, AND CHARACTERISTIC CLASSES 123 We end this section by describing two corollaries of this important theorem. Corollary 3.51.For any real vector bundle η, the deRham cocycle σ2k(FA)represent the cohomology class (2π)2kpk(η)∈H4k(M;R), while [σ2k+1(FA)]is zero inH4k+2(M;R). Proof. This just follows from the definition of the Pontrjagin classes in terms of the even Chern classes of the complexification, and the fact that the odd Chern classes of the complexification have order two and therefore represent the zero class in H∗(M;R). /square Recall that a flat connection is one whose curvature is zero. The following is immediate form the above theorem. Corollary 3.52.If a real (or complex) vector bundle has a flat connection, then all its Pon- trjagin (or Chern) classes with rational coefficients are zero. We recall that a bundle has a flat connection if and only if its structure group can be reduced to a discrete group. Thus a complex vector bundle with a discrete structure group has zero Chern classes with rational coefficients. This can be interpreted as saying that if ι:G⊂GL(n,C) is the inclusion of a discrete subgroup, then the map in cohomology, Q[c1,···,cn] =H∗(BU(n);Q) =H∗(BGLn(C);Q)ι∗ −−−−→H∗(BG;Q) is zero. CHAPTER 4 Homotopy Theory of Fibrations In this chapter we study the basic algebraic topological properties of fiber bundles, and their generalizations, “Serre fibrations”. We begin with a discussion of homotopy groups and their basic properties. We then show that fibrations yield long exact sequences in homotopy groups and use it to show that the loop space of the classifying space of a group is homotopy equivalent to the group. We then develop basic obstruction theory for liftings in fibrations, use it to interpret characteristic classes as obstructions, and apply them in several geometric contexts, including vector fields, Spin structures, and classification of SU(2) - bundles over four dimensional manifolds. We also use obstruction theory to prove the existence of Eilenberg - MacLane spaces, and to prove their basic property of classifying cohomology. We then develop the theory of spectral sequences and then discuss the famous Leray - Serre spectral sequence of a fibration. We use it in several applications, including a proof of the theorem relating homotopy groups and homology groups, a calculation of the homology of the loop space Ω Sn, and a calculation of the homology of the Lie groups U(n) and O(n). 1. Homotopy Groups We begin by adopting some conventions and notation. In this chapter, unless otherwise specified, we will assume that all spaces are connected and come equipped with a basepoint. When we write [X,Y] we mean homotopy classes of basepoint preserving maps X→Y. Suppose x0∈Xand y0∈Yare the basepoints. Then a basepoint preserving homotopy between basepoint preserving mapsf0andf1:X→Yis a map F:X×I→Y such that each Ft:X×{t}→Yis a basepoint preserving map and F0=f0andF1=f1. If A⊂XandB⊂Y, are subspaces that contain the basepoints, ( x0∈A, andy0∈B), we write [X,A;Y,B] to mean homotopy classes of maps f:X→Yso that the restriction f|AmapsAtoB. Moreover homotopies are assumed to preserve these subsets as well. That is, a homotopy defining this equivalence relation is a map F:X×I→Ythat restricts to a basepoint preserving homotopy F:A×I→B. We can now give a strict definition of homotopy groups. 125 126 4. HOMOTOPY THEORY OF FIBRATIONS Definition 4.1.Thenthhomotopy group of a space Xwith basepoint x0∈Xis defined to be the set πn(X) =πn(X,x 0) = [Sn,X]. Equivalently, this is the set πn(X) = [Dn,Sn−1;X,x 0] whereSn−1=∂Dnis the boundary sphere. Exercise. Prove that these two definitions are in fact equivalent. Remarks. 1. It will often helpful to us to use as our model of the disk Dnthen- cube In= [0,1]n. Notice that in this model the boundary ∂Inconsists ofn- tuples (t1,···,tn) with ti∈[0,1] where at least one of the coordinates is either 0 or 1. 2. Notice that for n= 1, this definition of the first homotopy group is the usual definition of the fundamental group. So far the homotopy “groups” have only been defined as sets. We now examine the group structure. To do this, we will define our homotopy groups via the cube In, which we give the basepoint (0 ,···,0). Let fandg: (In,∂In)−→(X,x 0) be two maps representing elements [ f] and [g]∈πn(X,x 0). Define f·g:In−→X by f·g(t1,t2,···,tn) =  f(2t1,t2,···,tn) fort1∈[0,1/2] g(2t−1,t2,···,tn) fort1∈[1/2,1] The mapf·g: (In,∂In)→(X,x 0) represents the product of the classes [f·g] = [f]·[g]∈πn(X,x 0). Notice that in the case n= 1 this is precisely the definition of the product structure on the funda- mental group π1(X,x 0). The same proof that this product structure is well defined and gives the fundamental group the structure of an associative group extends to prove that all of the homotopy groups are in fact groups under this product structure. We leave the details of checking this to the reader. We refer the reader to any introductory textbook on algebraic topology for the details. As we know the fundamental group of a space can be quite complicated. Indeed any group can be the fundamental group of a space. In particular fundamental groups can be very much noncommutative. However we recall the relation of the fundamental group to the first homology group, for which we again refer the reader to any introductory textbook: 1. HOMOTOPY GROUPS 127 Theorem 4.1.LetXbe a connected space. Then the abelianization of the fundamental group is isomorphic to the first homology group, π1(X)/[π1,π1]∼=H1(X) where [π1,π1]is the commutator subgroup of π1(X). We also have the following basic result about higher homotopy groups. Proposition 4.2.Forn≥2, the homotopy group πn(X)is abelian. Proof. Let [f] and [g] be elements of πn(X) represented by basepoint preserving maps f: (In,∂In)→(X,x 0) andg: (In,∂In)→(X,x 0), respectively. We need to find a homotopy between the product maps f·gandg·fdefined above. The following schematic diagram suggests such a homotopy. We leave it to the reader to make this into a well defined homotopy. f gf g f g f g f gg f /square Now assume A⊂Xis a subspace containing the basepoint x0∈A. Definition 4.2.Forn≥1 we define the relative homotopy group πn(X,A) =πn(X,A,x 0)to be homotopy classes of maps of pairs πn(X,A) = [(Dn,∂Dn,t0); (X,A,x 0)]. wheret0∈∂Dn=Sn−1andx0∈Aare the basepoints. 128 4. HOMOTOPY THEORY OF FIBRATIONS Exercise. Show that for n>1 the relative homotopy group πn(X,A) is in fact a group. Notice here that the zero element is represented by any basepoint preserving map of pairsf: (Dn,∂In)→(X,A) that is homotopic (through maps of pairs) to one whose image lies entirely inA⊂X. Again, letA∈Xbe a subset containing the basepoint x0∈A, and leti:A /arrowhookleft→Xbe the inclusion. This induces a homomorphism of homotopy groups i∗:πn(A,x 0)→πn(X,x 0). Also, by ignoring the subsets, a basepoint preserving map f: (Dn,∂Dn)→(X,x 0) defines a map of pairsf: (Dn,∂Dn,t0)→(X,A,x 0) which defines a homomorphism j∗:πn(X,x 0)→πn(X,A,x 0). Notice furthermore, that by construction, the composition j∗◦i∗:πn(A)→πn(X)→πn(X,A) is zero. Finally, if given a map of pairs g: (Dn,Sn−1,t0)→(X,A,x 0), then we can restrict gto the boundary sphere Sn−1to produce a basepoint preserving map ∂g: (Sn−1,t0)→(A,x 0). This defines a homomorphism ∂∗:πn(X,A,x 0)→πn−1(A,x 0). Notice here that the composition ∂∗◦j∗:πn(X)→πn(X,A)→πn−1(A) is also zero, since the application of this composition to any representing map f: (Dn,Sn−1)→ (X,x 0) yields the constant map Sn−1→x0∈A. We now have the following fundamental property of homotopy groups. Compare with the analogous theorem in homology. Theorem 4.3.LetA⊂Xbe a subspace containing the basepoint x0∈A. Then we have a long exact sequence in homotopy groups ···∂∗−−−−→πn(A)i∗−−−−→πn(X)j∗−−−−→πn(X,A)∂∗−−−−→πn−1(A)→···→π1(A)i∗−−−−→π1(X) Proof. We’ve already observed that j∗◦i∗and∂∗◦j∗are zero. Similarly, i∗◦∂∗is zero because an element in the image of ∂∗is represented by a basepoint preserving map Sn−1→Athat extends 1. HOMOTOPY GROUPS 129 to a mapDn→X. Thus the image under i∗, namely the composition Sn−1→A /arrowhookleft→Xhas an extension to Dnand is therefore null homotopic. We therefore have image (∂∗)⊂kernel (i∗) image (i∗)⊂kernel (j∗) image (j∗)⊂kernel (∂∗). To finish the proof we need to show that all of these inclusions are actually equalities. Consider the kernel of ( i∗). An element [ f]∈πn(A) is inker(i∗) if and only if the basepoint preserving composition f:Sn→A⊂Xis null homotopic. Such a null - homotopy gives an extension of this map to the disk F:Dn+1→X. The induced map of pairs F: (Dn+1,Sn)→(X,A) represents an element inπn+1(X,A) whose image under ∂∗is [f]. This proves that image (∂∗) =kernel (i∗). The other equalities are proved similarly, and we leave their verification to the reader. /square Remark. Even though this theorem is analogous to the existence of exact sequences for pairs in homology, notice that its proof is much easier. Notice that π0(X) is the set of path components of X. So a space is (path) - connected if and only ifπ0(X) = 0 (i.e the set with one element). We generalize this notion as follows. Definition 4.3.A spaceXis said to be m- connected if πq(X) = 0 for 0≤q≤m. We now do our first calculation. Proposition 4.4.Ann- sphere isn−1connected. Proof. We need to show that any map Sk→Sn, wherek < n is null homotopic. Now since spheres can be given the structure of simplicial complexes, the simplicial approximation theorem says that any map f:Sk→Snis homotopic to a simplicial map (after suitable subdivisions). So we assume without loss of generality that fis simplicial. But since k<n , the image of of flies in the k- skeleton of the n- dimensional simplicial complex Sn. In particular this means that f:Sk→Sn is not surjective. Let y0∈Snbe a point that is not in the image of f. Thenfhas image in Sn−y0 which is homeomorphic to the open disk Dn, and is therefore contractible. This implies that fis null homotopic. /square 130 4. HOMOTOPY THEORY OF FIBRATIONS 2. Fibrations Recall that in chapter 2 we proved that locally trivial fiber bundles satisfy the Covering Homo- topy Theorem 2.2. A generalization of the notion of a fiber bundle, due to Serre, is simply a map that satisfies this type of lifting property. Definition 4.4.A Serre fibration is a surjective, continuous map p:E→Bthat satisfies theHomotopy Lifting Property forCW - complexes. That is, if Xis anyCW - complex and F:X×I→Bis any continuous homotopy so that F0:X×{0}→Bfactors through a map f0:X→E, then there exists a lifting ¯F:X×I→Ethat extends f0onX×{0}, and makes the following diagram commute: X×I¯F−−−−→E =/arrowbt/arrowbtp X×I−−−−→ FB. A Hurewicz fibration is a surjective, continuous map p:E→Bthat satisfies the homotopy lifting property for all spaces. Remarks. 1. Obviously every Hurewicz fibration is a Serre fibration. The converse is false. In these notes, unless otherwise stated, we will deal with Serre fibrations, which we will simply refer to as fibrations. 2. The Covering Homotopy Theorem implies that a fiber bundle is a fibration in this sense. The following is an important example of a fibration. Proposition 4.5.LetXbe any connected space with basepoint x0∈X. LetPX denote the space of based paths in X. That is, PX={α:I→X:α(0) =x0}. The path space PX is topologized using the compact - open function space topology. Define p:PX→X byp(α) =α(1).ThenPX is a contractible space, and the map p:PX→Xis a fibration, whose fiber atx0,p−1(x0)is the loop space ΩX. Proof. The fact that PXis contractible is straightforward. For a null homotopy of the identity map one can take the map H:PX×I→PX, defined by H(α,s)(t) =α((1−s)t). 2. FIBRATIONS 131 To prove that p:PX→Xis a fibration, we need to show it satisfies the Homotopy Lifting Property. So let F:Y×I→Xandf0:X→PXbe maps making the following diagram commute: Y×{0}f0−−−−→PX ∩/arrowbt/arrowbtp Y×I−−−−→ FX Then we can define a homotopy lifting, ¯F:Y×I→PXby defining for ( y,s)∈Y×I, the path ¯F(y,s) :I→X ¯F(y,s)(t) =  f0(y)(2t 2−s) for t∈[0,2−s 2] F(y,2t−2 +s) fort∈[2−s 2,1] One needs to check that this definition makes ¯F(y,s)(t) a well defined continous map and satisfies the boundary conditions ¯F(y,0)(t) =f0(y,t) ¯F(y,s)(0) =x0 ¯F(y,s)(1) =F(y,s) These verifications are all straightforward. /square The following is just the observation that one can pull back the Homotopy Lifting Property. Proposition 4.6.Letp:E→Bbe a fibration, and f:X→Ba continuous map. Then the pull back,pf:f∗(E)→Xis a fibration, where f∗(E) ={(x,e)∈X×Esuch thatf(x) =p(e)} andpf(x,e) =x. The following shows that in the setting of homotopy theory, every map can be viewed as a fibration in this sense. Theorem 4.7.Every continuous map f:X→Yis homotopic to a fibration in the sense that there exists a fibration ˜f:˜X→Y and a homotopy equivalence h:X/similarequal−−−−→ ˜X 132 4. HOMOTOPY THEORY OF FIBRATIONS making the following diagram commute: Xh−−−−→ /similarequal˜X f/arrowbt/arrowbt˜f Y =Y. Proof. Define ˜Xto be the space ˜X={(x,α)∈X×YIsuch thatα(0) =x.} where here YIdenotes the space of continuous maps α: [0,1]→Ygiven the compact open topology. The map ˜f:˜X→Yis defined by ˜f(x,α) =α(1). The fact that ˜f:˜X→Yis a fibration is proved in the same manner as theorem 4.5, and so we leave it to the reader. Define the map h:X→˜Xbyh(x) = (x,/epsilon1x)∈˜X, where/epsilon1x(t) =xis the constant path at x∈X. Clearly ˜f◦h=fso the diagram in the statement of the theorem commutes. Now define g:˜X→Xbyg(x,α) =x.Clearlyg◦his the identity map on X. To see that h◦gis homotopic to the identity on ˜X, consider the homotopy F:˜X×I→˜X, defined by F((x,α),s) = (x,αs), where αs:I→Xis the path αs(t) =α(st). So in particular α0=/epsilon1xandα1=α. ThusFis a homotopy betweenh◦gand the identity map on ˜X. Thushis a homotopy equivalence, which completes the proof of the theorem. /square The homotopy fiber of a map f:X→Y,Ff, is defined to be the fiber of the fibration ˜f:˜X→Y defined in the proof of this theorem. That is, Definition 4.5.The homotopy fiber Ffof a basepoint preserving map f:X→Yis defined to be Ff={(x,α)∈X×YIsuch thatα(0) =f(x) andα(1) =y0.} wherey0∈Yis the basepoint. So for example, the homotopy fiber of the inclusion of the basepoint y0/arrowhookleft→Yis the loop space ΩY. The homotopy fiber of the identity map id:Y→Yis the path space PY. The homotopy fibers are important invariants of the map f:X→Y. The following is the basic homotopy theoretic property of fibrations. Theorem 4.8.Letp:E→Bbe a fibration over a connected space Bwith fiberF. So we are assuming the basepoint of E, is contained in F,e0∈F, and thatp(e0) =b0is the basepoint in B. Leti:F /arrowhookleft→Ebe the inclusion of the fiber. Then there is a long exact sequence of homotopy groups: ···∂∗−−−−→πn(F)i∗−−−−→πn(E)p∗−−−−→πn(B)∂∗−−−−→πn−1(F)→ ···→π1(F)i∗−−−−→π1(E)p∗−−−−→π1(B). 2. FIBRATIONS 133 Proof. Notice that the projection map p:E→Binduces a map of pairs p: (E,F)→(B,b0). By the exact sequence for the homotopy groups of the pair ( E,F), 4.3 it is sufficient to prove that the induced map in homotopy groups p∗:πn(E,F)→πn(B,b0) is an isomorphism for all n≥1. We first show that p∗is surjective. So let f: (In,∂In)→(B,b0) represent an element of πn(B). We can think of a map from a cube as a homotopy of maps of cubes of one lower dimension. Therefore by induction on n, the homotopy lifting property says that that f:In→Bhas a basepoint preserving lifting ¯f:In→E. Sincep◦¯f=f, and since the restriction offto the boundary ∂Inis constant at b0, then the image of the restriction of ¯fto the boundary ∂Inhas image in the fiber F. That is, ¯finduces a map of pairs ¯f: (In,∂In)→(E,F) which in turn represents an element [ ¯f]∈πn(E,F) whose image under p∗is [f]∈πn(B,b0). This proves that p∗is surjective. We now prove that p∗:πn(E,F)→πn(B,b0) is injective. So let f: (Dn,∂Dn)→(E,F) be a map of pairs that represents an element in the kernel of p∗. That means p◦f: (Dn,∂Dn)→(B,b0) is null homotopic. Let F: (Dn,∂Dn)×I→(B,b0) be a null homotopy between F0=fand the constant map /epsilon1:Dn→b0. By the Homotopy Lifting Property there exists a basepoint preserving lifting ¯F:Dn×I→E having the properties that p◦¯F=Fand ¯F:Dn×{0}→Eis equal to f: (Dn,∂Dn)→(E,F). Sincep◦¯F=Fmaps∂Dn×Ito the basepoint b0, we must have that ¯Fmaps∂Dn×Itop−1(b0) =F. Thus ¯Fdetermines a homotopy of pairs, ¯F: (Dn,∂Dn)×I→(E,F) with ¯F0=f. Now consider ¯F1: (Dn,∂Dn)×{1}→E. Nowp◦¯F1=F1=/epsilon1:Dn→b0.Thus the image of ¯F1lies inp−1(b0) =F. Thus ¯Fgives a homotopy of the map of pairs f: (Dn,∂Dn)→ (E,F) to a map of pairs whose image lies entirely in F. Such a map represents the zero element of πn(E,F). This completes the proof that p∗is injective, and hence is an isomorphism. As observed earlier, this is what was needed to prove the theorem. /square We now use this theorem to make several important calculations of homotopy groups. In par- ticular, we prove the following seminal result of Hopf. 134 4. HOMOTOPY THEORY OF FIBRATIONS Theorem 4.9. π2(S2)∼=π3(S3)∼=Z. πk(S3)∼=πk(S2)for allk≥3.In particular, π3(S2)∼=Z,generated by the Hopf map η:S3→S2. Proof. Consider the Hopf fibration η:S3→S2=CP1with fiberS1. Recall that S1is an Eilenberg - MacLane space K(Z,1) since it is the classifying space of bz. Thus πq(S1) =  Zfor q = 1 0 for all other q. (Remark. The fact that the classifying space Bπof a discrete groupπis an Eilenberg - MacLane spaceK(π,1) can now be given a simpler proof, using the exact sequence in homotopy groups of the universal bundle Eπ→Bπ.) Using this fact in the exact sequence in homotopy groups for the Hopf fibration η:S3→S2, together with the fact that πq(S3) = 0 forq≤2, one is led to the facts that π2(S2)∼=π1(S1) =Z, and thatη∗:πk(S3)→πk(S2) is an isomorphism for k≥3. To examine the case k= 3, consider the homomorphism (called the Hurewicz homomorphism ) h:π3(S3)→H3(S3) =Z defined by sending a class represented by a self map f:S3→S3, to the image of the fundamental class in homology, f∗([S3])∈H3(S3)∼=Z. Clearly this is a homomorphism (check this!). Moreover it is surjective since the image of the identity map is the fundamental class, and thus generates, H3(S3),H([id]) = [S3]∈H3(S3). Thusπ3(S3) contains an integral summand generated by the identity. In particular, since η∗:π3(S3)→π3(S2) is an isomorphism, this implies that π3(S2) contains an integral summand generated by the Hopf map [ η]∈π3(S2). The fact that these integral summands generate the entire groups π3(S3)∼=π3(S2) will follow once we know that the Hurewicz homomorphism is an isomorphism in this case. Later in this chapter we will prove the more general “Hurewicz theorem” that says that for any k>1, and any ( k−1) - connected space X, the Hurewicz homomorphism is an isomorphism in dimension k:h:πk(X)∼=Hk(X). /square Remark. As we remarked earlier in these notes. these were the first nontrivial elements found in the higher homotopy groups of spheres, πn+k(Sn), and Hopf’s proof of their nontriviality is commonly viewed as the beginning of modern Homotopy Theory [ 43] We end this section with an application to the “homotopy stability” of the orthogonal and unitary groups, as well as their classifying spaces. 3. OBSTRUCTION THEORY 135 Theorem 4.10.The inclusion maps ι:O(n)/arrowhookleft→O(n+ 1) and U(n)/arrowhookleft→U(n+ 1) induce isomorphisms in homotopy groups through dimensions n−2and2n−1respectively. Also, the induced maps on classifying spaces, Bι:BO(n)→BO(n+ 1) and BU(n)→BU(n+ 1) induce isomorphisms in homotopy groups through dimensions n−1and2nrespectively. Proof. The first two statements follow from the existence of fiber bundles O(n)/arrowhookleft→O(n+ 1)→Sn and U(n)/arrowhookleft→U(n+ 1)→S2n+1, the connectivity of spheres 4.4, and by applying the exact sequence in homotopy groups to these fiber bundles. The second statement follows from the same considerations, after recalling from 2.28 the sphere bundles Sn→BO(n)→BO(n+ 1) and S2n+1→BU(n)→BU(n+ 1). /square 3. Obstruction Theory In this section we discuss the obstructions to obtaining a lifting to the total space of a fibration of a map to the basespace. As an application we prove the important “Whitehead theorem” in homotopy theory, and we prove general results about the existence of cross sections of principal O(n) orU(n) - bundles. We do not develop a formal theory here - we just develop what we will need for our applications to fibrations. For a full development of obstruction theory we refer the reader to [42]. LetXbe aCW - complex. Recall that its cellular k- chains,Ck(X) is the free abelian group generated by the k- dimensional cells in X. The co-chains with coefficients in a group Gare defined by Ck(X,G) =Hom (Ck(X),G). 136 4. HOMOTOPY THEORY OF FIBRATIONS Theorem 4.11.Letp:E→Bbe a fibration with fiber F. Letf:X→Bbe a continuous map, whereXis aCW - complex. Suppose there is a lifting of the (k−1)- skeleton ˜fk−1:X(k−1)→E. That is, the following diagram commutes: X(k−1)˜fk−1−−−−→E ∩/arrowbt/arrowbtp X−−−−→ fB. Then the obstruction to the existence of a lifting to the k-skeleton, ˜fk:X(k)→Ethat extends ˜fk−1, is a cochain γ∈Ck(X;πk−1(F)). That is,γ= 0if and only if such a lifting ˜fkexists. Proof. We will first consider the special case where X(k)is obtained from Xk−1)by adjoining a singlek-dimensional cell. So assume X(k)=X(k−1)∪αDk whereα:∂Dk=S(k−1)→Xk−1)is the attaching map. We therefore have the following commuta- tive diagram: Sk−1α−−−−→ X(k−1)˜fk−1−−−−→E ∩/arrowbt ∩/arrowbt/arrowbtp Dk−−−−→ ⊂X(k−1)∪αDk−−−−→ fB Notice that ¯fk−1has an extension to X(k−1)∪αDk=X(k)that liftsf, if and only if the composition Dk⊂X(k−1)∪αDkf−−−−→Blifts toEin such a way that it extends ¯fk−1◦α. Now view the composition Dk⊂X(k−1)∪αDkf−−−−→Bas a a map from the cone on Sk−1to B, or in other words, as a null homotopy F:Sk−1×I→BfromF0=p◦¯fk−1◦α:Sk−1→X(k−1)→ E→Bto the constant map F1=/epsilon1:S(k−1)→b0∈B. By the Homotopy Lifting Property, Flifts to a homotopy ¯F:S(k−1)×I→E with ¯F0=¯fk−1◦α.Thus the extension fkexists onX(k−1)∪αDkif and only if this lifting ¯Fcan be chosen to be a null homotopy of ¯fk−1◦α. But we know ¯F1:Sk−1×{1}→EliftsF1which is the constant map /epsilon1:Sk−1→b0∈B. Thus the image of ¯F1lies in the fiber F, and therefore determines an element γ∈πk−1(F).The homotopy ¯F1can be chosen to be a null homotopy if and only if ¯F1:Sk−1→Fis null homotopic. (Because combining ¯Fwith a null homotopy of ¯F1, i.e an extension of ¯F1to a mapDk→F, is still a lifting of F, since the extension lives in a fiber over a point.) But this is only true if the homotopy class γ= 0∈πk−1(F). 3. OBSTRUCTION THEORY 137 This proves the theorem in the case when X(k)=X(k−1)∪αDk. In the general case, suppose thatX(k)is obtained from X(k−1)by attaching a collection of k- dimensional disks, indexed on a set, say J. That is, X(k)=X(k−1)/uniondisplay j∈J∪αjDk. The above procedure assigns to every j∈Jan “obstruction” γj∈πk−1(F). An extension ¯fkexists if and only if all these obstructions are zero. This assignment from the indexing set of the k- cells to the homotopy group can be extended linear to give a homomorphism γfrom the free abelian group generated by the k- cells to the homotopy group πk−1(F), which is zero if and only if the extension ¯fkexists. Such a homomorphism γis a cochain, γ∈Ck(X;πk−1(F)). This completes the proof of the theorem. /square We now discuss several applications of this obstruction theory. Corollary 4.12.Any fibration p:E→Bover aCW - complex with an aspherical fiber F admits a cross section. Proof. Sinceπq(F) = 0 for all q, by the theorem, there are no obstructions to constructing a cross section inductively on the skeleta of B. /square Proposition 4.13.LetXbe ann- dimensional CW - complex, and let ζbe anm- dimensional vector bundle over X, withm≥n. Thenζhasm−nlinearly independent cross sections. If ξis a d- dimensional complex bundle over X, thenξadmitsd−[n/2]linearly independent cross sections, where [n/2]is the integral part of n/2. . Proof. Letζbe classified by a map fm:X→BO(m). To prove the theorem we need to prove thatfmlifts (up to homotopy) to a map fnX→BO(n). We would then have that ζ∼=f∗ m(γm)∼=f∗ n(γn)⊕/epsilon1m−n whereγkis the universal k- dimensional vector bundle over BO(k), and/epsilon1jrepresents the j- dimensional trivial bundle. These isomorphisms would then produce the m−nlinearly independent cross sections of ζ. overX. Now recall there is a fibration O(m)/O(n)→BO(n)→BO(m). That is, the fiber of p:BO(n)→BO(m) is the quotient space O(m)/O(n). Now by a sim- ple induction argument using 4.10 shows that the fiber O(m)/O(n) isn−1 connected. That is, piq(O(m)/O(n)) = 0 forq≤n−1. By 4.11 all obstructions vanish for lifting the n- skeleton of X 138 4. HOMOTOPY THEORY OF FIBRATIONS to the total space BO(n). Since we are assuming Xisn- dimensional, this completes the proof. The complex case is proved similarly. /square Corollary 4.14.LetXbe a compact, n- dimensional CW complex. Then every element of the reduced real K- theory, ˜KO(X)can be represented by a n- dimensional vector bundle. Every element of the complex K- theory, ˜K(X)can be represented by an [n/2]- dimensional complex vector bundle. Proof. By 2.32 we know ˜KO(X)∼=[X,BO ] and ˜K(X)∼=[X,BU ]. But by the above proposition, any element α∈[X,BO ] lifts to an element αn∈[X,BO (n)] which in turn classifies an n- dimensional real vector bundle representing the ˜KO- classα. Similarly, any element β∈[X,BU ] lifts to an element αn∈[X,BU ([n/2])] which in turn classifies an [ n/2] - dimensional complex vector bundle representing the ˜K- classβ./square We now use this obstruction theory to prove the well known “Whitehead Theorem”, one of the most important foundational theorems in homotopy theory. Theorem 4.15.SupposeXandYareCW - complexes and f:X→Ya continuous map that induces an isomorphism in homotopy groups, f∗:πk(X)∼=−−−−→πk(Y)for allk≥0 Thenf:X→Yis a homotopy equivalence. Proof. By 4.7 we can replace f:X→Yby a homotopy equivalent fibration ˜f:˜X→Y. That is, there is a homotopy equivalence h:X→˜Xso that ˜f◦h=f. Sincefinduces an isomorphism in homotopy groups, so does ˜f. By the exact sequence in homotopy groups for this fibration, this means that the fiber of the fibration ˜f:˜X→Y, i.e the homotopy fiber of f, is aspherical. thus by 4.11 there are no obstructions to finding a lifting ˜ g:Y→˜Xof the identity map ofY. Thus ˜gis a section of the fibration, so that ˜f◦˜g=id:Y→Y. Now leth−1:˜X→Xdenote a homotopy inverse to the homotopy equivalence h. Then if we define g=h−1◦˜g:Y→X 3. OBSTRUCTION THEORY 139 we then have f◦g:Y→Yis given by f◦g=f◦h−1◦˜g =˜f◦h◦h−1◦˜g ∼˜f◦˜g =id:Y→Y. Thusf◦gis homotopic to the identity of Y. To show that g◦fis homotopic to the identity of X, we need to construct a homotopy X×I→Xthat lifts a homotopy X×I→Yfromf◦g◦ftof. This homotopy is constructed inductively on the skeleta of X, and like in the argument proving 4.11, one finds that there are no obstructions in doing so because the homotopy fiber of fis aspherical. We leave the details of this obstruction theory argument to the reader. Thus fandgare homotopy inverse to each other, which proves the theorem. /square The following is an immediate corollary. Corollary 4.16.An aspherical CW - complex is contractible. Proof. IfXis an aspherical CW - complex, then the constant map to a point, /epsilon1:X→pt induces an isomorphism on homotopy groups, and is therefore, by the above theorem, a homotopy equivalence. /square The Whitehead theorem will now allow us to prove the following important relationship between the homotopy type of a topological group and its classifying space. Theorem 4.17.LetGbe a topological group with the homotopy type of a CW complex., and BGits classiftying space. Then there is a homotopy equivalence between Gand the loop space, G/similarequalΩBG. Proof. It was shown in chapter 2 that there is a model for a universal G- bundle,p:EG→BG withEGaG- equivariant CW - complex. In particular, EGis aspherical, and hence by the Whitehead theorem, it is contractible. Let H:EG×I→EG be a contraction. That is, H0:EG×{0}→EGis the constant map at the basepoint e0∈EG, , andH1:EG×{1}→EGis the identity. Composing with the projection map, Φ =p◦H:EG×I→BG 140 4. HOMOTOPY THEORY OF FIBRATIONS is a homotopy between the constant map to the basepoint Φ 0:EG×{0}→b0∈BGand the projection map Φ 1=p:EG×{1}→BG. Consider the adjoint of Φ, ¯Φ :EG→P(BG) ={α:I→BGsuch thatα(0) =b0.} defined by ¯Φ(e)(t) = Φ(e,t)∈BG. Then by definition, the following diagram commutes: EG¯Φ−−−−→ (BG) p/arrowbt/arrowbtq BG =BG whereq(α) =α(1), forα∈P(BG). Thus Φ is a map of fibrations that induces a map on fibers φ:G→ΩBG. Comparing the exact sequences in homotopy groups of these two fibrations, we see that φinduces an isomorphism in homotopy groups. A result of Milnor [ 29] that we will not prove says that if X is aCW complex, then the loop space Ω Xhas the homotopy type of a CW - complex. Then the Whitehead theorem implies that φ:G→ΩBGis a homotopy equivalence. /square 4. Eilenberg - MacLane Spaces In this section we prove a classification theorem for cohomology. Recall that in chapter 2 we proved that there are spaces BGthat classify principal G- bundles over a space X, in the sense that homotopy classes of basepoint preserving maps, [ X,BG ] are in bijective correspondence with isomorphism classes of principal G- bundles. Similarly BO(n) andBU(n) classify real and complex n- dimensional vector vector bundles in this same sense, and BOandBUclassifyK-theory. In this section we show that there are classifying spaces K(G,n) that classify n- dimensional cohomology with coefficients in Gin this same sense. These are Eilenberg - MacLane spaces. We have discussed these spaces earlier in these notes, but in this section we prove their existence and their classification properties. 4.1. Obstruction theory and the existence of Eilenberg - MacLane spaces. In chapter 2 we proved that for any topological group Gthere is a space BGclassifyingGbundles. For G discrete, we saw that BG=K(G,1), an Eilenberg - MacLane space whose fundamental group is G, and whose higher homotopy groups are all zero. In this section we generalize this existence theorem as follows. 4. EILENBERG - MACLANE SPACES 141 Theorem 4.18.LetGbe any abelian group and nan integer with n≥2. Then there exists a spaceK(G,n)with πk(K(G,n)) =  G,ifk=n, 0,otherwise. This theorem will basically be proven using obstruction theory. For this we will assume the following famous theorem of Hurewicz, which we will prove later in this chapter. We first recall the Hurewicz homomorphism from homotopy to homology. Letf: (Dn,Sn−1)→(X,A) represent an element [ f]∈πn(X,A). Letσn∈Hn(Dn,Sn−1)∼=Z be a preferred, fixed generator. Define h([f]) =f∗(σn)∈Hn(X,A). The following is straightforward, and we leave its verification to the reader. Lemma 4.19.The above construction gives a well defined homomorphism h∗:πn(X,A)→Hn(X,A) called the “Hurewicz homomorphism”. The following is the “Hurewicz theorem”. Theorem 4.20.LetXbe simply connected, and let A⊂Xbe a simply connected subspace. Suppose that the pair (X,A)is(n−1)- connected, for n>2.That is, πk(X,A) = 0 ifk≤n−1. Then the Hurewicz homomorphism h∗:πn(X,A)→Hn(X,A)is an isomorphism. We now prove the following basic building block type result concerning how the homotopy groups change as we build a CW - complex cell by cell. Theorem 4.21.LetXbe a simply connected, CW - complex and let f:Sk→X be a map. Let X/primebe the mapping cone of f. That is, X/prime=X∪fDn+1 which denotes the union of Xwith a disk Dn+1glued along the boundary sphere Sk=∂Dk+1viaf. That is we identify t∈Skwithf(t)∈X. Let ι:X /arrowhookleft→X/prime be the inclusion. Then ι∗:πk(X)→πk(X/prime) is surjective, with kernel equal to the cyclic subgroup generated by [f]∈πk(X). 142 4. HOMOTOPY THEORY OF FIBRATIONS Proof. Letg:Sq→X/primerepresent an element in πq(X/prime) withq≤k. By the cellular approx- imation theorem, gis homotopic to a cellular map, and therefore one whose image lies in the q- skeleton of X/prime. But forq≤k, theq- skeleton of X/primeis theq- skeleton of X. This implies that ι∗:πq(X)→πq(X/prime) is surjective for q≤k. Now assume q≤k−1, then ifg:Sq→X⊂X/primeis null homotopic, any null homotopy, i.e extension to the disk G:Dq+1→X/primecan be assumed to be cellular, and hence has image inX. This implies that for q≤k−1,ι∗:πq(X)→πq(X/prime) is an isomorphism. By the exact sequence in homotopy groups of the pair ( X/prime,X), this implies that the pair ( X/prime,X) isk- connected. By the Hurewicz theorem that says that πk+1(X/prime,X)∼=Hk+1(X/prime,X) =Hk+1(X∪fDk+1,X) which, by analyzing the cellular chain complex for computing H∗(X/prime) isZif and only if f:Sk→X is zero in homology, and zero otherwise. In particular, the generator γ∈πk+1(X/prime,X) is represented by the map of pairs given by the inclusion γ: (Dk+1,Sk)/arrowhookleft→(X∪fDk+1,X) and hence in the long exact sequence in homotopy groups of the pair ( X/prime,X), ···→πk+1(X/prime,X)∂∗−−−−→πk(X)ι∗−−−−→πk(X/prime)→··· we have∂∗(γ) = [f]∈πk(X). Thusι∗:πk(X)→πk(X/prime) is surjective with kernel generated by [ f]. This proves the theorem. /square We will now use this basic homotopy theory result to establish the existence of Eilenberg - MacLane spaces. Proof. of 4.18 Fix the group Gand the integer n≥2. Let{γα:α∈A} be a set of generators ofG, whereAdenotes the indexing set for these generators. Let {θβ:β∈B} be a corresponding set of relations. In other words Gis isomorphic to the free abelian group FAgenerated byA, modulo the subgroup RBgenerated by{θβ:β∈B} . Consider the wedge of spheres/logicalortext ASnindexed on the set A. Then by the Hurewicz theorem, πn(/logicalordisplay ASn)∼=Hn(/logicalordisplay ASn)∼=FA. Now the group RBis a subgroup of a free abelian group, and hence is itself free abelian. Let/logicalortext BSn be a wedge of spheres whose nth- homotopy group (which by the Hurewicz theorem is isomorphic to its homology, which is free abelian) is RB. Moreover there is a natural map j:/logicalordisplay BSn→/logicalordisplay ASn 4. EILENBERG - MACLANE SPACES 143 which, on the level of the homotopy group πnis the inclusion RB⊂FA. LetXn+1be the mapping cone ofj: Xn+1=/logicalordisplay ASn∪j/uniondisplay BDn+1 where the disk Dn+1corresponding to a generator in RBis attached via the map Sn→/logicalortext ASn giving the corresponding element in πn(/logicalortext ASn) =FA. Then by using 4.21 one cell at a time, we see thatXn+1is ann−1 - connected space and πn(Xn) is generated by FAmodulo the subgroup RB. In other words, πn(Xn+1)∼=G. Now inductively assume we have constructed an space Xn+kwith πq(Xn+k) =  0 ifq<n, Gifq=nand 0 ifn<q≤n+k−1 Notice that we have begun the inductive argument with k= 1, by the construction of the space Xn+1above. So again, assume we have constructed Xn+k, and we need to show how to construct Xn+k+1with these properties. Once we have done this, by induction we let k→∞ , and clearly X∞ will be a model for K(G,n). Now suppose π=πn+k(Xn+k) is has a generating set {γu:u∈C}, whereCis the indexing set. LetFCbe the free abelian group generated by the elements in this generating set. Let/logicalortext u∈CSn+k u denote a wedge of spheres indexed by this indexing set. Then, like above, by applying the Hurewicz theorem we see that πn+k(/logicalordisplay u∈CSn+k u)∼=Hn+k(/logicalordisplay CSn+k)∼=FC. Let f:/logicalordisplay CSn+k→Xn+k be a map which, when restricted to the sphere Sn+k urepresents the generator γu∈π=πn+k(Xn+k). We defineXn+k+1to be the mapping cone of f: Xn+k+1=Xn+k∪f/uniondisplay u∈CDn+k+1. Then by 4.21 we have that πq(Xn+k)→πq(Xn+k+1) is an isomorphism for q<n +k, and πn+k(Xn+k)→πn+k(Xn+k+1) is surjective, with kernel the subgroup generated by {γu:u∈C}. But since this subgroup gener- atesπ=πn+k(Xn+k) we see that this homomorphism is zero. Since it is surjective, that implies πn+k(Xn+k+1) = 0. Hence Xn+k+1has the required properties on its homotopy groups, and so we have completed our inductive argument. /square 144 4. HOMOTOPY THEORY OF FIBRATIONS 4.2. The Hopf - Whitney theorem and the classification theorem for Eilenberg - MacLane spaces. We now know that the Eilenberg - MacLane spaces K(G,n) exist for every n and every abelian group G, and when n= 1 for every group G. Furthermore, by their construction in the proof of 4.18 they can be chosen to be CW - complexes. In this section we prove their main property, i.e they classify cohomology. In order to state the classification theorem properly, we need to recall the universal coefficient theorem, which says the following. Theorem 4.22.(Universal Coefficient Theorem) Let Gbe an abelian group. Then there is a split short exact sequence 0→Ext(Hn−1(X);G)→Hn(X;G)→Hom (Hn(X),G)→0. Corollary 4.23.IfYis(n−1)- connected, and π=πn(Y), then Hn(Y;π)∼=Hom (π,π). Proof. SinceYis (n−1) connected, Hn−1(Y) = 0, so the universal coefficient theorem says that Hn(Y;π)∼=Hom (Hn(Y),π). But the Hurewicz theorem says that the Hurewicz homomorphism h∗:π=πn(Y)→Hn(Y) is an isomorphism. The corollary follows by combining these two isomorphisms. /square For an (n−1) - connected space Yas above, let ι∈Hn(Y;π) be the class corresponding to the identity map id∈Hom (π,π) under the isomorphism in this corollary. This is called the fundamental class. Given any other space X, we therefore have a set map φ: [X,Y]→Hn(X,π) defined byφ([f]) =f∗(ι)∈Hn(X;π). The classification theorem for Eilenberg - MacLane spaces is the following. Theorem 4.24.Forn≥2andGπany abelian group, let K(π,n)denote an Eilenberg - MacLane space with πn(K(π,n)) =π, and all other homotopy groups zero. Let ι∈Hn(K(π,n);π)be the fundamental class. Then for any CW - complexX, the map φ: [X,K (π,n)]→Hn(X;π) [f]→f∗(ι) is a bijective correspondence. We have the following immediate corollary, giving a uniqueness theorem regarding Eilenberg - MacLane spaces. 4. EILENBERG - MACLANE SPACES 145 Corollary 4.25.LetK(π,n)1andK(π,n)2beCW - complexes that are both Eilenberg - MacLane spaces with the same homotopy groups. Then there is a natural homotopy equivalence betweenK(π,n)1andK(π,n)2. Proof. Letf:K(π,n)1→K(π,n)2be a map whose homotopy class is the inverse image of the fundamental class under the bijection φ: [K(π,n)1,K(π,n)2]∼=−−−−→Hn(K(π,n)1;π)∼=Hom (π,π). This means that f:K(π,n)1→K(π,n)2 induces the identity map in Hom (π,π), and in particular induces an isomorphism on πn. Since all other homotopy groups are zero in both of these complexes, finduces an isomorphism in homotopy groups in all dimensions. Therefore by the Whitehead theorem 4.15, fis a homotopy equivalence. /square We begin our proof of this classification theorem by proving a special case, known as the Hopf - Whitney theorem. This predates knowledge of the existence of Eilenberg - MacLane spaces. Theorem 4.26.(Hopf-Whitney theorem) Let Ybe any (n−1)- connected space with π=πn(Y). LetXbe anyn- dimensional CW complex. Then the map φ: [X,Y]→Hn(X;π) [f]→f∗(ι) is a bijective correspondence. Remark. This theorem is most often used in the context of manifolds, where it implies that if Mn is any closed, orientable manifold the correspondence [Mn,Sn]→Hn(Mn;Z)∼=Z is a bijection. Exercise. Show that this correspondence can alternatively be described as assigning to a smooth mapf:Mn→Snits degree,deg(f)∈Z. Proof. (Hopf - Whitney theorem) We first set some notation. Let Ybe (n−1) - connected, and have basepoint y0∈Y. LetX(m)denote the m- skeleton of the n- dimensional complex X. LetCk(X) =Hk(X(k),X(k−1)) be the cellular k- chains inX. Alternatively, Ck(X) can be thought of as the free abelian group on the k- dimensional cells in the CW - decomposition of X. LetZk(X) 146 4. HOMOTOPY THEORY OF FIBRATIONS andBk(X) denote the subgroups of cocycles and coboundaries respectively. Let Jkbe the indexing set for the set of k- cells in this CW - structure. So that there are attaching maps αk:/logicalordisplay j∈JkSk j→X(k) so that the ( k+ 1) - skeleton X(k+1)is the mapping cone X(k+1)=X(k)∪αk/uniondisplay j∈JkDk+1 j. We prove this theorem in several steps, each translating between cellular cochain complexes or cohomology on the one hand, and homotopy classes of maps on the other hand. The following is the first step. Step 1. There is a bijective correspondence between the following set of homotopy classes of maps of pairs, and the cochain complex with values in π: φ: [(X(n),X(n−1)),(Y,y0)]→Cn(X;π). Proof. A map of pairs f: (X(n),X(n−1))→(Y,y0) is the same thing as a basepoint preserving map from the quotient, f:X(n)/X(n−1)=/logicalordisplay j∈JnSn j→Y. So the homotopy class of fdefines and is defined by an assignment to every j∈Jn, an element [fj]∈πn(Y) =π.But by extending linearly, this is the same as a homomorphism from the free abelian group generated by Jn, i.e the chain group Cn(X), toπ. That is, this is the same thing as a cochain [f]∈Cn(X;π). /square Step 2. The mapφ: [X,Y]→Hn(X;π) is surjective. Proof. . Notice that since Xis ann- dimensional CW - complex, all n- dimensional cochains are cocycles, Cn(X;π) =Zn(X;π). So in particular there is a surjective homomorphism µ:Cn(X;π) =Zn(X;π)→Zn(X;π)/Bn(X;π) =Hn(X;π). A check of the definitions of the maps defined so far yields that the following diagram commutes: [(X(n),X(n−1)),(Y,y0)]φ−−−−→∼=Cn(X;π) ρ/arrowbt/arrowbtµ [X,Y]−−−−→ φHn(X;π) whereρis the obvious restriction map. By the commutativity of this diagram, since µis surjective andφ: [(X(n),X(n−1)),(Y,y0)]→Cn(X;π) is bijective, then we must have that φ: [X,Y]→ Hn(X;π) is surjective, as claimed. /square 4. EILENBERG - MACLANE SPACES 147 In order to show that φis injective, we will need to examine the coboundary map δ:Cn−1(X;π)→Cn(X;π) from a homotopy point of view. To do this, recall that the boundary map on the chain level, ∂: Ck(X)→Ck−1(X) is given by the connecting homomorphism Hn(X(k),X(k−1))→Hk−1(X(k−1),X(k−2)) from the long exact sequence in homology of the triple, ( X(k),X(k−1),X(k−2)). This boundary map can be realized homotopically as follows. Let c(X(k−1)) be the cone on the subcomplex X(k−1), c(X(k−1)) =X(k−1)×I/(X(k−1)×{1}∪{x0}×I), which is obviously a contractible space. Consider the mapping cone of the inclusion X(k−1)/arrowhookleft→X(k), X(k)∪c(X(k−1).By projecting the cone to a point, there is a projection map pk:X(k)∪c(X(k−1))→X(k)/X(k−1)=/logicalordisplay j∈JkSk j which is a homotopy equivalence. ( Note. The fact that this map induces an isomorphism in homology is straight forward by computing the homology exact sequence of the pair ( X(k)∪ c(X(k−1)),X(k)).The fact that this map is a homotopy equivalence is a basic point set topolog- ical property of CW - complexes coming from the so - called “Homotopy Extension Property”. However it can be proved directly, by hand, in this case. We leave its verification to the reader.) Let uk:X(k)→/logicalordisplay j∈JkSk j be the composition X(k)/arrowhookleft→X(k)∪c(X(k−1))pk−−−−→X(k)/X(k−1)=/logicalortext j∈JkSk j. Then the composition of ukwith the attaching map αk+1:/logicalordisplay j∈Jk+1Sk j→X(k) (whose mapping cone defines the ( k+ 1) - skeleton X(k+1)), is a map between wedges of k- spheres, dk+1:/logicalortext j∈Jk+1Sk jαk+1−−−−→X(k)uk−−−−→/logicalortext j∈JkSk j. The following is immediate from the definitions. Step 3. The induced map in homology, (dk+1)∗:Hk(/logicalordisplay j∈Jk+1Sk j)→Hk(/logicalordisplay j∈JkSk j) Ck+1(X)→Ck(X) is the boundary homomorphism in the chain complex ∂k+1:Ck+1(X)→Ck(X). Now consider the map [(X(n),X(n−1)),(Y,y0)]φ−−−−→∼=Cn(X;π) =Zn(X;π)µ−−−−→Hn(X;π). 148 4. HOMOTOPY THEORY OF FIBRATIONS We then have the following corollary. Step 4. A mapf:X(n)/X(n−1)=/logicalortext j∈JnSn j→Yhas the property that µ◦φ([f]) = 0∈Hn(X;π) if and only if there is a map fn−1:/logicalordisplay j∈J(n−1)Sn j→Y so thatfis homotopic to the composition /logicalortext j∈JnSn jdn−−−−→/logicalortext j∈Jn−1Sn jfn−1−−−−→Y. Proof. Sinceφ: [(X(n),X(n−1)),(Y,y0)]→Cn(X;π) =Zn(X;π) is a bijection, µ◦φ([f]) = 0 if and only if φ([f]) is in the image of the coboundary map. The result then follows from step 3. /square Step 5. The composition X(n)un−−−−→/logicalortext j∈JnSn jdn−−−−→/logicalortext j∈Jn−1Sn j is null homotopic. Proof. The mapunwas defined by the composition X(n)/arrowhookleft→X(n)∪c(X(n−1)pn−−−−→ /similarequal/logicalortext j∈JnSn j. But notice that if we take the quotient X(n)∪c(X(n−1)/X(n)we get the suspension X(n)∪c(X(n−1)/X(n)= ΣX(n−1). Furthermore, the map between the wedges of the spheres, dn:/logicalortext j∈JnSn j→/logicalortext j∈Jn−1Sn jis directly seen to be the composition dn:/logicalortext j∈JnSn j/similarequalX(n)∪c(X(n−1)proj.−−−−→X(n)∪c(X(n−1)/X(n)= ΣX(n−1)Σun−1−−−−→/logicalortext j∈Jn−1Sn j. Thus the composition dn◦un:X(n)→/logicalortext j∈JnSn j→/logicalortext j∈Jn−1Sn jfactors as the composition X(n)/arrowhookleft→X(n)∪c(X(n−1)proj.−−−−→X(n)∪c(X(n−1)/X(n)= ΣX(n−1)Σun−1−−−−→/logicalortext j∈Jn−1Sn j. But the composite of the first two terms in this composition, X(n)/arrowhookleft→X(n)∪c(X(n−1)proj.−−−−→X(n)∪c(X(n−1)/X(n) is clearly null homotopic, and hence so is dn◦un. /square We now complete the proof of the theorem by doing the following step. Step. 6. The correspondence φ: [X,Y]→Hn(X;π) is injective. 4. EILENBERG - MACLANE SPACES 149 Proof. Letf, g:X→Ybe maps with φ([f]) =φ([g])∈Hn(X;π).SinceY- is (n−1) - connected, given any map h:X→Y, the restriction to its ( n−1) - skeleton is null homotopic. (Exercise. Check this!) Null homotopies define maps ˜f,˜g:X∪c(X(n−1))→Y given byfandgrespectively on X, and by their respective null homotopies on the cones, c(X(n−1)). Using the homotopy equivalence pn:X(n)∪c(X(n−1))/similarequalX(n)/X(n−1)=/logicalortext j∈JnSn j, we then have maps ¯f,¯g:X(n)/X(n−1)→Y which, when composed with the projection X=X(n)→X(n)/X(n−1)are homotopic to fand grespectively. Now by the commutativity of the diagram in step 2, since φ([f]) =φ([g]), then µ◦φ([¯f]) =µ◦φ([¯g]). Or equivalently, µ◦φ([¯f]−[¯g]) = 0 where we are using the fact that [(X(n),X(n−1)),Y] = [/logicalordisplay j∈JnSn j,Y] =⊕j∈Jnπn(Y) is a group, and maps to Cn(X;π) is a group isomorphism. Letψ:X(n)/X(n−1)→Yrepresent [ ¯f]−[¯g]∈[/logicalortext j∈JnSn j,Y].Thenµ◦φ(ψ) = 0. Then by step 4, there is a map ψn−1:/logicalortext j∈Jn−1Sn j→Yso thatψn−1◦dnis homotopic to ψ. Thus the composition Xproj.−−−−→X/X(n−1)ψ−−−−→Y is homotopic to the composition X→X(n)/Xn−1=/logicalortext j∈JnSn jdn−−−−→/logicalortext j∈Jn−1Sn jψn−1−−−−→Y. But by step 5, this compostion is null homotopic. Now since ψrepresents [ ¯f]−[¯g], a null homotopy of the composition Xproj.−−−−→X/X(n−1)ψ−−−−→Y defines a homotopy between the compositions Xproj.−−−−→X/X(n−1)¯f−−−−→YandXproj.−−−−→X/X(n−1)¯g−−−−→Y. The first of these maps is homotopic to f:X→Y, and the second is homotopic to g:X→Y. Hencef/similarequalg, which proves that φis injective. /square We now know that the correspondence φ: [X,Y]→Hn(X;π) is surjective (step 2) and injective (step 6). This completes the proof of this theorem. /square We now proceed with the proof of the main classification theorem for cohomology, using Eilen- berg - MacLane spaces ( 4.24). 150 4. HOMOTOPY THEORY OF FIBRATIONS Proof. The Hopf Whitney theorem proves this theorem when Xis ann- dimensional CW - complex. We split the proof for general CW - complexes into two cases. Case 1.Xisn+ 1 - dimensional. Consider the following commutative diagram (4.1)[X,K (π,n)]φ−−−−→Hn(X;π) ρ/arrowbt/arrowbtρ [X(n),K(π,n)]φn−−−−→∼=Hn(X(n);π) where the vertical maps ρdenote the obvious restriction maps, and φndenotes the restriction of the correspondence φto then- skeleton, which is an isomorphism by the Hopf - Whitney theorem. Now by considering the exact sequence for cohomology of the pair ( X,X(n)) = (X(n+1),X(n)), one sees that the restriction map ρ:Hn(X,π)→Hn(X(n),π) is injective. Using this together with the fact that φnis an isomorphism and the commutativity of this diagram, one sees that to show thatφ: [X,K (π,n)]→Hn(X;π) is surjective, it suffices to show that for γ∈Hn(X,π) with ρ(γ) =φn([fn]), wherefn:X(n)→K(π,n), thenfncan be extended to a map f:X→K(π,n). Using the same notation as was used in the proof of the Hopf - Whitney theorem, since X= X(n+1), we can write X=X(n)∪αn+1/uniondisplay j∈Jn+1D(n+1) whereαn+1:/logicalortext j∈Jn+1Sn j→X(n)is the attaching map. Thus the obstruction to finding an extension f:X→K(π,n) of the map fn:X(n)→K(π,n), is the compostion /logicalortext j∈Jn+1Sn jαn+1−−−−→X(n)fn−−−−→K(π,n). Now since/logicalortext j∈Jn+1Sn jisn- dimensional, the Hopf - Whitney theorem says that this map is determined by its image under φ, φ([fn◦αn+1])∈Hn(/logicalordisplay j∈Jn+1Sn j;π). But this class is α∗ n+1(φ([fn])), which by assumption is α∗ n+1(ρ(γ)). But the composition Hn(X;π)ρ−−−−→Hn(X(n),π)α∗ n+1−−−−→Hn(/logicalortext j∈Jn+1Sn j;π) are two successive terms in the long exact sequence in cohomology of the pair ( X(n+1),X(n)) and is therefore zero. Thus the obstruction to finding the extension f:X→K(π,n) is zero. As observed above this proves that φ: [X,K (π,n)]→Hn(X;π) is surjective. We now show that φis injective. So suppose φ([f]) =φ([g]) forf, g:X→K(π,n). To prove thatφis injective we need to show that this implies that fis homotopic to g. Letfnandgnbe the restrictions of fandgtoX(n). That is, fn=ρ([f]) :X(n)→K(π,n) andgn=ρ([g]) :X(n)→K(π,n) 4. EILENBERG - MACLANE SPACES 151 Now by the commutativity of diagram 4.1 and the fact that φnis an isomorphism, we have that fn andgnare homotopic maps. Let Fn:X(n)×I→K(π,n) be a homotopy between them. That is, F0=fn:X(n)×{0}→K(π,n) andF1=gn:X(n)×{1}→ K(π,n).This homotopy defines a map on the ( n+ 1) -subcomplex of X×Idefined to be ˜F: (X×{0})∪(X×{1})∪X(n)×I→K(π,n) where ˜Fis defined to be fandgonX×{0}andX×{1}respectively, and FonX(n)×I. But sinceXis (n+ 1) - dimensional, X×Iis (n+ 2) - dimensional, and this subcomplex is its ( n+ 1) - skeleton. So X×Iis the union of this complex with ( n+ 2) - dimensional disks, attached via maps from a wedge of ( n+ 1) - dimensional spheres. Hence the obstruction to extending ˜Fto a mapF:X×I→K(π,n) is a cochain in Cn+2(X×I;πn+1(K(π,n)). But this group is zero since πn+1(K(π,n)) = 0. Thus there is no obstruction to extending ˜Fto a mapF:X×I→K(π,n), which is a homotopy between fandg. As observed before this proves that φis injective. This completes the proof of the theorem in this case. General Case. Since, by case 1, we know the theorem for ( n+1) - dimensional CW- complexes, we assume that the dimension of Xis≥n+ 2. Now consider the following commutative diagram: [X,K (π,n)]φ−−−−→Hn(X;π) ρ/arrowbt/arrowbtρ [X(n+1),K(π,n)]φn+1−−−−→∼=Hn(X(n+1);π) where, as earlier, the maps ρdenote the obvious restriction maps, and φn+1denotes the restriction ofφto the (n+ 1) skeleton, which we know is an isomorphism, by the result of case 1. Now in this case the exact sequence for the cohomology of the pair ( X,X(n+1)) yields that the restriction map ρ:Hn(X;π)→Hn(X(n+1),π) is an isomorphism. Therefore by the commutativity of this diagram, to prove that φ: [X,K (π,n)]→Hn(X;π) is an isomorphism, it suffices to show that the restriction map ρ: [X,K (π,n)]→[X(n+1),K(π,n)] is a bijection. This is done by induction on the skeleta X(K)ofX, withK≥n+ 1. To complete the inductive step, one needs to analyze the obstructions to extending maps X(K)→K(π,n) toX(K+1) or homotopies X(K)×I→K(π,n) toX(K+1)×I, like what was done in the proof of case 1. However in these cases the obstructions will always lie is spaces of cochains with coefficients in πq(K(π,n)) withq=KorK+ 1, and so q≥n+ 1. But then πq(K(π,n)) = 0 and so these obstructions will always vanish. We leave the details of carrying out this argument to the reader. /square 152 4. HOMOTOPY THEORY OF FIBRATIONS 5. Spectral Sequences One of the great achievements of Algebraic Topology was the development of spectral sequences. They were originally invented by Leray in the late 1940’s and since that time have become funda- mental calculational tools in many areas of Geometry, Topology, and Algebra. One of the earliest and most important applications of spectral sequences was the work of Serre [ 36] for the calculation of the homology of a fibration. We divide our discussion of spectral sequences in these notes into three parts. In the first section we develop the notion of a spectral sequence of a filtration. In the next section we discuss the Leray - Serre spectral sequence for a fibration. In the final two sections we discuss applications: we prove the Hurewicz theorem, calculate the cohomology of the Lie groupsU(n), andO(n), and of the loop spaces Ω Sn, and we discuss Spin andSpin C- structures on manifolds. 5.1. The spectral sequence of a filtration. A spectral sequence is the algebraic machinery for studying sequences of long exact sequences that are interelated in a particular way. We begin by illustrating this with the example of a filtered complex. LetC∗be a chain complex, and let A∗⊂C∗be a subcomplex. The short exact sequence of chain complexes 0−→A∗/arrowhookleft→C∗−→C∗/A∗−→0 leads to a long exact sequence in homology: −→···−→ Hq+1(C∗,A∗)−→Hq(A∗)−→Hq(C∗)−→Hq(C∗,A∗)−→Hq−1(A∗)−→··· This is useful in computing the homology of the big chain complex, H∗(C∗) in terms of the homology of the subcomplex H∗(A∗) and the homology of the quotient complex H∗(C∗,A∗). A spectral sequence is the machinery used to study the more general situation when one has a filtration of a chain complex C∗by subcomplexes 0 =F0(C∗)/arrowhookleft→F1(C∗)/arrowhookleft→···/arrowhookleft→Fk(C∗)/arrowhookleft→Fk+1(C∗)/arrowhookleft→···/arrowhookleft→C∗=/uniondisplay kFk(C∗). LetDk ∗be the subquotient complex Dk ∗=Fk(C∗)/Fk−1(C∗) and so for each kthere is a long exact sequence in homology −→Hq+1(Dk ∗)−→Hq(Fk−1(C∗))−→Hq(Fk(C∗))−→Hq(Dk ∗)−→··· By putting these long exact sequences together, in principle one should be able to use information about⊕kH∗(Dk ∗) in order to obtain information about H∗(C∗) = lim−→kH∗(Fk(C∗)). 5. SPECTRAL SEQUENCES 153 A spectral sequence is the bookkeeping device that allows one to do this. To be more specific, consider the following diagram. (5.1)0 0 /arrowbti/arrowbti Hq(F1(C∗)) Hq−1(F1(C∗))=−−−−→Hq−1(D1 ∗) /arrowbti/arrowbti ...... /arrowbti/arrowbti Hq(Fk−p(C∗))j−−−−→Hq(Dk−p ∗)∂−−−−→Hq−1(Fk−p−1(C∗))j−−−−→Hq−1(Dk−p−1 ∗ ) /arrowbti/arrowbti ... Hq−1(Fk−p(C∗))j−−−−→Hq−1(Dk−p ∗) /arrowbti/arrowbti ...... /arrowbti/arrowbti Hq(Fk−2(C∗)) Hq−1(Fk−3(C∗)) /arrowbti/arrowbti Hq(Fk−1(C∗))j−−−−→Hq(Dk−1 ∗)∂−−−−→Hq−1(Fk−2(C∗))j−−−−→Hq−1(Dk−2 ∗)) /arrowbti/arrowbti Hq(Fk(C∗))j−−−−→Hq(Dk ∗)∂−−−−→Hq−1(Fk−1(C∗))j−−−−→Hq−1(Dk−1 ∗)) /arrowbti/arrowbti ...... /arrowbti/arrowbti Hq(C∗) Hq−1(C∗) The columns represent the homology filtration of H∗(C∗) and the three maps ∂,j, andicombine to give long exact seqences at every level. Letα∈Hq(C∗). We say that αhasalgebraic filtration k, ifαis in the image of a class αk∈Hq(Fk(C∗)) but is not in the image of Hq(Fk−1(C∗)). In such a case we say that the image 154 4. HOMOTOPY THEORY OF FIBRATIONS j(αk)∈Hq(Dk ∗) is a representative ofα. Notice that this representative is not unique. In particular we can add any class in the image of d1=j◦∂:Hq+1(Dk+1 ∗)−→Hq(Dk ∗) toj(αk) and we would still have a representative of α∈Hq(C∗) under the above definition. Conversely, let us consider when an arbitrary class β∈Hq(Dk ∗) represents a class in Hq(C∗). By the exact sequence this occurs if and only if the image ∂(β) = 0, for this is the obstruction to β being in the image of j:Hq(Fk(C∗))→Hq(Dk ∗) and ifj(˜β) =βthenβrepresents the image i◦···◦i(˜β)∈Hq(C∗). Now∂(β) = 0 if and only if it lifts all the way up the second vertical tower in diagram 5.1 The first obstruction to this lifting, (i.e the obstruction to lifting ∂(β) toHq−1(Fk−2(C∗)) is that the composition d1=j◦∂:Hq(Dk)−→Hq−1Dk−1 ∗) mapsβto zero. That is elements of Hq(C∗) are represented by elements in the subquotient ker(d1)/Im(d1) ofHq(Dk ∗). We use the following notation to express this. We define Er,s 1=Hr+s(Dr ∗) and define d1=j◦∂:Er,s 1−→Er−1,s 1. ris said to be the algebraic filtration of elements in Er,s 1andr+sis the total degree of elements inEr,s 1. Since∂◦j= 0, we have that d1◦d1= 0 and we let Er,s 2=Ker(d1:Er,s 1→Er−1,s 1 )/Im(d1:Er+1,s 1→Er,s 1) be the resulting homology group. We can then say that the class α∈Hq(C∗) has as its representative, the classαk∈Ek,q−k 2 . Now let us go back and consider further obstructions to an arbitrary class β∈Ek,q−k 2 repre- senting a class in Hq(C∗). Represent βas a cycle in E1:β∈Ker(d1=j◦∂∈Hq(Dk ∗)).Again,β represents a class in Hq(C∗) if and only if ∂(β) = 0. Now since j◦∂(β) = 0,∂(β)∈Hq−1(Fk−1(C∗)) lifts to a class, say ˜β∈Hq−1Fk−2(C∗). Remember that the goal was to lift ∂(β) all the way up the vertical tower (so that it is zero). The obstruction to lifting it the next stage, i.e to Hq−1(Fk−3(C∗)) is thatj(˜β)∈Hq−1(Dk−2 ∗) is zero. Now the fact that a d1cycleβhas the property that ∂(β) lifts toHq−1Fk−2(C∗)) allows to define a map d2:Ek,q−k 2−→Ek−2,q−k+1 2 5. SPECTRAL SEQUENCES 155 and more generally, d2:Er,s 2−→Er−2,s+1 2 by composing this lifting with j:Hs+r−1(Fr−2(C∗))−→Hs+r−1(Dr−2 ∗). That is,d2=j◦i−1◦∂. It is straightforward to check that d2:Er,s 2−→Er−2,s+1 2 is well defined, and that elements of Hq(C∗) are actually represented by elements in the subquotient homology groups ofE∗,∗ 2: Er,s 3=Ker(d2:Er,s 2→Er−2,s+1 2 )/Im(d2:Er+2,s−1 2→Er,s 1) Inductively, assume the subquotient homology groups Er,s jhave been defined for j≤p−1 and differentials dj:Er,s j−→Er−j,s+j−1 j defined on representative classes in Hr+s(Dr ∗) to be the composition dj=j◦(ij−1=i◦···◦i)−1◦∂ so thatE∗,∗ j+1is the homology Ker(dj)/Im(dj). We then define Er,s p=Ker(dp−1:Er,s p−1→Er−p+1,s+p−2 p−1 )/Im(dp−1:Er+p−1,s−p+2 p−1→Er,s p−1). ThusEk,q−k p is a subquotient of Hq(Dk ∗), represented by elements βso that∂(β) lifts to Hq(Fk−p(C∗)). That is, there is an element ˜β∈Hq(Fk−p(C∗)) so that ip−1(˜β) =∂(β)∈Hq−1(Fk−1(C∗)). The obstruction to ˜βlifting toHq−1(Fk−p−1(C∗)) isj(β)∈Hq(Dk−p ∗). This procedure yields a well defined map dp:Er,s p−→Er−p,s+p−1 p given byj◦(ip−1)−1◦∂on representative classes in Hq(Dk ∗). This completes the inductive step. Notice that if we let Er,s ∞= lim−→pEr,s p thenEk,q−k ∞ is a subquotient of Hq(Dk ∗) consisting of precisely those classes represented by elements β∈Hq(Dk ∗) so that∂(β) lifts all the way up the vertical tower i.e ∂(β) is in the image of ipfor allp. This is equivalent to the condition that ∂(β) = 0 which as observed above is precisely the condition necessary for βto represent a class in Hq(C∗). These observations can be made more precise as follows. 156 4. HOMOTOPY THEORY OF FIBRATIONS Theorem 4.27.LetIr,s=Image (Hr+s(Fr(C∗))−→Hr+s(C∗)).ThenEr,s ∞is isomorphic to the quotient group Er,s ∞∼=Ir,s/Ir−1,s+1. Thus theE∗,∗ ∞determines H∗(C∗)up to extensions. In particular, if all homology groups are taken with field coefficients we have Hq(C∗)∼=/circleplusdisplay r+s=qEr,s ∞. In this case we say that {Er,s p,dp}is a spectral sequence starting at Er,s 1=Hr+s(Dr ∗), and converging to Hr+s(C∗). Often times a filtration of this type occurs when one has a topological space Xfiltered by subspaces, ∗=X0/arrowhookleft→X1/arrowhookleft→···/arrowhookleft→Xk/arrowhookleft→Xk+1/arrowhookleft→···/arrowhookleft→X. An important example is the filtration of a CW - complexXby its skeleta, Xk=X(k). We get a spectral sequence as above by applying the homology of the chain complexes to this topological filtration. This spectral sequence converges to H∗(X) withE1termEr,s 1=Hr+s(Xr,Xr−1). From the construction of this spectral sequence one notices that chain complexes are irrelevant in this case; indeed all one needs is the fact that each inclusion Xk−1/arrowhookleft→Xkinduces a long exact sequence in homology. Exercise. Show that in the case of the filtration of a CW - complexXby its skeleta, that the E1 -term of the corresponding spectral sequence is the cellular chain complex, and the E2- term is the homology of X, Er,s 2=  Hr(X),ifs= 0 0 otherwise Furthermore, show that this spectral sequence “collapses” at the E2level, in the sense that Er,s p=Er,s 2for allp≥2 and hence Er,s ∞=Er,s 2. Now ifh∗(−) is any generalized homology theory (that is, a functor that obeys all the Eilenberg - Steenrod axioms but dimension) then the inclusions of a filtration as above Xk−1/arrowhookleft→Xkinduce long exact sequences in h∗(−), and one gets, by a procedure completely analogous to the above, a spectral sequence converging to h∗(X) withE1term Er,s 1=hr+s(Xr,Xr−1). 5. SPECTRAL SEQUENCES 157 Again, for the skeletal filtration of a CW complex, this spectral sequence is called the Atiyah - Hirzebruch spectral sequence for the generalized homology h∗. Exercise. Show that the E2-term of the Atiyah - Hirzebruch spectal sequence for the generalized homology theory h∗is Er,s 2=hr+s(Sr)⊗Hr(X). Particularly important examples of such generalized homology theories include stable homotopy (∼=framed bordism ), other bordism theories, and K- homology theory. Similar spectral sequences also exist for cohomology theories. The reader is referred to [ 27] for a good general reference on spectral sequences with many examples of those most relevant in Algebraic Topology. 5.2. The Leray - Serre spectral sequence for a fibration. The most important example of a spectral sequence from the point of view of these notes is the Leray - Serre spectral sequence of a fibration. Given a fibration F→E→B, the goal is to understand how the homology of the three spaces (fiber, total space, base space) are related. In the case of a trivial fibration, E=B×F→B, the answer to this question is given by the Kunneth formula, which says, that when taken with field coefficients, H∗(B×F;k)∼=H∗(B;k)⊗kH∗(F:k), wherekis the field. Whenp:E→Bis a nontrivial fibration, one needs a spectral sequence to study the homology. The idea is to construct a filtration on a chain complex C∗(E) for computing the homology of the total space E, in terms of the skeletal filtration of a CW - decomposition of the base space B. Assume for the moment that p:E→Bis a fiber bundle with fiber F. For the purposes of our discussion we will assume that the base space Bis simply connected. Let B(k)be thek- skeleton ofB, and define E(k) =p−1(B(k))⊂E. We then have a filtration of the total space Eby subspaces ∗/arrowhookleft→E(0)/arrowhookleft→E(1)/arrowhookleft→···/arrowhookleft→E(k)/arrowhookleft→E(k+ 1)/arrowhookleft→···/arrowhookleft→E. To analyze the E1- term of the associated homology spectral spectral sequence we need to compute theE1- term,Er,s 1=Hr+s(E(r),E(r−1)). To do this, write the skeleta of Bin the form B(r)=B(r−1)∪/uniondisplay j∈JrDr j. Now since each cell Dris contractible, the restriction of the fibration Eto the cells is trivial, and so E(r)−E(r−1)∼=/uniondisplay j∈JrDr×F. 158 4. HOMOTOPY THEORY OF FIBRATIONS Moreover the attaching maps are via the maps ˜αr:/logicalordisplay j∈JrSr−1 j×F→E(r−1) induced by the cellular attaching maps αk:/logicalortext j∈JkSk−1 j→B(k−1). Using the Mayer - Vietoris sequence, one then computes that Er,s 1=Hr+s(E(r),E(r−1)) =Hr+s(/uniondisplay j∈JrDr×F,/uniondisplay j∈JrSr−1×F) =Hr+s(/logicalordisplay j∈JrSr×F,F) =Hr(/logicalordisplay j∈JrSr)⊗Hs(F) =Cr(B;Hs(F)). These calculations indicate the following result, due to Serre in his thesis [ 36]. We refer the reader to that paper for details. It is one of the great pieces of mathematics literature in the last 50 years. Theorem 4.28.Letp:E→Bbe a fibration with fiber F. Assume that Fis connected and B is simply connected. Then there are chain complexes C∗(E)andC∗(B)computing the homology of EandBrespectively, and a filtation of C∗(E)leading to a spectral sequence converging to H∗(E) with the following properties: (1)Er,s 1=Cr(B)⊗Hs(F) (2)Er,s 2=Hr(B;Hs(F)) (3)The differential djhas bidegree (−j,j−1): dj:Er,s j→Er−j,s+j−1 j. (4)The inclusion of the fiber into the total space induces a homomorphism i∗:Hn(F)→Hn(E) which can be computed as follows: i∗:Hn(F) =E0,n 2→E0,n ∞⊂Hn(E) whereE0,n 2→E0,n ∞is the projection map which exists because all the differentials djare zero onE0,n j. 5. SPECTRAL SEQUENCES 159 (5)The projection map induces a homomorphism p∗:Hn(E)→Hn(B) which can be computed as follows: Hn(E)→En,0 ∞⊂En,0 2=Hn(B) whereEn,0 ∞includes into En,0 2as the subspace consisting of those classes on which all differentials are zero. This is well defined because no class in En,0 jcan be a boundary for anyj. Remark. The theorem holds when the base space is not simply connected also. However in that case the E2-term is homology with “twisted coefficients”. This has important applications in many situations, however we will not consider this issue in these notes. Again, we refer the reader to Serre’s thesis [ 36] for details. We will finish this chapter by describing several applications of this important spectal sequence. The first, due to Serre himself [ 36], is the use of this spectral sequence to prove that even though fibrations do not, in general, admit long exact sequences in homology, they do admit exact sequences in homology through a range of dimensions depending on the connectivity of the base space and fiber. Theorem 4.29.Letp:E→Bbe a fibration with connected fiber F, whereBis simply connected andHi(B) = 0 for0<i<n , andHi(F) = 0 fori<i<m . Then there is an exact sequence Hn+m−1(F)i∗−−−−→Hn+m−1(E)p∗−−−−→Hn+m−1(B)τ−−−−→Hn+m−2(F)→···→H1(E)→0 Proof. TheE2-term of the Serre spectral sequence is given by Er,s 2=Hr(B;Hs(F)) which, by hypothesis is zero for 0 <r<n or 0<j <m . Letq<n +m. Then this implies that the composition series for Hq(E), given by the filtration defining the spectral sequence, reduces to the short exact sequence 0→E0,q ∞→Hq(E)→Eq,0 ∞→0. Now in general, for these “edge terms”, we have Eq,0 ∞=kernel{dq:Eq,0 q→E0,q−1 q}and E0,q ∞=coker{dq:Eq,0 q→E0,q−1 q}. But whenq<n +m, we haveEq,0 q=Eq,0 2=Hq(B) andE0,q−1 q =E0,q−1 2 =Hq−1(F) because there can be no other differentials in this range. Thus if we define τ:Hq(B)→Hq−1(F) 160 4. HOMOTOPY THEORY OF FIBRATIONS to bedq:Eq,0 q→E0,q−1 q , forq <n +m, we then have that p∗:Hq(E)→Hq(B) maps surjectively onto the kernel of τ, and ifq<n +m−1, then the kernel of p∗is the cokernel of τ:Hq+1(B)→Hq(F). This establishes the existence of the long exact sequence in homology in this range. /square Remark. The homomorphism τ:Hq(B)→Hq−1(F) forq<n +min the proof of this theorem is called the “transgression” homomorphism. 5.3. Applications I: The Hurewicz theorem. As promised earlier in this chapter, we now use the Serre spectral sequence to prove the Hurewicz theorem. The general theorem is a theorem comparing relative homotopy groups with relative homology groups. We begin by proving the theorem comparing homotopy groups and homology of a single space. Theorem 4.30.LetXbe ann−1- connected space, n≥2. That is, we assume πq(X) = 0 for q≤n−1. ThenHq(X) = 0 forq≤n−1and the previously defined “Hurewicz homomorphism” h:πn(X)→Hn(X) is an isomorphism. Proof. We assume the reader is familiar with the analogue of the theorem when n= 1, which says that for Xconnected, the first homology group H1(X) is given by the abelianization of the fundamental group h:π1(X)/[π1,π1]∼=H1(X) where [π1,π1]⊂π1(X) is the commutator subgroup. We use this preliminary result to begin an induction argument to prove this theorem. Namely we assume that the theorem is true for n−1 replacing nin the statement of the theorem. We now complete the inductive step. By our inductive hypotheses, Hi(X) = 0 fori≤n−2 andπn−1(X)∼=Hn−1(X).But we are assuming that πn−1(X) = 0. Thus we need only show that h:πn(X)→Hn(X) is an isomorphism. Consider the path fibration p:PX→Xwith fiber the loop space Ω X. Nowπi(ΩX)∼=πi+1(X), and soπi(ΩX) = 0 fori≤n−2. So our inductive assumption applied to the loop space says that h:πn−1(ΩX)→Hn−1(ΩX) is an isomorphism. But πn−1(ΩX) =πn(X). Also, by the Serre exact sequence applied to this fibration, using the facts that (1) the total space PXis contractible, and (2) the fiber Ω Xisn−2 - connected and the base space Xis (n−1) - connected 5. SPECTRAL SEQUENCES 161 we then conclude that the transgression, τ:Hn(X)→Hn−1(ΩX) is an isomorphism. Hence the Hurewicz map h:πn−1(ΩX)→Hn−1(ΩX) is the same as the Hurewicz map h:πn(X)→Hn(X), which is therefore an isomorphism. /square We are now ready to prove the more general relative version of this theorem 4.20 Theorem 4.31.LetXbe simply connected, and let A⊂Xbe a simply connected subspace. Suppose that the pair (X,A)is(n−1)- connected, for n>2.That is, πk(X,A) = 0 ifk≤n−1. Then the Hurewicz homomorphism h∗:πn(X,A)→Hn(X,A)is an isomorphism. Proof. . Replace the inclusion ι:A/arrowhookleft→X. by a homotopy equivalent fibration ˜ ι:˜A→Xas in 4.7. Let Fιbe the fiber. Then πi(Fι)∼= πi+1(X,A), by comparing the long exact sequences of the pair ( X,A) to the long exact sequence in homotopy groups for the fibration ˜A→X. So by the Hurewicz theorem 4.30 we know that πi(F) =Hi(F) = 0 fori≤n−2 and h:πn−1(F)→Hn−1(F) is an isomorphism. But as mentioned, πn−1(F)∼=πn(X,A) and by comparing the homology long exact sequence of the pair ( X,A) to the Serre exact sequence for the fibration F→˜A→B, one has thatHn−1(F)∼=Hn(X,A). The theorem follows. /square As a corollary, we obtain the following strengthening of the Whitehead theorem 4.15 which is quite useful in calculations. Corollary 4.32.SupposeXandYare simply connected CW - complexes and f:X→Ya continuous map that induces an isomorphism in homology groups, f∗:Hk(X)∼=−−−−→Hk(Y)for allk≥0 Thenf:X→Yis a homotopy equivalence. 162 4. HOMOTOPY THEORY OF FIBRATIONS Proof. Replacef:X→Yby the inclusion into the mapping cylinder ¯f:X /arrowhookleft→¯Y where ¯Y=Y∪fX×Iwhich is homotopy equivalent to Y, and ¯fincludesXinto ¯YasX×{1}. SinceXandYare simply connected, we have that π2(X)∼=H2(X) andπ2(Y)∼=H2(Y). Thus f∗:π2(X)→π2(Y) is an isomorphism. Again, since XandYare simply connected, this implies thatπq(¯Y,X ) = 0 forq= 1,2. Thus we can apply the relative Hurewicz theorem. However since f∗:Hk(X)∼=Hk(Y) for allk≥0, we have that Hk(¯Y,X ) = 0 for all k≥0. But then the Hurewicz theorem implies that πk(¯Y,X ) = 0 for all k, which in turn implies that f∗:πk(X)→πk(Y) is an isomorphism for all k. The theorem follows from the Whitehead theorem 4.15. /square 5.4. Applications II: H∗(ΩSn)andH∗(U(n)).In this section we will use the Serre spectral sequence to compute the homology of the loop space Ω Snand the cohomology ring of the Lie groups, H∗(U(n)). Theorem 4.33. Hq(ΩSn) =  Zifqis a multiple of n−1, i.eq=k(n−1) 0otherwise Proof. ΩSnis the fiber of the path fibration p:PSn→Sn. Since the total space of this fibration is contractible, the Serre spectral sequence converges to 0 in positive dimensions. That is, Er,s ∞= 0 for allr,s, except that E0,0 ∞=Z. Now since the base space, Snhas nonzero homology only in dimensions 0 and n(when it is Z), then Er,s 2=Hr(Sn;Hs(ΩSn)) is zero unless r= 0 orn. In particular, since dq:Er,s q→Er−q,s+q−1 q , we must have that for q<n , dq= 0. ThusEr,s 2=Er,s nand the only possible nonzero differential dnoccurs in dimensions dn:En,s n→E0,s+n−1 n. It is helpful to picture this spectral sequence as in the following diagram, where a dot in the ( r,s) - entry denotes a copy of the integers in Er,s n=Hr(Sn;Hs(ΩSn)). Notice that if the generator σn,0∈En,0 nis in the kernel of dn, then it would represent a nonzero class inEn,0 n+1. Butdn+1and all higher differentials on En,0 n+1must be zero, for dimensional reasons. That is,En,0 n+1=En,0 ∞. But we saw that En,0 ∞= 0. Thus we must conclude that dn(σn,0)/negationslash= 0. For 5. SPECTRAL SEQUENCES 163 dn dn dn n-12(n-1)3(n-1) r ns E nr,s the same reasoning, (i.e the fact that En,0 n+1= 0) we must have that dn(kσn,0)/negationslash= 0 for all integers k. This means that the image of dn:En,0 n→E0,n−1 n isZ⊂E0,n−1 n =Hn−1(ΩSn).On the other hand, we claim that dn:En,0 n→E0,n−1 n must be surjective. For if α∈E0,n−1 n is not in the image of dn, then it represents a nonzero class in E0,n−1 n+1=E0,n−1 ∞ . But as mentioned earlier E0,n−1 ∞ = 0. Sodnis surjective as well. In fact we have proven that dn:Z=Hn(Sn) =En,0 n→E0,n−1 n =E0,n−1 2 =Hn−1(ΩSn) is an isomorphism. Hence Hn−1(ΩSn)∼=Z, as claimed. Now notice this calculation implies a calculation of En,n−1 2 , namely, En,n−1 2 =Hn(Sn;Hn−1(ΩSn) =bz. Repeating the above argument shows that En,n−1 2 =Enn,n−1 and that dn:En,n−1 n→E0,2(n−1) n must be an isomorphism. This yields that Z=E0,2(n−1) 2 =H2(n−1)(ΩSn). Repeating this argument shows that for every q,Z=En,q(n−1) 2 =En,q(n−1) n and that dn:En,q(n−1) n→En0,(q+ 1)(n−1)∼=H(q+1)(n−1)(ΩSn) is an isomorphism. And so Hk(n−1)(ΩSn) =Zfor allk. 164 4. HOMOTOPY THEORY OF FIBRATIONS We can also conclude that in dimensions jnot a multiple of n−1, thenHj(ΩSn) must be zero. This is true by the following argument. Assume the contrary, so that there is a smallest j >0 not a multiple of n−1 withHj(ΩSn) =E0,j 2/negationslash= 0. But for dimensional reasons, this group cannot be in the image of any differential, because the only Er,s qthat can be nonzero with r >0 is whenr=n. So the only possibility for a class α∈E0,j 2to represent a class which is in the image of a differential isdn:En,s n→E0,s+n−1 n . Soj=s+n−1. But since jis the smallest positive integer not of the form a multiple of n−1 withHj(ΩSn) nonzero, then for s < j ,En,s n=Hn(Sn,Hs(ΩSn)) =Hs(ΩSn) can only be nonzero if sis a multiple of ( n−1), and therefore so is s+n−1 =j. This contradiction implies that if jis not a multiple of n−1, thenHj((ΩSn)) is zero. This completes our calculation ofH∗(ΩSn). /square We now use the cohomology version of the Serre spectral sequence to compute the cohomology of the unitary groups. We first give the cohomological analogue of 4.28. Again, the reader should consult [ 36] for details. Theorem 4.34.Letp:E→Bbe a fibration with fiber F. Assume that Fis connected and Bis simply connected. Then there is a cohomology spectral sequence converging to H∗(E), with Er,s 2=Hr(B;Hs(F)), having the following properties. (1)The differential djhas bidegree (j,−j+ 1) : dj:Er,s j→Er+j,s−j+1 j. (2)For eachj,E∗,∗ jis a bigraded ring. The ring multiplication maps Ep,q j⊗Ei,j j→Ep+i,q+j j. (3)The differential dj:Er,s j→Er+j,s−j+1 j.is an antiderivation in the sense that it satisfies the product rule: dj(ab) =dj(a)·b+ (−1)u+va·dj(b) wherea∈Eu,v j. (4)The product in the ring Ej+1is induced by the product in the ring Ej, and the product in E∞is induced by the cup product in H∗(E). We apply this to the following calculation. 5. SPECTRAL SEQUENCES 165 Theorem 4.35.There is an isomorphism of graded rings, H∗(U(n))∼=Λ[σ1,σ3,···,σ2n−1], the graded exterior algebra on one generator σ2k−1in every odd dimension 2k−1for1≤k≤n. Proof. We prove this by induction on n. Forn= 1,U(1) =S1and we know the assertion is correct. Now assume that H∗(U(n−1))∼=Λ[σ1,···σ2n−3].Consider the Serre cohomology spectral sequence for the fibration U(n−1)⊂U(n)→U(n)/U(n−1)∼=S2n−1. Then theE2- term is given by E∗,∗ 2∼=H∗(S2n−1;H∗(U(n−1)) =H∗(S2n−1)⊗H∗(U(n−1)) and this isomorphism is an isomorphism of graded rings. But by our inductive assumption we have that H∗(S2n−1)⊗H∗(U(n−1))∼=Λ[σ2n−1]⊗Λ[σ1,···σ2n−3] ∼=Λ[σ1,σ3,···,σ2n−1]. Thus E∗∗ 2∼=Λ[σ1,σ3,···,σ2n−1] as graded algebras. Now since all the nonzero classes in E∗,∗ 2have odd total degree (where the total degree of a class α∈Er,s 2isr+s), and all differentials increase the total degree by one, we must have that all differentials in this spectral sequence are zero. Thus E∗,∗ ∞=E∗,∗ 2∼=Λ[σ1,σ3,···,σ2n−1]. We then conclude that H∗(U(n)) =∼=Λ[σ1,σ3,···,σ2n−1] which completes the inductive step in our proof. /square 5.5. Applications III: Spin andSpin Cstructures. In this section we describe the notions ofSpin andSpin Cstructures on vector bundles. We then use the Serre spectral sequence to identify characteristic class conditions for the existence of these structures. These structures are particularly important in geometry, geometric analysis, and geometric topology. Recall from chapter 2 that an n- dimensional vector bundle ζover a space Xis orientable if and only if it has a SO(n) - structure, which exists if and only if the classifying map fζ:X→BO(n) has a homotopy lifting to BSO (n). In chapter 3 we proved the following property as well. Proposition 4.36.Then- dimensional bundle ζis orientable if and only if its first Stiefel - Whitney class is zero, w1(ζ) = 0∈H1(X;Z2). 166 4. HOMOTOPY THEORY OF FIBRATIONS ASpin structure on ζis a refinement of an orientation. To define it we need to further study the topology of SO(n). The groupO(n) has two path components, i.e π0(O(n))∼=Z2andSO(n) is the path component of the identity map. In particular SO(n) is connected, so π0(SO(n)) = 0. We have the following information about π1(SO(n)). Proposition 4.37.π1(SO(2)) = Z. Forn≥3, we have π1(SO(n)) =Z2. Proof.SO(2) is topologically a circle, so the first part of the theorem follows. SO(3) is topologically the projective space SO(3)∼=RP3 which has a double cover Z2→S3→RP3. SinceS3is simply connected, this is the universal cover ofRP3and hence Z2=π1(RP3) =π1(SO(3)). Now forn≥3, consider the fiber bundle SO(n)→SO(n+1)→SO(n+1)/SO(n) =Sn. By the long exact sequence in homotopy groups for this fibration we see that π1(SO(n))→π1(SO(n+ 1)) is an isomorphism for n≥3. The result follows by induction on n. /square Sinceπ1(SO(n)) =Z2, the universal cover of SO(n) is a double covering. The group Spin (n) is defined to be this universal double cover: Z2→Spin (n)→SO(n). Exercise. Show thatSpin (n) is a group and that the projection map p:Spin (n)→SO(n) is a group homomorphism with kernel Z2. Now the group Spin (n) has a natural Z2action, since it is the double cover of SO(n). Define the groupSpin C(n) using this Z2- action in the following way. Definition 4.6.The group Spin C(n) is defined to be Spin C(n) =Spin (n)×Z2U(1). where Z2acts onU(1) byz→−zforz∈U(1)⊂C. Notice that there is a principal U(1) - bundle, U(1)→Spin C(n) =Spin (n)×Z2U(1)→Spin (n)/Z2=SO(n). Spin C- structures have been recently shown to be quite important in the Seiberg - Witten theory approach to the study of smooth structures on four dimensional manifolds [ 21]. 5. SPECTRAL SEQUENCES 167 The main theorem of this section is the following: Theorem 4.38.Letζbe an oriented n- dimensional vector bundle over a CW - complexX. Letw2(ζ)∈H2(X;Z2)be the second Stiefel - Whitney class of ζ. Then (1)ζhas aSpin (n)structure if and only if w2(ζ) = 0 . (2)ζhas aSpin C(n)- structure if and only if w2(ζ)∈H2(X;Z2)comes from an integral cohomology class. That is, if and only if there is a class c∈H2(X;Z)which maps to w2(ζ) under the projection map H2(X;Z)→H2(X;Z2). Proof. The question of the existence of a Spin orSpin Cstructure is equivalent to the exis- tence of a homotopy lifting of the classifying map fζ:X→BSO (n) toBSpin (n) orBSpin C(n). To examine the obstructions to obtaining such liftings we first make some observations about the homotopy type of BSO (n). We know that BSO (n)→BO(n) is a double covering (the orientation double cover of the universal bundle). Furthermore π1(BO(n) =π0(O(n)) =Z2, so this is the universal cover of BO(n). In particular this says that BSO (n) is simply connected and πi(BSO (n))→πi(BO(n)) is an isomorphism for i≥2. Recall that for nodd, sayn= 2m+ 1, then there is an isomorphism of groups SO(2m+ 1)×Z2∼=O(2m+ 1). Exercise. Prove this! This establishes a homotopy equivalence BSO (2m+ 1)×BZ2∼=BO(2m+ 1). The following is then immediate from our knowledge of H∗(BO(2m+ 1);Z2)∼=Z2[w1,···,w2m+1] andH∗(BZ2;Z2)∼=Z2[w1]. Lemma 4.39. H∗(BSO (2m+ 1);Z2)∼=Z2[w2,···,w2m+1] wherewi∈Hi(BSO (2m+1);Z2)is theithStiefel - Whitney class of the universal oriented (2m+1) - dimensional bundle classified by the natural map BSO (2m+ 1)→BO(2m+ 1). Corollary 4.40.Forn≥3,H2(BSO (n);Z2)∼=Z2, with nonzero class w2. 168 4. HOMOTOPY THEORY OF FIBRATIONS Proof. This follows from the lemma and the fact that for n≥3 the inclusion BSO (n)→ BSO (n+1) induces an isomorphism in H2, which can be seen by looking at the Serre exact sequence for the fibration Sn→BSO (n)→BSO (n+ 1). /square This allows us to prove the following. Lemma 4.41.The classifying space BSpin (n)is homotopy equivalent to the homotopy fiber Fw2 of the map w2:BSO (n)→K(Z2,2) classifying the second Stiefel - Whitney class w2∈H2(BSO (n);Z2). Proof. The groupSpin (n) is the universal cover of SO(n), and hence is simply connected. This means that BSpin (n) is 2 - connected. By the Hurewicz theorem this implies that H2(BSpin (n);Z2) = 0. Thus the composition BSpin (n)p−−−−→BSO (n)w2−−−−→K(Z2,2) is null homotopic. Convert the map w2to a homotopy equivalent fibration, ˜ w2:˜BSO (n)→K(Z2,2). The mappdefines a map (up to homotopy) ˜ p:BSpin (n)→˜BSO (n), and the composition ˜ p◦w2is still null homotopic. A null homotopy Φ : BSpin (n)×I→K(Z2,2) between ˜ p◦w2and the constant map at the basepoint, lifts, due to the homotopy lifting property, to a homotopy ˜Φ :BSpin (n)×I→ ˜BSO (n) between ˜pand a map ¯ pwhose image lies entirely in the fiber over the basepont, Fw2, ¯p:BSpin (n)→Fw2. We claim that ¯ pinduces an isomorphism in homotopy groups. To see this, observe that the ho- momorphism pq:πq(BSpin (n))→πq(BSO (n)) is equal to the homomorphism πq−1(Spin (n))→ πq−1(SO(n)) which is an isomorphism for q≥3 becauseSpin (n)→SO(n) is the universal cover. But similarly πq(Fw2)→πq(BSO (n)) is also an isomorphism for q≥3 by the exact sequence in ho- motopy groups of the fibration Fw2→BSO (n)w2−−−−→K(Z2,2), sincew2induces an isomorphism onπ2.BSpin (n) andFw2are also both 2 - connected. Thus they have the same homotopy groups, and we have a commutative square for q≥3, πq(BSpin (n))¯p∗−−−−→πq(Fw2) p/arrowbt∼=/arrowbt∼= πq(BSO (n))−−−−→ =πq(BSO (n)). Thus ¯p:BSpin (n)→Fw2induces an isomorphism in homotopy groups, and by the Whitehead theorem is a homotopy equivalence. /square 5. SPECTRAL SEQUENCES 169 Notice that we are now able to complete the proof of the first part of the theorem. If ζis any oriented,n- dimensional bundle with Spin (n) structure, its classifying map fζ:X→BSO (n) lifts to a map ˜fζ:X→BSpin (n), and hence by this lemma, w2(ζ) =f∗ ζ(w2) =f∗ ζ◦p∗(w2) = 0. Conversely, ifw2(ζ) = 0, then the classifying map fζ:X→BSO (n) has the property that f∗ ζ(w2) = 0. This implies that the composition Xfζ−−−−→BSO (n)w2−−−−→K(Z2,2) is null homotopic. A null homotopy lifts to give a homotopy between fζand a map whose image lies in the homotopy fiber Fw2, which, by the above lemma is homotopy equivalent to BSpin (n). Thus fζ:X→BSO (n) has a homotopy lift ˜fζ:X→BSpin (n), which implies that ζhas aSpin (n) - structure. We now turn our attentiona to Spin C- structures. Consider the projection map p:Spin C(n) =Spin (n)×Z2U(1)→U(1)/Z2=U(1). pis a group homomorpism with kernel Spin (n).ptherefore induces a map on classifying spaces, which we call c, c:BSpin C(n)→BU(1) =K(Z,2) which has homotopy fiber BSpin (n). But clearly we have the following commutative diagram BSpin (n)⊂−−−−→B(Spin (n)×bz2U(1))=−−−−→BSpin C(n) =/arrowbt/arrowbt/arrowbtp BSpin (n)−−−−→ B(Spin (n)/Z2)−−−−→ =BSO (n) Therefore we have the following diagram between homotopy fibrations BSpin (n)−−−−→BSpin C(n)c−−−−→K(Z,2) =/arrowbt/arrowbt/arrowbtp BSpin (n)−−−−→BSO (n)w2−−−−→K(Z2,2) wherep:K(Z,2)→K(Z2,2) is induced by the projection Z→Z2. As we’ve done before we can assume that p:K(Z,2)→K(Z2,2) andw2:BSO (n)→K(Z2,2) have been modified to be fibrations. Then this means that BSpin C(n) is homotopy equivalent to the pull - back along w2of the fibration p:K(Z,2)→K(Z2,2): BSpin C(n)/similarequalw∗ 2(K(Z,2)). But this implies that the map fζ:X→BSO (n) homotopy lifts to BSpin C(n) if and only if there is a mapu:X→K(Z,2) such that p◦u:X→K(Z2,2) is homotopic to w2◦fζ:X→K(Z2,2). Interpreting these as cohomology classes, this says that fζlifts toBSpin C(n) (i.eζhas aSpin C(n) 170 4. HOMOTOPY THEORY OF FIBRATIONS - structure) if and only if there is a class u∈H2(X;Z) so that the Z2reduction of u,p∗(u) is equal tow2(ζ)∈H2(X;Z2). This is the statement of the theorem. /square Bibliography [1] J.F. Adams, On the non-existence of elements of Hopf invariant one , Annals of Math. 272, (1960), 20 - 104. [2] J.F. Adams, Vector fields on spheres , Annals of Math. 2, 75, (1962), 603-632. [3] M. Atiyah and R. Bott, The Yang - Mills equations over Riemann surfaces , Phil. Trans. Royal Soc. Lond. 308, (1982), 523-615. [4] M. Atiyah, and G. Segal, Equivariant K-theory and completion , J. Diff. Geom. 3(1969), 1 - 18. [5] R. Bott and L. Tu, Differential Forms in Algebraic Topology , Springer Grad. 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