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