basic topology of 3-manifolds

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1、NotesonBasic3-ManifoldTopologyAllenHatcherChapter1.CanonicalDecomposition1.PrimeDecomposition.2.TorusDecomposition.Chapter2.SpecialClassesof3-Manifolds1.SeifertManifolds.2.TorusBundlesandSemi-Bundles.Chapter3.HomotopyProperties1.TheLoopandSphereTheorems.Thesenotes,originallywritteninthe1

2、980’s,wereintendedasthebeginningofabookon3manifolds,butunfortunatelythatprojecthasnotprogressedveryfarsincethen.Afewsmallrevisionshavebeenmadein1999and2000,butmuchmoreremainstobedone,bothinimprovingtheexistingsectionsandinaddingmoretopics.ThenexttopictobeaddedwillprobablybeHakenmanifolds

3、inx3.2.Foranysubsequentupdateswhichmaybewritten,theinterestedreadershouldcheckmywebpage:http://www.math.cornell.edu/˜hatcherThethreechaptersherearetoacertainextentindependentofeachother.ThemainexceptionsarethatthebeginningofChapter1isaprerequisiteforalmostev-erythingelse,whilesomeofthela

4、terpartsofChapter1areusedinChapter2.x1.1PrimeDecomposition1Chapter1.CanonicalDecompositionThischapterbeginswiththefirstgeneralresulton3manifolds,Kneser’stheoremthateverycompactorientable3manifoldMdecomposesuniquelyasaconnectedsumMP1]


]Pnof3manifoldsPiwhichareprimeinthesensethattheycan3b

5、edecomposedasconnectedsumsonlyinthetrivialwayPiPi]S.Aftertheprimedecomposition,weturninthesecondsectiontothecanonicaltorusdecompositionduetoJaco-ShalenandJohannson.1WeshallworkintheCcategorythroughout.All3manifoldsinthischapterareassumedtobeconnected,orientable,andcompact,possiblywithbou

6、ndary,unlessotherwisestatedorconstructed.1.PrimeDecomposition3ImplicitintheprimedecompositiontheoremisthefactthatSisprime,other-wiseonecouldonlyhopeforaprimedecompositionmoduloinvertibleelements,asinalgebra.ThisisimpliedbyAlexander’stheorem,ourfirsttopic.Alexander’sTheoremThisquitefundame

7、ntalresultwasoneoftheearliesttheoremsinthesubject:T3heorem1.1.Everyembedded2sphereinRboundsanembedded3ball.3Proof:LetSRbeanembeddedclosedsurface,withh:S!Rtheheightfunctiongivenbythezcoordinate.AfterasmallisotopyofSwemayassumehisamorsefunctionwithallitscriticalpointsindis