## Cellular properties of nilpotent spaces

We show that cellular approximations of nilpotent Postnikov stages are always nilpotent Postnikov stages, in particular classifying spaces of nilpotent groups are turned into classifying spaces of nilpotent groups. We use a modified Bousfield-Kan homology completion tower z(k) X whose terms we prove are all X-cellular for any X. As straightforward consequences, we show that if X is K-acyclic and nilpotent for a given homology theory K, then so are all its Postnikov sections P-n X, and that any nilpotent space for which the space of pointed self-maps map(*) (X, X) is "canonically" discrete must be aspherical.

Published in:
Geometry & Topology, 19, 5, 2741-2766
Year:
2015
Publisher:
Coventry, Geometry & Topology Publications
ISSN:
1465-3060
Laboratories: