# Homology with local systems

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.