Calculation of Hirzebruch genera for manifolds act

arXiv:math/9909081v2[math.AT]29Dec2005CALCULATIONOFHIRZEBRUCHGENERAFORMANIFOLDSACTEDONBYTHEGROUPZ/pVIAINVARIANTSOFTHEACTIONTarasE.PanovAbstract.WeobtaingeneralformulaeexpressingHirzebruchgeneraofamanifoldwithZ/p-actionintermsofinvariantsofthisaction(thesetsofweightsoffixedpoints).Asanillustration,weconsidernumerousparticularcasesofwell-knowngenera,inparticular,theellipticgenus.Wealsodescribetheconnectionwiththeso-calledConner–Floydequationsfortheweightsoffixedpoints.IntroductionInthispaperweobtaingeneralformulaeexpressingHirzebruchgeneraofaman-ifoldactedonbyZ/pwithfinitelymanyfixedpointsorfixedsubmanifoldswithtrivialnormalbundleviainvariantsofthisaction.Wealsodescribetheconnectionwiththeso-calledConner–Floydequationsfortheweightsoffixedpoints.ActionsofZ/pwerestudiedin[12],[13],[11],[8],wheretheso-calledConner–Floydequationswerededucedwithincobordismtheory(seeformulae(31),(32)).TheseequationsformnecessaryandsufficientconditionsforasetsofelementsofZ/ptobethesetofweightsofsomeZ/p-action(see§3forthedefinition).TwoapproachesforthecalculationofHirzebruchgeneraofastablycomplexmanifoldwithaZ/p-actionwereproposedin[5].ThefirstapproachisbasedontheapplicationoftheAtiyah–Bott–Lefschetzfixedpointformula[1],andsoforitsrealizationitisnecessarytohaveanellipticcomplexofbundlesthatareassociatedtothetangentbundleofthemanifold.TheAtiyah–Bott–Lefschetzformulaobtainedin[1]generalizestheclassicalLefschetzformulaforthenumberoffixedpointsandenablesustocalculatetheequivariantindexofanellipticcomplexofbundlesoveramanifoldbymeansofcertaincontributionfunctionsofthefixedsubmanifolds(seethedetailsbelow).Inparticular,ifanoperatoractsonamanifoldwithfinitelymanyfixedpoints,thecorrespondingequivariantindexcanbeexpressedintermsofthefixedpointweights.Itwasshownin[5]howtoexpresstheToddgenus,whichistheindexofanellipticcomplex(namely,theDolbeaultcomplex)overamanifoldwithZ/p-action,viatheequivariantindexofthesamecomplexfortheactionofthegeneratorofZ/p.ThisequivariantindexentersintotheAtiyah–Bott–Lefschetzformula.InthiswayonededucestheformulaeexpressingtheToddgenusintermsoftheweightsof1991MathematicsSubjectClassification.57R20,57S25(Primary)58G10(Secondary).PartiallysupportedbytheRussianFoundationforFundamentalResearch,grantno.96-01-01414.TypesetbyAMS-TEX2TARASE.PANOVfixedpointsforthisZ/p-action.Theformulaeobtainedbythismethodcontainthenumber-theoreticaltraceofacertainalgebraicextensionoffieldsofdegree(p−1).Inthispaper,weusethesameapproachtoobtaintheformulaeforothergeneraofmanifoldswithZ/p-action:thesignature(ortheL-genus),theEulernumber,theˆA-genus,thegeneralχy-genus,andtheellipticgenus.Bythismethodwealsoobtainsomegeneralequations(see.§5)foranarbitraryHirzebruchgenushavingthepropertytobetheindexofacertainellipticcomplexofbundlesassociatedtothetangentbundleofamanifoldwithZ/p-action.InthispaperweusethegeneralizedLefschetzformulainthesomewhatdifferentformulation,statedin[3].Thisformulaandespeciallythe“recipe”itsuggestsforcalculatingtheequivariantindexofacomplexviacontributionfunctionsoffixedpoints(see.§5)aremoreconvenientforapplicationsthantheformulafrom[1]usedin[5].ThegeneralizedLefschetzformulawasdeducedin[3]fromthecohomologicalformoftheAtiyah–Singerindextheorem,whichwasalsoprovedthere.Weapplybothformulae:someoftheresults(see§4)weobtainarebasedonthe“old”Lefschetzformulaof[1],whileothersusethe“recipe”in§5basedontheformulafrom[3].AnotherapproachtotheequationsforHirzebruchgenera,alsoproposedin[5],isbasedonanapplicationofcobordismtheoryinthesamewayasinthederivationoftheConner–Floydequationsin[12],[13].In[5],theauthorsgiveaformulaexpressingthemodpcobordismclassofastablycomplexmanifoldwithaZ/p-actionintermsofinvariantsoftheaction.ThentheyshowthatthedifferencebetweenthetwoformulaefortheToddgenus(obtainedbythefirstandsecondmethods)isexactlythesumoftheConner–FloydequationsfortheToddgenus.Inthispaperweshow(seeTheorem7.1)thatthedifferencebetweenthetwoseeminglydifferentformulaededucedbythesetwomethodsforanarbitrarygenusisaweightedsum(withintegercoefficients)oftheConner–Floydequationsforthisgenus.Acaseofparticularinterestisthatoftheso-calledellipticgenus.InWitten’spapers,acertaininvariantwasassignedtoeachoriented2n-dimensionalmani-foldM2n.ThisinvariantistheequivariantindexoftheDirac-likeoperatorforthecanonicalactionofcircleS1onmanifold’sloopspace.S.Ochanine[14]showedthatthisindexisaHirzebruchgenuscorrespondingtotheellipticsine;whichledtotheterm“ellipticgenus”.In[2],[9]andotherpapers,therigiditytheoremfortheellipticgenusofmanifoldswithS1-actionwasproved.ThistheoremstatesthatifweregardtheequivariantellipticgenusϕS1(M)ofsuchamanifoldasacharacterofthegroupS1,thenϕS1(M)isthetrivialcharacterandisequaltotheellipticgenusϕ(M).Atthesametime,theellipticgenustakesitsvaluesintheringZ12[δ,ε],anditsvalueonanymanifoldM2nisamodularformofweightnonthesubgroupΓ0(2)⊂SL2(Z)(cf.[7]).InthispaperweobtainformulaefortheellipticgenusofamanifoldwithaZ/p-actionhavingfinitelymanyfixedpoints.Asummaryofresultsofthispartofthepaperhasalreadybeenpublishedin[15].AsanapplicationwededucecertainrelationsbetweentheLegendrepolyno