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Introduction

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

Abstract

In Part Two our aim is to define minimal CW-complexes X k , Y k+1 homotopy equivalent to \({\mathbb{F}_k}({\mathbb{R}^{n + 1}})\) and \({\mathbb{F}_{k + 1}}({S^{n + 1}})\), respectively. In Chapter V we determine the structure of \({H^*}({\mathbb{F}_k}(M);\mathbb{Z})\), as an algebra, when M is ℝ n+1 or S n+1. We view the generators α rs of the group \({\pi _n}({\mathbb{F}_k}({\mathbb{R}^{n + 1}}),q)\), defined in Chapter II, §2, as spherical homology classes and introduce the elements \(\left\{ {\alpha _{rs}^* \in {H^n}({\mathbb{F}_k}({\mathbb{R}^{n + 1}};\mathbb{Z})|1 \leqslant s < r \leqslant r} \right\}\) dual to the α rs . These elements generate the group \({H^n}({\mathbb{F}_k}({\mathbb{R}^{n + 1}}),\mathbb{Z})\) and are invariant, set-wise, up to sign, under the action of the symmetric group ∑ k . Moreover, they satisfy the cohomological version of the Y-B relations of Chapter II, §3. We show that \({H^*}({\mathbb{F}_k}({\mathbb{R}^{n + 1}}),\mathbb{Z})\) is the universal, commutative, graded algebra generated by the set of all α rs * modulo the ideal generated by the Y-B relations. The proof is by induction on the natural filtration in diagram F k (ℝ n+1) of Chapter II. The rest of Chapter V is devoted to determining the cohomology algebra of \(\mathbb{F}_{k + 1} (S^{n + 1} )\). These results lead to cohomology bases consisting of multifold products of the elements of \(\left\{ {\alpha _{rs}^*|1 \leqslant s < r \leqslant k} \right\}\).

Keywords

Symmetric Group Configuration Space Loop Space Factor Table Cohomology Algebra 
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.

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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