The Alexander–Whitney chain map.pdf

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1、TheAlexander–WhitneychainmapTheformulaThereisawell-knownnaturaltransformationα,whichconsistsofhomomorphismsα(X,Y):C(X×Y)−−→C(X)⊗C(Y)forallspacesXandY,whereC(X)denotesthesingularchaincomplexofX.Indegreen,itisdefinedonanysingularsimplexσ:∆n→X×Y,withcoordinatesσ:∆n→Xandσ:∆n→Y,bytheAlexander–Wh

2、itneyformula:12Xnnαnσ=σ1◦λk⊗σ2◦ρlinC(X)⊗C(Y),(1)k+l=nwhereλn:∆k⊂∆nandρn:∆l⊂∆narethelinearmapsdefinedontheverticesbyklλn(e)=e(for0≤i≤k)andρn(e)=e(for0≤i≤l).Themapλnembedskiilin−l+ik∆kasthe“lowest”k-dimensionalfaceof∆nandρnembeds∆lasthe“highest”ll-dimensionalfaceof∆n.Proposition2Thehomomorphi

3、smsαndoinfactformanaugmentedchainmap.IteratedfacemapsWecaneasilyexpressthemapsλnandρnintermsoftheklstandardfaceinclusionsη:∆n⊂∆n+1(for0≤i≤n+1),whereηomitsthevertexiieifromitsimage,asfollows:nknλk=ηn◦ηn−1◦...◦ηk+2◦ηk+1:∆⊂∆andnlnρl=η0◦η0◦...◦η0:∆⊂∆(withn−lfactors).Weneedtoknowhowtheyinteract

4、withthemapsηi.[Thefollowingresultsmayalsobeprovedbyinduction,usingthestandardformulann+2ηi◦ηj=ηj◦ηi−1:∆⊂∆for0≤jk.kandn+1n+1λ◦ηk+1=λ.(6

5、)k+1k(b)Formaps∆l→∆n+1involvingρ:n+1nρl,ifi≤n−l;ηi◦ρl=n+1(7)ρ◦ηi−n+l,ifi>n−l.l+1andn+1n+1ρ◦η0=ρ.(8)l+1l110.616AlgebraicTopologyJMBFile:alexwhit,RevisionA;10Apr2009;Page12TheAlexander–WhitneychainmapProofIn(a),(5)isclearfori>k,astheimageofλnisspannedbye,e,...e,k01kwhichdoesnotincludee.Ifi≤

6、k,theimageofηλnomitseandisspannedbyii◦kie,e,...,e,e,...e;wegetthesameeffectbyfirstapplyingη:∆k→∆k+101i−1i+1k+1in+1n+1n+1andthenλk+1.In(6),ηk+1leavese0,e1,...,ekfixed,andsodoλk+1andλk.Part(b)canbeprovedanalogously.Alternatively,itisusefultointroducetheinversionlinearmapu:∆n→∆n(notpartofthestan

7、dardsimplicialstructure)nnngivenbyun(ei)=en−iforalli.Weobservethatρl=un◦λl◦ul,un+1◦ηi◦un=ηn+1−i,andun◦un=id;then(b)followsimmediatelyfrom(a).ProofofthechainmapProofofProposition2Givenasingularsimplexτ:∆n+1→X×Y,withcoordinatesτ:∆n+1→Xandτ:∆n+1→Y,wehavetoshowtha