(Co)homology of the Eilenberg-MacLane spaces K(G,n)

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.

• 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
• 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
• 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
• 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

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
• @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
• 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:

http://doc.rero.ch/record/482/files/Clement_these.pdf

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
• 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.