高等数学(8)

926A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGERsequences

+1(M;Zm) (M) (M)KO→KO→KO

f f f

+1(X;Zm) (X) (X)KO→KO→KO

thatf (α)=0.TheL-theorystatementinTheorem2nowfollowsfromthe

1factthatKO[1]=L(e)[].

4.TheproofofTheoremA

Webeginbystatingsomede nitions.

De nition4.1.

(i)Amapf:X→YbetweenmetricspacesissaidtobecoarseLipschitz

ifthereareconstantsCandDsothatdY(f(x),f(x ))<CdX(x,x )wheneverdX(x,x )>D.NoticethatcoarseLipschitzmapsarenotnecessarilycontinuous.Infact,ifdiamX<∞,everymapde nedonXiscoarseLipschitz.

(ii)Amapf:X→YiscoarselyproperifforeachboundedsetB Y,

f 1(B)hascompactclosureinX.

ThefollowingcorollaryconstructstheRiemannianmanifoldsappearinginallofourmaintheorems.

Proposition4.2.IfXisthecell-likeimageofacompactmanifoldandnisgiven,thenforsomesuitablechoiceofweights,cXisuniformlyn-connected.

Proof.TheCEimageofanycompactANR(absoluteneighborhoodre-tract)islocallyn-connectedforalln,sothepropositionfollowsfromLemma

2.2.See[19]forreferences.

Corollary4.3.Letf:Sk 1→Xbeacell-likemap.ThenRkhasauniformlycontractibleRiemannianmetricwhichiscoarselyequivalenttocX,wheretheconeisweightedasinProposition4.2.

Proof.Considercf:cSk 1→cX.ThisinducesapseudometriconRk.Thebasicliftingpropertyforcell-likemaps(see[19])showsthatRkwiththispseudometricisuniformlyn-connectedifandonlyifcXis.Ifn≥k 1,thismeansthattheinducedpseudometriconRkisuniformlycontractible.Adding anysu cientlysmallmetrictothispseudometric—themetricfromRk～=Dk

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