高等数学(6)

924A.N.DRANISHNIKOV,STEVENC.FERRY,ANDSHMUELWEINBERGER

Remark3.4.In[5],[6]theanalogousresultwasprovenforcohomologytheory.Theproofissimilarforhomologytheory.See[9].

Theorem3.3alsohasarelativeversion:

(K(Z,n))=0.Thenforany niteTheorem3.3 .Supposethath (K,L)thereexistacompactumpolyhedralpair(K,L)andanyelementα∈h

Yandamapf:(Y,L)→(K,L)sothat

(i)c-dimZ(Y L)≤n.

(ii)α∈Im(f ).

(iii)f|L=idL.

Theproofisessentiallythesame.Hereisthekeylemmaintheproofof willrefertoreducedcomplexK-homologyTheorem3.2.Inwhatfollows,K willrefertoreducedrealK-homology.andKO

Lemma3.5.LetMnbea2-connectedn-manifold,n≥7,andletαbe (M;Zm),m∈Z.ThenthereexistcompactaZ MandanelementinKO

Y MalongwithaCEmapg:(Z,M)→(Y,M)sothat

(1)g|M=idM.

(2)dim(Z M)=3.

(3)j (α)=0,wherej:M→Yistheinclusion.

(K(Zk,n);Zm)=0forn≥3.WecannowapplyThe-Proof.By[26],KO +1(Cone(M),M)¯∈KOorem3.3 tothepair(Cone(M),M)andtheelementα

with α¯=αinthelongexactsequenceof(Cone(M),M),obtainingaspace +1(Y,M)withY Mwithcdim(Y M)=3sothatthereisaclassα¯ ∈KO

α¯ =αandaCEmapg:(Z,M)→(Y,M)withdim(Z M)=3.Theexactsequence:

+1(Y,M)→KO (M)→KO (Y)KO j

showsthatj (α)=0.

Next,weconstructaparticularlyniceretractionZ→M.

Lemma3.6.Let(Z,M)beacompactpairwithdim(Z M)=3andMa2-connectedn-manifold,n≥7.Thenthereisaretractionr:Z→Mwithr|(Z M)one-to-one.

Proof.Theexistenceoftheretractionfollowsfromobstructiontheoryappliedtothenerveofa necoverofZ.TherestisstandarddimensiontheoryusingtheBairecategorytheorem.

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