Let $X$ be a finite CW-complex of $n$. We can consider $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module ($i\geq 2$). Moreover, if $\tilde{X}$ is the universal covering of $X$, then $H_i (\tilde{X})$ is a finitely generated $\mathbb{Z}\pi_1 (X)$-module for $i\geq 2$.

My question: is that is there relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module or $H_i (\tilde{X})$ as $\mathbb{Z}\pi_1 (X)$-module for $2\leq i\leq n$?

I know that the connection between homotopy groups and homology groups is given by the Hurewicz-Theorem. But I want a relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module or  $H_i (\tilde{X})$ as $\mathbb{Z}\pi_1 (X)$-module particularly when they are free for $2\leq i\leq n$.

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).

Let $X$ be a finite CW-complex of $n$. We can consider $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module ($i\geq 2$). My question is that is there any direct relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module for $2\leq i\leq n$?

I know that the connection between homotopy groups and homology groups is given by the Hurewicz-Theorem. But I want a relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module particularly when they are free for $2\leq i\leq n$.

Let $X$ be a finite CW-complex of $n$. We can consider $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module ($i\geq 2$). Moreover, if $\tilde{X}$ is the universal covering of $X$, then $H_i (\tilde{X})$ is a finitely generated $\mathbb{Z}\pi_1 (X)$-module for $i\geq 2$.

My question: is that is there relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module or $H_i (\tilde{X})$ as $\mathbb{Z}\pi_1 (X)$-module for $2\leq i\leq n$?

I know that the connection between homotopy groups and homology groups is given by the Hurewicz-Theorem. But I want a relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module or $H_i (\tilde{X})$ as $\mathbb{Z}\pi_1 (X)$-module particularly when they are free for $2\leq i\leq n$.

Let $X$ be a finite CW-complex of finite dimension. We can consider $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module ($i\geq 2$). My question is that is there any direct relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module? For example, if $\pi_i (X)$ is free as $\mathbb{Z}\pi_1 (X)$-module, then $H_i (Z)$ is a free $\mathbb{Z}$-module or conversely?

I know that the connection between homotopy groups and homology groups is given by the Hurewicz-Theorem. But I want a relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module particularly when there are free. Thanks in advance.

Let $X$ be a finite CW-complex of $n$. We can consider $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module ($i\geq 2$). My question is that is there any direct relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module for $2\leq i\leq n$?

I know that the connection between homotopy groups and homology groups is given by the Hurewicz-Theorem. But I want a relation between $\pi_i (X)$'s as $\mathbb{Z}\pi_1 (X)$-module and $H_i (X)$'s as $\mathbb{Z}$-module particularly when they are free for $2\leq i\leq n$.