# Reference: Betti Numbers of the free loop space are finite

let $M$ be a compact, simply connected Riemannian manifold with dimension $< \infty$. I'm looking for a reference that $$\dim H_k(\Lambda M, \mathbb{Z}) < \infty,$$

is true in that case. Here $$\Lambda M= \{c :S^1 \rightarrow M| c\text{ is square summable and absolutely continous}\}$$ is the free loop space.

Thanks,

Mick

• Is $k$ fixed here? Sep 16, 2015 at 10:42
• This can be proved using the Serre spectral sequences for the fibrations $\Omega M\to\Lambda M\to M$ and $\Omega M\to PM\to M$, but I do not know where this is spelled out. Sep 16, 2015 at 10:58
• No, it isn't fixed. Sep 16, 2015 at 11:49

As Neil Strickland points out, for a fixed $k$ the Betti number of the free loop space is indeed finite. However it is worth pointing out that the sequence of Betti numbers $b_k(\Lambda M)$ is unbounded whenever the rational cohomology algebra of $M$ requires at least two generators. This is the main result of
• In the non-simply connected case you can easily have infinite $0$-th and $1$-st Betti numbers, e.g. if $\pi_1(M)=\mathbb{Z}$ then $\Lambda M$ has infinitely many path components. Not sure about the higher Betti numbers. Feb 28, 2022 at 7:47