M382D NOTES: DIFFERENTIAL TOPOLOGY ARUNDEBRAY MAY16,2016 ThesenotesweretakeninUTAustin’sMath382D(DifferentialTopology)classinSpring2016,taughtbyLorenzoSadun. I live-TEXedthemusingvim,andassuchtheremaybetypos;pleasesendquestions,comments,complaints,andcorrectionsto [email protected]. ThankstoAdrianClough,ParkerHund,MaxReistenberg,andThérèseWuforfindingand correctingafewmistakes. CONTENTS 1. TheInverseandImplicitFunctionTheorems: 1/20/16 2 2. TheContractionMappingTheorem: 1/22/16 4 3. Manifolds: 1/25/16 6 4. AbstractManifolds: 1/27/16 8 5. ExamplesofManifoldsandTangentVectors: 1/29/16 9 6. SmoothMapsBetweenManifolds: 2/1/16 11 7. ImmersionsandSubmersions: 2/3/16 13 8. Transversality: 2/5/16 15 9. PropertiesStableUnderHomotopy: 2/8/16 17 10. MaytheMorseBeWithYou: 2/10/16 19 11. PartitionsofUnityandtheWhitneyEmbeddingTheorem: 2/12/16 21 12. Manifolds-With-Boundary: 2/15/16 22 13. RetractsandOtherConsequencesofBoundaries: 2/17/16 24 14. TheThomTransversalityTheorem: 2/19/16 26 15. TheNormalBundleandTubularNeighborhoods: 2/22/16 27 16. TheExtensionTheorem: 2/24/16 28 17. IntersectionTheory: 2/26/16 30 18. TheJordanCurveTheoremandtheBorsuk-UlamTheorem: 2/29/16 32 19. GettingOriented: 3/2/16 33 20. OrientationsonManifolds: 3/4/16 35 21. OrientationsandPreimages: 3/7/16 36 22. TheOrientedIntersectionNumber: 3/9/16 37 23. ExamDebriefing: 3/21/16 39 24. (Minus)SignsoftheTimes: 3/23/16 41 25. TheLefschetzNumber: 3/25/16 43 26. MultipleRootsandtheLefschetzNumber: 3/28/16 45 27. VectorFieldsandtheEulerCharacteristic: 3/30/16 46 28. : 4/1/16 49 29. TheHopfDegreeTheorem: 4/4/16 49 30. TheExteriorDerivative: 4/6/16 50 31. PullbackofDifferentialForms: 4/8/16 53 32. DifferentialFormsonManifolds: 4/11/16 55 33. Stokes’Theorem: 4/13/16 57 34. Tensors: 4/15/16 59 35. ExteriorAlgebra: 4/18/16 61 1 2 M382D(DifferentialTopology)LectureNotes 36. TheIntrinsicDefinitionofPullback: 4/20/16 63 37. deRhamCohomology: 4/22/16 64 38. HomotopyInvarianceofCohomology: 4/25/16 65 39. ChainComplexesandtheSnakeLemma: 4/27/16 68 40. TheSnakeLemma: 4/29/16 70 41. GoodCoversandCompactlySupportedCohomology: 5/2/16 72 42. TheDegreeinCohomology: 5/4/16 75 Lecture1. The Inverse and Implicit Function Theorems: 1/20/16 “Themostimportantlessonofthestartofthisclassistheproperpronunciationofmyname[Sadun]: itrhymeswith‘balloon.’ ” We’rebasicallygoingtomarchthroughthetextbook(GuilleminandPollack),withalittlemoreinthebeginning andalittlemoreintheend;however,we’regoingtobeabitmoreabstract,talkingaboutmanifoldsmoreabstractly, ratherthanjustembeddingthemin(cid:82)n,thoughthetheoremsaremostlythesame. Atthebeginning,we’lldiscuss theanalyticunderpinningstodifferentialtopologyinmoredetail,andattheend,we’llhopefullyhavetimeto discussdeRhamcohomology. Suppose f :(cid:82)n→(cid:82)m. Itsderivativeisdf;whatexactlyisthis? Thereareseveralpossibleanswers. • It’sthebestlinearapproximationto f atagivenpoint. • It’sthematrixofpartialderivatives. Whatweneedtodoismakegood,rigoroussenseofthis,moresothaninmultivariablecalculus,andrelatethetwo notions. Definition1.1. Afunction f :(cid:82)n→(cid:82)misdifferentiableatana∈(cid:82)nifthereexistsalinearmap L:(cid:82)n→(cid:82)msuch that |f(a+h)−f(a)−L(h)| lim =0. (1.2) h→0 |h| Inthiscase, L iscalledthedifferentialof f ata,writtendf| . a Sinceh∈(cid:82)n,butthevectorinthenumeratorisin(cid:82)m,soit’squiteimportanttohavethemagnitudesthere,or elseitwouldmakenosense. Anotherwaytorewritethisisthat f(a+h)= f(a)+L(h)+o(small),i.e.alongwithsomesmallerror(whatever thatmeans). Thismakessenseofthefirstnotion: L isalinearapproximationto f neara. Now,let’smakesense ofthesecondnotion. Theorem1.3. If f isdifferentiableata,thendf isgivenbythematrix(cid:128)∂fi(cid:138). ∂xj Proof. Theidea: if f isdifferentiableata,then(1.2)holdsforh→0alonganypath! Solet’stakee beaunitvectorandh=te as t→0in(cid:82). Then,(1.2)reducesto j j L(te )= f(a1,a2,...,aj+t,aj+1,...,an)−f(a), j t andas t→0,thisshows L(e )i = ∂fi. (cid:130) j ∂xj Inparticular,if f isdifferentiable,thenallpartialderivativesexist. Theconverseisfalse: thereexistfunctions whosepartialderivativesexistatapointa,butarenotdifferentiable. Infact,onecanconstructafunctionwhose directionalderivativesallexist,butisnotdifferentiable! Therewillbeanexampleonthefirsthomework. The ideaisthatdirectionalderivativesrecordlinearpaths,butdifferentiabilityrequiresallpaths,andsomakingthings failalong,say,aquadratic,willproducethesestrangecounterexamples. Nonetheless,ifallpartialderivativesexist,thenwe’realmostthere. 1 TheInverseandImplicitFunctionTheorems: 1/20/16 3 Theorem1.4. Supposeallpartialderivativesof f existataandarecontinuousonaneighborhoodofa;then, f is differentiableata. In calculus, one can formulate several “guiding” ideas, e.g. the whole change is the sum of the individual changes,thewholeisthe(possiblyinfinite)sumoftheparts,andsoforth. Oneparticularoneis: onevariableata time. Thisprinciplewillguidetheproofofthistheorem. Proof. Theproofwillbegivenform=2andn=1,butyoucanfigureoutthesmalldetailsneededtogeneralize it;forlargern,justrepeattheargumentforeachcomponent. Wewanttocompute f(a +h ,a +h )−f(a ,a ) 1 1 2 2 1 2 = f(a +h ,a +h )−f(a +h ,a )+f(a +h ,a )−f(a ,a ) 1 1 2 2 1 1 2 1 1 2 1 2 Regrouping,thisistwosingle-variablequestions. Inparticular,wecanapplythemeanvaluetheorem: thereexist c ,c ∈(cid:82)suchthat 1 2 = ∂ f (cid:12)(cid:12)(cid:12) h + ∂ f (cid:12)(cid:12)(cid:12) h ∂x2(cid:12) 2 ∂x1(cid:12) 1 (a1+h1,a2+c2) (a1+c1,a2) =(cid:130) ∂ f (cid:12)(cid:12)(cid:12) − ∂ f (cid:12)(cid:12)(cid:12) (cid:140)h +(cid:130) ∂ f (cid:12)(cid:12)(cid:12) − ∂ f (cid:12)(cid:12)(cid:12) (cid:140)h +(cid:18) ∂ f (cid:12)(cid:12)(cid:12) , ∂ f (cid:12)(cid:12)(cid:12) (cid:19)(cid:129)h1(cid:139), ∂x1(cid:12)a1+c1,a2 ∂x1(cid:12)a 1 ∂x2(cid:12)a1+h1,a2+c2 ∂x2(cid:12)a 2 ∂x1(cid:12)a ∂x2(cid:12)a h2 butsincethepartialsarecontinuous,thelefttwotermsgoto0,andsincethelasttermislinear,itgoesto0as h→0. (cid:130) We’lloftentalkaboutsmoothfunctionsinthisclass,whicharefunctionsforwhichallhigher-orderderivatives existandarecontinuous. Thus,theydon’thavetheproblemsthatonecounterexamplehad. B·C Sincewe’regoingtobemakinglinearapproximationstomaps,thenweshoulddiscusswhathappenswhen youperturblinearmapsalittlebit. Recallthatif L:(cid:82)n→(cid:82)m islinear,thenitsimageIm(L)⊂(cid:82)m anditskernel ker(L)⊂(cid:82)n. Supposen≤m;then, L issaidtohavefullrankifrankL=n. Thisisanopencondition: everyfull-ranklinear functioncanbeperturbedalittlebitandstaylinear. Thiswillbeveryuseful: ifa(possiblynonlinear)function’s differentialhasfullrank,thenonecansaysomeinterestingthingsaboutit. If n≥m,thenfullrankmeansrank m. Thisisonceagainstable(anopencondition): onecanwritesucha linearmapas L=(A|B),whereAisaninvertiblem×mmatrix,andinvertibilityisanopencondition(sinceit’s givenbythedeterminant,whichisacontinuousfunction). To actually figure out whether a linear map has full rank, write down its matrix and row-reduce it, using Gaussianelimination. Then,youcanreadoffabasisforthekernel,determiningthefreevariablesandtherelations determiningtheothervariables. Ingeneral,forak-dimensionalsubspaceof(cid:82)n,youcanpickkvariablesarbitrarily andtheseforcetheremainingn−kvariables. Thepointis: thesubspaceisthegraphofafunction. Now,wecanapplythistomoregeneralsmoothfunctions. Theorem1.5. Suppose f :(cid:82)n→(cid:82)m issmooth,a∈(cid:82)n,anddf| hasfullrank. a (1) (Inversefunctiontheorem)Ifn=m,thenthereisaneighborhoodU ofasuchthat f| isinvertible,witha U smoothinverse. (2) (Implicitfunctiontheorem)Ifn≥m,thereisaneighborhoodU ofasuchthatU∩f−1(f(a))isthegraph ofsomesmoothfunction g:(cid:82)n−m→(cid:82)m (uptopermutationofindices). (3) (Immersion theorem) If n≤ m, there’s a neighborhood U of a such that f(U) is the graph of a smooth g:(cid:82)n→(cid:82)m. Thistime,theresultsarelocalratherthanglobal,butonceagain,fullrankmeans(local)invertibilitywhen m=n,andmoregenerallymeansthatwecanwriteallthepointssentto f(a)(analogoustoakernel)asthe graphofasmoothfunction. It’spossibletosharpenthesetheoremsslightly: insteadofmaximalrank,youcanusethatifdf| hasblock a formwiththesquareblockinvertible,thensimilarstatementshold. 4 M382D(DifferentialTopology)LectureNotes Thecontentofthesetheorems,thewaytothinkofthem,isthatinthesecases,smoothfunctionslocallybehave likelinearones. Butthisisnottoomuchofasurprise: differentiabilitymeansexactlythatafunctioncanbelocally wellapproximatedbyalinearfunction. Thepointoftheproofisthatthehigher-ordertermsalsovanish. Forexample,ifm=n=1,thenfullrankmeansthederivativeisnonzeroata. Inthiscase,it’sincreasingor decreasinginaneighborhoodofa,andthereforeinvertible. Ontheotherhand,ifthederivativeis0,thenbad thingshappen,becauseit’scontrolledbythehigher-orderderivatives,soonecanhaveanoninvertiblefunction (e.g.aconstant)oraninvertiblefunctionwhoseinverseisn’tsmooth(e.g. y =x3 at x =0). Thisisnotthelasttimeinthisclassthatmaximalrankimpliesniceanalyticresults. We’regoingtoprove(2);then,aslinear-algebraiccorollaries,we’llrecovertheothertwo. Lecture2. The Contraction Mapping Theorem: 1/22/16 Today,we’regoingtoprovethegeneralizedinversefunctiontheorem,Theorem1.5. We’llstartwiththecase wherem=n,whichisalsothesimplestinthelinearcase(fullrankmeansinvertible,almosttautologically). Theorem2.1. Let f :(cid:82)n→(cid:82)n besmooth. Ifdf| isinvertible,then a (1) f isinvertibleonaneighborhoodofa, (2) f−1 issmoothonaneighborhoodofa,and (3) d(f−1)|f(a)=(df|a)−1. Proofofpart(1). Withoutlossofgenerality,wecanassumethata= f(a)=0bytranslating. Wecanalsoassume thatdf| =I,byprecomposingwithdf|−1: a a (cid:82)(cid:79)(cid:79)n f (cid:47)(cid:47)(cid:61)(cid:61)(cid:82)n df|−1 a (cid:82)n Ifweprovetheresultforthediagonalarrow,thenitisalsotruefor f. Sincethedomainandcodomainof f are differentinthisproof,we’regoingtocalltheformerX andthelatterY,so f :X →Y. Now,since f issmooth,itsderivativeiscontinuous,sothere’saneighborhoodofainX givenbythe x such that(cid:107)df| −I(cid:107)<1/2.1 Andbyshrinkingthisneighborhood,wecanassumethatitisaclosedballC. x OnC, f isinjective: if x ,x ∈C,thensinceC isconvex,thenthere’salineγ(t)=x +tv (where v=x −x ) 1 2 1 2 1 joining x1 to x2,and ddft =(df|γ(t))v. Therefore (cid:130)(cid:90) 1 (cid:140) f(x2)−f(x1)= df|γ(t)dt v 0 (cid:90) 1 = (cid:0)(df|γ(t)−I)+I(cid:1)vdt 0 (cid:90) 1 =x2−x1+ (df|γ(t)−I)vdt. 0 Wecanboundtheintegral: (cid:12) (cid:12) (cid:12)(cid:12)(cid:12)(cid:90) 1(cid:0)df|γ(t)−I(cid:1)v(cid:12)(cid:12)(cid:12)≤(cid:90) 1(cid:12)(cid:12)(df|γ(t)−I)v(cid:12)(cid:12)dt≤(cid:90) 1 21|v|dt= |2v|. (cid:12) 0 (cid:12) 0 0 Thus,since x −x =v,then f(x )−f(x )hasmagnitudeatleast v/2,soinparticularitcan’tbezero. Thus, f is 2 1 2 1 injectiveonC. Thepointis,sincedf isclosetotheidentityonC,wegetanerrortermthatwecanmakesmall. 1Therearemanydifferentnormsonthespaceofn×nmatrices,butsincethisisafinite-dimensionalvectorspace,theyareallequivalent. However,forthisproofwe’regoingtotaketheoperatornorm(cid:107)A(cid:107)= sup |Av|. v∈Sn−1 2 TheContractionMappingTheorem: 1/22/16 5 Toconstructaninverse,weneedtomakeitsurjectiveonaneighborhoodof f(a)inY. Thewaytodothisis calledthecontractionmappingprinciple,butwe’lldoitbyhandfornowandrecoverthegeneralprinciplelater. To be precise, we’ll iterate with a “poor-man’s Newton’s method:” if y ∈ Y, then given xn, let xn+1 = x0− (f(x )− y)= y+x − f(x )(sincewe’reusingthederivativeattheorigininsteadofat x,andthisisjustthe 0 0 0 identity). Afixedpointofthisiterationisapreimageof y. Specifically,we’llwant x =a,sincewe’retryingto 0 boundthedistanceofourfixedpointfroma. Since xn+1−xn= y+xn−f(xn)−(y+xn−1−f(xn−1))=(xn−xn−1)−(f(xn)−f(xn−1)), then|xn+1−xn|<(1/2)|xn−xn−1|,soinparticular,thisisaCauchysequence! Thus,itmustconverge,andtoa valuewithmagnitudenomorethan2|y|(since f(x )= f(a)=0). Thus,ifC hasradiusR,thenforany y inthe 0 ballofradius1/2fromtheorigin(inY), y hasapreimage x,so f issurjectiveonthisneighborhood. (cid:130) Now,wecandiscussthecontractionmappingprinciplemoregenerally. Definition2.2. LetX beacompletemetricspaceand T :X →X beacontinuousmapsuchthatd(T(x),T(y))≤ cd(x,y)forall x,y ∈X andsomec∈[0,1). Then, T iscalledacontractionmapping. Theorem2.3(Contractionmappingprinciple). IfX isacompletemetricspaceand T acontractionmappingonX, thenthere’sauniquefixedpoint x (i.e. T(x)=x). Proof. Uniquenessisprettysimple: if T hastwofixedpoints x and x(cid:48) suchthat x (cid:54)= x(cid:48),then d(T(x),T(x(cid:48)))≤ cd(x,x(cid:48))=d(T(x),T(x(cid:48))),andc<1,sothisisacontradiction,so x =x(cid:48). Existenceisbasicallytheproofwejustsaw: pickanarbitrary x0∈X andlet xn+1=T(xn). Then,d(xm,xn)≤ c|n−m−1|d(xn,xn−1),sothissequenceisCauchy,andhasalimit x. Then,since T iscontinuous, T(x)=x. (cid:130) Now,backtothetheorem. ProofofTheorem2.1,part(2). Onceagain,weassume f(0)=0. Bythefundamentaltheoremofcalculus,onour neighborhoodof0, (cid:90) 1 y = f(x)= df| (x)dt. tx 0 Sinceweassumeddf| =I,and f issmooth,thendf iscontinuous,soforany(cid:34)>0,there’saneighborhoodU of 0 0suchthatforall x ∈U,df| =I+A,where(cid:107)A(cid:107)<(cid:34). Whenweintegratethis,thismeans y =x+o(|x|): df is x “smallin x.” Hence,|x|−(cid:34)<|y|<|x|+(cid:34),sosinceU isbounded,thisputsaboundon x intermsof y,too;in otherwords, x = y+o(|y|)(thisislittle-o,becausewecandothisforany(cid:34)>0,thoughtheneighborhoodmay change). Thisisexactlywhatitmeansfor f−1 tobedifferentiableat y = f(0),anditsderivativeistheidentity! In general,ifdf|0(cid:54)=I,butisstillinvertible,thenwegetthatdf−1|f(0)=(df|0)−1. We’dlikethistoextendtoaneighborhoodoftheorigin. Sincedf| isinvertible,anddf iscontinuous,then 0 locallyaneighborhoodof0correspondstoaneighborhoodofdf| inthespaceofn×nmatrices,andviceversa. 0 Butthesetofinvertiblematricesisopeninthespaceofmatrices,sothere’saneighborhoodV of0suchthatdf| x isinvertibleforall x ∈V,soforeach x ∈V,df−1|f(x)=(df|x)−1. Then,matrixinversionisacontinuousfunction onthesubspaceofinvertiblematrices,sothismeansdf−1 iscontinuousinaneighborhoodof f(0). Thisgivesusonederivative;wewantedinfinitelymany. Usingthechainrule, ∂(df−1) ∂(df)−1 ∂x = , ∂ y ∂x ∂ y and ∂x = (df)−1. So we want to understand derivatives of matrices. Let Abe some invertible matrix-valued ∂y function,sothatAA−1=I. Thus,usingtheproductrule,A(cid:48)A−1=A(A−1)(cid:48)=0,sorearranging,(A−1)(cid:48)=A−1A(cid:48)A−1. Thatis,thederivativeinversecanbespecifiedintermsoftheinverseandthederivativeofA. Inparticular,this means ∂(df−1) isaproductofcontinuousfunctions(∂(df) and(df)−1),soitiscontinuous. Bythesameargument, ∂y ∂x soisthepartialderivativeinthe x-direction,sobyTheorem1.4,df−1 isdifferentiable. Thiscanberepeatedasan inductiveargumenttoshowthatdf−1 isdifferentiableasmanytimesasdf is,andbysmoothness,thisisinfinitely often. (cid:130) WecanusethistorecovertherestofTheorem1.5ascorollaries. 6 M382D(DifferentialTopology)LectureNotes ProofofTheorem1.5,part(2). First, for the implicit function theorem, let n> m and f :(cid:82)n →(cid:82)m be smooth withfullrank,andchooseabasisinwhichdf| =(A|B)inblockform,whereAisaninvertiblem×mmatrix. a Thetheoremstatementisthatwecanwritethefirst mcoordinatesasafunctionofthelast n−mcoordinates: specifically,thatthereexistsaneighborhood U of a suchthat U∩ f−1(f(a))=U∩{g(y),y}forsomesmooth g:(cid:82)n−m→(cid:82)m.2 Now,theproof. Let x ∈(cid:82)m and y ∈(cid:82)n−m,andlet (cid:129)x(cid:139) (cid:129)f(x,y)(cid:139) F = . y y Hence, (cid:18) (cid:19) A B dF| = . a 0 I Thisisinvertible,sinceAis: det(dF| )=det(A)(cid:54)=0. Thus,weapplytheinversefunctiontheoremtoF toconclude a thatasmooth F−1 exists,andsoifπ denotesprojectionontothefirstcomponent, x =π ◦F−1(0,y)=g(y). (cid:130) 1 1 Lecture3. Manifolds: 1/25/16 “Eraseanynotesyouhaveofthelasteightminutes! Butthefirst40minuteswereokay.” Recallthatwe’vebeendiscussionTheorem1.5,acollectionofresultscalledtheinversefunctiontheorem,the implicitfunctiontheorem,andtheimmersiontheorem. Thesearelocal(notglobal)results,andgeneralizesimilar resultsforlinearmaps: notallmatricesaresquare,butifamatrixhasfullrank,itcanbewrittenintwoblocks, oneofwhichisinvertible. Usingthiswithdf| asourmatrixistheideabehindprovingTheorem1.5: thefirst a severalvariablesdeterminetheremainingvariables. However,wedon’tknowwhichvariablestheyare: youmayhavetopermute x ,...,x togetthelastvariables 1 n assmoothfunctionsofthefirstones. Forexample,foracircle,thetangentlineishorizontalsometimes(sowe can’talwaysparameterizeintermsof x )andverticalatothertimes(sowecan’tonlyuse x ). 2 1 Beforeweprovetheimmersiontheorem(part(3)ofTheorem1.5),let’srecallwhattoolsweusetotalkabout curvesintheplane. (1) Acommontechniqueisusingaparameterizedcurve,theimageofasmoothγ(t):(cid:82)→(cid:82)2whosederivative isneverzero(toavoidsingularities). Forexample, f(t)=(t2,t3)hasazeroattheorigin,butthecurve oneobtainsis y =±x3/2,whichhasacuspat(0,0). Thisisthecontentoftheimmersiontheorem. (2) Anotherwaytodescribecurvesisaslevelsets: f(x,y)=c,mostfamouslythecircle. Thisisthecontent oftheimplicitfunctiontheorem: thislookslikeagraph-likecurvelocally. (3) Thisbringsustothemostsimplemethod: graphsoffunctions,justlikeincalculus. AndthepointofTheorem1.5isthatthesethreeapproachesgiveyouthesamesets,uptopermutationofvariables (andthatacurveisthegraphofafunctiononlylocally). Wehavethesethreepicturesofwhathigher-dimensional surfaceslooklike. Andthatmeansthatwhenwetalkaboutmanifolds,whicharetheanalogueofhigher-dimensionalsurfaces,we shouldkeepthesethingsinmind: amanifoldmaybedefinedabstractly,butweunderstandmanifoldsthrough thesethreevisualizations. ProofofTheorem1.5,part(3). We’regoingtoprovetheequivalentstatementthatifthefirstnrowsofdf| are a linearlyindependent,thentheremainingm−nvariablesaresmoothfunctionsinthefirstn. 2Forexample,ifn=2andm=1,consider f(x)=|x|2−1,anda=(cosθ,sinθ). Then, f−1(f(a))istheunitcircle,sotheimplicit functionistellingusthatlocally,thecircleisafunctionofx intermsofx ,orviceversa. 1 2 4 Manifolds: 1/25/16 7 Since f : (cid:82)n → (cid:82)m, then let π denote projection onto the first n coordinates, so we have a commutative 1 diagram (cid:82)n f (cid:47)(cid:47)(cid:82)m π π ◦f (cid:33)(cid:33) (cid:15)(cid:15) 1 1 (cid:82)n. Inblockform,df| =(cid:0)A(cid:1),whereAisinvertible,andtherefored(π ◦ f)| =A. Thisisinvertible,so(π ◦ f)−1 a B 1 a 1 hasaninverseinaneighborhoodofa,bytheinversefunctiontheorem. Thus,ifπ denotesprojectionontothe 2 last m−ncoordinates,then g =π ◦ f ◦(π ◦ f)−1 writesthelast m−ncoordinatesintermsofthefirst n,as 2 1 desired. (cid:130) Now,we’rereadytotalkaboutsmoothmanifolds. Definition3.1. Ak-manifoldX in(cid:82)n isasetthatlocallylookslikeoneofthedescriptions(1),(2),or(3)fora smoothsurface. Thatis,itsatisfiesoneofthefollowingdescriptions. (1) Forevery p∈X,there’saneighborhoodU of pwhereonecanwriteN−kvariablesinsmoothfunctions oftheremainingkvariables,i.e.thereisaneighborhoodV ⊂(cid:82)k andasmooth g:V →(cid:82)N−k suchthat X ∩U ={(x,g(x)):x ∈V}(uptopermutation). (2) X islocallytheimageofasmoothmap,i.e.forevery p∈X,there’saneighborhoodU of pandasmooth f :(cid:82)k →(cid:82)N with full rank such that the image of f in U is X ∩U. This is the “parameterized curve” analogue. (3) Locally,X isthelevelsetofasmoothmap f :(cid:82)N →(cid:82)N−k withfullrank. Ifkisunderstoodfromcontext(ornotimportant),X willalsobecalledamanifold. Thebigtheoremisthatthesethreeconditionsareequivalent,andthisfollowsdirectlyfromTheorem1.5. Forexample,supposewehavethegraphofasmoothfunction y =x2. Howcanwewritethisastheimageofa smoothmap? Well,(x,y)=(t,t2)hasnonzeroderivative,andwecandoexactlythesamething(locally)fora manifoldingeneral. Andit’sthelevelset f(x,y)=0,where f(x,y)= y−x2,andthesamethingworks(locally) formanifolds: forageneralgraphy= g(x),thisisthelevelsetof f(y,x)=y−g(x),whosederivativedf has blockmatrixform(I | −dg),whichhasfullrank. Neat. Andperhapsmostusefulfornow,somethingthat’slocallyagraphisreallyeasytovisualize: it’sthebedrockon whichonefirstdefinedcurvesandsurfaces. Now,that’samanifoldin(cid:82)n. AsfarasGuilleminandPollackareconcerned,that’stheonlykindofmanifold thereis,butwewanttotalkaboutabstractmanifolds,butthatmeanswe’llneedonemoreimportantproperty. SupposeX ⊂(cid:82)N isamanifold,and p∈X. We’regoingtolookataneighborhoodof pastheimageofasmooth g :(cid:82)k→(cid:82)N;thisisthemostcommonandmostfundamentaldescriptionofamanifold. However,thisisnotin 1 generalunique;suppose g :(cid:82)k→(cid:82)N landsinadifferentneighborhoodof p—though,byrestrictingtotheir 2 intersection,wecanassumewehavetwosmoothmaps(sometimescalledcharts)intothesameneighborhood, andtheybothhaveinverses,sowehaveawell-definedfunction g−1◦g :(cid:82)k→(cid:82)k. Isitsmooth? 2 1 Theorem3.2. g−1◦g issmooth. 2 1 Thekeyassumptionhereisthatdg anddg bothhavemaximalrank. 1 2 Definition3.3. ThetangentspacetoX at p,denoted TpX,isIm(dg1|g−1(p));itisak-dimensionalsubspaceof(cid:82)N. 1 Thisisthesetofvelocityvectorsofpathsthrough p,whichmakessense,becausesuchapathmustcomefroma pathdownstairsin(cid:82)k,since g islocallyinvertible. 1 Lemma3.4. Thetangentspaceisindependentofchoiceof g . 1 TheideaisthatanyvelocityvectormustcomefromapathinbothIm(dg1|g−1(p))andIm(dg2|g−1(p)),sothese 1 2 twoimagesarethesame. Then,we’llpunttheproofofTheorem3.2tonextlecture. 8 M382D(DifferentialTopology)LectureNotes Lecture4. Abstract Manifolds: 1/27/16 Lasttime,weweretalkingaboutchangeofvariables,butweweremissingalemmathat’simportantforthe proof,butnotreallytherightwaytoviewmanifolds. LetX beak-dimensionalmanifoldin(cid:82)n,soforanyp∈X,there’samapφfromtheneighborhoodoftheorigin in(cid:82)k toaneighborhoodof p in X,whereφ(0)=p anddφ| hasrank k. We’dlikealocalinversetoφ,which 0 we’llcall F;it’samapfromaneighborhoodof(cid:82)n toaneighborhoodof(cid:82)k. We’dlike F tobesmooth,andwe want F◦φ=id|(cid:82)k. Bypermutingcoordinates,wecanassumethatthefirstkrowsofdφ arelinearlyindependent. Thatis,dφ| 0 hasblockform(cid:0)A(cid:1),whereAisinvertible. Then,defineφ(cid:101):(cid:82)k×(cid:82)n−k→(cid:82)n sending(x,y)T→φ(x)+(0,y)T,3so B thatφ(cid:101)(x,0))=φ(x). φ andφ(cid:101)fitintothefollowingdiagram. (cid:82)k(cid:31)(cid:127) x(cid:55)→(x,0) (cid:47)(cid:47)(cid:82)n φ(cid:101) (cid:47)(cid:47)(cid:52)(cid:52)(cid:82)n φ Thus,bythechainrule, (cid:18) (cid:19) A 0 dφ(cid:101)|0= B I , sodφ(cid:101)|0 hasfullrank! Thus,inaneighborhoodof p,ithasaninverse,andcertainlytheinclusion(cid:82)k(cid:44)→(cid:82)n hasa leftinverseπ(projectionontothefirstkcoordinates),sowecanlet F =π◦φ(cid:101)−1,because F◦φ(x)=F◦φ(cid:101)(x,0)=π◦φ(cid:101)−1◦φ(cid:101)((x,0))=π(x,0)=x. Likewise,φ◦F =id| ,sinceeverypointinourneighborhoodisintheimageofφ. X Thisishowwetalkaboutsmoothnessonmanifolds: wedon’tknowwhatsmoothnessmeansonsomearbitrary submanifold,sowe’llusethefactthatwecanlocallypretendwe’rein(cid:82)n totalkaboutsmoothness. Supposeφ,ψ:(cid:82)k⇒X aretwosuchsmoothcoordinatemaps;we’dliketofindasmoothfunction g froma neighborhoodin(cid:82)k toaneighborhoodin(cid:82)k relatingthem(again,locally). Butwehavealocalinversetoψcalled F,sosincewewantψ=φ◦g,thendefine g=F◦ψ,becauseφ◦g=φ◦F◦ψ=ψ. And g isthecompositionof twosmoothfunctions,soit’ssmooth(thisisTheorem3.2). Thisisourchange-of-coordinatesoperation. Theorem4.1. Afunction g:X →(cid:82)m canbeextendedtoasmoothmapG onaneighborhoodof pin(cid:82)n iff g◦φ is smooth. Thisisanothernotionofsmooth: thefirstonedeterminessmoothnessbycoordinates,andthesecondsays thatsmoothfunctionsonasubmanifoldarerestrictionsofsmoothfunctions(cid:82)n→(cid:82)m. Butthetheoremsaysthat they’retotallyequivalent. Proof. SupposesuchasmoothextensionG exists;sinceG| =g andIm(φ)⊂X,thenG◦φ=g◦φ. G andφ are X smooth,soG◦φ=g◦φ issmooth. Conversely,if g◦φ issmooth,thenletG=g◦φ◦F,whichisasmoothmap(sinceit’sacompositionoftwo smoothfunctions)outofaneighborhoodof pin(cid:82)n. (cid:130) ThisextrinsicdefinitionistheoneGuilleminandPollackusethroughouttheirbook;theothernotiondoesn’t dependonanembeddinginto(cid:82)n,butwehadtocheckthatitwasindependentofchangeofcoordinates(which byTheorem3.2issmooth,sowe’reOK).Thismeanswecanmakethefollowingdefinition. Definition4.2. • Achart(cid:82)k→X foratopologicalspaceX isacontinuousmapthat’sahomeomorphismontoitsimage. • An(abstract)smoothk-manifoldisaHausdorffspaceX equippedwithchartsϕα:(cid:82)k→X suchthat (1) everypointinX isintheimageofsomechart,and 3φ(cid:101)ispronounced“phi-twiddle.” 5 AbstractManifolds: 1/27/16 9 (2) foreverypairofoverlappingchartsϕα andϕβ,thechange-of-coordinatesmapϕβ−1◦ϕα:(cid:82)k→(cid:82)k issmooth. Thedefinitionissometimeswrittenintermsofneighborhoodsin(cid:82)k,soeachchartisamap U →X,where U ⊂(cid:82)k,butthisiscompletelyequivalenttothegivendefinition,sincetan:(−π/2,π/2)→(cid:82)isadiffeomorphism (andtherearemanyothers,e.g.ex/(1+ex)). Thepointisthateverypointhasaneighborhoodhomeomorphicto (cid:82)k,evenifwethinkofneighborhoodsaslittleballsmuchofthetime. Therearelotsofdifferentcategoriesofmanifolds: aCn manifoldhasthesamedefinition,butwerequirethe change-of-coordinatesmapstomerelybeCn (ntimescontinuouslydifferentiable);ananalyticmanifoldrequires thechange-of-coordinatesmapstobeanalytic;andinthesamewayonecandefinecomplex-analyticmanifolds (holomorphicchange-of-coordinatesmaps)andalgebraicmanifolds. Foratopologicalmanifoldwejustrequirethe change-of-coordinatesmapstobecontinuous,whichisalwaystrueforacoveringofcharts. Butinthisclass,the degreeofregularitywecareaboutissmoothness. Definition 4.3. Let X be a manifold and f : X → (cid:82)n be continuous. Then, f is smooth if for every chart ϕα:(cid:82)k→X,thecomposition f ◦ϕα issmooth. Thisisjustlikethedefinitionofsmoothnessformanifoldslivingin(cid:82)n. Example4.4. LetX bethesetoflinesin(cid:82)2(notjustthesetoflinesthroughtheorigin). Thisisamanifold,butwe wanttoshowthis. Usingpoint-slopeform,wecandefineamapφ :(cid:82)2→X sending(a,b)(cid:55)→{(x,y): y =ax+b}, 1 whichcoversalllinesthataren’tvertical. Weneedtohandletheverticallineswithanotherchart,φ :(cid:82)2→X 2 sending(c,d)(cid:55)→x =cy+d. Thesechartsintersectforalllinesthatareneitherverticalnorhorizontal,sothechange-of-coordinatesmap describesc=1/aandd=−b/a,i.e. g(a)=(1/a,−b/a). Andsincewe’rerestrictedtonon-verticallines,a(cid:54)=0, sothisissmooth,and g−1(c,d)=(1/c,−d/c),whichisalsosmooth(sincewe’renotlookingathorizontallines). Thus,we’redescribedX asamanifold. It turns out that X is a Möbius band. A line may be described by a direction (an angle coordinate) and an offset(intersectionwiththe x-axis,headinginthespecifieddirection). However,therearetwodescriptions,given byflippingthedirection: (θ,D)∼(θ+π,−D). Thus,thisisthequotientofaninfinitelylongcylinderbyhalfa rotationandatwist,givingusaMöbiusband. Onethingwehaven’ttalkedmuchaboutis: whydomanifoldsneedtobeHausdorff? Thismakesourexample muchlessterrible: here’sjustonecreatureweavoidwiththiscondition. Example4.5(Linewithtwoorigins). Taketwocopiesof(cid:82)2,andidentify(x,1)∼(x,2)forall x (cid:54)=0. Thus,we seemtohaveonecopyof(cid:82),buttwodifferentcopiesoftheorigin. Thechartsareperfectlynice: anyintervalon eithercopyof(cid:82)isachartforthisspace,buteveryneighborhoodofoneoftheoriginscontainstheother,soitisn’t Hausdorff(itis T ,though). SeeFigure1fora(notperfectlyaccurate)depictionofthisspace. Wedon’twantto 1 • • FIGURE 1. Depiction of the line with two origins. Note, however, that the two origins are technicallyinfinitelyclosetogether. havespaceslikethisone,sowerequiremanifoldstobeHausdorff. TuneinFridaytolearnhowtodeterminewhentwomanifoldsareequivalent. Isthesamespacewithdifferent chartsadifferentmanifold? Lecture5. Examples of Manifolds and Tangent Vectors: 1/29/16 “Howdoyoumaketheunitdiscintoamanifold? Withpiecharts.” 10 M382D(DifferentialTopology)LectureNotes Today,we’regoingtomakethenotionofamanifoldmorefamiliarbygivingsomemoreexamplesofwhatstructures canarise: specifically,the2-sphereS2 andtheprojectivespaces(cid:82)(cid:80)n and(cid:67)(cid:80)n. Then,we’llmovetodiscussing tangentvectorsandhowtodefinesmoothmapsbetweenmanifolds. Example5.1(2-sphere). Theconcrete2-sphereisS2={x∈(cid:82)3:|x|2=1}. Whyisthisamanifold? FIGURE2. The2-sphere,anexampleofamanifold. (cid:112) Wecanputchartson(cid:112)thissurfaceasfollows: ifz>0,thenwehave(cid:112)achart(u,v, 1−u2−v2),andifz<0, thenthechartis(u,v,− 1−u2−v2). Similarly,if y >0,wehave(u, 1−u2−v2,v),andsimilarlyfor y <0 andfor x. However,since0(cid:54)∈S2,thenthiscoversallofS3,andonecancheckthatthetransitionmapsaresmooth andthechartmapshavefullrank. Anotherwaytorealizethisisthatif f :(cid:82)3→(cid:82)isdefinedby f(x,y,z)=x2+y2+z2,then f issmoothand S2= f−1(1). Thus,S2 isthelevelsetofasmoothfunctionwhosederivativedf =(2x,2y,2z)hasfullrank,soby theimplicitfunctiontheorem,itmustbeamanifold. Thatis,youcanseeS2 isamanifoldusingmapsintoit,ormapsoutofit. Example5.2(Realprojectivespace). (cid:82)(cid:80)n,realprojectivespace,isdefinedtobethesetoflinesthroughtheorigin in(cid:82)n+1. Anynonzeropointin(cid:82)n+1 definesalinethroughtheorigin,andscalingapointdoesn’tchangethisline. Thus,(cid:82)(cid:80)n={r∈(cid:82)n+1\0}/(r∼λrforλ∈(cid:82)\0). Wehavecoordinates(x ,...,x )for(cid:82)n+1,andwanttomake 0 n coordinateson(cid:82)(cid:80)n. ThesetU ={x:x (cid:54)=0}isopen,and(x ,x ,...,x )∼(1,x /x ,...,x /x )in(cid:82)(cid:80)n,sowegetachartonU . 0 0 0 1 n 1 0 n 0 0 We’re parameterizing non-horizontal lines by their slope (or, well, the reciprocal of it). Thus, we have a map ψ :(cid:82)n→(cid:82)(cid:80)n sending(x ,...,x )(cid:55)→[(1,x ,...,x )](wherebracketsdenotetheequivalenceclassin(cid:82)(cid:80)n). 0 1 n 1 n Wecandothiswitheverycoordinate: letψ :(cid:82)n→(cid:82)(cid:80)n send(x ,...,x )(cid:55)→[(x ,1,x ,...,x )],andsoforth. 1 1 n 1 2 n Then,sinceeverypointin(cid:82)(cid:80)n hasanonzerocoordinate,thenthiscovers(cid:82)(cid:80)n. Arethetransitionmapssmooth? (cid:82)(cid:80)2 willillustratehowitworks: if[1,a,b]=[c,1,d],thenc=1/aandd = b/a,whichissmooth(becausein thesecharts,aandc arenonzero). Bytheway,(cid:82)(cid:80)1 isjustacircle. Moregenerally,onecanalsorealize(cid:82)(cid:80)n astheunitspherewithoppositepoints identified(everyvectorcanbescaledtoaunitvector,butthenx∼−x). However,(cid:82)(cid:80)2,etc.,aremoreinteresting spaces. Example5.3(Complexprojectivespace). Wecanalsorefertocomplexprojectivespace,(cid:67)(cid:80)n. Theideaof“lines throughtheorigin”isthesame,but,despitewhatalgebraicgeometerscallit,aone-dimensionalcomplexsubspace looksalotmorelikea(real)planethanarealline. Inanycase,one-dimensionalcomplexsubspacesof(cid:67)n+1 are givenbynonzerovectors,sowedefine(cid:67)(cid:80)n ={r∈(cid:67)n+1\0}/(r∼λr,λ∈(cid:67)\0). Now,thesamedefinitionsof chartsgiveusψ :(cid:67)n→(cid:67)(cid:80)n,butsinceweknowhowtomap(cid:82)2n→(cid:67)n,thisworksjustfine. k Inthiscase,thefirstinterestingcomplexprojectivespaceis(cid:67)(cid:80)1. Ourtwochartsare[1,a]and[b,1],andtheir overlapiseverythingbutthetwopoints[1,0]and[0,1]. Inotherwords,everypointisoftheform[z,1]forsome z ∈(cid:67)or[1,0]: thatis[1,0]isa“pointatinfinity”∞,whosereciprocalis0! So(cid:67)(cid:80)1 isthecomplexnumbers plusoneextrapoint. WecanactuallyrealizethisasS2 usingamapcalledstereographicprojection: thespheresits inside(cid:82)3,andthe xy-planecanbeidentifiedwith(cid:67). Then,thelinebetweenthenorthpole(0,0,1)andagiven (u,v,0)(correspondingto[u+vi,0])intersectsthesphereatasinglepoint,whichisdefinedtobetheimageof theprojection(cid:67)(cid:80)1→S2. However,thepointatinfinityisn’tidentifiedinthisway,andneitheristhenorthpole;