Loop Products and Closed Geodesics

arXiv:0707.3486v1[math.AT]24Jul2007LOOPPRODUCTSANDCLOSEDGEODESICSMARKGORESKY1ANDNANCYHINGSTON2Abstract.ThecriticalpointsofthelengthfunctiononthefreeloopspaceΛ(M)ofacompactRie-mannianmanifoldMaretheclosedgeodesicsonM.ThelengthfunctiongivesafiltrationofthehomologyofΛ(M)andweshowthattheChas-SullivanproductHi(Λ)×Hj(Λ)∗-Hi+j−n(Λ)iscompatiblewiththisfiltration.WeobtainaverysimpleexpressionfortheassociatedgradedhomologyringGrH∗(Λ(M))whenallgeodesicsareclosed,orwhenallgeodesicsarenondegenerate.WealsoconstructanewbutrelatedcohomologyproductHi(Λ,Λ0)×Hj(Λ,Λ0)⊛-Hi+j+n−1(Λ,Λ0)(whereΛ0=Mistheconstantloops),alsocompatiblewiththelengthfiltration,andweobtainasimilarexpressionfortheringGrH∗(Λ,Λ0)inthesetwocases.Thenon-vanishingofproductsσ∗n∈H∗(Λ)andτ⊛n∈H∗(Λ,Λ0)isshowntoberelatedR.Bott’sanalysisoftherateatwhichtheMorseindexgrowswhenageodesicisiterated.Contents1.Introduction22.Thefreeloopspace63.ThefinitedimensionalapproximationofMorse84.Support,Criticalvalues,andlevelhomology105.TheChas-SullivanProduct116.Indexgrowth157.Levelnilpotence188.Cohomologyproducts209.Supportandcriticallevels2910.Levelnilpotenceforcohomology3111.Levelproductsinthenondegeneratecase3112.Homologyproductwhenallgeodesicsareclosed3613.Cohomologyproductswhenallgeodesicsareclosed4014.Relatedproducts45AppendixA.ˇCechhomologyandcohomology47AppendixB.Thomisomorphisms49AppendixC.TheoremsofMorseandBott52AppendixD.ProofofTheorem14.254AppendixE.Associativityof⊛58References60Keywordsandphrases.Chas-Sullivanproduct,loopproduct,freeloopspace,Morsetheory,energy.1.SchoolofMathematics,InstituteforAdvancedStudy,PrincetonN.J.ResearchpartiallysupportedbyDARPAgrant#HR0011-04-1-0031.2.Dept.ofMathematics,CollegeofNewJersey,EwingN.J..11.Introduction1.1.LetMbeasmoothcompactmanifoldwithoutboundary.In[CS],M.ChasandD.SullivanconstructedanewproductstructureHi(Λ)×Hj(Λ)∗-Hi+j−n(Λ)(1.1.1)onthehomologyH∗(Λ)ofthefreeloopspaceΛofM.In[CKS]itwasshownthatthisproductisahomotopyinvariantoftheunderlyingmanifoldM.Incontrast,theclosedgeodesicsonMdependonthechoiceofaRiemannianmetric,whichwenowfix.InthispaperweinvestigatetheinteractionbeweentheChas-SullivanproductonΛandtheenergyfunction,orrather,itssquareroot,F(α)=pE(α)=Z10|α′(t)|2dt1/2,whosecriticalpointsareexactlytheclosedgeodesics.Foranya,0≤a≤∞wedenotebyΛ≤a,Λa,Λ=a,Λ(a,b](1.1.2)thoseloopsα∈ΛsuchthatF(α)≤a,F(α)a,F(α)=a,aF(α)≤b,etc.(Whena=∞wesetΛa=Λ≤a=Λ.)InthispaperwewillusehomologyH∗(Λ≤a;G)withcoefficientsintheringG=ZifMisorientableandG=Z/(2)otherwise.In§5weprovethefollowing.1.2.Theorem.TheChas-Sullivanproductextendstoafamilyofproducts1ˇHi(Λ≤a)סHj(Λ≤b)∗−→ˇHi+j−n(Λ≤a+b)ˇHi(Λ≤a,Λ≤a′)סHj(Λ≤b,Λ≤b′)∗−→ˇHi+j−n(Λ≤a+b,Λ≤max(a+b′,a′+b))ˇHi(Λ≤a,Λa)סHj(Λ≤b,Λb)∗−→ˇHi+j−n(Λ≤a+b,Λa+b)(1.2.1)whenever0≤a′a≤∞and0≤b′b≤∞.TheseproductsarecompatiblewithrespecttothenaturalinclusionsΛ≤c′→Λ≤cwheneverc′≤c.WerefertoˇHi(Λ≤a,Λa)asthelevelhomologygroup,orthehomologyatlevela,withitsassociatedlevelhomologyproduct(1.2.1).ItiszerounlessaisacriticalvalueofF.1.3.In§8weconsideranalogousproductsinthecohomologyofthefreeloopspace.(Wediscussthecupproductbrieflyin§9.4.)ItispossibletomimictheconstructionoftheChas-Sullivanproduct,wordforword,incohomology,butthisresultsinatrivialproduct,cf.§8.1.Howeverbyutilizingacertainoneparameterfamilyofreparametrizations,itispossibletoconstructanontrivialproductincohomology.1Here,ˇHi(Λ≤a)denotesˇCechhomology.InLemmaA.4weshowthatthesingularandˇCechhomologyagreeif0≤a≤∞isaregularvalueorifitisanondegeneratecriticalvalueinthesenseofBott.21.4.Theorem.Let0≤a′a≤∞and0≤b′b∞.ThereisafamilyofproductsHi(Λ,Λ0)×Hj(Λ,Λ0)⊛−→Hi+j+n−1(Λ,Λ0)(1.4.1)ˇHi(Λ≤a,Λ≤a′)סHj(Λ≤b,Λ≤b′)⊛−→ˇHi+j+n−1(Λ≤min(a+b′,a′+b),Λ≤a′+b′)ˇHi(Λ≤a,Λa)סHj(Λ≤b,Λb)⊛−→ˇHi+j+n−1(Λ≤a+b,Λa+b)(1.4.2)whichareassociativeand(sign-)commutative,andarecompatiblewiththehomomorphismsinducedbytheinclusionsΛ≤c′→Λ≤cwheneverc′c.Theproduct(1.4.1)isindependentoftheRiemannianmetric.Thesameconstructiongives(cf.§8.4)a(possiblynoncommutative)product⊛onthecohomologyofthebasedloopspaceΩsuchthath∗(a⊛b)=h∗(a)⊛h∗(b)wherea,b∈H∗(Λ)andh:Ω→Λdenotestheinclusion.In§13.9wecalculatesomenon-zeroexamplesofthisproduct.1.5.Ifthering(H∗(Λ,Λ0),⊛)isfinitelygeneratedthentheexistenceoftheproduct⊛isalreadyenoughtoansweraquestionofY.Eliashberg,cf.§9.5:themaximaldegreeofan“essential”homologyclassoflevel≤tcangrowatmostlinearlywitht.1.6.Thereisawell-knownisomorphismbetweentheFloerhomologyofthecotangentbundleofMandthehomologyofthefreeloopspaceofM,whichtransformsthepair-of-pantsproductintotheChas-Sullivanproductonhomology,see[AS1,AS2,SaW,Vi,CHV].ThecohomologyproductdescribedaboveshouldthereforecorrespondtosomegeometricallydefinedproductontheFloercohomology;itwouldbeinterestingtoseeanexplicitconstruc-tionofthisproduct.(Theobviouscandidatewouldbesome1-parametervariationofthecoproductonhomologygivenbytheupside-downpairofpants.)Itwouldalsobeinterestingtocomparethecohomologyproductdescribedabovewiththecoproductinhomologythatisoutlinedin[Su].(Forodddimensionalspherestheproductin[Su]iszerowhilethe⊛productisnon-zero.)1.7.Thecriticalvalue(see§4)ofahom