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. 

General idea for computation of $H(\Omega X)$ (due to Serre, AFAIK) is to consider a (Serre) fibration $\Omega X\to PX\cong pt\to X$ and use Leray-Serre spectral sequence (it allows, in particular, to compute easily (at least, in simply-connected case) $H(\Omega X;\mathbb Q)$; cohomology with integer coefficients are, indeed, more complicated). It's discussed, I believe, in any textbook covering LSSS — e.g. in Hatcher's.
