# loop space of a finite CW-complex

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.

• I think the serre spectral sequence gives this immediately when X is simply connected – Thomas Rot Sep 4 at 13:16
• I guess if $X = S^1$ this is false, so definitely some $\pi_1$ conditions need to be made – Dylan Wilson Sep 4 at 13:32
• $H_0(\Omega X)$ 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 $H_1$. – 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).