Algebraic Topology 0203

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1、Course421:HilaryTerm2003HomologyTheoryDavidR.WilkinsCopyrightcDavidR.Wilkins1988{2003Contents5SimplicialHomologyGroups25.1TheChainGroupsofaSimplicialComplex...........25.2BoundaryHomomorphisms....................55.3TheHomologyGroupsofaSimplicialComplex........6

2、5.4SimplicialMapsandInducedHomomorphisms.........85.5ConnectednessandH0(K)....................86IntroductiontoHomologicalAlgebra116.1ExactSequences.........................116.2ChainComplexes.........................146.3TheMayer-VietorisSequence................

3、..187TheTopologicalInvarianceofSimplicialHomologyGroups197.1ContiguousSimplicialMaps...................197.2TheHomologyofBarycentricSubdivisions...........217.3ContinuousMapsandInducedHomomorphisms........237.4HomotopyEquivalence......................2615Simplic

4、ialHomologyGroups5.1TheChainGroupsofaSimplicialComplexLetKbeasimplicialcomplex.Foreachnon-negativeintegerq,let
q(K)betheadditivegroupconsistingofallformalsumsoftheform111222sssn1(v0;v1;:::;vq)+n2(v0;v1;:::;vq)+


+ns(v0;v1;:::;vq);wheren;n;:::;nareintegersandvr;

5、vr;:::;vrare(notnecessarilydis-12s01qtinct)verticesofKthatspanasimplexofKforr=1;2;:::;s.(Inmoreformallanguage,thegroup
q(K)isthefreeAbeliangroupgeneratedbythesetofall(q+1)-tuplesoftheform(v0;v1;:::;vq),wherev0;v1;:::;vqspanasimplexofK.)Werecallsomebasicfactsconc

6、erningpermutations.ApermutationofasetSisabijectionmappingSontoitself.ThesetofallpermutationsofsomesetSisagroup;thegroupmultiplicationcorrespondstocompositionofpermutations.AtranspositionisapermutationofasetSwhichinterchangestwoelementsofS,leavingtheremainingelem

7、entsofthesetxed.IfSisniteandhasmorethanoneelementthenanypermutationofScanbeexpressedasaproductoftranspositions.Inparticularanypermutationofthesetf0;1;:::;qgcanbeexpressedasaproductoftranspositions(j1;j)thatinterchangej1andjforsomej.Associatedtoanypermutation

8、ofanitesetSisanumber,knownastheparityorsignatureofthepermutation,whichcantakeonthevalues1.Ifcanbeexpressedastheproductofanevennumberoftranspositions,then=+1;i