# Cellular Models

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

## Abstract

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 = ℝ n+1, S n+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.

## Keywords

Cellular Structure Homotopy Class Cellular Model Homology Class Factor Table
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.