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1、NotesonBasic3-ManifoldTopologyAllenHatcherChapter1.CanonicalDecomposition1.PrimeDecomposition.2.TorusDecomposition.Chapter2.SpecialClassesof3-Manifolds1.SeifertManifolds.2.TorusBundlesandSemi-Bundles.Chapter3.HomotopyProperties1.TheLoopandSphereTheorems.Thesenotes,originally
2、writteninthe1980’s,wereintendedasthebeginningofabookon3manifolds,butunfortunatelythatprojecthasnotprogressedveryfarsincethen.Afewsmallrevisionshavebeenmadein1999and2000,butmuchmoreremainstobedone,bothinimprovingtheexistingsectionsandinaddingmoretopics.Thenexttopictobeaddedwi
3、llprobablybeHakenmanifoldsinx3.2.Foranysubsequentupdateswhichmaybewritten,theinterestedreadershouldcheckmywebpage:http://www.math.cornell.edu/˜hatcherThethreechaptersherearetoacertainextentindependentofeachother.ThemainexceptionsarethatthebeginningofChapter1isaprerequisitefo
4、ralmostev-erythingelse,whilesomeofthelaterpartsofChapter1areusedinChapter2.x1.1PrimeDecomposition1Chapter1.CanonicalDecompositionThischapterbeginswiththefirstgeneralresulton3manifolds,Kneser’stheoremthateverycompactorientable3manifoldMdecomposesuniquelyasaconnectedsumMP1]]
5、Pnof3manifoldsPiwhichareprimeinthesensethattheycan3bedecomposedasconnectedsumsonlyinthetrivialwayPiPi]S.Aftertheprimedecomposition,weturninthesecondsectiontothecanonicaltorusdecompositionduetoJaco-ShalenandJohannson.1WeshallworkintheCcategorythroughout.All3manifoldsinthischa
6、pterareassumedtobeconnected,orientable,andcompact,possiblywithboundary,unlessotherwisestatedorconstructed.1.PrimeDecomposition3ImplicitintheprimedecompositiontheoremisthefactthatSisprime,other-wiseonecouldonlyhopeforaprimedecompositionmoduloinvertibleelements,asinalgebra.Thi
7、sisimpliedbyAlexander’stheorem,ourfirsttopic.Alexander’sTheoremThisquitefundamentalresultwasoneoftheearliesttheoremsinthesubject:T3heorem1.1.Everyembedded2sphereinRboundsanembedded3ball.3Proof:LetSRbeanembeddedclosedsurface,withh:S!Rtheheightfunctiongivenbythezcoordinate.Aft
8、erasmallisotopyofSwemayassumehisamorsefunctionwithallitscriticalpointsindis
