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Let suppose that we are given a connected CW-complex $X$, such that we know

1. All its homology groups.
2. All its homotopy groups, in particular we know $\pi_{1}(X)$.

As far as I know there is no spectral sequence converging to the homology of the universal covering $\tilde{X}$ of $X$. Why it is so difficult (I guess there is no a clear method in general) to compute the homology groups of $\tilde{X}$ ?

Let suppose that we are given a connected CW-complex $X$, such that we know

1. All its homology groups.
2. All its homotopy groups, in particular we know $\pi_{1}(X)$.

As far as I know there is no spectral sequence converging to the homology of the universal covering $\tilde{X}$ of $X$. Why it is so difficult (I guess there is no a clear method in general) to compute the homology groups of $\tilde{X}$ ?

Edit: I would like to thank all the authors for their answers, I did learn a lot. I had to choose one answer. I am aware that my question was vague enough, but at the and it seems that the answer I was looking for corresponds more to the one given by M. Rivera.

Let suppose that we are given a connected CW-complex $X$, such that we know

1. All its homology groups.
2. we now all its homotopy groups, in particular we know $\pi_{1}(X)$.

As far as I know there is no spectral sequence converging to the homology of the universal covering $\tilde{X}$ of $X$. Why it is so difficult (I guess there is no a clear method in general) to compute the homology groups of $\tilde{X}$ ?

Let suppose that we are given a connected CW-complex $X$, such that we know

1. All its homology groups.
2. All its homotopy groups, in particular we know $\pi_{1}(X)$.

As far as I know there is no spectral sequence converging to the homology of the universal covering $\tilde{X}$ of $X$. Why it is so difficult (I guess there is no a clear method in general) to compute the homology groups of $\tilde{X}$ ?

Let suppose that we are given a connected CW-complex $X$, such that we know

1. All its homology groups.
2. we now all its homotopy groups, in particular we know $\pi_{1}(X)$.

As far as I know there is no spectral sequence converging to the homology of the universal covering $\tilde{X}$ of $X$. Why it is so difficult (I guess there is no a clear method in general) to compute the homology groups de $\tilde{X}$ ?

Let suppose that we are given a connected CW-complex $X$, such that we know

1. All its homology groups.
2. we now all its homotopy groups, in particular we know $\pi_{1}(X)$.

As far as I know there is no spectral sequence converging to the homology of the universal covering $\tilde{X}$ of $X$. Why it is so difficult (I guess there is no a clear method in general) to compute the homology groups of $\tilde{X}$ ?