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

Homology and cell structure of nilpotent spaces


Author: Robert H. Lewis
Journal: Trans. Amer. Math. Soc. 290 (1985), 747-760
MSC: Primary 55P99; Secondary 20C07, 57M99
MathSciNet review: 792825
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Abstract: Let $ A$ and $ X$ denote finitely dominated nilpotent $ {\text{CW}}$ complexes. We are interested in questions relating the homology groups of such spaces to their cell structure and homotopy type. We solve a problem posed by Brown and Kahn, that of constructing nilpotent complexes of minimal dimension. When the fundamental group is finite, the three-dimensional complex we construct may not be finite; we then construct a finite six-dimensional complex.

We investigate the set of possible cofibers of maps $ A \to X$, and find a severe restriction. When it is met and the fundamental group is finite, $ X$ can be constructed from $ A$ by attaching cells in a natural way. The restriction implies that the classical notion of homology decomposition has no application to nilpotent complexes.

We show that the Euler characteristic of $ X$ must be zero. Several corollaries are derived to the theory of finitely dominated nilpotent complexes.

Several of these results depend upon a purely algebraic theorem that we prove concerning the vanishing of homology of nilpotent modules over nilpotent groups.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0792825-4
PII: S 0002-9947(1985)0792825-4
Article copyright: © Copyright 1985 American Mathematical Society