On the 𝑝-adic completions of nonnilpotent spaces

Abstract

This paper deals with the p p -adic completion F p ∞ X {F_{p\infty }}X developed by Bousfield-Kan for a space X X and prime p p . A space X X is called F p {F_p} -good when the map X → F p ∞ X X \to {F_{p\infty }}X is a mod - p \bmod \text {-}p homology equivalence, and called F p {F_p} -bad otherwise. General examples of F p {F_p} -good spaces are established beyond the usual nilpotent or virtually nilpotent ones. These include the polycyclic-by-finite spaces. However, the wedge of a circle with a sphere of positive dimension is shown to be F p {F_p} -bad. This provides the first example of an F p {F_p} -bad space of finite type and implies that the p p -profinite completion of a free group on two generators must have nontrivial higher mod - p \bmod \text {-}p homology as a discrete group. A major part of the paper is devoted to showing that the desirable properties of nilpotent spaces under the p p -adic completion can be extended to the wider class of p p -seminilpotent spaces.