Why is the cohomology of a $K(G,1)$ group cohomology?

Question

Let $G$ be a (finite?) group. By definition, the Eilenberg-MacLane space $K(G,1)$ is a CW complex such that $\pi_1(K(G,1)) = G$ while the higher homotopy groups are zero. One can consider the singular cohomology of $K(G,1)$, and it is a theorem that this is isomorphic to the group cohomologies $H^*(G, \mathbb{Z})$. According to one of my teachers, this can be proved by an explicit construction of $K(G, 1)$.

On the other hand, it seems like there ought to be a categorical argument. $K(G, 1)$ is the object that represents the functor $X \to H^1(X, \mathbb{Z})$ in the category of pointed CW complexes, say, while the group cohomology consists of the universal $\delta$-functor that begins with $M \to M^G$ for $M$ a $G$-module. In particular, I would be interested in a deeper explanation of this "coincidence" that singular cohomology on this universal object happens to equal group cohomology.

Is there one?

Answer 1

Akhil, you're thinking of this the opposite of how I think group cohomology was discovered. The concept of group cohomology originally centered around the questions about the (co)homology of $K(\pi,1)$-spaces, by people like Hopf (he called them aspherical rather than $K(\pi,1)$ spaces, and Hopf preferred homology to cohomology at that point). I think the story went that Hopf observed his formula for $H_2$ of a $K(\pi,1)$, which was a description of $H_2$ entirely in terms of the fundamental group of the space.

This motivated people to ask to what extent (co)homology is an invariant of the fundamental group of a $K(\pi,1)$-space. This was resolved by Eilenberg and Maclane. Eilenberg and Maclane went the extra step to show that one can define cohomology of a group directly in terms of a group via what nowadays would be called a "bar construction" (ie skipping the construction of the associated $K(\pi,1)$-space). Bar constructions exist topologically and algebraically and they all have a similar feel to them. On the level of spaces, bar constructions are ways of constructing classifying spaces. For groups they construct the cohomology groups of a group. The latter follows from the former -- if you're comfortable with the concept of the "nerve of a category", this is how you construct an associated simplicial complex to a group (a group being a category with one object). The simplicial (co)homology of this object is your group (co)homology.

Dieudonne's "History of Algebraic and Differential Topology" covers this in sections V.1.D and V.3.B. I don't think that answers all your questions but it answers some.

Answer 2

Well, you'll probably want a more conceptual proof, but one thing you can do is check they are computed by the same chain complex: for $K(G,1)$ take the simplicial construction of the classifying space $BG$ and compute its cohomology in the usual way for simplicial sets (using the dual to the complex of formal linear combinations of simplices); for the group cohomology take the free resolution of the trivial $\mathbb{Z}[G]$-module $\mathbb{Z}$ where the nth term is $\mathbb{Z}[G^n]$, which amounts to the same complex as before.

Answer 3

This is really a comment on ryan'sanswer:

I have to disagree with Ryan. Group cohomology was in its early stages before Eilenberg and Maclane came along. There are awful and ugly formulations of just $H^1$ and $H^2$ that lead me to believe that they must have been formulated before E&M did their work. I am thinking of factor sets and cocycle conditions that people wrote down a long time ago. I think this is a matter of history though.

Edit: I just checked wikipedia http://en.wikipedia.org/wiki/Group_cohomology the history section near the end. It seems to agree with what I said above.