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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 ℝn+1 or S n+1. The simpler spaces are bouquets of n-dimensional spheres when M = ℝn+1; when M = S n+1, they include the Stiefel manifold O n+2,2 of orthonormal 2-frames in ℝn+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

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2001 Springer-Verlag Berlin Heidelberg

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Fadell, E.R., Husseini, S.Y. (2001). Computation of H *(Λ(M)). In: Geometry and Topology of Configuration Spaces. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56446-8_15

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  • DOI: https://doi.org/10.1007/978-3-642-56446-8_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63077-4

  • Online ISBN: 978-3-642-56446-8

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