Construction of generalized Eilenberg-MacLane spaces

Question

The Eilenberg-MacLane spaces $K(G,q)$ are readily generalized to study cohomology with local coefficients.The generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ are spaces with only two nnvanishing homotopy groups, one of them the fundamental group is $\pi$, the other is $G$ in dimension $q$. And the fundamental group $\pi$ acts on $G$ by representation $\rho:\pi\rightarrow Aut(G)$, where $Aut(G)$ is the group of automorphisms of $G$.

It has been given the construction of Eilenberg-MacLane space K(G,n) in the category of CW complex in All Hathcer's book 'ALGEBRAIC TOPOLOGY'. In Samuel Gitler's article 'Cohomology Operations with Local Coefficients' . The generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ are constructed in the category of Kan Complex, but I have no idea about Kan complex and I have not learned them before. Is it possible to reference me some books or articles which talk about $K_{\pi}(G,q)$ in the category of CW complex?

And I have some idea about construction in the follow. But I am not sure if I am right.

Let $K(\pi,1)$ be the base space and homomorphism $\rho:\pi\rightarrow Aut(G)$ be the representation. The homomorphism $\rho$ can induce representation $\rho_{*}:\pi\rightarrow Homeo(K(G,q);K(G,q))$, where $Homeo(K(G,q);K(G,q))$ is the group of homeomorphisms of $K(G,q)$ itself. Then we can construct a fiber bundle $K(G,q)\rightarrow E\rightarrow K(\pi,1)$ by the representation $\rho_{*}$.

I want know if the total space $E$ is the generalized Eilenberg-MacLane space $K_{\pi}(G,q)$ we need?

Answer 1

In some sense, you are right, but as noted in the comments you need to be a bit careful. I am not really sure about references, but I'll reformulate Gitler's construction for you:

Take $G$ a group. The first thing is to take the classifying space $K(G,q)$. This can be done functorially, see this MO-discussion. (Unfortunately, the best way to do this is via realization from simplicial sets - I do not know a nice way which does everything strictly with CW-complexes only.) For $\pi$ acting on $G$, this provides a representation of $\pi$ on $K(G,q)$ (or better, as noted in Qiaochu Yuan's comment - on the model of $K(G,q)$ as realization of some explicitly given simplicial set). The generalized Eilenberg-Mac Lane-space $L_{\pi}(G,q)$ is then defined as the Borel construction $E\pi\stackrel{\pi}{\times}K(G,q)$, where $E\pi$ is now any contractible CW-complex with a free action of $\pi$, and $\stackrel{\pi}{\times}$ means taking the product and dividing out the diagonal $\pi$-action. This is the construction in Gitler's paper.

It is then a general fact in equivariant cohomology, associated fiber bundles etc. that there is a fiber bundle $K(G,q)\to L_{\pi}(G,q)\to K(\pi,1)$ as written in the question.