Relations Between Homology and Homotopy Groups of Spaces

Relations Between Homology and Homotopy Groups of Spaces Author(s): Samuel Eilenberg and Saunders MacLane Reviewed work(s): Source: Annals of Mathematics, Second Series, Vol. 46, No. 3 (Jul., 1945), pp. 480-509 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1969165 . Accessed: 23/02/2013 12:14

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to Annals of Mathematics.

http://www.jstor.org

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions ANNALs OF MATHEMATICS Vol. 46, No. 3, July, 1945

RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY GROUPS OF SPACES*

BY SAMUEL EILENBERG AND SAUNDERS MACLANE (Received February 16, 1945)

CONTENTS

PAGE Introduction...... 480 Chapter I. Constructions on Groups ...... 484 Chapter II. The Main Theorems ...... 491 Chapter III. Products...... 502 Chapter IV. Generalization to Higher Dimensions ...... 506 Bibliography...... 509

INTRODUCTION

A. This paper is a continuationof an investigation,started by H. Hopf [5][6],studying the influenceof the fundamentalgroup 7r,(X) on the homology structureof the space X. We shall consideran arcwiseconnected topological space' X and the following groups derived fromX: 7r,(X)-the nthhomotopy group of X constructedrelative to some point xo e X as base point. In particular,the 1Wthomotopy group 7r,(X)is the fundamental group of X, see [71]. Hn(X, G)-the nthhomology group of X with coefficientgroup G. Both G and H'(X, G) are discreteabelian groups. If G = I is the additive group of integers,we writeHn(X) instead of H'(X, I). ,n(X)-the sphericalsubgroup of Hn(X); this is the image of the group 7rn(X)under the natural homomorphismvn: n(X) -* Hn(X). Hn(X, G)-the nthcohomology group of X withG as coefficientgroup. Both G and Hn(X, G) are topologicalabelian groups.2 The homologyand cohomologygroups of X are defined(Ch. II) usingsingular simplexes in X, with ordered vertices,as recentlyintroduced by one of the authors [1].

* Presented to the American Mathematical Society, April 23, 1943. Most of the results were published without proof in a preliminary report [3]. The numbers in brackets refer to the bibliography at the end of the paper. (Added in proof) After this paper had been submitted for publication, a paper by H. Hopf, (iber die BettischenGruppen, die zur einer beliebigenGruppe gehdren,Comment. Math. Helv. 17 (1944), pp. 39-79, came to the authors' attention. Although the methods em- ployed are quite different,the two papers overlap considerably. l A topological space is a set with a family of subsets called "open sets" subject to the following axioms: The union of any number of open sets is open, the intersection of two open sets is open: the empty set and the whole space are open. 2 A topological group is one which carries a topology with respect to which the group operations are continuous. No separation axioms are assumed. 480

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 481

B. We list the knownresults dealing with the relationsbetween the homology and homotopygroups. 10) The groupH'(X) is isomorphicwith the factorgroup of 7r,(X)by its commutatorsubgroup. 2?) If 7r.(X) = 0 for0 < n < r then Hr(X) _ 7rr(X);Hurewicz [72]. 30) A space X is called asphericalif 7ri(X) = 0 fori > 1. In an arcwisecon- nectedaspherical space the fundamentalgroup 7r,(X) determines all the homology and cohomologygroups of X; Hurewicz [74]. The algebraic mechanismof this determinationwas unknown. 40) The group 7ri(X) determinesthe group H2(X)/22(X). This was proved by Hopf T5] only in the case when X is a connectedpolyhedron. The word "determines"is used in the followingsense. Given any group II = F/IRrepre- sented as a factorgroup of a (non abelian) freegroup F by an invariantsub- group R, considerthe group (1) h2(HI)= R n [F, F]/[F, R] where n stands for set theoreticintersection and [A, B] is the subgroupgener- ated by all elementsof the formaba-'b-', for a e A, b e B. It was shown by Hopf that h2(HI) depends only on II and not on the representationII = FIR, and that

H2(X)/2;2(X) _- h2[71(X)].

50) If 7r.(X) = 0 for 1 < n < r then the group 7r,(X)determines the group HT(X)/1T(X). This was proved by Hopf [61only in the case when X is a connected polyhedron. The proofgives no algebraic procedurefor the determina- tion. The theoremsof this paper include and generalizeall these results. More- over, we succeed in gettinga completealgebraic formulationof the group con- structionsneeded forthe various "determinations." C. The task of handlinga varietyof groupconstructions, which, judging from the complexityof formula(1), is likely to become quite involved,is simplified by the followingdevice developed in Chapter I. Given a (non abelian) group II we constructan abstract complex K(HJ.). The homology and cohomology groups of this complex,denoted by H'(HI, G) and Hn(H, G), turn out to be preciselythe groups needed for the various descriptions. The cohomology groupsHn(H, G) can be describeddirectly (without the complexK(HJ.)) as follows. A functionf of n variables fromthe group II with values in the topological abelian group G will be called an n-cochain. The coboundarybf of f is the (n + 1)-cochaindefined by

(6if)(X1, * *, Xn+l) = f(X2 , ...* Xn+l) (2) + 2 (-1)if(xl, i, Xx xi+1, , xn+1) + (_-1) n+lf(xl, *.- xn).

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 482 SAMUEL EILENBERG AND SAUNDERS MACLANE

The cochainsform an additive group Cn(H, G).3 The cocycles (i.e., cochains f wsith3f = 0) forma subgroupZn(H, G). Since 33f= 0 it followsthat the co- boundaries(i.e., cochainsf of the formf = ag forg e Cn,1(H,G)) forma subgroup Bn(r(HG) of Z,(H, G). The cohomologygroups of HI are then definedas the factorgroups4 H,(H, G) = Zn(H, G)/Bn(HT,G).

For nt= 0, 1, 2 the cohomologygroups H.(H, G) furnishwell knowninvariants Ho(H1,G) - G HIJ(IH,G) - Hom (IH,G) H2(11, G) - Extcent (G, IH) where Hom (IH,G) stands for the group of all homomorphismsp: IH - G, and Extcent (G, II) forthe group of all centralgroup extensionsof the group G by the group HI.5 The homologygroups H'(H1) withintegral coefficients have the followingvalues for n = 0, 1, 2. H0(IH) -I H'(H) H/[H, HT] H2(II)-CharExtcent (P. II), whereP denotesthe group of real numbersreduced mod 1, while Char G is the group of all charactersof the group G, i.e., Char G = Hom (G, P). The developmentof the algebraic ideas of this paper was purposelylimited to the needs of the topological applications. Consequently,we have entirely omittedthe discussionof the algebraicallyimportant case when the group HI acts as a group of operatorson the coefficientgroup G. We will returnto this subject in anotherpaper. 6 D. Proposition30) can now be formulatedas follows: THEOREM I. If X is arcwiseconnected and asphericalthen the homology and

3 C,(r, G) is topologizedas follows. Given an n-tuple(xl, **n, xn) and givenan open set U in G, consider the set of n-cochainsf such thatf(xi, * , xn)EU as a basic open set in Cn(r, G). Arbitrary open sets in Cn(r, G) can be obtained from the basic ones using finite intersections and arbitrary unions. It is easy to verify that the homomorphism a is continuous with respect to this topology. I Since no separation axioms in topological groups are postulated, the factor group Zq/Bq is topological even if B, is not a closed sub-group of Zq 5 For more details see ?4 below. This is the second application of the group of group extensions to problems in topology. In a previous paper [2] the authors have studied the group of abelian extensions in connection with the problem of classifying and computing the homology and cohomology groups for various coefficientgroups. Both groups Hom (II, G) and Extcent (G, 11) carry a topology; see [2], p. 762 and p. 770. 6 See Bull. Amer. Math. Soc. 50 (1944), p. 53.

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 483 cohomologygroups of X are determinedby thefundamental group iri(X). More precisely, - H-(Xy G) H n(7 (X) 2G)

Hn(XI G) - Hn(71(X), G). In order to formulateour generalizationof Hopf's propositions40) and 50) we need the followinggroups derived fromthe sphericalsubgroup ;n(X): Zn(X, G)-the subgroup of Hn(X, G) consistingof all elements of the form Igizi where gi cG, zi E n(X). An(X, G)-the subgroup of Hn(X, G) consistingof those cohomologyclasses that annihilateevery element of ;n(X), when the Kroneckerindex is the multiplication. THEOREM II. If X is arcwiseconnected and 7n(X) = 0 for 1 < n < r then Hn(X, G) Hn(iri(X), G) for n < r H,(X, G) Hn(7ri(X),G) forn < r

H (X, G)/: (X, G) H ((X), G) Ar(X, G) Hr(7ri(X),G). Both TheoremsI and II are derivedin Chapter II. E. It was shown by Hopf [5] that the group xrj(X)not only determinesthe group H2/22 but also has a bearing upon the products (i.e., the cup and cap products)in X. In Chapter III we defineproducts in the complexK(HI) and show that the isomorphismsconstructed in provingTheorems I and II preserve the products. F. In Chapter IV we considerthe case when 7rn(X)= 0 forn < q and study the influenceof the group xrq(X)upon the homologystructure of X. We obtain a theoremsimilar to the precedingTheorem II. The group constructions are also derivedfrom a suitably definedabstract complex; however,we do not have the algebraicinterpretations as in the precedingcase. G. Theorem II impliesthat

H 2(X)/2;2(X) -- H 2(rl(X)) forevery arcwise connectedspace X. On the otherhand Hopf [5] has shown that

H 2(X)/2;2(X) -h 2(7rl(X)) forevery connected polyhedron, with the grouph2 defined as above. Comparing these two results we obtain an isomorphism (3) h2(I) _-H2(H) _ CharExteent(P, H)

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 484 SAMUEL EILENBERG AND SAUNDERS MACLANE for all groups II that are fundamentalgroups of connectedpolyhedra; i.e., for groups that can be described by means of a finitenumber of generatorsand relations. The fact is that the isomorphism(3) can be establishedpurely alge- braicallyfor all discretegroups H. The proofis not given in this paper.

CHAPTER I

CONSTRUCTIONS ON GROUPS 1. The complexK(H) For any discrete (multiplicative)group H we define an abstract complex K(H) as follows. An n-dimensionalcell an (n > 0) in K(H) is to be an ordered array

(1.1) an = [X0 * n] of n + 1 elementsof H, subject to the equivalence relation

(1.2) [xO, * * *, Xn] = [XO, *. , xxn] forall x e H. Consequentlythere is only one 0-cell,namely [x]. The boundaryrelation is n (1.3) ([oX0 * X n]I E: (- 1)i [oX ** Xxi X***X n] i20 whereXi on the rightmeans that the argumentxi is to be omitted. Aftercollecting terms, (1.3) can be rewrittenas a sum over all the (n - 1)- cells an-1 of K(H) withintegral coefficients [an; an-1], in the form v r On = al[a.fl n; 7n-1]07n-1fla] whichdefines the incidencenumbers [an; on-1]. Because of possiblerepetitions on the rightside of (1.3) the incidencenumbers may be integersothers than 0, I or -1. We verifyat once that ad = 0; and consequentlythe cells an withthe incidencenumbers [on; n-1l]define a closure finiteabstract complexK(H); see [1, ?1]. In this complexwe may considerfinite chains over a discreteabelian coefficientgroup G; theylead to discretehomology groups Hn(H, G) = Hn(K(HI) G). Similarlywe may consider (infinite)cochains in K(H) with coefficientsin a topologicalabelian group G; they lead to topologizedcohomology groups7 Hn(HXG) = Hn(K(H), G).

7 It is easy to see that the topology of C,(,I, G) as defined in 3 is the usual topology of the group of cochains in an abstract complex; see [1, ?31.

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOL 485

In generalthe homologyand cohomologygroups of the complexK(H) will be referredto as the homologyand cohomologygroups of the group H over a suitable abelian group G as coefficientgroup. Similarlywe shall understandthe groups C'(H, G), Z'(H, G), and B'(H, G) of the chains, cycles,and bounding cycles of H to be the appropriategroups associated with the complex K(H). The topologicalgroups Cn(H, G), Zn(H, G), and B(I(H, G) ofthe cochains,cocycles, and coboundariesare interpretedin analogous fashion. Since an n-dimensionalcochain over a group G is a functionwhich with each n-cella' associates an elementof G, a cochain of Cn(H, G) can be interpretedas a functionF of n + 1 variables on H, withvalues in G, subject to the conditions

(1.4) F(xxo , * * , xx1n)= F(xo,X * * , x1,) forall x e H. The coboundaryis then given by the formula n+1 (1.5) (6F)(xo , , xn+) = E (-1) F(xo, * , , ,xn+) i20 In the followingsections two alternativedefinitions of the complexK(H) will be given. For purposes of identificationthe method adopted in the present section will be designatedas the homogenousmethod. In the complexK(H) we have the well knownduality between homology and cohomology,expressed as an isomorphism(see [8, p. 129]) (1.6) Hn(H1,G) _ Char Hn(H, Char G) forany discreteabelian group G. This formulagives the homologygroups of HI once the cohomologygroups of H with compact coefficientsare known. 2. Matric definitionof K(H) For the purpose of the applicationsof the complexK(H) it is convenientto have an alternativedefinition based on certainmatrices. We shall consider(n + 1) X (n + 1) matrices

A = 11dij 11)i) j = 0, 1, ... n, with elementsdij in H such that (2.1) dijdjk = dik, i, j, k = O. 1, n. This conditionimplies that

dii = 1, dij = d7!, dik = dldok. With each such matrix A we will associate an n-cell of the complex K(H) accordingto the correspondence (2.2) A - WooX*d-, dol ,n

Conversely, given an n-cell [XO, -, x-n] (if K(H) we construct a matrix A = II xi- xj 1 . This matrixsatisfies condition (2.1). We verifyat once that this proceduregives a 1-1-correspondencebetween the matrices A describedabove

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 486 SAMUEL EILENBERG AND SAUNDERS MACLANE and the n-cellsof K(H). Consequentlywe may regardeach n-cellof K(l) as a matrixand vice-versa. In orderto translatethe boundary relation (1.3) into the matricterminology we notice that if

[X* Xn]A then

[X Xin )xi) whereA"i) is the matrixobtained from A by erasingthe ithrow and the ithcolumn. Consequently the boundary relation becomes n (2.3) = Z (-1)'lA i=O 3. The non-homogenousdescription of K(l) For the algebraicapplications of the complexK(H) it is convenientto have a third,"non-homogenous", description of K(H). We shall considern-tuples (xi, ** *,n) n ofelements xs e H. Witheach n-tuple associate an n-cell of K(H) as follows:

(XI , Xn) X[1 X, XIX2 .XXi Xn]; converselywith each n-cell we can associate an n-tupleby the rule

-1 X (3.1) [ so , ** Xn] -slx,* ,xl* 0XI I _. x^-1In This gives a 1-1 correspondencebetween the n-cells of K(H) and the n-tuples (xi, *- *, Xn),so that each n-cellof K(H) may be regardedas an n-tuple(xl, * - , Xn) and vice versa. The O-cellof K(H) becomesthen the (vacuous) 0-tuple( ). In orderto translatethe boundaryrelation (1.3) we noticethat

9[1, xI, xix2, *, xlI *... x . [xI, xIx2, * X*n]Xi n + E (-1) [1 X, XX2 * X XI Xi * Xi Xn] ill and that

. [xiXx2, *** Xi *n] = X2X3 X Xn] (X2, Xn)

[1,xI),XiX2, XI,X1 xi, ...,xi ... x (X,(x *- ixi+ *- ,x,")fori

... [11, Xi, X1X2, XI X n-11X (X1, *X *n-O, so that n-I

(3X. Xn) = (X2X Xn) + E (-1) (XI Xi Xi+ * Xn) (3.2) ire

+r (-lBn(X, . . -Xn 1).

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 487

This is the boundaryrelation in the non-homogenousdefinition of the complex K(H). Usingthis definition,an n-cochainof H over a coefficientgroup G is a function f of n variables on II to G, whilethe coboundaryaf is definedby

(6f)(xI, , Xn+I) = f(x2, I, xni)

... + E (-1)f(X * Xi+,Xi X Xn+ 1) + (1)n+lf(XI, XXn)

Each non-homogeneouscochain can be translated into a homogeneousone and vice-versa,by means of the formulae

F(xo,X * X1n) = f(x iXI, xl x2, i, xn-LXn) ... fAXI, *** Xn) = F(1j X Il X1X2j X* xl**n).- We shall also exhibitdirectly the connectionbetween the matricand the non- homogeneousdescription of K(HI). Given an (n + 1) X (n + 1) matrix

A-= | dij| satisfyingcondition (2.1), we have by (2.2)

A [d00X dOlX * don] and by (3.1)

X [doo dOl X don] -* (d'ldo0, doj1d02 ,d-1Idon) But

d-~_Idoi = di-,,odoi = di-,,i and consequentlythe formula

(3.5) A --> (do. , d12, ...* dn-1,n) gives the desired connection. 4. The cases n = 0, 1, 2 Usingthe non-homogenousdescription of the complexK(H) we shall now give a group theoreticinterpretation of the cohomologygroups Hn(H, G) for n = 0 1,2. n = 0. Since K(H) containsonly one 0-cell,written as the 0-tuple ( ), each 0-cochain is a constant in G so that Co(H, G) = G. The coboundary of a 0-cochainf is (Wf)(x)= f( ) + (-1)'f( ) = 0, so that 5f = 0. Hence Zo(H, G) = G. Since thereare no cells of dimension-1 we have Bo(II, G) = 0 and therefore (4.1) Ho(H1,G) = G.

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 488 SAMUEL EILENBERG AND SAUNDERS MACLANE

From (1.6) we get forthe 0th homologygroup

H0(11, G) _ Char Ho(II, Char G) _ Char Char G _ G and therefore

(4.2) H11(11G) __ G. n = 1. A 1-cochainf eCi(H, G) is a functionof 1 variable on H to G. The coboundary of f is

(Of)(XI, X2) = f(X2) - f(XlX2) + f(Xl) so thatf is a cocycleif and only iff satisfiesthe identity

f(xlX2) = f(Xl) + f(X2); i.e., if and only if f is a homomorphicmapping of H into G. Consequently Z1(H, G) is the group Hom(HI,G) of all the homomorphicmappings of HIinto G. Since every0-cochain is a cocycle,we have B1(HI,G) = 0 so that (4.3) Hi(H, G) = Hom (HI,G). In particularif G = P is the additivegroup of reals reducedmod 1 we get (4.4) H1(H, P) = Char II. From the duality formula(1.6) we get (4.5) H1(ll G) _ Char Hom (H, Char G) or, in termsof the tensorproduct (see [2, p. 788]) (4.6) H'(H, G) _ HoG. For integralcoefficients we findfrom (4.4) and (1.6) that H'I(H) _ Char H1(II, P) _ Char Char II and therefore (4.7) H'(H) _ II/[H, H] where [HI,II] is the commutatorsubgroup of I. n = 2. A 2-cochainf e C2(l, G) is a functionof 2 variables on H to G. The coboundaryof f is

5f(XI , X2 , X3) = f(X I, X2) - f(XIX2 , X3) + f(X I, X2X3) - f(XI , X2), so that f is a cocycleif and only iff satisfiesthe identity

(4.8) f(XI, X2) + f(X1, X2X3) = f(lX2 I X3) + f(x1 I X2). This identitymeans that f is a central"factor set" of HIin G. The group of all such factorsets may be writtenas Factcent (HI, G). A 2-cochainf is a co- boundaryif it can be writtenas

(X1, X2) = h(xi) - h(Xlx2) + h(X2)

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 489 forsome functionh on H to G. Factor sets of this special formare known as "transformationsets"; the group of all such may be writtenas Trans (II, G). The second cohomologygroup may thus be writtenas (4.9) H2(11,G) = Factcent (II, G)/Trans (II, G). This group occurs in the theoryof group extensions. We call the group E a centralgroup extension of G by II ifthe centerof E containsG and if E/G = IT. Given any factorset f, we may definea correspondinggroup extension Ef as the set of all pairs (g, x), withg e G, x e II, and withthe multiplicationrule

- (91, xI) (92, x2) = (91 + 92 + f(xIx2), xIx2). One observesthat this multiplicationis associative if and only iff satisfiesthe condition(4.8) above. The group Es constructedfrom f is indeed an extension of G by II, forthe correspondence(g, x) -*+ x is a nomomorphismof Ef onto I. The kernelof this homomorphismis the set Go of all pairs (g, 1). This set is isomorphicto the group G under the correspondenceG *-* (g -f(1, 1), 1). Furthermore,Go lies in the centerof E,. Every centralgroup extensionE of G by II may be representedin the form E1, fora suitable factorset f. The factorset f is not unique; two factorsets f and h determine"equivalent" groups extensionsif and onlyif f - h is a trans- formationset. Therefore,the cosets of Factcent (11, G)/Trans (11,G) are in one-to-onecorrespondence with the central group extensionsof G by II; this factorgroup consequentlyis known as the group Extcent (II, G) ofsuch group extensions. We may thus write (4.9) as (4.10) H2(11,G) _ Extcent (II, G) S. Examples In this sectionwe shall computethe cohomologygroups Hn(ni, G) forvarious groups II. Except in one trivial case the computationwill be carried out by constructinga space X whose fundamentalgroup 7ri(X) is isomorphicwith HI and for which suitable higherhomotopy groups vanish, and then applying to this space the main theoremof this paper as stated in the introduction. A) II is thenull group. The complex K(H) has only one n-cell a' foreach n > 0. Formula (1.3) shows that

da = an forn even, dO" = 0 forn odd. Consequentlyall the homologyand cohomologygroups of II are null groups. B) II is a free (non abelian) group. Let X be a connectedgraph with HI as fundamentalgroup. All the homotopyand all the cohomologygroups of X are null forthe dimensionsn > 1. Consequently,by TheoremI Hn(1,G)=O forn>1,

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 490 SAMUEL EILENBERG AND SAUNDERS MACLANE whilefrom (4.3) we have H1(fl,G) = Hom (II, G), so that H1(IH,G) is the direct sum of G withitself m times,where m is the numberof freegenerators of II. C) H is a free abelian groupwith r generators.Let X be the r-dimensional torus; i.e., the cartesianproduct of r circumferences.We have 7ri(X) _ II, 7r,(X) = O forn > 1. Since Z'(X) = 0 forn > 1 we have An(X,G) = Hn(X, G) forn > 1. Since Hn(X, G) = 0 forn > r

Hn(X, G) = G forn < r whereG' standsfor the directproduct G X G X * X G j times,we have Hn(H, G) = 0 forn > r

Hn(H,G) = G forn < r. D) H is a cyclicgroup of orderm. For each n > 0 thereis a rotationX of the (2n + 1)-sphereS2n+1 such that 10) Xhas the periodm; 20) X preservesthe orien- tation of S2n+ ; 30) none of the rotationsX, X2,* *Xm-1 has a fixedpoint. By identifyingthe pointsx, X(x), **, X W1(x)we thereforeget an orientablemani- fold X, oftencalled a generalizedlens space, whose universalcovering space is S2'"+. Consequently we have 7r1(X)= I, 7ri(X) = O for 1 < i < 2n + 1. Hence the application of the Theorem II gives H2n(H) G) '--H2n(X) G) j~ffn+,(11IXG) e-aA2nf+l(X,G). From the Poincare dualityrelation we get H2n(X,G) _ H1(X, G). Since H1(X) is cyclicof orderm we get H1(X, G) = G/mG. Consequently, H2n(H,G) - G/mG. Since X is an orientablemainfold of dimension2n + 1 we have - H2n+l(x) I_ H2n+l(X, G) G. The sphere S2n+l is an m-foldcovering of X, so the sphericalhomology classes are the ones that are divisibleby m:

y t1(X) = mI. From this it followsthat a cohomologyclass f2n+lwill annihilate2;2n+1(X) if and only if mf2I+1 = 0. ConsequentlyA2n+l(X, G) _ mGand H2n+1(H,G) _ mUG where mGis the subgroupthe elementsg e G such that mg = 0.

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 491

Using the Cartesian product of a suitable number of circumferencesand generalizedlens spaces we could in a similarfashion compute the cohomology groupof any abelian groupH witha finitenumber of generators.

CHAPTER II

THE MAIN THEOREMS8 6. The singularcomplex S(X) We brieflyreview the basic conceptsof the singularhomology theory as de- velopedin [1]. Let s = < po, ***, p, > be a simplex,in some euclidean space, taken with a definiteorder of verticespo < pi < .. < pn. A continuousmapping T:s -* X will be called a singularn-simplex in the topologicalspace X. If s1 is another simplexwith orderedvertices of the same dimensionas s, then thereis a unique linear mapping B: si - s preservingthe order of the vertices. The singular simplexes T:s - X TB:s1 -X willbe called equivalent (notation: T _ TB). The equivalence classes obtained thisway are the cells of the abstractcomplex S(X). We shall use the same sym- bol to denote an individualsingular simplex and its equivalence class. We denoteby ski)the face of s oppositeto the ithvertex and by

T():,s(i)- X the partial mapping T(t) = T | 8(i). The boundary in S(X) is definedby n AT = E (_ 1)i T(t).

The homologyand cohomologygroups of the complex S(X) will be called the singularhomology and cohomologygroups of the space X, and will be written as Hn(X, G) and Hn(X, G). Let xo e X be a fixedpoint of X chosen as base point. We denote by S1(X) the subcomplexof S(X) obtained by consideringonly those singularsimplexes T: s -* X whichmap all the verticesof s into xo. Since S1(X) is a closed subcomplex of S(X) the identitymapping 711(T) = T is a chain transformation (see [1, ?4])

1: S -(X) + S(X). It was shownin [1, ?31] that if the space X is arcwiseconnected then the chain transformationn' is a chain equivalence; i.e., that thereis chain transformation

8 This chapter and the following ones lean rather heavily on the methods and the results of [1].

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 492 SAMUEL EILENBERG AND SAUNDERS MACLANE

s1:S(X) -+SJ(X) such that the chain transformations

- vlxql:Si(X) SJ(X) 77mi: S(X) S(X)S are both chain homotopicto the identitychain transformationof the complex S1(X) into itselfand of S(X) into itself,respectively. In particular,the homology and cohomologygroups of the complexesS(X) and S1(X) are isomorphic. "Weshall always choose the base pointXO E X used in the definitionof S1(X) as the base pointfor the definitionof the fundamentalgroup 7ri(X) and ofthe higher homotopygroups irn(X). It is clear that the complex S1(X) is more closely connectedwith the fundamentalgroup 7r1(X)than is the largercomplex S(X). In fact, every singular 1-simplexin S1(X) uniquely determinesan elementof iri(X). Consequentlyin studyingthe influenceof the group iri(X) upon the homologystructure of X we shall use the complex S1(X) almost exclusively. We shall also have occasion to considerthe subcomplexesSm(X) of S(X) consistingof those T:s -, X whichmap all the faces of s of dimensionless than m into the point o0. The identitymapping furnishesthe chain transformation 77m:S m,(X) -- S (X). Clearly S(X) = SO(X). A cyclez' in the complexSm(X) (m < n) will be called sphericalif it is homologous in Sm(X) to a cyclein the subcomplexSn(X). In otherwords zn is spherical if z' - 0 in Sm(X) mod Sn(X). In each of the homologygroups Hn(Sm(X), G) (or H'(X, G) form = 0) we thus get a subgroupZn(Sm(X) G) (or 22(X, G) form 0) of the sphericalhomology classes. It was provedin [1, ?33] that the subgroup jn(X) of the integralhomology group Hn(X) is the image of the homotopygroup 7rn(X)under its natural mappinginto Hn(X). In the cohomologygroup Hn(Sm(X), G) of Sm(X) the correspondingsubgroup An(Sm(X), G) consistsof all the cohomologyclasses that annihilatethe subgroup ,n (Sm (X)), under the Kronecker index as multiplication. In particular An(X, G) is the annihilatorof 2n(X). If X is arcwise connected,the transformation-i: S1(X) -- S(X) is a chain equivalenceand underthe isomorphismsof the homologyand cohomologygroups inducedby -qlthe subgroups2n(S1(X), G) correspondto ?n(X, G) and similarly An(S1(X), G) correspondto An(X, G).

7. The chain transformationK The basic tool used in this paper forthe study of the influenceof the fundamentalgroup xri(X)upon the structureof the space X is a chain transformation

K:Si(X) -* K(7r1(X)) The definitionof this transformationproceeds as follows. Let s = < po ... p, > be an n-dimensionalordered simplex and let T:s

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 493

X be a singularsimplex in S1(X). Since everyvertex pi of s is mapped into the base point xo e X each edge pipi of s maps into a closed path in X and therefore determinesuniquely an elementdij of 7r1(X). If i = we definedij = 1. It is clear that dij = d7,1. For any threevertices pi, pi, pk the interiorof the tri- angle PiPjpk is mapped into X and therefore

di,djkdki= 1. This implies that dil,djk=dik, i, jI = 0, 1, ... ,n, so that the matrix A = IIdjll satisfiesthe conditions(2.1) and thus is an n-cellof the complexK(7r,(X)). We define

K(T) = A. This gives a mapping of the cells of the complex S1(X) into those of K(7r(X)) and leads to homomorphismsof the integralchains in S1(X) into those of K(Xi (X)). These homomorphismswill also be denotedby the letterK.. In ordertc show that K is a legitimatechain transformationwe must show that K and the boundaryoperator d commute. Let aT = (-l)'T(" whereT(') is the face of T opposite the ith vertex. Let A = K(T) and let A"' be the matrixobtained fromA by erasingthe ith row and the ith column. As remarkedin ?2, we have OA = :_1ii

It is immediatelyclear from the definitionof K that K(T(')) = A('). Consequently,

aKT = dA = 2(- 1)tA(t) = :(-1)tKT(t) = K2(-l)T() = KaT which proves that OK = KC. The chain transformationK induces homomorphisms(see [1, ?4])

K:H'(Sl (X), G) H'(7r,(X), G)

K*:H,,(7r,(X) G) Hn,(Si (X) G). These homomorphismswill be used to comparethe groups of the space X with those of the complex K(7r1(X)).

8. Propertiesof K Before we proceed with the proof of our main theoremwe shall list a few propertiesof the chain transformationK. LEMMA 8.1. If z' is a cycle (with any coefficients)in the subcomplexS2(X) of S1(X) thenthe cycle KZ' boundsin K(7r1(X)).

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 494 SAMUEL EILENBERG AND SAUNDERS MACLANE

PROOF. Let Andenote the n-cellof K(7ri(X)) representedby the matrixwith n di = 1. Clearly dA = 0 if n is odd and 3An = An-1 if n is even. Now let T':s -> X be a singularn-simplex in X belongingto S2(X). This means that all the edges of S are mapped into the base point xo of X. Conse- quently,by the definitionof K, it followsthat K(T) = to . If now zn is any cycle in S2(X) then K(zn) = kA wherek is some elementof the coefficientgroup. If n is odd then 3An+l = An and K(zn) = a(kAn+1). If n is even then 30K(Z') = kOAn= kAn-1. Since zn is a cycle,we have aK(Zn) = 0, hence k = 0 and K(Zn) = 0. In either case K(Zn) bounds in K(7r1(X)), q.e.d. LEMMA 8.2. Under the homomorphism

K:Hn(Si(X), G) - H n(7ri(X), G) thesubgroup ..(S1(X), G) maps into zero. PROOF. Let zn be a cycle in S1(X) belonging to a homology class in ;n(S1(X), G). By the definition of a spherical cycle in S1(X), the cycle zn is homologous in S1(X) to a cycle z2 in S2(X). Hence K(Zn) is homologous to K(Zn). But K(Z2) bounds in K(7r,(X)) in virtue of the preceding lemma. Hence K(Z') bounds, q.e.d. LEMMA 8.3. Under the homomorphism

K*:Hn(7rl(X), G) -* Hn(Sl(X), G) theimage of Hn(71 (X), G) is containedin thesubgroup An(S1(X), G). PROOF. Let fn 6 Hn(7ri(X), G) and let zn e 2n(S1(X)). We have the Kron- ecker intersection

KI(K*fn, Zn) = KIfn, JCZn), but KZn = 0 by Lemma 8.2. Hence K*fn annihilates the group ,n(S1(X)) and consequently K*f E An(Sl(X), G). 9. Proofof Theorem I

We shall now study the transformation K in the case when X is aspherical; i.e., when iri(X) = 0 for i > 1. THEOREM Ia. If X is asphericalthen the chain transformation

K:S1(X) -4 K(7r,(X)) is a chiainequivalence. This result contains Theorem I as a corollary. For, if X is arcwise connected, the chain transformation

1:Sl(X) -+ S(X)

also is a chain equivalence; it follows that the complexes S(X), S1(X), and K(1rT(X)) are all chain equivalent. Consequently the complexes S(X) and K(7r1(X)) have isomorphic homology and cohomology groups, as asserted by Theorem I.

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 495

PROOF OF THEOREM Ia. We shall definea chain transformation

:K(7ri(X)) >- S1(X) subject to the followingconditions: (9.1) for each cell A of K(iri(X)), K(A) is a singularsimplex of S1(X),

(9.2) KK(A) = A,

(9.3) if K(A) = T, thenk(A(i)) = T") Since K(7r1(X)) and S1(X) each have only one 0-cell AOand To, we define K(O) = To. Let Al be a 1-cellof K(7r,(X)), so that 1d1 whered e iri(X). Take a 1-simplexsl with orderedvertices and let T:sl5> X be a continuousfunction mapping sl into a closed path about xobelonging tothe elementd of the fundamentalgroup. DefineK(A) = T. Next, let 1 d de A= d-1 1 e (de)-' e-1 1 be a 2-cellof K(7ri(X)) and let s = < VOVlV2> be a 2-simplexwith ordered vertices. The faces of A are

A = 1 - e '(1) 1 de A(2)= 1 d e' 1 A (de)f' 1 d-1d 1 We consider the 1-simplexes

S = < V1V2 > S < VOV2 > s(2) = < VoVi > and since K has already been definedfor the 1-cellsof K(7r,(X)) we have three mappings

K( )) :s -> X, i = 0, 1, 2, whichgive closed paths about xobelonging to the elmentse, de, and d of iri(X) respectively. Jointlythese threemappings give a mapping T of the boundary b(s) of into X, and this mapping is nullhomotopic. Consequently T can be extendedto a mapping T:s -* X. We defineK(A) = T. From now on, we proceedby induction. Suppose that R(A) has been defined forall cells A of dimension< k(k > 2). Let A be a k-cellof K(7ri(X)). Let s be a k-simplexwith ordered vertices. If ski)is the ith face ofs, wvehave mappings

_(AUi))S) X and in view of (9.3) these mappingsagree on the commonparts of any two faces of s. Consequentlythey combine and give a mapping T:b(s) -- X of the

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 496 SAMUEL EILENBERG AND SAUNDERS MACLANE boundary b(s) of s. Since b(s) is homeomorphicto a (k - 1)-sphere and lXk-l(X) = 0 because k - 1 > 1, the mapping T can be extendedto a mapping T:s -> X. We defineK(A) = T. From the constructionit is clear that (9.1)- (9.3) are satisfied,so that the definitionof Kis complete. It followsfrom (9.3) that Kand a commuteso that Kis a chain transformation. From (9.2) it followsthat

KK = 1.

In orderto show that K is a chain equivalence it is now sufficientto show that

(9.4) KK - 1 which means that the chain transformationiK is chain homotopic with the identitymapping of S1(X) into itself. We shall now definefor each singularsimplex T:s -> X of S1(X) a "singular prism" RT:S X I -X, wheres X I is the Cartesianproduct of the simplexs with the closed interval [0, 1], subject to the followingconditions: (9.5) RT(P, 0) = T(p),

(9.6) RT(P, 1) = (kKT)(p),

(9.7) RT(P, t) = RT(i)(p, t) forpe s (9.8) If T, T2, then RT1 RT2

Here RT1 RT2 means that these two singularprisms are equivalent, as defined in [1, ?11]. If T has dimension0 we defineRT by settingRT: s X I -o X. If T has dimension 1 thens v I is a rectanglewith b(s X I) as boundary. We definea mapping R:b(s X I) -> X by setting R(p, o) = (p),

R(p, 1) =(KT)(p), R(p,t) =xo for peb(s). This mappingis continuous. Accordingto (9.2) we have

KKKT = KT and thereforeby the definitionof K the paths T:s -* X and iKT:s -> X represent the same elementof the fundamentalgroup 7r,(X). Consequentlythe mapping R is nullhomotopicand can be extendedto a mapping RT:S X I -X.

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 497

From now on we proceedby induction. Suppose that RT has been definedfor singularsimplexes T of dimension< k(k > 1). Let T:s -> X be a singular k-simplexof S1(X). We definea mapping

R: b(s X I) -> X by setting R(p, 0)= T(p),

R(p, 1) = (4KT) (p), R(p, t) = RT(i)(p, 1) forp es This mappingis continuous. Since b(s X I) is homeomorphicwith the k-sphere and since 7rk(X)= 0 because k > 1, the mappingR can be extendedto a mapping RT:S X I -+X. It is easy to see that conditions(9.5)-(9.8) are still satisfied, and thereforethe definitionof RT is completed. We shall now considerthe standard subdivisionof the prisms X I and the basic chaind(s X I) ofthis subdivision, as definedin [1, ?16]. For each T e S1(X) the continuousmapping RT applied to the chain d(s X I) in the polyhedron s X I yields a chain in S(X) whichwe shall denote by -IT (this is the chain (s X I, d(s X I), RT) in the "triple" notationof [1, ?15]). Since the standard subdivisionof s X I introducesno new verticesin addition to those of s X 0 and s X 1 and since RT maps all those verticesinto xo it followsthat OJTis a chain in S1(X). Furthermoreit followsfrom (9.5)-(9.8) and fromthe properties of d(s X I) [1, ?16] that 0T =iKT - T - T. This proves (9.4) and completesthe proofof TheoremIa. 10. Proofof TheoremII We now proceedto examinethe case when

(10.1) 7rn(X) = 0 for 1 < n < r.

If we examinethe constructionof - in the previousparagraph we noticethat in orderto define;A for a k-dimensionalcell of K(7r,(X)) we used the fact that 7rk4(X) O. Consequentlythe definitionof K can be duplicated for all cells A of dimension ? r. We summarizethe propertiesof - which will be used: (10.2) i = d3,

(10.3) KK = 1. (10.4) If A is an (r + 1)-cell of K(7r,(x)) theni(6A) is a spherical cycle in S1(X). Next we shall examinethe definitionof the operator0 in the last part of the precedingsection. In definingthe continuousmappings RT we have used the fact that 7rk(X)= 0 wherek is the dimensionof the singularsimplex T. Con-

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 498 SAMUEL EILENBERG AND SAUNDERS MACLANE sequentlythe definitionof RT and thereforealso that of 9DTcarries over without change,provided that the singularsimplex T has dimension

(10.5) agT = RKT- T- aT dim T < r. If dim T = r, then 9iT is not defined,but the definitionof ?IXT readilyimplies that (10.6) If T is a singularr-simplex of S1(X), thenthe chain

KKT - T -Z)aT is a sphericalr-cycle in S1(X). These factswill be used in the proofof the followingtheorem. THEOREM Iha. If 7r,(X) = 0 for 1 < n < r thenthe chain transformation

K:AS(X) -* K(ir(X)) inducesthe following isomorphisms

(10.7) K: Hn(S1(X) G-U) H n(7rl(X),G) forn < r, (10.8) K*:Hn(7rl(X), G) Hn(Sl(X), G) for n < r,

(10.9) K:Hr(Si(X), G)/ r(S1(X), G) +-+ Hr(7rl(X), G),

(10.10) K*:Hr(lrl(X), G) +-+ Ar(Sl(X)X G).

If X is arcwiseconnected, these isomorphismscombined with the isomorphisms induced by the chain equivalence qj: S1(X) -* S(X) produce the isomorphisms requiredfor Theorem II. PROOF OF THEOREM Ila. We shall denote by

[S1(X)]r and [K(7r (X))]r the subcomplexesof S1(X) and K(7ri(X)) whichconsist of all cells of dimension

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 499

(10.11) K:H r(Sl) (X) I G) -*H'(7r,(X), G) is a homomorphismonto. In orderto completethe proofof (10.9) we mustshow that the kernelof the homomorphism(10.11) is the subgroup 2r(S1(X), G) of Hr(S(X), G). We have alreadyseen fromLemma 8.2 that the group 2r(S1(X), G) is containedin the kernel. Now let z' be a cycle in SI (X) with coefficients in G such that KZT- 0. Let C =EgijAs?l be an (r + 1)-chainin K(ir1(X)) such that

dOr+l = KZr and consider the r-chain in Si(X):

KKZ = K(aCr+l) = kxgjaAr+' = 2K(aAr+l)

Accordingto (10.4), KKZT is a sphericalcycle in S1(X). From (10.6) we deduce that KKZ _ Zr is a sphericalcycle. ConsequentlyzT is a sphericalcycle in S1(X). This concludes the proof of (10.9). Before we proceed with the proofof (10.10) we should recall the definition of K*. Given any cochain f in the complex K(7r,(X)) we can definea cochain K*f in S1X) by setting

(K*f)(T) = f(KT). This maps the cohomologygroups of K(7r,(X)) homomorphicallyinto those of S1(X). In particular,we have

(10.12) K*:Hr(K(7rl(X)), G) -* Hr(Sl(X), G). We shall show firstthat the kernelof this homomorphismis zero. To this end, let f be an r-cocylein K(7r,(X), G) such that

K*f = 6h, whereh is an (r - 1)-cochainin S1(X). Define

h' (r1) = h(,Ar-1) for every (r - 1)-cell Ar-1 of K(7r,(X)). Clearly h' is an (r - 1)-cochain in K(iri(X)). We have

(bh') (Ah((aAr) = = h(kaAr) = h(aRAr)

(6h)(RAr) = (K*f) (,Ar) = f(KKAT) = f(Ar) since by (10.3) KAT = Ar. Consquently6h' = f, which proves that the kernel of (10.12) is zero. We have shown already (Lemma 8.3) that the homomorphism(10.12) maps Hr(K(7rT(X)),G) into the subgroup A,(S1(X), G). In order to complete the

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 500 SAMUEL EILENBERG AND SAUNDERS MACLANE proofof (10.10) it is sufficienttherefore to show that K* maps H,(K(iri(X)), G) onto A,(Sj(X), G). Let f be a cocyclebelonging to a cohomologyclass in Ar(Si(X), G). Definea cochainf' of K(7r1(X)) by setting

ft(Ar) = Aid) for Ar in K(7r1(X)). We have

(Sf,)(Ar+l)_f'(OaT+) = f(Oa'Ar+l) = 0 since by (10.4) aOAr+lis a sphericalcycle in S1(X) and sincef annihilatesall the sphericalcycles by the definitionof the subgroupAr It followsthat f' is a cocycle in K(iri(X)). We can furtherdefine an (r - 1)-cochainh in S1(X) by setting h(Trl) = f(OTrl)

Accordingto (10.6) KKTr - Tr- OaT7 is a sphericalcycle in S1(X), and sincef belongsto a cohomologyclass in Ar , f annihilatesall the sphericalcycles, consequently

f(RKTr - Tr _ @aTr) = 0. WVenow compute the coboundaryAh of h as (6h)(Tr) = h(OTr) = f(a T r)

= f(KKTr) - f(Tr) = f'(KTr) - f(Tr)

= (K*f')(Tr) - f(Tr) and therefore

Ah= Kefy - f This shows that K*f' and f are cohomologous,and thereforethe cohomology class of f is the image under K* of the cohomologyclass of f'. This concludes the proofof Theorem Ha. 11. Discussion of naturality In the case whenX = P is a locally finiteconnected polyhedron P, given in a definitesimnplicial decomposition, it is usefulto have an explicitconnection between the complex K(wri(P)) and the homologygroups of P definedfrom the simplicialdecomposition. In [1, Ch. II] the complexesk(P) and K(P) were defined;k(P) was the abstract complexobtained fromP by assigningan orientationto any simplex of P, whileK(P) was the much larger abstractcomplex in whichevery simplex of P is taken with a definiteorder of verticesand two differentorderings of vertices lead to two distinctcells in K(P). The two complexesare comparedby means of chain transformations a:K(P) -+ k(P) &:k(P) -K(P)

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 501 whichform an equivalence pair. The chain transformationa was definedin a natural and, in a sense, in a unique fashion,while the definitionof a involved an elementof choice and was unique onlyto withina chainhomotopy. Next we shall considerthe singular complex S(P) and the natural chain transformation 3:K(P) -S(P), whichwas provedin [1, Chap IV] to be a chain equivalence. Furtherwe have the subcomplexS1(P) of S(P) and the chain transformations

771:Si -* 8(P) -1:S(P) -* Si(P) whichform an equivalencepair. Again,as in the case of (a, a) thetransforma- tion ql is natural,while the definitionof Xlinvolves some choicesand is unique only to withina chain homotopy. Finally, we have the natural chain transformation K:Si(P) - K(-i(P)). All told we have the followingnatural chain transformations k(P) - K(P) -* S(P) S1(P) K(7r,(P)). This diagramshows that the naturalchain transformationson hand do not per- mit the comparisonof the complexesk(P) and K(7r1(X)) directly. However,if we use a and t1, we have

k(P) K(1P) -e S(P) aj S1(P) K(7r1(P)), so that we get a chain transformation - (11 .1) #cjj,: k(P) K(7r,(P)) . We shall shownow how the transformation(11.1) can be obtaineddirectly without the use of the complexesK(P), S(P), and S1(P). The direct definitionof (11.1) will involve two choices: 10 We choose a definiteorder of the verticesof the polyhedronP. 2? For each vertexv of P we choose a definitepath xovleading from the base point x0oP to v. Given a simplexs in P with the verticesvo < v, < * < v,n,we denoteby dij the elementof 7ri(P)determined by the closed path

(Xo0v)(viv j) (XOVj) wherevivi denotes the rectilinearpath fromvi to vi. It is apparent that the matrixA = 11dij 11is an n-cellin the complexK(ir1(P)). By setting0(s) = A, we can get the requiredchain transformation: O:k(P) -- K(7r1(P)).

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 502 SAMUEL EILENBERG AND SAUNDERS MACLANE

An analysis of the definitionsof x and nj shows readily that a and i can be chosen so as to have

a= Kjfla This impliesthat 0 is unique to withina chain homotopy,a factwhich could be proved directly. The conceptof naturalityis used herein a vague sense,which, however, could be made quite rigoroususing the theoryof functors[4].

CHAPTER III PRODUCTS

12. Productsin K(H) Let Fo be the 0-dimensionalcochain in K(H) withintegral coefficients defined as

Fo(x) = 1 forall X E H. Since, as we have remarkedin ?4, every 0-cochainis a cocycle, it followsthat Fo is a cocycle. Consequently,in the terminologyof [1, ?24], the complex 17(11)is augmentable. We shall follownow the pattern of [1, Ch. V] and defineproducts in the complex K(fl). Let two homogeneouscochains

Fp E Cp(H, G1), Fq E Cq(H, G2) be given with the coefficientgroups G1 and G2 paired to a thirdgroup G. We define

(12.1) (Fp U Fq)j(Xo, * , Xp+q) = Fp(xo , * *, xp)Fq(Xp X *, Xp+q). We verifythat this definitiongives a homogeneouscochain Fp U Fq E Cp+q(H, G), and that the cup product so definedsatisfies axioms (Ul) -(U5) of [1, ?25]. This establishesK(Hl) as a complexwithyproducts. Let fp, f, , and fp U fq be the non homogeneouscochains correspondingto the homogeneouscochains Fp , Fq, and Fp U Fq. Accordingto (3.4) we have fp U fq(Xl *, Xp+q) = Fp U Fq(l, X1, X1X2, , X1 Xp+q)

= Fp(1, x1, ** , X1 . Xp)Fq(Xl .. Xp, **, xl ** Xp+q)

= FP(1, x1, *** , X1 ... xp)Fq(l, Xp+l, Xp+lXp+2, *** , Xp+1 ... Xp+q)

= fp(xl, ***, Xp)fq(Xp+l, ***, Xp+q). Hence we have the formula

(12.2) fp U fq(Xl, * * * , Xp+q) = fp(Xl, , Xp)fq(Xp+l, , Xp+q) forthe cup productin the non-homogeneousdescription.

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 503

We followwith the matricdescription. Let a (p + q)-cell of K(fl) be given as a (p + q + 1) X (p + q + 1) matrix A = II dj. We shall denoteby pAthe matrixobtained from A by erasingthe last q rowvsand columnsof A, and by Aq the matrixobtained by erasingthe firstp rowvsand columns of A. Accordingto (3.5) we have

A (do,) d12, dp+q-l,p+q)

-* , (do,) d2 ., dp-,p) Aq (dp,p+?,dp+l?,+2, * * * , dp+qi1,p+q). Consequently(12.2) impliesthe followingdefinition of the cup productin the matricdescription (12.3) fp U fq(A) = fp(pA)fq(Aq) The definitionof a cup productcarries with it [1, ?26] a correspondingdefini- tion forthe cap product fqnCfP+q e CP(H,G) of a cochainfq E Cq(H,GD) and a chain cP+qe C"P?(f, GO, providedthat G1 and G2 aie suitably paired to G. The cap productcorresponding to our definition of the cup productin K(H) can be describedas follows. Let cp+q = 9A, where g e G2, then fq nfcP+q= [f (Aq)g]pA. 13. Productsin S(X) and S1(X) We shall brieflyreview the definitionof the cup product in the complexes S(X) and S1(X), as introducedin [1]. Let

T:s -> X wheres = < po, X... pp+q > be a singularsimplex in X of dimensionp + q. We shall considerthe faces

Ps = < Po, ***, pp > Sq = < Pp,).. * pp+q > of s and the correspondingsingular simplexes defined by the partial mappings PT- TIps Tq= Tjsq. For two cochains

fp e Cp(X, G1), fq e Cq(X, G2) whereG1 and G2 are paired to G, the cup product

fp'U fq e Cp+q(X, G) is definedby the formula

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 504 SAMUEL EILENBERG AND SAUNDERS MACLANE

(13.1) (fpU fq)(T) = fp(pT)fq(Tq). The correspondingcap productcan be definedas follows. Let

fq E Cq(X, G1), cP+q e Cp+q(X G2),

whereG1 and G2are pairedto G. Furtherlet CP+q be ofthe formgT, whereg EG2 then

fq n Cp+q = [fq(Tq)g]pT. If the simplex T is the subcomplexS1(X), then both PT and Tq are in S1(X). Formula (13.1) can thereforebe used to definethe cup productin the complex S1(X). The chain transformation

771:Sl(X) S(X)

was definedas the identitymapping i71T = T of the complexS1(X) into S(X). Given a cochainf of S(X), the cochain iAif of S1(X) is definedby

(iif)(T) = f(i1T) = f(T). This means that the cochain t~f is obtainedfrom f by cuttingdown the domain of definitionof f to include just simplicesof S1X). This implies that for the cup productswe have

(UpU ffq) = fU ql fq

so that in the terminologyof [1, ?27], the chain transformation77 preservesthe products. Since 77, is a chain equivalence (when X is arcwise connected),it followsthat the parings which the cup and the cap products induce on the homologyand cohomologyclasses of S1(X) will be isomorphicwith the corre- spondingpairings in S(X).

14. Reduced products It was shown in [1, ?35] that if G1 and G2 are paired to G, and if p > 0 and q > 0, then for any two cohomologyclasses

fp e Hp(X, G1), fq e Hq(X, G2) we have

fp U fq E Ap+q(X,G)

wherethe subgroupAp+q of Hp+q is definedas in ?8 above. Consequently,by a trivial restrictionof the range,we may considerthe product as a pairingof the groups Hp and Hq to the group Ap+q. This pairingwill be called the reduced cup product.

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 505

For the cap productit was shownin [1, ?35) that if p > 0, q > 0, and

fq E Hq(X, G1), zP+ E (X G2), then fq nZP+q = 0, wherethe subgroup Vp+qof HP+q is definedas in ?8 above. Consequentlythe pairing of the groups Hq(X, GI) and HfP+q(X, G2) with values in HP(X, G) induces a pairingof the groups Hq(X, G1) and HP+G(X, Q2)I/ P+d(X, G2) with valules in HP(X, G). This pairing will be called the reduced cap product. The reduced products are defined in the complex S1(X) in an identical fashion. They lead to pairings isomorphic with the corresponding pairings in S(X).

15. Comparisonof productsin X and K (iri(X)) We shall show firstthat the chain transformation

K: Si (X) -+K (7ri(X)) preserves the products. In fact, if K(T) = A, then it is clear fromthe definitions involved that K(JT) = pA and that K(J'q) = A,. Consequently,

K*(fpU fq)(T) = (fpU fq)(KT) =fp U fq(A) = fp(pA)fq(Aq)

= fp(KpT) fq(KTq) K*fp(pT)K*fq(Tq)

= (K*fp U K*fq)(T), hence we get

(15.1) K*CfpU fq) = K*fpU K*fq for any two cochains fp e Cp(iri(X), G1) and fq e Cq(7r1(X), G2), provided that GI and G2 are paired to some group G. This implies that if, for the dimensions p, q, and p + q, the transformation K establishes the isomorphisms

H,(S,(X), G,) _ Hp(7ri(X), G,)

Hq(Sl(X), G2) _ HQ,(,7r(X), G2)

Hp+q(Sl(X), G) -_ Hp+q(7rl(X), G), then for these groups the cup products also will be mapped isomorphically. A similar statement applies to the cap product. Since the products in the complexes S(X) and S1(X) are isomorphic, Theorem Ia implies the following: ADDENDUM TO THEOREM I. If X is aspherical,then the cup and thecap products of thehomology and cohomologyclasses in X are determinedby thefundamental group iri(X). More precisely,the productsin X are isomorphicwith the corre- spondingproducts in thecomplex K(7r,(X)).

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 506 SAMUEL EILENBERG AND SAUNDERS MACLANE

In orderto get a similaraddendum to TheoremII wvenotice that if K induces the isomorphisms

H,(S1(X), Gj) fIp(7ri(X), G0) p > 0 Hq(Si(X), G2) Hq(ri(X), G2) q > 0 Ap+q(Si(X), G) Hp+q(7r(X), G), then the reduced cup product of the groups on the left is isomorphicwith the cup productof the groupson the right. Similarly,if K induces the isomorphisms IIq(Si(X), GU) - Hq(7r,(X),G,) q > 0

Hp+q(S,(X), G2)/1jP+q(S(X), G2) - Hp+'(7r1(X), G2) HP(S,(X), G) - HP(ri(X), G) p > 0, then the reduced cap product of the groups on the left is isomorphicwith the cap product of the groups on the right. Since the products in the complexes S,(X) and S(X) are isomorphic,Theorem Ila impliesthe following: ADDENDUM TO THEOREM II. If

Trn(X) =0 for 1 0, q > 0, and p + q = r, thenthe reduced cup and cap productsin X are isomorphicwith the corre- spondingproducts in the complexK (7ri(X)).

CHAPTER IV. GENERALIZATION TO HIGHER DIMENSIONS 16. Preliminaries In the precedingchapters we have studied the influenceof the fundamental group 7r,(X)of a space upon the homologystructure of X. A similardiscussion can be carriedout, replacingthe group7ri(X) by a higherhomotopy group 7rm(X). The group constructionsinvolved are rather complicated, and consequently the finalresults have less interest. We shall treatthe entiretopic verysketchily; we shall give all the necessarydefinitions and shall state the theorems,but we shall omitthe proofs,since theyare strictlyanalogous to the ones in Chapter II. Let X be a topologicalspace with base point xo . We shall considerthe total singular complex S(X) and the subcomplex Sm(X) (m > 1) consistingof all those singularsimplexes

T = sX s = < Po, Pn > whichmap all faces of s of dimension

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 507

P, thenthe definition4(ao, a,a) = a gives a function 0 of m + 1 variables as = 0.* , n withvalues in 7rm(X). This definesq5 only when a0, a,, ** *, amare distinct;we set

(16.1) c(ao , * * * , am) = 0 if a, X , am are notdistinct. This function0 has the followingimportant property: m+1

(16.2) E-o for any systemof m + 2 numbersai = 0 *.*.*,n. If a0,o , a.1+ are distinct,the verticesPao, .. *, Pa.+i determinean (m + 1)- dimensionalface s' of the simplexs, and (16.2) asserts that the mapping T of the boundaryof s' gives the zero elementof the homotopygroup. If ai = aj fori < j, thenin view of (16.1) formula(16.2) reducesto .. (-)"O(ao, ,A,*di) am+,) + (- 1)'(ao, * * ailj* am-,l) = ?1 whichmeans that the function4 is alternatingin its variables. 17. The complexesK(HI, m) Let II be an abelian group writtenadditively. Guided by the precedingdig- cussionwe shall definean (n, m)-cellof the group II to be a functionof m + 1 variables 4)(ao, **,a) e II wherea = O***, n, subject to the conditions(16.1) and (16.2). Given an (n, m)-cell4 we shall definethe (n- 1, m)-,cell+(i fori-O0 ***, n, as follows

(bo ***, b) =(ao X am) where aj = bj if bj < i aj =1 + b if bj > i. The numberm will be fixedthroughout the discussion. The (n, m)-cellswill be consideredas the n-dimensionalcells of an abstract complex K(H, m) with the boundaryA0 of an (n, m) cell definedas n ao = E(-1)$+(z)) i=O the summationbeing understoodin the freegroup of the (n - 1)-chainsgener- ated by the (n - 1, m)-cells. The fact that 00o = 0 is an easy consequenceof the followingfact [,)(i)]O) = [O(+1)](i) fori < j which followsdirectly from the definitionof +).

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions 508 SAMUEL EILENBERG AND SAUNDERS MACLANE

If m = 1, each (n, m)-cellis a functionof two variables and thereforecan be writtenas a (n + 1) X (n + 1) matrix. It is then obvious that (16.2) reduces to the previous condition (2.1), and that the complex K(H, 1) coincideswith the complexK (11),as definedin Chapter I. We remarkthat whilethe complex K(H) was definedfor all groupsII, the definitionof K(H, m) assumes that H is abelian, because of the additive formof (16.2).

18. The chain transformationsKm Let T be a singularn-simplex in Sm(X). The function4 constructedin ?16 is an (n, m)-cellof 7rm(X)and thereforeis an n-dimensionalcell of the complex K(7rm(X), m). We shall write

KmT = 4).

If T(i) is the face of T oppositethe ith vertex,then it is apparentthat KmT(') = ( .

ConsequentlyKm commuteswith the boundaryoperator, and we have a chain transformation:

Km:Sm, (X) ---K (7m (X) m)X . The chain transformationKm is a generalizationof the chain transformation K: Si(X) -- K(7ri(X)); in fact, if we regardthe complexesK(7r,(X), 1) and K(7r,(X)) as identical,we we have K = Ki. In particular,the proofsof TheoremsIa and Ila carryover withoutessential changes,and lead to the followingtheorems: THEOREM Ia. If 7r,(X) = 0 for all n > m, then the chain transformation

Km Sm(X) -K (rm (X), m) is a chain equivalence. THEOREM IIa. If 7r(X) = O form < n < r, then the chain transformation

Km Sm(X) - K(7rm(X), m) = K induces the following isomorphisms:

Km:Hn(Sm(X), G) ?-+ Hn(K, G) for n < r Kr:Hn(K, G) H1,(Si(X), G) for n < r Km:H'(Sm(X), G)/Tr(Sm(X), G) +-+ Hr(K, G)

K :Hr(K, G) +-+ Ar(Sm(X), G). The subgroups 2rand Ar are definedin ?6. Let us now assume that X is arcwise connected,and that 7rn(X)= 0 forall n < m. It was shownin [1, ?31] that the identitychain transformation

*. _qnSmX SY > (X)

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions RELATIONS BETWEEN HOMOLOGY AND HOMOTOPY 509 is a chain equivalence. Consequently-qm induces isomorphisms of the homology and cohomologygroups of Sm(X) and S(X). Under these isomorphismsthe groups 2(S.m(X), G) and 2;(X, G) correspondto one another. Similarly,the groups Ar(Sm(X), G) and AT(X, G) correspond. Theorems I' and IIa there- forelead to the followingtheorems: THEOREM tm. If X is arcwiseconnected and

irn(X) = 0 forn < mandm < n, thenthe homology and cohomologygroups of X are determinedby the homotopy group rm(X). More precisely, Hn(X, G) - Hn(K, G),

Hn(X7 G) Hn(K7 G) whereK = K(7rm(X), m). THEOREM 1rn. If X is arcwiseconnected and

7rn(X)=0 forn < m and m < n < r then

Hn(X G) ? Hn(K G) forn < X

H,(X, G) Hn(K, G) forn < i

HT(X, G)/2r(X, G) - HT(K G)

Ar(X, G) _ Hr(K, G) whereK = K(7rm(X), m). The resultsof Chapter III are subject to a similargeneralization.

BIBLIOGRAPHY

[11 EILENBERG, S., Singular homologytheory, Ann. of Math. 45 (1944), pp. 407-447. [21EILENBERG, S., AND MACLANE, S., Group extensions and homology,Ann. of Math. 43 (1942), pp. 757-831. [31 , Relations betweenhomology and homotopygroups, Proc. Nat. Acad. Sc. U. S. A. 29 (1943), pp. 155-158. [41 , General theory of natural equivalences, Trans. Amer. Math. Soc. 58 (1945), pp. 231-294. [51 HOPF, H., Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1942), pp. 257-309. [61- -, Nachtrag zu der Arbeit "Fundamentalgruppe und zweite Bettische Gruppe," Comment. Math. Helv. 15 (1942), pp. 27-32. [71.2.3. 4] HUREWICZ, W., Beitrage zur Topologie der Deformationen,Proc. Akad. Amsterdam, 38 (1935), pp. 112-119 and 521-528; 39 (1936), pp. 117-125 and pp. 215-224. [81 LEFSCHETZ, S., Algebraic Topology, Amer. Math. Soc. Colloquium Series, vol. 27, New York, 1942.

THE UNIVERSITY OF MICHIGAN HARVARD UNIVERSITY

This content downloaded on Sat, 23 Feb 2013 12:14:35 PM All use subject to JSTOR Terms and Conditions