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1、MORSETHEORYBYJ.MilnorBasedonlecturenotesbyM.SPIVAKandR.WELLSANNALSOFMATHEMATICSSTUDIESPRINCETONUNIVERSITYPRESSAnnalsofMathematicsStudiesNumber51MORSETHEORYBYJ.MilnorBasedonlecturenotesbyM.SPIVAKandR.WELLSPRINCETON,NEWJERSEYPRINCETONUNIVERSITYPRESSCopyright©1963,©1969,byPrincetonUniversityPressAl
2、lRightsReservedL.C.Card63-13729ISBN0-691-08008-9ThirdPrinting,withcorrectionsandanewPreface,1969FourthPrinting,1970FifthPrinting,1973PrintedintheUnitedStatesofAmericaPREFACEThisbookgivesapresent-dayaccountofMarstonMorse'stheoryofthecalculusofvariationsinthelarge.However,therehavebeenim-portantde
3、velopmentsduringthepastfewyearswhicharenotmentioned.LetmedescribethreeoftheseR.PalaisandS.SmalenavestudiedMorsetheoryforareal-valuedfunctiononaninfinitedimensionalmanifoldandhavegivendirectproofsofthemaintheorems,withoutmakinganyuseoffinitedimensionalap-proximations.Themanifoldsinquestionmustbel
4、ocallydiffeomorphictoHilbertspace,andthefunctionmustsatisfyaweakcompactnesscon-dition.Asanexample,tostudypathsonafinitedimensionalmanifoldMoneconsiderstheHilbertmanifoldconsistingofallabsolutelycon-tinuouspathsw:(0,11-Mwithsquareintegrablefirstderivative.Ac-countsofthisworkarecontainedinR.Palais
5、,MorseTheoryonHilbertManifolds,Topology,Vol.2(1963),pp.299-340;andinS.Smale,MorseTheoryandaNon-linearGeneralizationoftheDirichletProblem,AnnalsofMathematics,Vol.8o(1964),pp.382_396.TheBottperiodicitytheoremswereoriginallyinspiredbyMorsetheory(seepartIV).However,moreelementaryproofs,whichdonotin-
6、volveMorsetheoryatall,haverecentlybeengiven.SeeM.AtiyahandR.Bott,OnthePeriodicityTheoremforComplexVectorBundles,Acts,Mathematica,Vol.112(1964),pp.229_247,aswellasR.Wood,BanachAlgebrasandBottPeriodicity,Topology,4(1965-66),pp.371-389.Morsetheoryhasprovidedtheinspirationforexcitingdevelopmentsindi
7、fferentialtopologybyS.Smale,A.Wallace,andothers,includingaproofofthegeneralizedPoincarehypothesisinhighdimensions.IhavetriedtodescribesomeofthisworkinLecturesontheh-cobordismtheorem,notesbyL.SiebenmannandJ.Sondow,PrincetonUn
