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Computation of H*(Λ(M))

  • Edward R. Fadell
  • Sufian Y. Husseini
Part of the Springer Monographs in Mathematics book series (SMM)

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 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Edward R. Fadell
    • 1
  • Sufian Y. Husseini
    • 1
  1. 1.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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