高等数学(14)

932A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

6.TheproofofTheoremC

Choosea2-connected(n 1)-manifoldMn,n≥7,andaCEmapρ:CE→XwhichisnotinjectiveonK-homology.SuchmapswereconstructedM

inSection3.Constructauniformlyn-connectedweightedconecXasabove¯beauniformlycontractiblesingularRiemannianmanifoldcoarselyandletM

isomorphictocX.Theassemblymap

lf¯¯))(M)→K (C (MK

factorsthrough2anaturalmap

lf¯¯)～(M)→KX (MK =KX (cX).

BythemaintheoremofSection7,KX (cX)～=K 1(X),sotheassemblymap f¯¯)haskernel.(M)→KX (MK

Remark6.1.Inparticular,Conjecture6.28of[20]isincorrect.Themapcdoesinducerationalisomorphisms,sotherationalversionoftheconjectureisstillopen.

Thesameprocedureshowsthatbothinjectivityandsurjectivityasser-tionsinaboundedanalog(foruniformlycontractiblespaces)oftheGeneral-izedBorelConjectureofFerry-Rosenberg-Weinberger[13]arefalse.AregularneighborhoodofasuitablesuspensionofaMoorespaceprovidesamanifoldwithboundarywhichhasoddtorsioninitsL(e)-homology.ThiscanbekilledbyaCEmapasinSection3andtherestoftheconstructionproceedsasabove.Wecouldgetacounterexampletotheintegralisomorphismconjectureonamanifolddi eomorphictoeuclideanspacebyperformingourconstructionstartingwithaCEmapρ:Sn→XwheretheinducedmaponK wasmulti-plicationbyk.SuchaCEmapwouldbetheresultofapplyingtheprocedureofSection3toakillthemodkreductionofanintegralclassinK (Sn).Usingreal,ratherthancomplexK-theory,wecouldgetcounterexamplesofthesamesorttotheanalogousinjectivityconjecture.

Remark6.2.Allofourexamplesarebasedonthedi erencebetweenKlfandKX.Consequently,ifoneiscarefultoassertallconjecturesforgeneralmetricspacesintermsofKXratherthanKlf,oneobtainsstatementswhicharenotcontradictedbytheseexamples.IncasethemanifoldZhasboundedgeometry—inparticular,ifZistheuniversalcoverofclosedmanifold—it f(Z)→KX (Z)isamonomorphism,sonoisnotdi culttoshowthatK

contradictiontoConjecture6.28arisesfromourconstructioninthatcase.

2See[18].

搜索“diyifanwen.net”或“第一范文网”即可找到本站免费阅读全部范文。收藏本站方便下次阅读，第一范文网，提供最新高中教育高等数学(14)全文阅读和word下载服务。