Let $X$ be a finite CW-complex of $n$.


1. For $i\geq 2$, $\pi\_i (X)$ is a $\mathbb{Z}\pi\_1 (X)$-module.
2. for $i\geq 2$, $H\_i (\tilde{X})$ is a finitely generated $\mathbb{Z}\pi\_1 (X)$-module, where $\tilde{X}$ is the universal covering of $X$.


I know that the connection between homotopy groups and homology groups is given by the Hurewicz-Theorem. But my question is that:


**My question**: Can we express $\pi\_i (X)$ (as a $\mathbb{Z}\pi\_1 (X)$-module) in terms of $H\_i (X)$'s (as $\mathbb{Z}$-module) or $H\_i (\tilde{X})$ (as $\mathbb{Z}\pi\_1 (X)$-module) for $2\leq i\leq n$? (particularly when they are free).



Relation between $\mathbb{Z}\pi_1 (X)$-module $\pi_n (X)$ and $\mathbb{Z}$-module $H_n (X)$ or $\mathbb{Z}\pi_1 (X)$-module $H_n (\tilde{X})$
Not sure if you're already aware of an exact sequence which implies Hurewicz's theorem, but if $X$ is a space with $\pi\_i(X)=0$ for $1<i<n$ then we have
$$H\_{n+1}(X)\to H\_{n+1}(\pi)\to \pi\_n(X)\_{\pi}\to H\_n(X)\stackrel{\psi}{\to} H\_n(\pi)\to0$$ where $\pi=\pi\_1(X)$ is the fundamental group and $H\_\*(\pi)$ is discrete group cohomology, and $\pi\_n(X)\_{\pi}$ denotes the coinvariants. See e.g. Ken Brown "Cohomology of Groups" exercise VII.7.6.


Generally, without assumptions on the homotopy groups, that map $\psi$ exists (in all degrees) and is canonical: You take a projective resolution of $\mathbb Z$ over $\mathbb Z\pi$, and you take the complex of free $\mathbb Z\pi$-modules $C\_\*(\text{universal cover of }X)$, to build a canonical chain map between those resolutions that induces $\psi$. This is also explained in the above reference somewhere in Chapter II.


