高等数学(18)

936A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

ItfollowsfromallofthisthatthenerveofthecoverORconstructedaboveisproperhomotopyequivalenttothemappingtelescopeofthenervesoftheUkwhichcorrespondtoarcsconnectinglevelskRand(k+1)RincX.WewillshowthatforallR,thistelescopeisproperhomotopyequivalenttothecomplementofXintheHilbertcube.Itfollowsthatforanylocally niteSteenrodhomologytheory,thelocally nitehomologyofthetelescopeisequaltothehomologyofXwithadimensionshift,asdesired.

¯iisasequenceofLemma7.8.IfXisacompactmetricspace,andU¯i+1re nesU¯i,andmesh(U¯i)→0,¯0={X},UopencoversofXsuchthatU¯i)isproperhomotopyequivalentthenthemappingtelescopeofthenervesN(U

toQ X,whereXisembeddedintheHilbertcubeQasaZ-set.

Proof.ThisisaformofChapman’sComplementTheorem[22],whichsaysthatthehomeomorphismtypeofthecomplementofaZ-embeddedcompactumXintheHilbertcubedependsonlyontheshapeofX.ThepointisthatthemappingtelescopecanbecompletedtoacontractibleANRbyaddingacopy¯i)’satin nity.CrossingwithQgivesacopyofoftheinverselimitoftheN(U

QcontainingaZ-setX whichisshapeequivalenttoX.ThecomplementofX istheproductofthetelescopewithQ.

Oneshouldbecarefulhere,sincealittlebitofthoughtgivesexampleswhereXistheunitintervalandX istheHilbertcube.Theargumentof[22]showsthatif{Ki,αi}isaninversesystemwithK0=pt,thenthemappingtelescopeof{Ki,αi}isproperhomotopyequivalent(evenin nitesimpleequiv-alent!)tothemappingtelescopeofanysystem{Li,βi}equivalentto{Ki,αi}inpro-homotopy.IfX=lim{Ki,αi},itiseasytoconstructasequenceof →coversUiofXsothatN(Ui)isPLhomeomorphictoKiandsothatthemapsinducedbyre nementarehomotopictotheαi’s.Sinceallsuchsequencesareeasilyseentobepro-equivalent,Lemma7.8follows.

Finally,wenotethatthesequenceofnervesN(Uk)andbondingmapsaboveisco nalwithasequenceasinthestatementofLemma7.8.ThiscompletestheproofofTheorem7.2.

RutgersUniversity,Piscataway,NJ

E-mailaddress:sferry@math.rutgers.edu

UniversityofFlorida,Gainesville,FL

E-mailaddress:dranishn@u .edu

TheUniversityofChicago,Chicago,IL

E-mailaddress:shmuel@math.uchicago.edu

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