Semi Simplicial Complexes and Singular Homology

7/27/2019 Semi Simplicial Complexes and Singular Homology

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Annals of Mathematics

Semi-Simplicial Complexes and Singular HomologyAuthor(s): Samuel Eilenberg and J. A. ZilberSource: The Annals of Mathematics, Second Series, Vol. 51, No. 3 (May, 1950), pp. 499-513Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1969364 .

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ANNALS OF MATHEMATICSVol. 51, No. 3, May, 1950

SEMI-SIMPLICIAL COMPLEXES AND SINGULAR HOMOLOGYBY SAMUEL EILENBERG AND J. A. ZILBER(Received anuary7,1949)

A simplicialcomplexhas the following wo properties: a) each q-simplexdetermines + 1 facesof dimension - 1, (b) thefacesof a simplex eterminethe simplex.Recent work n singular omology heory ftopological paces [1]and homol-ogyand cohomologyheory fabstractgroups 2]haveledto abstract omplexeswhich atisfy a) without atisfyingb). We shallgive a generaldefinitionfthisclass ofcomplexes nd showhow the variousconstructionsfhomology heory(includinghomologywith ocal coefficients,up-products, tc.) can be carriedout just as forsimplicial omplexes.The chief xampleofsuch a "semi-simplicial"omplex s the singular omplexS(X) of a topological pace X. Although hiscomplex s very"large" it is pos-sibleto find ubcomplexes fS(X) whichcontain ll theinformationhat S(X)carriesbut which re stripped feverythinguperfluousrom hepointofviewof homotopy.The existence nd uniquenessof such minimalsubcomplexes sestablished.Theseminimal omplexes re themain tool inthepaperof S. Eilen-berg and S. MacLane [4] immediately ollowing.

1. Semi-simplicial omplexesA semi-simplicialomplexK is a collection felements f called simplexestogetherwith two functions. he first unction ssociates with each simplexa- n integer _ 0 calledthedimension fa; we thensay thata is a q-simplex.The secondfunction ssociateswith each q-simplex - q > 0) of K and witheach integer ? i ? q a (q - 1)-simplex a() called the th faceofa, subjecttothe condition

(1.1) [0()J0(i) = [OfW]U-4)fori < j and q > 1.We observe hat thisdefinition oesnotexcludethepossibility f two distinctq-simplexes a and Xwith (i) = 7(i) for = 0, q.We maypass to lowerdimensional acesofar y iteration.f 0 ? ij < ...


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500 SAMUEL EILENBERG AND J. A. ZILBERA subcomplex ofK is a subcollection fsimplexes f K with the propertythatifa e L then all the facesofa are inL.The group of q-dimensional integral)chains Cq(K) is defined s the free

abeliangroupwith theq-simplexes fK as freegenerators. he boundary omo-morphism0: CQ(K) >+ C2_(K)

is defined y setting or ach generatoro = Ej (-1)ta.i=O

One verifies eadily hat0a = 0. This leads directly o thedefinition f cycles,boundaries, nd of the homologygroup Hq(K). Following he usual procedurewe may also definehomology roupsHq(K, G) with an arbitrarybelian coeffi-cientgroupG as well as relativehomology roupsHq(K, L, G) moduloa sub-complexL.A q-dimensionalochainf C2(K, G) maybe defined ither s a homomorphismf:Cq(K) -* G or as a function (o) defined n the set ofq-simplexes f K withvalues in G. The coboundary s defined y

q+1(f)(o) E= E (-_)tf'f('(i))i=Ofor each (q + 1)-simplex ofK. This leads to cohomology roupsH'(K, G).Relative cohomology roupsHq(K, L, G) are obtainedby considering ochainswhich re zeroon everysimplexofthesubcomplex .The cup-products or cohomologymay be defined y the Alexanderformulajust as in the case ofa simplicial omplexwith ordered ertices.Let the groupsG1 nd G2be pairedto thegroupGand letcochains ' e C'(K, G1), 2 e Cq(K,G2)be given.Definethe cochainfi


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SEMI-SIMPLICIAL COMPLEXES AND SINGULAR HOMOLOGY 5012. Local coefficients

Steenrod's theoryof homology nd cohomologywith local coefficients5]can be builtveryconvenientlyn a semi-simplicial omplex.Let K be such acomplex.For thesake ofbrevitywe shall discuss cohomology nly.A local systemG = {G(a), y(Q) of groups n K consistsof two functions;the first ssignsto each vertex i.e. O-simplex)a fK a groupG(a), the secondassignsto each edge (i.e. 1-simplex)fi fK an isomorphism

,y0O: G(0(1,) ---G@(O?))subjectto the condition7(0f(0, 1))Y(0(l, 2)) = 7(a(o, 2))

for ach 2-simplex a.Let G = {G(a), -yQ() be a local systemofabelian groups nK. A q-cochainf ofK over G is a functionwhichto each q-simplex ofK assignsan elementf(o-) f hegroupG(o-(o))ssociatedwith he eadingvertex (o)of . The q-cochainsform n abeliangroupC'(K, G). The coboundary f is a (q + 1)-cochain efinedby q+1(6f)(0) = 7(O(o.1))f(0Aa) + X (-i)%f(ofti).iP=l

It is easyto verifyhatb5f= 0. The groupZ'(K, G) ofcocyles s thendefinedas the kernel f 5:Cq - C+ while thegroupBq(K, G) of coboundaries s theimage groupof 5:Cl _ Cqa. he qth cohomology roupofK overG isHq(K, G) = Z(K G)/B(K, G).

Let T: K1 * K be a simplicialmap. From thegiven ocal systemG in K wedefine local systemT*G ofgroups nK1as follows:T*G = {G(T(a)), -y(T I))forvertices and edges inK1 Iff e C(K, G) wedefine(T*f)(ar)= f(Ta)

for -simplexes ofK1 and find hatT*fEC(K, T*G).Clearly6(T*f) T*(af)so that a homomorphismT* H'(K, G) --)Hq(Ki2T*G)is obtained.

3. SingularhomologyTypical examples of semi-simplicial omplexesare encountered n singularhomology heory.Select foreach dimension a fixedEuclidean q-simplexA, withorderedver-tices do < * < do. Considerthe simplicialmaps

e :,Aq^-_* i=O-=0, qwhich re orderpreservingnd map ,- ontothe face ofAqopposited .


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502 SAMUEL EILENBERG AND J. A. ZILBERA map T: A,-*>X ofAqintoa topological paceX is called a singular -simplexin X. The faces of T are defined s

T() = Te':4.1- X.The singularsimplexesof X thus make up a semi-simplicial omplex S(X)called the total singular omplexofX. If A is a subspace ofX, S(A) is a sub-complexofS(X); thehomology nd cohomology roupsof thepair (X, A) aredefined o be those of thepair (S(X), S(A)).Fromnow on we shall assume thatX is arcwiseconnected nd that a fixedpointx* ofX has been selected as base point.A singular simplexT: Aq + Xsuch that T(Aq) = (x*) will be called collapsed.We denoteby Sn(X) the sub-complexof S(X) consisting f all singular implexesT such that all faces ofT of dimension n are collapsed.Thus fordimensions < n, S(X) containsonly one q-simplex, amely hecollapsedone.With referenceo local coefficientst shouldbe remarked hat a local systemofgroupson the space X yieldsa local systemofgroupson the complexS(X)and vice-versa. t also determines local coefficientystem n each of the com-plexes S,(X). On the complexS1(X) the local systemreduces to one groupG- G(x*), and each 1-simplex f S1(X) defines n automorphismfthisgroup.These automorphismsetermine,nd are determined y, thefashion n whichthefundamental roup7r1(X) withx* as base point)operatesonG. In thecom-plexes S,(X), n > 1, the automorphismsre all identitymaps and the localcoefficientystem ollapses.

4. MinimalcomplexesTwo singularq-simplexesTo and T1 in a space X are called compatibleftheirfacescoincide:T"t) = T(t)for = 0, **, q. If in additionToand T1 aremembersof a continuousone parameterfamilyTt, 0 < t < 1, of singularq-simplexes,ll ofwhich re compatible,we say that Toand T1are homotopic.For q = 0 anytwosimplexes re compatible, nd sinceX is assumedto be arc-

wise connected, heyare also homotopic.A subcomplexM ofS(X) willbe calledminimalprovided:(4.1) For eachq the ollapsed -simplex :A, --* x* is in M.(4.2) If T is a singular -simplexll ofwhose acesare in M, thenM containsa uniquesingular -simplex ' compatible ith nd homotopicoT.To showthatminimal ubcomplexes xistwe proceedby induction.Assumethat a subcomplexM(') of S(X) has been defined ontaining nly simplexes fdimension


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SEMI-SIMPLICIAL COMPLEXES AND SINGULAR HOMOLOGY 503Since everytwo 0-simplexesn S(X) are compatible nd homotopic,M con-tains only one 0-simplex, amelythe collapsed one. This implies(4.3) Everyminimal ubcomplex of S(X) is a subcomplex f S1(X).More generallywe have(4.4) If thehomotopyroup r,-1(X) vanishes,henor veryminimal ubcomplexM of S(X) we haveM n Sn1(X) C Sn X).PROOF. Let T be an (n - 1)-simplex ofM n S.-1(X). Since all the facesofT are collapsed,T is compatiblewith he collapsed n - 1)-simplex To, andsince

srn-l(X) = 0, T is homotopic o To Thus by (4.2) T = Toand T is collapsed.As a corollary f (4.4) we have(4.5) If thehomotopy roups7ri(X)vanish or < n then veryminimal ub-complexM of S(X) is a subcomplexfS,(X).5. The mainhomotopyWe shall considerprismslIqA,-, X I, q > 0

whereA,, is the (q - 1)-simplex sedto define ingular q - 1)-simplexeswhileI is the closed interval < t ? 1. The mapseql: AQ-2 Aq1, 1 0i , q -1

definemapsPg I11q-1 -> 11g

by settingp(x, t) = (ea l(x), t).We further ave themapsqb:Aq-, IIq, I 0 < t _1

defined y b'(x) (x, t). Themapsb' andb' are of pecial nterest.A continuousmapping P:llq -* X

is called a singularq-prism n X. The singular q - 1)-prismp(i) = Pp:I,-,-* X

is calledthe ill faceofP, i = 0.**, q - 1. The singular q - 1)-simplexesP(t)=Pbq:Aq1--X, 0 t? 1

will be considered,n particular (O) and P(1) willbe called the lower and theupperbase ofthesingular rismP.(5.1) LetX bean arcwise onnectedpace and let M bea minimal ubcomplexofX relativeosomebasepointxs eX. There s then functionPTI which oeach


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504 SAMUEL EILENBERG AND J. A. ZILBERsingular -simplex in X assignsa singular q + 1)-prism T in X subject othe ollowing onditions(i) PT($ = PT(ii) PT(O) = T.(iii) PT(1) is in the ubcomplex,(iv) If T e M then T(t) = T forallO ? t ? 1.A function PT} satisfying onditions i)-(iv) will be called a homotopye-forming (X) into M.We begin the constructionf PT with the dimension ero.A 0-simplexT inX is represented y a point x e X; we then selectPT to be a path joiningxtand x,with heprovision hatPT is the collapsedpath if x = x*.Suppose, by induction, hat PT is defined or simplexesT of dimensionXso thatf2 = fi = f on A and f2bq+l= T2 . The map f2 is thenhomotopicwithfi (considered nlyon B). Sincefi is defined ll over 7I+1, there s an extensionf : Hq+1- X of 2. Define PT = f3 ; conditions (i)-(iv) are then easily verified.If we denote

eptT PT(t), 0 _. t < I,then for every singular q-simplex T in X, ((otT) (p) is continuous simultaneouslyin 0 ? t _ 1 and p e Ag , and conditions (i)-(iv) can be rewritten as follows.(i)' 'pt: S(X) -- S(X) is simplicial,(ii)' po is the identity,(iii)' =M0TM,(iv)' VptT =T for T e Mk and 0 -< t '_ 1.


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SEMI-SIMPLICIAL COMPLEXES AND SINGULAR HOMOLOGY 505Thus in a sense ,. is a retraction fS(X) onto M while JOt} is a homotopy on-nectingthis retractionwith the identitymap. Thus we may say that M is adeformation etractof S(X).

6. Applications f the mainhomotopyWe shall use themainhomotopy o comparethehomology nd cohomologyofthe complexS(X) withthose ofa minimal ubcomplexM.By a suitable implicial ubdivision fH. one can definesee [1, ?16]) a functionwhich to each singularprismP: Hq -> X assigns a q-dimensional hain c(P)in S(X) such thatOc(P) = P(1) - P(O) - E (-1)i c(P(")).Now letPT be themainhomotopy f?5 and define T = C(PT) e C+'(S(X))foreach q-simplex ofS(X). ThereresulthomomorphismsD: Cq(S(X)) *C+(S(X))such that

IDT + DdT =o1T - T.Consequentlywe have(6.1) The inclusion simplicial map i:M -- S(X) and thesimplicialmap8P1:(X) -* M are suchthat hecompositionpji:M * M is the dentity hile

thecompositionp1: S(X) -* S(X) is chainhomotopico the dentity.A corollary f (6.1) is(6.2) The inclusionmap i:M -* S(X) induces somorphismsf thehomologyand cohomologyroups fthe paceX with hose ftheminimal omplexM.Fromtheproperties fPT and C(PT) it follows asilythat if T is a q-simplex-in n(X) thenD(T) is a (q + 1)-chain n S,(X). Thus (6.1) and (6.2) may berestatedwithS(X) and M replaced by Sn(X) and M n Sn(X) respectively.(6.3) If thehomotopy roup 7r,,(X)vanishes hen he nclusionmapj:S., (X)-+ S8-(X) induces somorphismsfthehomologynd cohomologyroups fSn(X)-withhose fSn_1(X). n particular his lways ppliesto themapS1(X) -* S(X).PROOF. Considerthe inclusionmaps

M n S,(X) 4Sn(X) 4 Sn-1Since by (4.4) we have 1M1 Sn(X) = M n SnI(X), it follows hat jin = in-lis the inclusionmap in1 M n Sni(X) -* Snl(X). Since both inand in-iinduce isomorphismsfthehomologynd cohomology roups, hesameappliesto j.As a corollary f (6.3) wehave

(6.4) If thehomotopy roups7ri(X)vanishfor < n then he nclusionmapsS,(X) - S(X) induces somorphismsf thehomologynd cohomologyroups fS(X) withthose f S,(X).All the isomorphismssserted n (6.2)-(6.4) are also valid forcohomologygroupswith local coefficients. s an examplewe shall indicate the reasoningleadingto theanaloguLef (6.2).


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506 SAMUEL EILENBERG AND J. A. ZILBERLet G be a local coefficientystemnX (i.e. in S(X)) and let G' be the nducedlocal system n M. The inclusionmap i:M -- S(X) induceshomomorphisms:

i*:H'(X, G) --.H'(M, G').To define n inversemap proceed as follows.Let T be anyq-simplex n X withleadingvertexT(o) Then PT(o) may be regarded as a 1-simplexn X whichyieldsan isomorphism

p(T) = y[ X be a continuousmap and let M be a minimal ubcomplex fS(X). The map f will be calledminimal ffor very implex ofK the singularsimplex T8 s inM.(6.5) Everymap f:K -* X is homotopico minimalmap.FurtherhehomotopyAt maybe so chosen hat f L is a subcomplexfK on which is minimalthenot(y) = f(y) for every ointy e L.PROOF. Let {ept} be the homotopy etracting (X) ontoM as defined n ?5.For each simplexs of K consider he homotopy

ot,e = (SOtfT8)T'.The homotopies t a defined n each simplex , together ieldthedesiredhomo-topyAt.


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SEMI-SIMPLICIAL COMPLEXES AND SINGULAR HOMOLOGY 5077. Uniqueness ofminimal omplexes

Suppose that in the arcwise connected pace X we have, in addition to theminimal omplexM, anotherminimal omplexM1 (constructed elative o somebase point 4).(7.1) The simplicialmap(p of ?5 mapstheminimal omplexMl isomorphicallyontotheminimal omplexM.Use will be made ofthe following lementaryemma,the proofofwhich sleft to the reader.(7.2) If P1 and P2 are two -prismsn X such that "i) = P(K) for = 0, ...q - 1, thenPI(0) and P2(0) are homotopicf and only f Pi(1) and P2(1) arehomotopic.In order o establish 7.1) we shallproveby nduction hat501 apstheq-skele-ton M(') ofM1 isomorphically ntothe q-skeletonM(') of M. For q = 0 theproposition s obvious. Suppose inductively hat the proposition s valid forq - 1.Let T1 and T2be two q-simplexesnM, and supposethat AT1= (PT2 = T.Then sp(T(s) = T(") = v1(T(t)).Thus by the inductivehypothesis (t) - T(")fori = 0,. , q i.e., T1 andT2arecompatible. inceP(Tj)= P") fori = 0,*, qand PT1(1)=PT2(1) it follows rom7.2) thatPT,(0) and PT2(0) are homotopic.Thus T1 and T2are homotopic, nd sincetheyareboth ntheminimal omplexM, it follows hatT1 = T2 .

Let nowT be any q-simplex fM. By the nductivehypothesis here s foreach i = 0., *- , q a unique (q - l)-simplexTXn M1 suchthat sp1Tj-=).Consider hesubsetA = Aq X (1) u AqX I ofHl+q where , istheboundary fAq The singular risms T, and the singular implexT together efine mapf:A- X

suchthatfb'41 = T,;fp'? = PTi fori = 0... ,q.

Since A is a retract f ll+, the mapf can be extended o a singularprismP.Consider the singularq-simplexT = P(o). Since Tji) = Ti, the faces of Tare in M1 and therefore here is in M1 a (unique) q-simplexT' compatibleand homotopicwith T. For theprisms and PT' we thenhave p"i) = P(T?)fori = 0, , q and P(O) and PT'(0) arehomotopic. t follows hatP(1) = T andPT'(1) =pT' are homotopic.Since they are both in M, we conclude thatT = (p1T'.This concludes heproof.8. Complete emi-simplicialomplexes

We write mlforthe ordered et (0, 1, , n), wherem is an integer? 0.By a map a: [mi] > In]willalwaysbe meant a weaklymonotoneunction rom[m] to [n].A map which s not strictlymonotonewill be calleddegenerate.A map a: [m] -* [n] induces a simplicial map &:Am * An.Thus foreverysingular simplexT:A, -An the composition a is defined;we shall write Ta


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508 SAMUEL EILENBERG AND J. A. ZILBERinstead of T&. This operationTa could be used, in place of the conceptoftheith face, as a starting oint ofthedefinitionfS(X). The correspondingbstracttheorywillnow be outlined.

A completeemi-simplicialomplex is a collection f"simplexes"o-, o eachofwhich s attacheda dimension > 0, suchthat for ach q-simplex and eachmap a: [m] - [q],where m > 0, there s defined n m-simplex a ofK, subjectto theconditions(8.1) If ev s the dentity ap [q] [q],then E, = a.(8.2) IfA: [n] * [m], henata): = c(ac).Whenever herelation = ra subsists,we say thata lies on r.A completesemi-simplicial omplex s semi-simplicial?1) in the followingsense. LetEq:[ql[q] ~0i [q] s defined yp = ii 0 < i K1 s simplicial fT(aa) = (Ta)a. Here a is anysimplex, nd a any map of m] nto q],where ima = q.For every integerm >- 0 introduce a complete semi-simplicial omplexK[m] as follows.A q-simplex fK[m] is anymap a: [q] * [m]. For everymapa: [n] -> [q] he simplex a is defined s the compositemap.A q-simplex isdegeneratef thas a factorizationa,where is degenerate.(8.3) A q-simplex rof a complete emi-simplicial omplexK has a unique"minimal"factorization ra,where is a map onto, nd r is a non-degeneratesimplex.Thedimension fr willbe calledthe ankofa.PROOF. Because of (8.1), a has at least one factorization 'a' where ': [q][m'],m' < q and r' is an m'-simplex. et m = ma) be thesmallest uchm' andlet Ta be thecorrespondingactorization.We assert hatr isnon-degeneratenda isonto.ndeedfr sdegeneratehen = r'/where: [m] > [m'] sdegenerate.Then /3 ay be factored nto3 = &ywherey: m] [i"],[ : m"] -* ['] withIn" < m. Consequently= Tra = -r'/a (WS) ya) where a: q] -+ [m], contraryto the definition fm. Similarly fa is not onto then a = To where/: q] >

[m'], y:[m'] -> [m]and m' < m. Hence a- = r(yy) - (ry)/with/3:[q] + [m']form' < m.This shows heexistence fa factorizationssertedn (8.3).To provethe uniquenesssof thisfactorization, upposethat a = rial withT1non-degeneratend a, a mappingof [q]onto [ml].We shallshow that 1 = rand a, = a. Since both a and a, are onto there xistmaps/3:[m]> [q], /3:[ml]* [q]


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SEMI-SIMPLICIAL COMPLEXES AND SINGULAR HOMOLOGY 509with

af3 Cm? au,1 - mi ESincerjax= irawe have -iaiI3= ra = TECm=r andsimilarlya(39= ri . SincebothX and r1are non-degeneratehemapsal3 and a,43 renon-degenerate.Consequentlyml = m and al343o = Cm . This implies = ri- Supposethatforsome i X [q]we have a(i) 5 ai(i). We mayselect 3so thatfla(i) = i, thenaigai = ali 5 ai contradicting, = CEm . Consequently=a, .Notice hat fa: [ml * [q], hen E K[q] and .(7 a- is the d' faceof .Therefore or ny q-simplex -we have (aa) -r =aC (aCm) = abet). Thus(8.4) (o -) () )Suppose ow hat = ia is theminimalactorizationf with : [q] + [ml ntoand rnon-degenerate.f (') is onto hen o(t) = o-( is theminimalactorizationofa() and rankaa = ranka. If a) : [q - 1] [ml s notonto henaci) = Emwhere3: q - 1] * [m 11s onto.Thus

(i) (2:) __ ai (at)ra = i-a - -Cat ai-pand since is onto tfollowshat ank ") < dim -(a) < dim = rank -.Sum-marizing ehave(8.5) If a = -a is theminimal actorizationiven y 8.3) then ither (" is onto,and then a(t) is theminimal actorizationf t) or a(') is notonto nd then anka'W< rank .Asanapplicationf 8.5)weprove(8.6) If a and a, aredegenerate-simplexesuchthat t a-_j) for = 0, ...* q,then - = a.

PROOF. Let a = ia, a1 = i-1a1e minimalfactorizationswitha:[q] - [m],a,: [q] > [ml]. incea isdegeneratendontothere xist t least two ndices suchthat (") s onto.Thenby 8.5)m = ranka = rank (J( =rank a(-' < rank ai = i.

Similarlyweprovethatmi < m.Thus m = im, Consequently ankaW- ranka- andtherefore(i) isonto or he amevalues f forwhich2(:)s onto. or ucha value of thesimplex ) = a() has twominimal actorizationsa 3 andTlal'). Thus r = ir and a( a(i). This implies (i) = ao(i) for F j. Sincethis strue or t east wo ndices, itfollowshat = a, . Thus = -19. Complete inimal omplexesAswasalreadyemarkedarlyn?8, he otal ingularomplex(X) of space

X maybe regardeds a completeemi-simplicialomplex. urningominimalsubcomplexest snaturalorequirehat heminimalomplex , be a subcom-plexofS(X) regardeds a completeemi-simplicialomplex. hus anadditionalconditionas tobe imposed:(9.1) If T is a q-simplexfM and a: [ml * [qi henTa is an m-simplexfM.It willbe shown hat a slight hangentheconstructionf M describedn


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510 SAMUEL EILENBERG AND J. A. ZILBER?4 will insurethis additionalproperty. irstobservethat in viewof (8.6) twodegenerateimplexes fS(X) thatarecompatible re equal. Thustheequivalenceclassesusedindefining (n+l) in ?4eachcontains tmostonedegenerateimplex.We shall require hatthe degenerate implexbe selectedwhenever here s onein the equivalence class. We mustnowprovethat (9.1) holds. The case of anon-degenerates trivial incethen Ta = T't ...m) for a suitable choice ofil < i2 ... < is-m. Assumethen that a and thereforelso Ta are degenerate.The propositions valid form = 0 sincethena is notdegenerate. uppose,byinduction, hat theproposition olds form - 1. Then (Ta)(") = Tac(0)s inMand thereforenM(ml). Thus Ta is a degenerate implexwith all of ts faces nM(m-1). By ourmodificationf the constructiont follows hat Ta is inM.Turningto the main homotopyof ?5, we modify he constructiono as toinsure hateachmap cpt e a simplicialmap in the sense ofcomplete emi-simpli-cial complexes.Thus we mustreplacethe condition(*) Pt(T() = ('ptT)(t)bythestrongerondition(**) f t(Ta) = (9eT)a,whereT is a singular -simplexnda: [m]* [p].Suppose thenthat ptThas beendefined or ll singular implexesT of dimen-sion < q and that it satisfies he conditions f ?5 as well as condition **) form < q andp < q. First consider hedegenerate ingular -simplexes . Let T =Tra e the "minimal" factorization iven by (8.3) wherer is a non-degeneratem-simplexnd a: [q] -* [ml s ontowithm < q. Define

=pjT= ('ptr)a.AfterptThas beendefined or ll degenerate -simplexes efinePtTfor he non-degenerate -simplexesxactly s in ?5.Thus (*) is assuredfor ll non-degenerateq-simplexes.We shall showthat (*) holds also for the degenerate -simplexes.Indeed we have

(Pt(T"') = 'pt(Ta() = ('ptr)a( = (ptT)Wenowprove **) for -simplexes a, a: [m]-> [p],m _< ,p < q. First onsiderthe case whena is non-degenerate.n thiscase (**) follows rom *) since Tais T".i - q-m) or uitable ndices i < i2 < ... < i-m . Next consider hecasewhena is onto. f T = ,r1stheminimal actorizationfT thenTa = r (Oea) stheminimal actorizationf Ta and therefore

'Pt(Ta)= pt(TrIa) ((ptr)13a ('pT)a.Since any a: [q]-> [m]maybe factoredntoa = ala2 where l is nondegenerateand a2 is onto, t followshat **)holdsfor ll a.


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SEMI-SIMPLICIAL COMPLEXES AND SINGULAR HOMOLOGY 51110. Normalization

Let K be a complete emi-simplicialomplex nd G = {G(a), ey(j) a localsystem f abelian groups n K. A cochainf e Cq(I, G) will be called normalizedprovided hatf(a) = 0 for verydegenerate -simplex ofK. It willbe shown nthe sequel that the coboundaryof a normalized ochain is again normalized.Let Hq (K, G) denote the cohomology roup obtained using only normalizedcochains.Explicitlyet H' (K, G) = Zq (K, G)/B' (K, G), whereZ' is the groupof normalized -cocycleswhileB' is the groupof coboundaries f normalized(q - 1)-cochains.The inclusionsZ' C Zq and B' C Bq then yield a naturalhomomorphism ', H- . The mainresultof this section s(10.1) The homomorphism'(K, G) -+ H(K, G) is an isomorphismnto:

nqK, G) Hq(K,G)As willbe shown n [4]thistheorem s a generalization f the normalizationtheorem n cohomology heoryof groups [3, ?6] [2]. The proofof (10.1) thatfollows s a directgeneralizationf that of[3].In additionto the identitymap cq: [q] -- [q] and its faceseq: q [q],i = 0, * , we shallalso considerhemaps77q [q]- [q -1 i = O. -- q- 1

defined y 7qj = j for _ i and yj = j - 1 for < j. We note that is} Sthetotality f ll themapsof q]onto q - 11.The followingdentitieswillconstantly e used?q4q= ?Eq-7aq- for j < i7q =-?-1 =77q Eq71E = E for i + 1 G((o)).PROOF. By (10.2) 1musthave the form = a-q where is a 0-simplex. heidentities hen mply1(0) = a = #(l) Consider he2-simplex q2 = W702 * Bydefinitionfa local system,wehave

y I I2)ay(avji?0e?) = Y(C 0 1e).


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512 SAMUEL EILENBERG AND J. A. ZILBERIt follows romhe dentitieshat he xpressionsnparenthesesll give cl = 1.Thus yj) = identity.A cochain eC'(K, G) willbe called -normalizedi = 0, * , q) if

f(AanI) = 0 for ll < i.Every ochain s 0-normalized.he q-normalizedochainsre n viewof 10.2)thenormalizedochains.(10.4) f f is i-normalized,hen o is 8f.PROOF. Let k < i. Then

q+1( )= /3)f(cq+4Q+l)+ E (- 1)'f(WOt+Ii+l)ji1where istheeading dge f r*+i Forj < kwehavef(anT+ieq+i) f((ftCrl 1) - 0since isi-normalized.imilarly(u_7k+lic+1) f(Cer'-nI) = 0 for + 1 < j.For = k,k + 1 we have

f(u_0+,4+i) = f(or) = fo k+ .Thuswe obtain hat 6f)akil+) = 0 ifk > 0. Fork = 0 there emains heexpression(f)f(f)- f(o),where isthe eading dge f +1. Since 3sdegen-erate,10.3) mplieshaty(Q) is the dentitynd thetwo ermsancel ut.(10.5)Foreveryochain eCq(K, G) such hat f s normalized,heres a cochaing e C'-(K, G) such hatf Sg s normalized.PROOF. The proof epends n thefollowinglgorithm.tartingwith hecochainf e C' construct ochains o, , fE CE and go, *, 1 e C'-' byinductions

o = f, fi+1 = A - 69 gj(o) = (-1)if(aq)for = 0, * , q -1. Then clearly ftifo = 6ffor ll i. Sincefq = f -g,where = go+ + gq-lit sufficeso show hat q is normalized. e shallprovebyinductionhat j is i-normalized.his s clearly alidfor = 0. Weproceed y nductionnd assume hat iis i-normalized.ince or < i we havegi(ang'-) =) (-1)( fa(un 1-q) = 0,gi s -normalized.husby 10.4)6gis -normalizedndtherefore1 s -normal-ized.To show hat ig is (i + 1)-normalizedeusethe dentity

flfv)= (-1)isfi('fqv~)which ollowsystraightforwardomputationsing he dentitiesndthefactthatfi s i-normalized.ince fc= Bf snormalized,heright and ide s zero.Thusfi+j(a-q') 0 andfj+j s (i + 1)-normalized.his concludes heproof f(10.5).Theorem10.1) is an immediateonsequencef 10.5). ndeed f e Z then5f= 0 is normalizednd (10.5) yields cochain e CG suchthat - bg snormalized.ince - geZ? this hows hat hehomomorphism' -I is


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SEMI-SIMPLICIAL COMPLEXES AND SINGULAR HOMOLOGY 513onto.Assume owthath e Z' and thath = bf or ome e Cq'-.Since5) =his normalizedheres a g e c2 suchthat - 6g is normalized.hus h is thecoboundaryfthenormalizedochain - 6g, howinghatH' -+ H' is an iso-morphism.Ifwe imit urselveso simple oefficients,e can give heorem10.1) a moreintuitiventerpretation.alla chain fK degeneratef t s a linear ombinationofdegenerateimplexes.hesame rguments in (10.4)shows hat hebound-aryof a degenerateimplexs a degeneratehain.Thus the degeneratehainsform chain omplex whichsa subcomplexfK (regardeds a chain omplexand not as a semi-simplicialomplex).The normalizedohomologyroupsH' (K,G)arethen othinglse han he elative ohomologyroups '(K, D, 6).If we examine hecohomologyequence

... -* H"(K, D, G) - H(K G) -> H(D G) -> H+'(K D, G) -theexactnesskernel image roperty)mplieshatH'(K, D, G) >-(K, G)is an isomorphismntofor ll valuesofq if andonly fH(D, G) = 0 for lldimensions. It followsrom ellknown niversaloefficientheoremshat hisholdsfor ll G ifand only f all the ntegral omologyroupsHq(D) vanish.SinceD containso non-trivialero hains,his sequivalent ith he tatementthatD isacyclic. hus 10.1) tated nly or impleoefficientssequivalent ith(10.6) The degeneratehain complexD of a completeemi-simplicialomplexis acyclic.A direct rooff 10.6)couldbegiven, utthen 10.1)would tillhaveto beproved or ocalcoefficients.

COLUMBIA UNIVERSITYBIBLIOGRAPHY

ill S. EILENBERG, Singularhomologyheory,nn. of Math. 45 (1944),407-447.[2] , Topologicalmethodsn abstract lgebra;Cohomologyheoryf groups,Bull.Amer.Math. Soc. 55 (1949),3-37.[3] S. EILENBERG and S. MACLANE, Cohomologyheoryn abstract roups. , Ann.ofMath.48 (1947),51-78.14] , Relationsbetweenomologyndhomotopyroups f spaces. II, Ann.ofMath.51 (1950),514-533.[5] N. E. STEENROD, Homology ith ocalcoefficients,nn.ofMath.44 (1943),610-627.[6] , Productsof cocycles nd extensionsfmappings,Ann. of Math. 48 (1947)290-320.

Semi Simplicial Complexes and Singular Homology

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