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1、CHAPTER14SelectedtopicsintheMorse-Novikovtheory1.HomologywithlocalcoefficientsandthedeRhamframeworkfortheMorse-NovikovtheoryForareal-valuedMorsefunctionf:M→RonaclosedmanifoldMtheMorseinequalitiesintheirsimplestformreadasmk(f)bk(M)wheremk(f)isthenumberofcriticalpointsoffofindexkan
2、dbk(M)isthek-thBettinumberofM,acommonalgebro-topologicalinvariant:bk(M)=dimHk(M,Q).Thesimplestlowerboundforthenumberofcriticalpointsofacircle-valuedMorsefunctionisasfollows:mk(f)bk(M,ξ)wheretheNovikovBettinumberbk(M,ξ)istherankoverZ((t))ofthecompletedmoduleHk(M¯)⊗Z((t)).Z[t,t−1
3、]Thisrankisnotsoeasytocalculate,sinceitinvolvestheZ[t,t−1]-modulestructureofthehomologyofthecycliccoveringM¯.ItturnsouthoweverthatthereisasimplerwaytocomputetheNovikovBettinumber.Namelybk(M,ξ)isequaltodim(Hk(M,L))whereHk(M,L)standsfortheho-mologyofMwithcoefficientsinagenericlocals
4、ystemassociatedwithξ.InthenextsubsectionweexplainwhatitmeansandhowtoobtainthecorrespondinginequalitieswithoutusingtheNovikovcomplex(theWittendeformationmethod).1.1.Homologywithlocalcoefficients.LetXbeafiniteconnectedCWcomplex,Gitsfundamentalgroupandρ:G→GL(1,k)≈k∗beahomomorphism,whe
5、rekisafield.ConsidertheuniversalcoveringX→X.ThecellularchaincomplexC∗(X)isacomplexoffreefinitely414Chapter14.TheNovikovcomplexgeneratedleftZG-modules.ConsiderthechaincomplexC∗(X,ρ)=k⊗C∗(X)ZGwherekisendowedwiththestructureofZG-moduleviatherepresentationρ.Thehomologyofthiscomplexisc
6、alledthehomologywithlocalcoeffi-cientswithrespecttoρanddenotedH∗(X,ρ).Inasimilarwayonedefinesthecohomologywithlocalcoefficients.Ifρisthetrivialrepresentation,thenH∗(X,ρ)isisomorphictotheordinaryhomologyH∗(X,k).Nowletξ∈H1(X,C)beacohomologyclass,andt∈C.Considerahomomorphismρ:π(X)=G→C∗;
7、ρ(γ)=etξ,γ¯,t1twhere¯γistheimageofγinH1(X),and·,·standsfortheKroneckerpairingbetweenhomologyandcohomology.LetuswriteBk(ξ,t)=dimHk(X,ρt),βk(ξ)=minBk(ξ,t).t∈CProposition1.1.Letξ∈H1(X,C).ThereisadiscretesubsetΔ∈Csuchthatforeveryt∈CΔwehaveBk(ξ,t)=βk(ξ).Proof.Itfollowsfromthenex
8、tlemma,whichiseasilydeducedfromtheprincipleofis
