Let $X$ be a finite connected pointed CW-complex and $H_{\ast}(\Omega X)$ the integral homology of the loop space on $X$. Are the homology groups $H_{n}(\Omega X)$ finitely generated abelian groups for any $n$ ?

If the answer is negative, what are the sufficient conditions to impose on $\pi_{1}(X)$ such that the homology groups $H_{n}(\Omega X)$ turns out to be finitely generated ?

My goal is to collect different sufficient conditions on the fundamental group for which a positive answer holds.

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    $\begingroup$ I think the serre spectral sequence gives this immediately when X is simply connected $\endgroup$ – Thomas Rot Sep 4 at 13:16
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    $\begingroup$ I guess if $X = S^1$ this is false, so definitely some $\pi_1$ conditions need to be made $\endgroup$ – Dylan Wilson Sep 4 at 13:32
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    $\begingroup$ $H_0(\Omega X)$ is the free abelian group on $\pi_1(X)$. You might ask if that is the only problem; you might ask about the higher homology of the connected component of $\Omega X$: $\Omega_0X=\Omega \tilde X$, but this doesn't help. If you take any $X$ with infinite $\pi_1$ and glue on a sphere $S^n$ at a point, bad things happen. For example, $S^1\vee S^2$ has universal cover a line with infinitely many $S^2$ glued on, thus infinitely generated $H_2$; and so its loop space has infinitely generated $H_1$. $\endgroup$ – Ben Wieland Sep 4 at 13:51

This is true for finite $\pi_1$ and false for infinite $\pi_1$: Let $\widetilde{X}$ denote the universal cover of $X$, then $\Omega\widetilde{X}$ is the unit connected component of $\Omega X$, and $\Omega X = \coprod_{\pi_1(X)} \Omega\widetilde{X}$. So if $\pi_1$ is infinite, then certainly $H_0(\Omega X)$ is not finitely generated as others have noted in the comments, and indeed if $\Omega\widetilde{X}$ has any nontrivial homology group (which is true unless $\widetilde{X}$ is contractible), some higher homology group of $\Omega X$ will be an infinite direct sum of nontrivial abelian groups, so also not finitely generated.

If $\pi_1$ is finite, on the other hand, $\widetilde{X}$ is again a finite CW complex, so in that case it suffices to look at the simply-connected case. For a simply-connected finite CW-complex $X$, $H_*(\Omega X)$ indeed consists of finitely generated abelian groups, which goes back to Serre (and is proved easily using the spectral sequence named after him).

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