高等数学(15)

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE933

7.KX ofweightedopencones

JohnRoe[20]hasintroducedthefollowingnotionofcoarsehomology:De nition7.1.IfXisacompletelocallycompactmetricspace,ase-quence{Ui}oflocally nitecoversofXbyrelativelycompactopensetsisˇcalledanAnti-CechsystemiftherearenumbersRi→∞suchthat

(i)diam(U)<RiforallU∈Ui.

(ii)RiisaLebesguenumberforUi.

ThecoarsehomologyofXwithcoe cientsinSis

lf(N(Ui);S),HX (X;S)=limH →

lf(P;S)istheSteenrodwhereN(Ui)isthenerveoftheopencoverUiandH S-homologyofthe1-pointcompacti cationofP,relin nity.S,ofcourse,is

aspectrum.

ˇItisnotdi culttoconstructanti-Cechsystemsofcovers,atleastwhenX

isacompletelocallycompactmetricspace.Forsome >0,chooseamaximalcollectionofdisjointopen ballsinXandconsiderthecollectionofR-ballsonthesamecenters.ForR>2 ,thisisacoverwithLebesguenumberatleastR 2 .If{Ri}isanymonotonesequenceapproachingin nity,thisallowsustoconstructasequenceofcoarsecovers{Ui}withdiameters<Ri.Anyˇanti-Cechsystemisco nal,soHX iswell-de ned.

Aninterestingquestioninmetrictopologyisto ndconditionsunderwhichHX (X;S)isequaltotheS-homologyofXatin nity.InsuchcasesHX isatopologicalinvariant,ratherthanametricinvariant.TheusualsortofnerveargumentgivesapropermapX→N(Ui)foreachiandthereforeamaplf(X;S)→HX (X;S).EvenwhenXisuniformlycontractible,theresultsH

ofthispapershowthatthismapneedbeneitheranintegralmonomorphismnoranintegralepimorphism.Wedo,however,havethefollowing:

Theorem7.2.IfXiscompactmetricandcXisaweightedconeonX,

lf(cX;S)→HX (cX;S)isanisomorphismforanyspectrumS.thenH

TheanalogousresulthasbeenprovenbyHigsonand/orRoeformanyuniformlycontractiblespaces.Wewillgiveaproofforweightedconeswhichincludesthein nite-dimensionalcase.

Proof.LetRbegivenandconsiderlevelskRintheweightedconecX.Packeachoftheselevelswith k-ballsoncentersckiasabove, ksmall,anddrawradialarcsfromeachckitothepointbelowitinlevel(k 1)R.Now

take(1+)R-neighborhoodsofthearcs.

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