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Introduction

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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 ℝn+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 (ℝn+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.

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© 2001 Springer-Verlag Berlin Heidelberg

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Fadell, E.R., Husseini, S.Y. (2001). Introduction. In: Geometry and Topology of Configuration Spaces. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56446-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-56446-8_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63077-4

  • Online ISBN: 978-3-642-56446-8

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