光滑闭流形上的Lefschetz不动点定理

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1、Chapter1Introduction1IntroductionLetMbeacompacttopologyspace，assumeMhomeomorphismfinitesimplecom·plexes，amapf：M_÷Minducesalinearmapofthehomologygroup^：绋(肘，Q)--t,炜(肘，Q)，thenLefschetznumberisdefinedbyL(，)=∑(一1)pTr(y,onHp(M，Q))PTheLefschetzfixedpointtheoremis：ifL(f)≠0，thenfhas

2、fixedpoint．In【7】，thereisaproofbytopologymethod．Inthispaper'weassumeMtobeasmoothclosedmanifold，meansasmoothcompactmanifoldwithoutboundary,andf：M---}Mbeasmoothmap．广definesamaponthecohomologyofMandtheLefschetznumberoffisdefinedby：L(，)=∑,(-])PTr(f"onnp(肘；c))WegetthemainresultTh

3、eorem1．1．(LefichetzFixedPointTheorem)矿己(力≠O,thenfhasfocedpoint．i．ethereexistx∈m，suchthatf(x)=工Theorem1．2．(ClassicalLefschetzFixedPointFormula)Letf：M—+Mbesmoothwithisolatednon-degeneratefixedpoints．Then：￡(，)=E(-1)pTr(f。onHp∽；c))=Esigndet(1--df)(xi)pfWecanconsideramorcgeneral

4、case，anypartialellipticcomplex．Weget：Lemma1．3．Let僻E)beanpartialdifferentialellipticcomplexoverMandletf：M_MbesmoothoCinducethemap广：r(e)_÷r(广陋))，where广陋)=工勘b)×％伍)bethepull-backbundle．That西?仃∈r(E)，(厂a)Cx)=仃(厂(工))Weassumegivenlinearmaps靠：局，，0)--+目加∞wedefineF=妒io广：r(e)--+r(E)，su

5、chthat毋F=F毋aboutthevectorbundle岛．Thenfand咖induceamaponH‘(只y)，wedefineL(f)p=∑(一1)‘rr(F+onH‘∽y))ThenwecomputeL(f)p=∑(一1)‘Tr(F+P叫令)whereAi=芹毋+毋一l肇listheassociatedLaplacian．WeusetheheatequationmethodtocomputetheLefschetznumber．1．Wechangethenumbertobethetraceofoperatorf*e一岱2．Com

6、putethetraceofoperatorbythetraceofthekerneloftheoperator3．UsetheinitialoperatorA'ssymbol，togettheoperator(A一九)-1bythemethodofpseodo-differentialoperators，thengetP叫△-一壶上e-tZ(△一Z)-1dZWeusethefollowingtheorem：Lemma1．4．LetPbeanellipticpartialdifferentialoperatoroforderd>0w办耙办is

7、self-adjointandwhichhasapositivedefinitele口dingsymbolfor专≠O．Letf：M_Mbeasmoothnon-degeneratemapandlet驴：E厂仅)-÷晟beasmoothvectorbundlehomomorphism．WedenoteF。=咖of*：r(e)--+r(广(E))—÷r(E)．IfK捃thekerneloye舯thenTr(f*e郇)=厶Tr(F+K)(f，工，x)dvol(x)．(a)lffhasnofvcedpoints，ITr(广e-tP)l≤Gt万asy

8、o，．anynyo，．0