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fibre Hicks Drexel Philly

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Lecture-style notes on fibre bundles, apparently from a Drexel/Philadelphia source judging by the file name. They cover top-down and bottom-up constructions, tangent and normal bundles, Seifert's fibred spaces, fibrations and sheaves, and applications in physics. Later sections treat group actions (orbits, stabilizers, transitive, faithful, free), the orbit-stabilizer theorem, Lie group bundles, and the Hopf fibration S1-S3-S2 via SU(2)/U(1). A reference list is included.

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FibreBundles Aﬁbrebundle isa6-tuple EBFpGVi φi . Eisthetotal space ,Bisthebase space andFistheﬁbre.p:E B istheprojection map andp 1 x F.Thelasttwoelements ofthis tuple relate these ﬁrstfour objects. The idea isthatateach point ofBacopyoftheﬁbreFisglued, making upthetotal spaceE. Onewaytodothisisjusttakeaproduct ofBandF,i.e.letEBF andp x y x.Forexample R2R R.This may betheﬁrstﬁbre bundle thatanyone everthought of!(Descartes supposedly thought thisway,according toS.S.Chern.) Itisnotthecase thateveryﬁbre bundle isofthisform, i.e.trivial. Toseethisconsider theMoebius strip andthecylinder asEwhere S1isthebaseBandaninterv alIistheﬁbreF. 1 TopDown One wayﬁbre bundles arise istopdown,i.e.youstart withEand break itupintoﬁbres. Forexample ifyouhaveaLiegroupGanda subgroup HyoucanformG H.Then theﬁbre isHandthespace is decomposed intocopies ofH,namely ,theleftcosets. Example RR Z S1 Example R2R2 Z2 S2,thetorus. Example S3S3 Zn lensspace. Example S3S3 S1 S2. These arespecial cases ofwhat isknownasagroupaction . 2 BottomUp Wecanalso build bundles bytaking aspace andattaching things toit.Themost natural example ofthisisthetangent bundle ofa differentiable manifold. Ateach point ofthemanifold weglue all possible velocity vectors. This istheﬁrstexample ofavector bun- dle,i.e. aﬁbre bundle where theﬁbre isavector space (and the groupGisthefulllinear group). Another natural vector bundle occurs ifwehaveamanifold embed- dedinaEuclidean space (oranother manifold !).Then ateach point wecanattach allthevectors thatarenormal tothatpoint. This forms thenormal bundle . Once youhaveonevector bundle youcanmaketons more. Suppose youhaveanoperation thattakesavector space andgivesyouback anewone, like .Then youcanapply thatoperation toeach ﬁbre of youbundle togetanewone! 3 ThePrimordial Bundle-Soup AsIsaid above,Chern thinks thatDescartes insome sense knew about ﬁbre bundles. Who elsedidbefore the20th century ?Probably Gauss hadsome idea (Ihavenohard evidence tosupport that). The reason Isaythisisbecause Gauss certainly thought alotabout tan- gent vectors, normal vectors andruled surfaces ,likex2 y2z2 1. Ingroup theory theidea ofa”twisted product” occurs pretty natu- rally andmanypeople were certainly awareofitinthe19th cen- tury .The development ofthetheory ofﬁber bundles istied to thethetheory ofgroups partly because ofthis, butalso because homotopy theory hastodowith groups. Sometime soon after thedisco veryofgroups, itwasobserv edthat onecould takethecartesian products oftwogroups tomakeathird. But, givenagroupGandanormal subgroup HGitwasn’talways thecase thatG G H H.Itfactthisispretty rare. SoGwas someho wbrokenupintocopies ofH,butthose copies weren’ tglued together inanobvious way. Nowallofthisiswell understood because wehavethenotion ofa semi directproduct ofgroups. 4 FibreBundlesinthePre-Cambrian In1934, Herbert Seifert published The Topology of 3 Dimensional Fiber edSpaces ,which contained adeﬁnition of anobject thatisakind ofﬁbre bundle. Seifert wasonly considering circles asﬁbres and3-manifolds forthetotal space. Thepoint was that2-manifolds hadbeen classiﬁed andnoweveryone wastrying toclassify 3-manifolds. The idea wastodecompose a3-manifold into circles overan”orbit surface”. This seemed likeareasonable approach since surfaces were classiﬁed. Hisdeﬁnition ofaﬁber edspace isthatitisa3-manifold satisfying thefollowing: 1)Themanifold canbedecomposed intoﬁbers, where each ﬁber is asimple closed curve. 2)Each point liesinexactly oneﬁbre. 3)Foreach ﬁberHthere exists afiberneighborhood ,that is,a subset consisting ofﬁbres andcontaining H,which canbemapped under aﬁber-preserving map onto aﬁbered solid torus, whereHis mapped onto the”middle ﬁbre”. 5 OtherthingsrelatedtoFibreBundles After theinvention ofLiegroups people were taking quotients like crazy (and stillare). Theyfound alotoffunnythings happening. Forexample, Hopf found aninteresting mapS3S2which washo- motopically non-tri vial. This map, knownastheHopf ﬁbration, has manyinteresting properties andstillplays amajor roleindifferen- tialgeometry andalgebraic topology .Itisinfactaﬁbre bundle with baseS2,total spaceS3andﬁbreS1!Infactthemap could beviewed asaquotient mapS3S3 S1. Thereason thattheHopf ﬁbration iscalled theHopf ﬁbration andnot theHopf ﬁbre bundle isthatthere issomething called aﬁbration , which isalotlikeaﬁbre bundle butmore general. Aﬁbration con- sists oftwospacesEandBandamapp:EBwith aproperty called thehomotop ylifting property .There isarough notion ofaﬁ- brethatcanbemade forﬁbrations, buttheyareahard todocalculus on. There isalsoobject knownasasheaf which alsofollowsthisidea of gluing aspace toeach point ofanother space. These were invented inthe1950’ sanddeveloped extensi velyinthe1960’ s,solving many problems inalgebraic geometry .The example ofasheaf ofgerms offunctions onadifferentiable manifold isafamiliar example ofa sheaf. 6 SomeApplications There aretoomanymathematical applications tolisthere!. Inphysics theidea ofatangent bundle isveryuseful. Inmechanics, forexample, often theconﬁguration space ofasystem isamanifold, andthephase space isthetangent bundle. There havebeen other uses ofﬁbre bundles inphysics. Theyare used forlotsofthings physics, such asgauge theory ,which wasin- vented byHermann Weyl. Inthelate1970’ sﬁbre bundles became quite fashionable incon- densed matter physics forexplaining certain properties ofdefects inordered matter ,likeacrystal. Itwasapparently quite successful inexplaining some ofthebehaviorofSuperﬂuid helium 3.Some people thatIknowstudy liquid crystals andsometimes useﬁbre bun- dles. Ihaveheard itsaid thatthestrength ofcertain steels wasin- creased duetodefects, andthatthiscould beexplained with bundles. Vibrational modes ofmolecules (buckybullsworkwell duetotheir symmetries) havearebeing analyzed using vector bundles. There aresome intriguing relations here between continuous anddiscrete mathematics (graph theory). 7 References JohnC.BaezandJavierP.Muniain. -Gaugefields knots andgravity. GlenBredon-IntroductiontoCompactTransformationGroups. Shiing-shen Chern-GlobalDifferentialGeometry. B.Dubrovin,A.Fomenko,S.Novikov-ModernGeometry PartII. JeanDieudonne -AHistoryofAlgebraicandDifferentialGeometry. DaleHusemoller -FibreBundles. KatsuoKawakubo-TheTheoryofTransformationGroups. DavidMermin-TheTopologicalTheoryofDefectsinOrderedMedia,ReviewsofModern Physics,Vol.51,No.3,July1979. JohnMilnor-CharacteristicClasses WalterPoor-DifferentialGeometricStructures. JosephRotman-HomologicalAlgebra. HerbertSeifertandWilliamThrelfall-SeifertandThrelfall:ATextbookofTopology. NormanSteenrod -TheTopologyofFibreBundles. ShlomoSternberg-GroupTheoryinPhysics.(Formoreonvectorbundlesappliedtomolecular vibrations ,seehttp://www .math.upenn.edu/ chung/.) 8 GroupActions IfXisasetandGisagroup then agroupaction (ontheleft) isa map :GXXthatsatisﬁes (i)g1 g2 x g1g2 xforallg1g2 G,x X, and (ii)e xxforallx X,whereeistheidentity ofG. Additionally wedeﬁne theorbit ofapointxofXas orb x gx g G andthestabilizer ofxas Stab x g G gxx. These ideas areEXTREMEL Yuseful. Remark Theorbits areequivalence classes andoneshould trytopicture the corresponding quotient space. 9 TransitiveActions Example Thesymmetry group ofacube actsonthecube. Thesym- metry group isisomorphic toA4,agroup of24elements. What are theorbits andstabilizers ? Remarks 1)Orbits aredisjoint. 2)Xistheunion oftheorbits. Example LetGO nandXSn. Deﬁnition The action ofGonXissaid tobetransiti veifforall xy Xthere isag Gsuch thatgxy.Inother words, there is only oneorbit. Example R R2R2, x a b a x b. Remark If”dimG” ”dimX”then oneexpects theaction nottobetransiti ve. 10 FaithfulActions Deﬁnition Theaction ofGonXissaidtobefaithful ifgxhxfor allx Ximplies thatg h. Or:theaction deﬁnes amapG XX.Then theaction isfaithful if thismap is1-1. Example C S1S1,where reiθz eiθz. Remark IfG XXandGactsby f x f xthen theaction isfaithful. 11 FreeActions Deﬁnition Theaction ofGonXissaid tobefreeifforallg G, g e,andforallx X,gx x. Example R R2R2asaboveisafreeaction. Example Thegroup ofdeck transformations ofacovering actsfreely onthetopspace ofthecovering. Example LetG O nandX Sn.Then theobvious action isnot free. Example Theaction ofthesymmetry group ofacube onthecube. This action isnotfree. Butitisﬁxed point free,i.e.there isno point thatisﬁxedbyalltheelements ofthegroup. 12 TheFundamental Theorem Theor emSuppose thatGactsonasetXandx X.Then there isa bijection Φ:G Stab x orb x. Remarks 1)Itisoften thecase thatΦwillhavemanygood properties. 2)Iftheaction istransiti vethenorb x X.This isofparticular interest. 13 Examples ofTransitiveactions Example O n 1 SnSn. Example SO n 1 SnSn Example U n S2n1S2n1 Example SU n S2n1S2n1 Example TakeG O nacting onthesetoflines through theorigin inRn,i.e.RPn. Example TakeGU nacting onthesetoflines through theorigin inCn,i.e.CPn Deﬁnition LetGn k Vbethesetofk-dimensional subspaces ofa vector spaceV. Example TakeG O nacting onGn k Rn.The stabilizer ofa point isisomorphic toO n k 14 Non-transiti veactions Example ForanygroupGandsubgroup HG,letHactbyusual leftmultiplication. Inparticular wecould takeGtobetheadditi ve group ofthereals andHtobetheintegers Z. Example AnygroupGactsonitself byconjug ation. Ifasubgroup isinvariant under thisaction, then itisnormal. Inthatcase theaction ofGonitself induces anaction ofGonthenormal subgroup. Example S1actsonS2byrotation. What isthequotient space ? 15 LieGroupsandBundles Theor emLetGbeaLiegroup andletHbeaclosed subgroup. The projection π:G G Hgivesrisetoaﬁbre bundle whereHisthe ﬁbreandthestructure group. Remarks 1)This isanexample ofaprinciple bundle . 2)This isaspecial case ofagroup acting transiti velyonaspace and thus itgivesrisetoaﬁbre bundle. HereGactsonG Htransiti vely andtheisotrop ygroup (stabilizer) isH. 16 Asamplebundle Example SU n CPn1CPn1.This action istransiti ve.What istheisotrop ygroup ?Ifamatrix inSU nﬁxesalinethen ithas theform0 A 0 α whereAisinU n 1anddet A α.Therefore wehavethebun- dle U n 1 SU n CPn1 17 TheHopfFibration Inthespecial case n=2ofthetheabovebundle wehave U 1 SU 2 CP1 Buteach ofthethree spaces ishomeomorphic toasphere :U 1 S1SU 2 S3andCP1 S2.Sowhat does thisbundle havetodo with S1S3S3 S1 (where weviewS3asagroup viathequaternions), ?Itturns outthat theyarethesame bundle. 18 AnotherView There isyetanother waytolook attheHopf ﬁbration: U 1 SU 2 SU 2 U 1 Lets look atthiscarefully .U(1) isagroup of1 1matrices each of whose determinant hasabsolute value 1,sothatisclearlyS1. SU 2 zw¯w ¯z z 2 w 21 , andifwewritez x iyandw a biitiseasy toseethatx2 y2 a2 b21,soSU 2 S3.WewillviewU 1asasubgroup ofSU 2bytheimbedding z z0 0¯z . 19 GettingtoS2 LetT A SU 2 tr A 0 Thetypical element ofThasthe form xiw¯w xi ,wherexisrealandw a biiscomple x.Since thismatrix isspecial unitary ,x2 a2 b21,i.e.T S2.Deﬁne π:SU 2 Tbytheequation π P PEP whereE i0 0 i .Theﬁbers ofπareexactly thecosets ofSU 2 U 1 !Sowehave U 1 SU 2 T S2 SU 2 U 1 There isalsoadirect waytowrite downamap fromS3toS2. 20 BacktoBundles Abundle with ﬁbreF,total spaceEandbase spaceMisamap π:E M where each pointphasanopen neighborhood UMsuch thatπ: π 1 U Uistheprojection pr1uptodiffeomorphism. Wesaythe bundle islocally trivialandthatΦisalocal trivialization .Also, in general itisassumed thatallofthespaces involvedareC∞manifolds andthatallthemaps aresmooth. 21 Changing Coordinates Observ ethatΦislikeacoordinate chart onamanifold. Suppose that Φ1andΦ2arelocal trivializations forUaandUbwhereUa Ub φ. Then eachp Ua Ubdetermines adiffeomorphism fromFtoF: ϕba:Ua Ub Diff F whereϕbaisdeﬁned by ΦbΦ 1 a p f p ϕba p f with p f Ua Ub F. Remarks 1)ϕaa idF. 2)Ifp Ua Ub Ucthenϕcb p ϕba p ϕca p. 3)Theϕbaareknownastransition functions . 22 BuildingaBundle One may reconstruct π:EMgiventheopen setsandthetran- sitions functions and theﬁbre. First consider thedisjoint union αUα F.Then glue together pa faand pb fbifpa pb p andfb ϕba p fa. 23 TwoBundles Ifthetransition functions alllieinagroupG Diff FthenGis saidtobethestructur egroupofthebundle. Remark Abundle canhavemanystructure groups. Deﬁnition π:EMisavector bundle ifFisavector spaceVand thestructure groupG Gl V,i.e.Gisasubgroup ofthegroup of invertible linear maps. Deﬁnition π:P Misaprinciple G-bundle iftheﬁbreFisaLie groupGandtheaction ofthestucture group onGcoincides with the action ofasubgroup ofGonGbyleftmultiplication. Example S1S1z z2isaprinciple bundle with ﬁbreZ2. 24 NewBundlesfromold Example IfEMisavector bundle then wecanform thebundle ofbasesB E M.Howdowedothis? Ingeneral, givenabundleE Monemay construct anewbundle, where theﬁbre oftheoriginal bundle,E,isjustreplaced bythestruc- turegroupG.Wedothisbytaking thedisjoint unionαUα Gand glueing together pa ha and pb hb ifpa pb pandhb ϕba ha. 25 PrincipleBundles1 AG-principle bundle ,orsimply aprinciple bundle isaﬁbre bun- dleπ:P Ewith ﬁbreFequal tothestructure groupGandhaving theproperty thatforallUaandUbwithUa Ub φ, ϕba:Ua Ub Left F Diff F whereLeft F Lg Lg h gh h Gg G.Inother words, changing coordinates corresponds tomultiplying theﬁbre ontheleft bysome element ofG. Lemma ForeveryG-principle bundle,Gactsnaturally onPonthe right. Proof Givenu P,wewanttodeﬁneug,foreachg G.LetU beaneighborhood aboutπ uthathasatrivialization. Using these coordinates, represent uas π u hwhereh G.Then deﬁneugto bethepoint ofPthathasthecoordinates π u hg.Itisnothard to check thatthisdeﬁnition isindependent ofcoordinates, andthen it isclear thatitisaright action. 1These notes aretakenessentially from ”Metric Differential Geometry” byKarsten Grove. 26 ExampleofPrincipleBundles Example Theprojection SnRPn.HereG O 1 Z2. Example TheHopf mapS2n 1CPn.Recall thatCPnistheset ofalllines inCn.Apoint onS2n 1getssenttothecomple xlinethat contains it.HereG U 1 S1. Example ˜X Xwhere ˜Xistheuniversal covering space ofX. ThenGπ1 X. Example The covering need notbeuniversal. TakeS1S1by z z2.This isaprinciple bundle with ﬁbreZ2.What isthegeneral situation ? 27 Pullbacks ofBundles LetE Mbeabundle with ﬁbreFandstructure groupG.Con- sider amapf:N M,whereNisanysmooth manifold. Deﬁne thepullback ofE Mbyftobethebundlef E Nwith ﬁbre Fwheref E q u f q π u andtheprojection map isde- ﬁned by q u π u.Interms ofdiagrams wehave Parallel transport along acurveγ: a b Mmay bedescribed using thepullback γ E. 28 Connections inBundles Letπ:E Mbeabundle. Thesubbundle (adistrib ution)V E ofthetangent bundleTE Pdeﬁned by V X TE π X 0 iscalled thevertical bundle ofE. Atthispoint weintroduce thenotationEptomeanπ1 p,p M. Withthisinmind,Vu TuEandVuisactually thetangent space atu oftheﬁbreEπ u .Butthere isnocanonical complement toHutoVu, i.e.asubspace Hu TuEsuch that Vu Hu TuE Such aspace iscalled ahorizontal space atu. Deﬁnition Aconnection inEisasubbundle ofthetangent bundle ofEsuch thateach ﬁbre isahorizontal space. IfEisaprinciple bundle with groupGwerequire thattheconnection beG-invariant, i.e. Hug Rg Hu u E g G 29 Connections inPrincipleBundles Ifwehaveaconnection inaprinciple bundleP Mthen achoice ofahorizontal subspace atuisequivalent toachoice ofaprojection proju:TuP Vu Then vertical space (i.e.aﬁbre ofthevertical bundle) mybeviewed asthetangent space totheﬁbre,G.Inturn, thiscanbeviewed asg, theliealgebra ofG,sowereally have proju:TuP g Therefore aconnection isequivalent tohaving agvalued 1-form on Pwhich isinvariant under theright action ofG. 30 Connections inVectorBundles LetE Mbeavector bundle. Wedeﬁne adifferential operator ∇asfollows. IdentifyVuwith theﬁbreEπ u .Letη Γ Ebea section, i.e.η:M Eandπη idM.Then let ∇η:TpM Ep bedeﬁned foreachp Mby ∇η X ∇Xη η Xvη X η Xh 31