Cellular properties of nilpotent spaces

Abstract

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 $\mathcal{K}$–acyclic and nilpotent for a given homology theory $\mathcal{K}$, then so are all its Postnikov sections $P_{n}X$, and that any nilpotent space for which the space of pointed self-maps ${map}_{\ast }\left(X,X\right)$ is “canonically” discrete must be aspherical.