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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.

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2 Answers 2

up vote 9 down vote accepted

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.

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Thank you. I will look into this. I'll leave the question open for a little while longer, to see if I get some more references. –  Thomas Rot Jul 2 '11 at 14:01

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.

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