Homology of the loop space Let $X$ be a nice space (manifold, CW-complex, what you prefer). I was wondering if there is a computable relation between the homology of $\Omega X$, the loop space of $X$, and the homology of $X$. I know that, almost by definition, the homotopy groups are the same (but shifted a dimension). Because the relation between homotopy groups and homology groups is very difficult, I expect that the homology of $\Omega X$ is very hard to compute in general. References would be great. 

Adams and Hilton gave a functorial way to describe the homology ring $H_\ast(\Omega X)$ in terms of the homology $H_\ast(X)$, at least when $X$ is a simply-connected CW complex with one $0$-cell and no $1$-cells. You'll find a more modern discussion of their construction here.
