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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Generalized group presentation and formal deformations of CW complexes


Author: Richard A. Brown
Journal: Trans. Amer. Math. Soc. 334 (1992), 519-549
MSC: Primary 57M05; Secondary 20F05, 57Q05, 57Q10
MathSciNet review: 1153010
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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.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1992-1153010-3
PII: S 0002-9947(1992)1153010-3
Keywords: Extended Nielsen transformations, simple homotopy equivalence, relative homotopy groups, homotopy systems, $ \Sigma $-systems, characteristic maps, upper semicontinuous decomposition, transiency
Article copyright: © Copyright 1992 American Mathematical Society