Let $(G, n)$ be a pair such that $n$ is a natural number, $G$ is a finite group which is abelian if $n \geq 1$. It is well-known that $\pi_n(K(G,n)) = G$ and $\pi_i (K(G,n)) = 0$ if $i \neq n$.

Also it is known that these spaces $K(G,n)$ play a very important role for cohomology. For any abelian group $G$, and any CW-complex $X$, the set $[X, K(G,n)]$ of homotopy classes of maps from $X$ to $K(G,n)$ is in natural bijection with the $n^{\mathrm{th}}$ singular cohomology group $H^n(X; G)$ with coefficients in $G$.

But what is known about the cohomology of the $K(G,n)$ themselves? It is interesting in the light of the above. Here I mean the singular cohomology with integral coefficients.

  • 5
    The (co)homology of $K(G,1)$ is well-known to equal the group (co)homology of $G$ with integer coefficients. I don't know what happens for $n>1$. – Robin Chapman May 15 '10 at 14:23
  • 9
    See an introductory algebraic topology text like Hatcher or May. The (co)homology of Eilenberg-Maclane spaces are heavily studied. In a "stable range" this cohomology is called the Steenrod Algebra. – Ryan Budney May 15 '10 at 14:29
  • 6
    Hatcher's spectral sequence text computes only cohomologies of Eilenberg-MacLane spaces with Z/2-coefficients, but her refers to J. P. May, A general approach to Steenrod operations, Springer Lecture Notes 168 (1970), 153–231 for an integral computation. – Lennart Meier May 15 '10 at 21:18
  • 3
    Another reference for the $\mathbb{F}_p$-homology of the $K(\mathbb{F}_p, n)$ (following Cartan's method) is in the thesis of Alain Prouté: logique.jussieu.fr/~alp/these_A_Proute-TAC.pdf – Bruno Stonek Feb 10 '17 at 13:09
up vote 16 down vote accepted

Computing the integral cohomology of $K(\pi,n)$'s is feasible but a bit tricky. In fact the only reference I know is exposé 11 of H. Cartan's seminar, year 7. I'd be interested if there are other sources that cover that.

  • I just noticed this answer. I asked for other sources as a question: mathoverflow.net/questions/50417/… I'm trying to get hold of Schafer's thesis, to see if it fills this niche. – Daniel Moskovich Jan 11 '11 at 13:37
  • 1
    @algori: Cartan seems to have also written up his calculations in some papers: "Sur les groupes d'Eilenberg-Mac Lane. I,II." (French) Proc. Nat. Acad. Sci. U. S. A. 40, (1954). I haven't looked at them yet, but presumably the presentation is cleaner. – Mark Grant Aug 18 '11 at 14:25
  • The original papers of Eilenberg and MacLane have quite a few computations in them. Not so much of specific groups, but still quite a few things seem to be worked out, if you can get past their notation. – Jonathan Beardsley Jul 10 '13 at 21:12
  • 3
    In a recent paper, Breen, Mikhailov and Touzé gave a functorial description of some integral homology groups of Eilenberg Mac Lane spaces, see Derived functors of the divided power functors, arxiv.org/abs/1312.5676 – ACL Feb 25 '15 at 10:17

You might find some useful information in the Lausanne thesis of Alain Clément:


In particular, he gives an account of Cartan's results in Chapter 2, then describes a C++ program for computing integral (co)homology of certain ($2$-local) Eilenberg-Mac Lane spaces in Chapter 3. An appendix lists the integral (co)homology groups of $K(\mathbb{Z}_2,2)$, $K(\mathbb{Z}_2,3)$, $K(\mathbb{Z}_4,2)$ and $K(\mathbb{Z}_4,3)$ up to degree $200$.

Expanding slightly on Ryan's comment: it's an easy fact (often attributed to Serre) that the set of cohomology operations $H^k(-;G)\to H^r(-;H)$ (i.e. natural transformations) is in 1-1 correspondence with $[K(G,k),K(H,r)]=H^r(K(G,k);H)$ (for any abelian groups $G,H$). There are tons of these, some easy, some not so easy to understand, corresponding to how easy the calculation of $H^r(K(G,k);H)$ is. A nice subset are the stable operations (compatible with a certain suspension $[K(G,k),K(H,r)]\to [K(G,k+1),K(H,r+1)]$ which come in families, the most common family being the Steenrod algebra, corresponding to $G=H=Z/p$. There are non-stable operations also, eg the Pontryagin square $H^k(-;Z/2)\to H^{2k}(-;Z/4)$.

If you don't mind a little notational pain, you can look at the original papers where Eilenberg and Mac Lane worked many cases out: On the homology of groups $H(\pi,n)$ (I, II, III). Annals of Mathematics ~1953.

  • Scott, is the ring structure in the cohomology computed there as well? – algori May 15 '10 at 15:26
  • 1
    I don't have the papers in front of me, but I will tentatively say "no". – S. Carnahan May 15 '10 at 16:13

In Q-subalgebras, Milnor basis, and cohomology of Eilenberg – Mac Lane spaces Tamanoi gives explicit polynomial generators of $H^*(K(\mathbb Z/p^k,n);\mathbb Z/p)$ and $H^*(K(\mathbb Z,n);\mathbb Z/p)$ for all primes $p$, using Milnor basis of the dual Steenrod algebra.

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.