Cellular Models

Our objective here is to describe cellular structures of $$\mathbb{F}_k (M)$$ naturally associated with the bases of $$H^* (\mathbb{F}_k (M))$$ for M = ℝn+1, Sn+1 of Theorem 4.2 and §6 of Chapter V. The basic ideas are the following: first, that the twisted product representation $$\mathbb{F}_k (\mathbb{R}^{n + 1} ) \simeq \,\mathbb{R}_1^{n + 1} \propto \cdots \propto \mathbb{R}_{k - 1}^{n + 1} $$ introduced in Chapter II, §4 leads to a twisted product $$ H_* (\mathbb{R}_1^{n + 1} ) \otimes \cdots \otimes H_* (\mathbb{R}_{k - 1}^{n + 1} ) \cong H_* (\mathbb{F}_k (\mathbb{R}^{n + 1} );\mathbb{Z}) $$ on homology, which we write as $$ \alpha _{r_1 s_1 } \otimes \cdots \otimes \alpha _{r_p s_p } \mapsto \omega = \alpha _{r_1 s_1 } \, \propto \cdots \propto \alpha _{r_p s_p } r_i < r_{i + 1} $$ for all 1 ≤ p ≤ (k - 1); and, second, that each p-fold twisted product ω leads to an imbedding $$ \varphi _\omega \,:\,S_1^n \, \times \cdots \times \,S_p^n \, \to \,\mathbb{F}_k (\mathbb{R}^{n + 1} ) $$ of a certain kind. These maps provide us with the cells and attaching maps of the desired complex.