高等数学(9)

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE927

willdo—producesauniformlycontractiblemetriconRkwhichisquasi-equivalenttocX.SinceXislocallyconnected,atheoremofBing[1]saysthatXhasapathmetric.IfwestartwithapathmetriconX,themetriconcXisalsoapathmetricandtheresultsof[11]allowustoconstructaRiemannianmetriconcSk 1whichisuniformlycontractibleandcoarseLipschitzequivalenttocX.

WehaveconstructedaRiemannianmanifoldZnhomeomorphictoRnsothatZiscoarselyisometrictoaweightedopenconeona“Dranishnikovspace”X.ByTheorem3.2,wecanchoosec:Sn 1→XsothatcdoesnotinduceamonomorphisminK(;Zk)-homologyandsuchthatthemapc×id:Rn→cXisacoarseisometry,whereweareusingpolarcoordinatestothinkofRnastheconeonSn 1.Inthisnotation,“c×id”referstoamapwhichpreserveslevelsintheconestructureandwhichisequaltoconeachlevel.

WeneedtoseethatZisnothypereuclidean.Thenextlemmashouldbecomfortingtoreaderswho ndthemselveswonderingaboutthe“degree”ofamapwhichisnotrequiredtobecontinuous.

Lemma4.4.IfZisanymetricspaceandf:Z→Rn(withtheeuclidean¯:Z→Rnwhichmetric)iscoarseLipschitz,thenthereisacontinuousmapf

isboundedlyclosetof.IffiscontinuousonaclosedY Z,thenwecan¯|Y=f|Y.choosef

Proof.ChooseanopencoverUofXbysetsofdiameter<1.ForeachU∈U,choosexU∈U.Let{φU}beapartitionofunitysubordinatetoUandlet ¯(x)=φU(x)f(xu).f

U∈U

BythecoarseLipschitzcondition,thereisaKsuchthat

d(x,x )<1 d(f(x),f(x ))<K.

¯(x)∈BK(f(x)),sod(f,f¯)Sinced(xU,x)<1forallUwithφU(x)=0,f

<K.

ContinuingwiththeproofofTheoremA,letf :Z→RnbeacoarselypropercoarseLipschitzmap.SinceZiscoarselyisomorphictocX,thereisacoarseLipschitzmapf:cX→Rn.Bytheabove,wemayassumethatfiscontinuous.

Sincefiscoarselyproper,f 1(B)isacompactsubsetofcX,whereBistheunitballinRn.ChooseTsolargethat

(X×[T,∞))∩f 1(B)= .

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