Generalized group presentation and formal deformations of CW complexes
Richard A. Brown · Transactions of the American Mathematical Society · 1992
A Peiffer-Whitehead word system W \mathcal {W} , or generalized group presentation, consists of generators, relators (words of order 2 2 ), and words of higher order n n that represent elements of a free crossed module ( n = 3 ) (n = 3) or a free module ( n > 3 ) (n > 3) . The P n {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 CW {\text {CW}} complex K K by removing a maximal tree and selecting one generator or word per cell, via relative homotopy. Given homotopy readings W 1 {\mathcal {W}_1} and W 2 {\mathcal {W}_2} of finite CW {\text {CW}} complexes K 1 {K_1} and K 2 {K_2} respectively, we show that W 1