I'm interested in the calculation of $H_n (G,\mathbb{Z})$ for G a finitely generated abelian group, particularly for $n=3$. It's carried out in the 1954/55 Séminaire Henri Cartan, titled "Algèbres d'Eilenberg-MacLane et homotopie". It does everything I need to do very nicely, but it's old, it's in French, it's original research and so not organized in a pedagogical way; and I really wish I had a reference in English which did the same thing in modern language.

Is there a textbook reference, or at least a more modern reference, for the calculation of the integral homology of a finitely generated abelian group?

EDIT: I'm interested primarily in a reference for the ring structure- I want to be able to write the homology in terms of exterior algebras, divided polynomial algebras... that sort of thing.

  • $\begingroup$ You are looking the the ring structure you get by applying the functor $H_\bullet(\mathord-,\mathbb Z)$ to the addition map $G\times G\to G$, I guess? $\endgroup$ Dec 26, 2010 at 14:16
  • $\begingroup$ Yes. The addition map induces the Pontryagin product on homology, making $H_\ast(-,\mathbb{Z})$ into a graded ring. I want to know what that graded ring is explicitly. $\endgroup$ Dec 26, 2010 at 14:19

2 Answers 2


You can try with Brown's book

Cohomology of groups,

Chapter V, Section 6, p. 121 ("Application: calculation of the homology of an abelian group").

Maybe take also a look at the following papers by Baumslag, Dyer and Groves:

The integral homology of finitely presented metabelian groups I

Amer. Journal of Math. 104 (1982), 173-182

The integral homology of finitely presented metabelian groups II

Amer. Journal of Math. 109 (1987), 133-156

and at the references therin.

EDIT. J. Schafer's thesis

J. Schafer, On the homology ring of an abelian group, Dissertation, University of Chicago, Chicago, Ill., 1965

seems strictly related to what you are looking for. However, I could not find any published paper with this title. The only Shafer's paper related to homology of abelian groups seems to be

J. Schafer: Abelian groups with a vanishing homology group

Canad. J. Math. 21(1969), 406-409.

  • $\begingroup$ Brown was where I began. But he doesn't do the integral homology, saying only that "the situation is more complicated" and refering the reader to Cartan. I glanced through the other two papers, and didn't find a calculation of $H_n(A,Z)$ or a reference to one. They seem to treat it basically as a parameter, and to work in terms of it. If I'm overlooking something, could you give me a page reference for where they make such a calculation? $\endgroup$ Dec 26, 2010 at 14:01
  • $\begingroup$ I've looked at these papers some years ago. I remember that they consider the structure of the whole homology sequence $H_1(G,\mathbb{Z})$, $H_2(X, \mathbb{Z})$, $H_3(X,\mathbb{Z})$, $\ldots$, but I cannot remember whether they contain any explicit calculation of a single homology group. Unfortunately, at the moment I cannot access to JSTOR to check... $\endgroup$ Dec 26, 2010 at 15:09
  • $\begingroup$ I added a further reference. Schafer's thesis seems strictly relate to what you are looking for, unfortunately it seems that it was never published (at least with this title) $\endgroup$ Dec 26, 2010 at 15:22
  • $\begingroup$ Hmmm... I wonder whether it's possible to get hold of Schafer's thesis. Thanks! $\endgroup$ Dec 26, 2010 at 16:11

I don't know exactly what information you need, but if that includes the actual calculation of the homology:

Look at Hilton and Stammbach's A Course in Homological Algebra. In Theorem 15.2 they prove Künneth's theorm for homology of groups with coefficients in $\mathbb Z$ which reduces the computation of the homology of a direct product of groups to the computation of the homology of the factors and certain $\mathrm{Tor}$s between them. Using the structure theorem for finitely generated abelian groups, the computation of homology for cyclic groups (Proposition VI.7.1 for the finite ones, section VI.4 and corollary VI.5.6 for the infinite ones), and straightforward computations of $\mathrm{Tor}$s between cyclic abelian groups, you are done.

  • $\begingroup$ Thanks! I'm in the position of having to give the maximally annoying comment of "you answered what I asked but not what I meant." I apologise! I didn't write the question very well (I will now edit the question)... I suppose I'm primarily after the ring structure. How do I calculate that? (a modern reference is a good answer as well.) $\endgroup$ Dec 26, 2010 at 14:14

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