Morse Theory for Cell Complexes

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1、AdvancesinMathematicsAI1650AdvancesinMathematics134,90145(1998)ArticleNo.AI971650MorseTheoryforCellComplexes*RobinFormanDepartmentofMathematics,RiceUniversity,Houston,Texas77251ReceivedJuly20,19950.INTRODUCTIONInthispaperwewillpresentaverysimplediscreteMorsetheoryforCWcomplexes.Inadditiontoprovin

2、ganaloguesofthemaintheoremsofMorsetheory,wealsopresentdiscreteanaloguesofsuch(seemingly)intrinsicallysmoothnotionsasthegradientvectorfieldandthegradientflowassociatedtoaMorsefunction.Usingthis,wedefineaMorsecomplex,adifferentialcomplexbuiltoutofthecriticalpointsofourdiscreteMorsefunctionwhichhasthe

3、samehomologyastheunderlyingmanifold.ThisMorsetheorytakesonparticularsignificanceinthecontextofPLmanifolds.Toclarifythisstatement,wetakeasmallhistoricaldigression.In1961,Smaleprovedtheh-cobordismtheoremforsmoothmanifolds(andhenceitscorollary,thePoincareconjectureindimension5)usingacombinationofhan

4、dlebodytheoryandMorsetheory[Sm2].In[Mi2],MilnorpresentedacompletelyMorsetheoreticproofoftheh-cobordismtheorem.In([Ma],[Ba],[St])Mazur,BardenandStallingsgeneralizedSmale'stheorembyreplacingSmale'shypothesisthatthecobordismbesimply-connectedbyaweakersimple-homotopycondition.Alongtheway,thismoregenera

5、ltheorem(thes-cobordismtheorem)wasextendedtoothercategoriesofmanifolds.Inparticular,aPLs-cobordismtheoremwasestablished.Inthiscase,itwasnecessarytoworkcompletelywithinthecontextofhandlebodytheory.TheMorsetheorypresentedinthispapercanbeusedtogiveaMorsetheoreticproofofthePLs-cobordismtheorem,alongthe

6、linesoftheproofin[Mi2].Intheremainderofthisintroduction,wepresentaninformalexpositionofthecontentsofthepaper.Toavoidminorcomplicationswewillrestrictattention,inthisintroduction,tosimplicialcomplexes.LetMbeanyfinitesimplicialcomplex,KthesetofsimplicesofM,andKpthesimplicesofdimensionp.AdiscreteMorsef

7、unctiononMwillactuallybeafunctiononK.Thatis,weassignasinglerealnumbertoeach*PartiallysupportedbytheNationalScienceFoundation.900001-87089825.00Copyright1998byAcademicPressAllrightsofreproductioninanyform