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.



loop space of a finite CW-complex
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).


