高等数学(5)

LARGERIEMANNIANMANIFOLDSWHICHAREFLEXIBLE923

SinceXislocallyn-connected,thereisanin nitedecreasingpositive

d(x) Bd(x) sequence{ri}suchthatforeveryxtheinclusions... Brrii+1dBri 1(x)arenullhomotopiconn-skeleta.Re nethesequencesothatactuallyd(x) BdinclusionsBirri 1(x)arenullhomotopiconn-skeleta.WesetNi=iri 1i.ρρ(x,i) cX.First,wenotethatB1(x,i) NowconsidertheballB1ρρd(x,i)B1(x)×[i 1,i+1]andthatB1(x,i)contractsinitselftoB1

∩(X×[i 1,i]) Bdi 1

ρρ(x,i)n-contractsinB3(x,i)bypushingdowntothe(i 2)-levelandsoB1performingthen-contractionthere.

Forballsofradius2thesamereasoningappliesifthecenterisatleast3awayfromthevertex.Wecontinueinthiswayandobservethatforanygivensizeball,centeredsu cientlyfarout,oneobtainsan-contractibilityfunctionoff(r)=r+2asrequired.Thewholespaceisthereforeuniformlycontractible.1i 1ρ(x)×[i 1,i].ButB3(x,i) Bd1i 2(x)×{i 2}

3.Designercompacta

De nition3.1.Amapf:M→Xfromaclosedmanifoldontoacompactmetricspaceiscell-likeorCEifforeachx∈XandneighborhoodUoff 1(x)thereisaneighborhoodVoff 1(x)inUsothatVcontractstoapointinU.

ThepurposeofthissectionistogiveexamplesofCEmapsf:M→Xsothatf :Hn(M;L(e))→Hn(X;L(e))hasnontrivialkernel.Theargumentgivenbelowisamodi cationofthe rstauthor’sconstructionofin nite-dimensionalcompactawith nitecohomologicaldimension.HereistheresultwhichwewilluseinprovingTheoremsA,B,andCoftheintroduction.

Theorem3.2.LetMnbea2-connectedn-manifold,n≥7,andlet (M;Zm).ThenthereisaCEmapq:M→XwithαbeanelementofKO (M;Zm)→KO (X;Zm)).Itfollowsthatifα∈H (M;L(e))α∈ker(q :KO

isanelementoforderm,modd,thenthereisaCEmapc:M→Xsothatc (α)=0inH (X;L(e)).

Webegintheproofofthistheorembyrecallingthestatementofamajorstepintheconstructionofin nite-dimensionalcompactawith nitecohomo-logicaldimension.

(K(Z,n))=0forsomegeneralizedho-Theorem3.3.Supposethath (L)mologytheoryh .Thenforany nitepolyhedronLandanyelementα∈h

thereexistacompactumYandamapf:Y→Lsothat

(1)c-dimZY≤n.

(2)α∈Im(f ).

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