Homology of the mapping class group for surfaces o

arXiv:0712.4254v1[math.AT]27Dec2007HomologyoftheMappingClassGroupΓ2,1forSurfacesofGenus2withaBoundaryCurveJochenAbhau,Carl-FriedrichB¨odigheimerandRalfEhrenfriedDedicatedtothememoryofHeinerZieschangAbstractWereportonthecomputationoftheintegralhomologyofthemappingclassgroupΓmg,1ofgenusgsurfaceswithoneboundarycurveandmpunctures,when2g+m≤5,inparticularΓ02,1.1IntroductionLetMod=Modmg,ndenotethemodulispaceofconformalequivalenceclassesofRiemannsurfacesF=Fmg,nofgenusg≥0withn≥1boundarycurvesandm≥0permutablepunctures.Oneobtainsan(m!)-foldcoveringspacefMod=fModmg,nofMod=Modmg,nifthepuncturesaredeclarednotpermutable.Likewise,letΓ=Γmg,nbethecorrespondingmappingclassgroupofisotopyclassesoforientation-preservingdiffeomorphismsfixingtheboundarypointwiseandpossiblypermutingthepunctures.Themappingclassgroupwherethepuncturesaretobefixedisasubgroupofindexm!inΓandisdenotedbyeΓ=eΓmg,n.ThemainresultofthisarticleisthecomputationoftheintegralhomologyH∗(Γ02,1;Z)andH∗(Γ12,1;Z)ofthemappingmappingclassgroupforsurfacesofgenus2withoneboundarycurveandwithnoresp.onepuncture.Thesecomputationsweredonesomeyearsagobythethirdauthorin[Eh]andredonebythefirstauthorin[Ab],bothbasedonworkbythesecondauthorin[B-1]and[B-3].Wegivethisbelatedreport,becauseveryfewhomologygroupsofmappingclassgroupsareknown;priortoourcomputationstheintegralhomologywasknownonlyfortheeasycaseg=1.SeeSection2formoreremarksonpreviouslyknownresults.ΓandeΓaretorsion-free,sincethediffeomorphismsarefixingatleastoneboundarycurve.TheyactthereforefreelyonthecontractibleTeichm¨ullerspacesTeich=Teichmg,nresp.eTeich=eTeichmg,n.Itfollows,thatthespaceModisanon-compact,connected(topological)manifoldofdimensiond=6g−6+3n+2m;themanifoldfModisalwaysorientable,butModisorientableonlyinthecasesm=0orm=1.Furthermore,theyhavethehomotopytypeoftheclassifyingspaceofthecorrespondingmappingclassgroup,1(1.1)fModeTeich/eΓ≃//BeΓModTeich/Γ≃//BΓThekeypointofourmethodistouseanewdescriptionofthemodulispaceModresp.fMod;thisisthespaceofparallelslitdomains,thespaceofconfigurationsofhpairsofparallel,semi-infiniteslitsinncomplexplanes;hereh=2g+m+2n−2.Moreprecisely,thereisavectorbundleHarm→Modresp.eHarm→fModofdimensiond∗=m+2n,anditsone-point-compactificationwillbedescribedbyparallelslitdomains.TodescribethesebundleswefirstreplacethemodulispacesabovebythemodulispaceofclosedRiemannsurfacesFofgenusg,onwhichnso-calleddipolepointsQ1,...,Qnwithnon-zerotangentvectorsX1,...,Xn,andmpermutableresp.non-permutablepuncturesP1,...,Pmarespecified.Aconformalequivalenceclassisgivenby[F,Q,P,X],withQ=(Q1,...,Qn),P=(P1,...,Pm),andX=(X1,...,Xn).ThesenewmodulispaceshavethesamehomotopytypeasModresp.fMod(orinotherwords,themappingclassgroupsareisomorphic).Notethattheirdimensionisd=6g−6+4n+2m,sinceforthesakeofsimplicitywespecifiedtangentvectorsandnotmerelytangentdirections.InthevectorbundleHarm→Modresp.eHarm→fMod,thefibreoverapoint[F,Q,P,X]inModresp.infModconsistsofallharmonicfunctionsu:F→¯R=R∪∞withasimplepoleateachQihavingdirectionXiplusalogarithmicsingularity,andwithalogarithmicsingularityateachPj.ThebundleHarmisflat,morepreciselyHarm∼=(fMod×SmRm)×R3n,andthebundleeHarmistrivial,beingthepullbackofHarmalongthecoveringfMod→fMod/Sm=Mod,whereSmisthem-thsymmetricgroup.In[B-3]wehave—followinganearlierversionin[B-1]—introducedafinitecellcomplexPar=Par(h,m,n)ofdimensiond+d∗=3hasacompactificationofHarm.ThecomplementofHarmisasubcomplexPar′⊂Parofcodimension1anditspointsrepresentdegeneratesurfaces.AsimilarstatementholdsforeHarm.Themainresultin[B-3]and[Eb-2]isPar−Par′∼=Harm≃Mod,(1.2)ePar−ePar′∼=eHarm≃fMod.(1.3)Welatergivesomedetailswhenweexhibitthecellularchaincomplexes.Thepairs(Par,Par′)resp.(ePar,ePar′)arerelativemanifolds,andviaPoincar´edualityweobtaintheisomorphisms:H∗(Par,Par′;O)∼=H3h−∗(Mod;Z),(1.4)H∗(ePar,ePar′;Z)∼=H3h−∗(fMod;Z)(1.5)Theseisomorphismsarethecap-productwitha(relative)fundamentalororientationclass[μ]inH3d(Par,Par′;O)resp.inH3d(ePar,ePar′;Z),whereOisthelocalcoefficientsysteminducedbytheorientationcovering.2Inthefollowingsectionsweconcentrateforthesakeofsimplicityonthecaseofasingleboundarycurve(n=1),althoughthecomputationscanbedoneinthegeneralcase.TocomputethehomologyofModandfModweactuallycomputedthecellularhomologyofthepairs(Par,Par′)withintegral,withmod2andwithrationalcoefficients,andwithcoefficientsintheorientationsystemO.Theactualcalculationsweredoneusingacomputerprogram;theprogramisnottrackingthegeneratorsofthehomologygroups,butsomegeneratorscaneasilybedetermined.InSection2wewilllistourresults.Wewillgivecommentsonpreviousandsimilarhomologycomputations.WegiveadescriptionofthespacesHarmandeHarminSection3,includingfiguresofconfigurations,i.e.parallelslitdomains;inparticularwegiveacompletelistofcellsforthecaseg=1andm=0,whichwewilluseinSection6forademonstration.InSection4wedescribethecellularchaincomplexes.SomepropertiesofthesechaincomplexesaredescribedinSection5.Thecalculationforthecaseg=1andm=0isdonebyhandinSection6todemonstratethemethod,inparticularthespectralsequenceused.Forthefundamentalclassμandforsomeotherhomologyclasseswecangiveformulas(fortheirPoi