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 and denote finitely dominated nilpotent 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 , and find a severe restriction. When it is met and the fundamental group is finite, can be constructed from 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 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:
https://doi.org/10.1090/S0002-9947-1985-0792825-4

Article copyright:
© Copyright 1985
American Mathematical Society