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Relations Between Homology and Homotopy Groups of Spaces Annals of Mathematics 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 funda- mental 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 com- mutatorsubgroup. 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 con- nected 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 con- tinuous 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 mul- tiplication. 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).
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