Introduction

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