Geometry and Topology of Configuration Spaces pp 243-267 | Cite as

# Computation of *H*_{*}(*Λ*(*M*))

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## Abstract

We have seen in the previous chapters that the space \(\mathbb{F}_k (M)\) can be described as a twisted product of simpler spaces when *M* is ℝ^{ n+1} or *S* ^{ n+1}. The simpler spaces are bouquets of *n*-dimensional spheres when *M* = ℝ^{ n+1}; when *M* = *S* ^{n+1}, they include the Stiefel manifold *O* _{ n+2,2} of orthonormal 2-frames in ℝ^{ n+2}, as well. We have also seen that the space \(\Omega \mathbb{F}_k (M)\) of based loops splits as a product of the loop spaces of the split factors as spaces, but not as loop spaces. A natural question to ask is whether the space of free loops \(\Lambda \mathbb{F}_k (M)\) splits, at the homology level, as a tensor product of the homology of the split factors of \(\mathbb{F}_k (M)\). We shall see in this chapter that this is the exception: it is true for *k* = 3, but not in general.

## Keywords

Spectral Sequence Betti Number Loop Space Chapter VIII Split Factor## Preview

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