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?
Best Answer
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.