高等数学(2)

920A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

jecturewasdisproved[17].InthispaperwewillshowthatthemonomorphismpartofthecoarseBaum-Connesconjecture(i.e.thecoarseNovikovconjec-ture)doesnotholdtruewithouttheboundedgeometrycondition.WewillconstructauniformlycontractiblemetriconR8forwhichµisnotamonomor-phism.Thus,acoarseformoftheintegralNovikovconjecturefailsevenfor nite-dimensionaluniformlycontractiblemanifolds.Infactwewillprovemore:ouruniformlycontractibleR8isnotintegrallyhypereuclidean,whichistosaythatitdoesnotadmitadegreeonecoarseLipschitzmaptoeuclideanspace.Alsointhispaper,wewillproduceauniformlycontractibleRiemannianman-ifold,abstractlyhomeomorphictoRn,n≥11,whichisboundedlyhomotopyequivalenttoanothersuchmanifold,butnotboundedlyhomeomorphictoit.Thisdisprovesonecoarseanalogoftherigidityconjectureforclosedasphericalmanifolds.Wewillalsoshowthatforeachk∈Zsomeofthesemanifoldsarenotmodkhypereuclidean.

OurconstructionisultimatelybasedonexamplesofDranishnikov[5],[6]ofspacesforwhichcohomologicaldimensiondisagreeswithcoveringdimension,andtheconsequentphenomenon,usingatheoremofEdwards(see[25]),ofcell-likemapswhichraisedimension.

De nition1.1.WewillusethenotationBr(x)todenotetheballofradiusrcenteredatx.Ametricspace(X,d)isuniformlycontractibleifforeveryrthereisanR≥rsothatforeveryx∈X,Br(x)contractstoapointinBR(x).ThemainexamplesofthisaretheuniversalcoverofacompactasphericalpolyhedronandtheopenconeinRnofa nitesubpolyhedronoftheboundaryoftheunitcube.Thereisasimilarnotionofuniformlyn-connectedwhichsaysthatanymapofann-dimensionalCWcomplexintoBr(x)isnullhomotopicinBR(x).

De nition1.2.WewillsaythataRiemannianmanifoldMnisintegrally(modk,orrationally)hypereuclideanifthereisacoarselypropercoarseLips-chitzmapf:M→Rnwhichisofdegree1(ofdegree≡1modk,orofnonzerointegraldegree,respectively).SeeSection4forde nitionsandelaborations.

Hereareourmainresults:

TheoremA.Foranygivenkandn≥8,thereisaRiemannianmani-foldZwhichisdi eomorphictoRnsuchthatZisuniformlycontractibleandrationallyhypereuclideanbutisnotmodk(orintegrally)hypereuclidean.

De nition1.3.(i)Amapf:X→YisacoarseisometryifthereisaKsothat|dY(f(x),f(x )) dX(x,x )|<Kforallx,x ∈Xandsothatforeachy∈Ythereisanx∈XwithdY(y,f(x))<K.

(ii)WewillsaythatuniformlycontractibleRiemannianmanifoldsZandZ areboundedlyhomeomorphicifthereisahomeomorphismf:Z→Z whichisacoarseisometry.

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