Rational Computations of the Topological K-Theory

arXiv:math/0507237v2[math.KT]15Jul2005RationalComputationsoftheTopologicalK-TheoryofClassifyingSpacesofDiscreteGroupsWolfgangL¨uck∗FachbereichMathematikUniversit¨atM¨unsterEinsteinstr.6248149M¨unsterGermanyFebruary2,2008AbstractWecomputerationallythetopological(complex)K-theoryoftheclas-sifyingspaceBGofadiscretegroupprovidedthatGhasacocompactG-CW-modelforitsclassifyingspaceforproperG-actions.Forinstanceword-hyperbolicgroupsandcocompactdiscretesubgroupsofconnectedLiegroupssatisfythisassumption.TheanswerisgivenintermsofthegroupcohomologyofGandofthecentralizersoffinitecyclicsubgroupsofprimepowerorder.Wealsoanalyzethemultiplicativestructure.Keywords:topologicalK-theory,classifyingspacesofgroups.MathematicsSubjectClassification2000:55N15.0.IntroductionandStatementsofResultsThemainresultofthispaperis:Theorem0.1(Mainresult).LetGbeadiscretegroup.DenotebyK∗(BG)thetopological(complex)K-theoryofitsclassifyingspaceBG.SupposethatthereisacocompactG-CW-modelfortheclassifyingspaceEGforproperG-actions.∗email:[e-mail]://:4925183383701ThenthereisaQ-isomorphismchnG:Kn(BG)⊗ZQ∼=−→Yi∈ZH2i+n(BG;Q)!×YpprimeY(g)∈conp(G)Yi∈ZH2i+n(BCGhgi;Qbp)!,whereconp(G)isthesetofconjugacyclasses(g)ofelementsg∈Goforderpdforsomeintegerd≥1andCGhgiisthecentralizerofthecyclicsubgrouphgigeneratedbyg.TheclassifyingspaceEGforproperG-actionsisaproperG-CW-complexsuchthattheH-fixedpointsetiscontractibleforeveryfinitesubgroupH⊆G.IthastheuniversalpropertythatforeveryproperG-CW-complexXthereisuptoG-homotopypreciselyoneG-mapf:X→EG.RecallthataG-CW-complexisproperifandonlyifallitsisotropygroupsarefinite,andisfiniteifandonlyifitiscocompact.TheassumptioninTheorem0.1thatthereisacocompactG-CW-modelfortheclassifyingspaceEGforproperG-actionsissatisfiedforinstanceifGisword-hyperbolicinthesenseofGromov,ifGisacocompactsubgroupofaLiegroupwithfinitelymanypathcomponents,ifGisafinitelygeneratedone-relatorgroup,ifGisanarithmeticgroup,amappingclassgroupofacompactsurfaceorthegroupofouterautomorphismsofafinitelygeneratedfreegroup.FormoreinformationaboutEGwereferforinstanceto[8]and[23].WewillproveTheorem0.1inSection4.WewillalsoinvestigatethemultiplicativestructureonKn(BG)⊗ZQinSection5.Ifoneiswillingtocomplexify,onecanshow:Theorem0.2(Multiplicativestructure).LetGbeadiscretegroup.Sup-posethatthereisacocompactG-CW-modelfortheclassifyingspaceEGforproperG-actions.ThenthereisaC-isomorphismchnG,C:Kn(BG)⊗ZC∼=−→Yi∈ZH2i+n(BG;C)!×YpprimeY(g)∈conp(G)Yi∈ZH2i+n(BCGhgi;Qbp⊗QC)!,whichiscompatiblewiththestandardmultiplicativestructureonK∗(BG)andtheoneonthetargetgivenbya,up,(g)
·b,vp,(g)
=a·b,(a·vp,(g)+b·up,(g)+up,(g)·vp,(g))
for(g)∈conp(G);a,b∈Yi∈ZH2i+∗(BG;C);up,(g),vp,(g)∈Yi∈ZH2i+∗(BCGhgi;Qbp⊗QC),2andthestructuresofagradedcommutativeringonQi∈ZH2i+∗(BG;C)andQi∈ZH2i+∗(BCGhgi;Qbp⊗QC)comingfromthecup-productandtheobviousQi∈ZH2i+∗(BG;C)-modulestructureonQi∈ZH2i+∗(BCGhgi;Qbp⊗QC)comingfromthecanonicalmapsBCGhgi→BGandC→Qbp⊗QC.InSection6wewillproveTheorem0.1andTheorem0.2underweakerfinitenessassumptionsthanstatedabove.IfGisfinite,wegetthefollowingintegralcomputationofK∗(BG).Through-outthepaperR(G)willbethecomplexrepresentationringandIGbeitsaug-mentationideal,i.e.thekerneloftheringhomomorphismR(G)→Zsend-ing[V]todimC(V).IfGp⊆Gisap-Sylowsubgroup,restrictiondefinesamapI(G)→I(Gp).LetIp(G)bethequotientofI(G)bythekernelofthismap.Thisisindependentofthechoiceofthep-Sylowsubgroupsincetwop-SylowsubgroupsofGareconjugate.ThereisanobviousisomorphismfromIp(G)∼=−→im(I(G)→I(Gp)).WewillproveinSection3Theorem0.3.(K-theoryofBGforfinitegroupsG).LetGbeafinitegroup.Foraprimepdenotebyr(p)=|conp(G)|thenumberofconjugacyclasses(g)ofelementsg∈Gwhoseorder|g|ispdforsomeintegerd≥1.ThenthereareisomorphismsofabeliangroupsK0(BG)∼=Z×YpprimeIp(G)⊗ZZbp∼=Z×Ypprime(Zbp)r(p);K1(BG)∼=0.TheisomorphismK0(BG)∼=−→Z×QpprimeIp(G)⊗ZZbpiscompatiblewiththestandardringstructureonthesourceandtheringstructureonthetargetgivenby(m,up⊗ap)·(n,vp⊗bp)=(mn,(mvp⊗bp+nup⊗ap+(upvp)⊗(apbp))form,n∈Z,up,vp∈Ip(G)andap,bp∈ZbpandtheobviousmultiplicationinZ,Ip(G)andZbp.TheadditiveversionofTheorem0.3hasalreadybeenexplainedin[16,page125].Inspecting[15,Theorem2.2]onecanalsoderivetheringstruc-ture.In[18]theK-theoryofBGwithcoefficientsinthefieldFpofpelementshasbeendeterminedincludingthemultiplicativestructure.TheproofofTheo-rem0.3wewillpresenthereisbasedontheideasofthispaper.Wewillandneedtoshowastrongerstatementaboutthepro-group{IG/(IG)n+1}inTheorem3.5(b).AversionofTheorem0.1fortopologicalK-theorywithcoefficientsinthep-adicintegershasbeenprovedbyAdem[1],[2]usingtheAtiyah-Segalcomple-tiontheoremforthefinitegroupG/G′providedthatGcontainsatorsionfreesubgroupG′offiniteindex.Ourmethodsallowtodropthiscondition,todealwithK∗(BG)⊗ZQdirectlyandstudysystematicallythemultiplicativestruc-tureforK∗(BG)⊗ZC.TheyarebasedontheequivariantcohomologicalCherncharacterof[22].3ForintegralcomputationsoftheK-theoryandK-homologyofclassifyingspacesofgroupswereferto[17].Thepaperisorganizedasfollows:1.BorelCohomologyandRationalization2.SomePreliminariesaboutPro-Modules3