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


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 \(\left\{ {\alpha _{rs}^*|1 \leqslant s < r \leqslant k} \right\}\).


Symmetric Group Configuration Space Loop Space Factor Table Cohomology Algebra 
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© 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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