Homology and cell structure of nilpotent spaces

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.