Let $X$ be a connected topological space with abelian fundamental group. Let $\mathcal{L}$ be a $\mathbb{Z}$-valued local system on $X$.

Suppose that I know the full homology $H_*(X;\mathbb{Z})$. Are there any tools which could allow me to compute (some part of) the local-coefficient homology $H_*(X; \mathcal{L})$? For example, if I know that the rank of $H_*(X;\mathbb{Z})$ becomes unbounded as a function of the degree, can I conclude the same about $H_*(X; \mathcal{L})$?


One approach is to use mod $2$ homology. You know that

$H_i(X;\mathbb Z/2)$ is isomorphic to both $ H_i(X)\otimes \mathbb Z/2\oplus Tor(H_i(X),\mathbb Z/2)$ and $H_i(X;\mathcal L:)\otimes \mathbb Z/2\oplus Tor(H_i(X;\mathcal L),\mathbb Z/2)$. If the integral homology groups are finitely generated, then this gives you what you want.

But if $2$ is invertible in the integral homology then I don't think there's much you can say. $X$ has a $2$-sheeted covering space $\tilde X$, and if $H_i(\tilde X)$ is a $\mathbb Z[1/2]$-module then it splits as a direct sum of the $+1$ and $-1$ ``eigenspaces'' for the action of the covering transformation, which are then $H_i(X)$ and $H_i(X;\mathcal L)$. These two parts need have nothing to do with each other.

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