# Homology of $K(\mathbb{Z}/n,1)$ via Fibrations.

I would like to calculate the $$\mathbb{Z}$$-Homology of $$K(\mathbb{Z}/n,1)$$ via the Fibration $$K(\mathbb{Z},1)\hookrightarrow K(\mathbb{Z},1)\twoheadrightarrow K(\mathbb{Z}/n,1)$$ and the Serre-Spectral-Sequence.

If we knew, that the local coeffitient system in $$H_p(K(\mathbb{Z}/n,1);H_q(K(\mathbb{Z},1)))$$ was trivial, then we could easily apply the Serre-Spectral sequence and derive $$H_p(K(\mathbb{Z}/n,1);\mathbb{Z})=\begin{cases}\mathbb{Z}&p=0\\ \mathbb{Z}/n &p=2k+1 \\ 0& else\end{cases}$$ The Problem here is obviously the non-trivial fundamental group of $$K(\mathbb{Z}/n,1)$$. This Problem is easily resolved for odd $$n$$ as there is no non-trivial morphism $$\mathbb{Z}/n\rightarrow Aut(\mathbb{Z})=\lbrace\pm 1\rbrace$$ for odd $$n$$. For even $$n$$ there is a non-trivial morphism, so we can not proceed using only formal arguments. Using Künneth one can assume $$n=2^k$$ if one wishes, but this also does not solve the real problem.

Is there some way to deduce the triviality of the local coeffitients (and are they trivial at all)?

## 1 Answer

Let's use the definition of $$BG$$ from Hatcher, built out of simplicies $$[g_0,\ldots, g_n]$$ whose vertices are in $$G$$, glued together along faces $$[g_0,\ldots, \hat g_i,\ldots, g_n]$$, and taken modulo the group action $$g[g_0,\ldots, g_n] = [gg_0,\ldots, gg_n]$$.

In order to compute the monodromy action on the fibers, we should find explicit paths in $$BG$$ representing the classes of $$\pi_1(G)$$. For $$g\in G$$, there is the linear path $$\gamma: I \to EG$$ along the 1-simplex $$[1,g]$$, whose projection in $$BG$$ is the loop in $$\pi_1BG$$.

Now, if we have a surjection of groups $$H \to G$$ and we wish to understand the monodromy of the fibration $$BH \to BG$$, then we need to lift the loop $$[1,g] \in \pi_1(BG)$$ to a path in $$BH$$. But it is easy to make this lift, simply take $$\tilde g \in H$$ and use $$[1,\tilde g]$$ which is a loop in $$BH$$. Since it's a loop, the induced action on the fiber is trivial.

The action on the homology groups is induced by the monodromy action, so the corresponding local coefficient system is trivial too.