Generalized Group Presentation and Formal Deformations of CW Complexes

Abstract

A Peiffer-Whitehead word system $\mathcal {W}$, or generalized group presentation, consists of generators, relators (words of order $2$), and words of higher order $n$ that represent elements of a free crossed module $(n = 3)$ or a free module $(n > 3)$. The ${P_n}$-equivalence relation on word systems generalizes the extended Nielsen equivalence relation on ordinary group presentations. Word systems, called homotopy readings, can be associated with any connected ${\text {CW}}$ complex $K$ by removing a maximal tree and selecting one generator or word per cell, via relative homotopy. Given homotopy readings ${\mathcal {W}_1}$ and ${\mathcal {W}_2}$ of finite ${\text {CW}}$ complexes ${K_1}$ and ${K_2}$ respectively, we show that ${\mathcal {W}_1}$ is ${P_n}$-equivalent to ${\mathcal {W}_2}$ if and only if ${K_1}$ formally $(n + 1)$-deforms to ${K_2}$. This extends results of P. Wright (1975) and W. Metzler (1982) for the case $n = 2$. For $n \geq 3$, it follows that ${\mathcal {W}_1}$ is ${P_n}$-equivalent to ${\mathcal {W}_2}$ if and only if ${K_1}$ and ${K_2}$ have the same simple homotopy type.