On the $p$-adic completions of nonnilpotent spaces

Abstract:

This paper deals with the $p$-adic completion ${F_{p\infty }}X$ developed by Bousfield-Kan for a space $X$ and prime $p$. A space $X$ is called ${F_p}$-good when the map $X \to {F_{p\infty }}X$ is a $\bmod \text {-}p$ homology equivalence, and called ${F_p}$-bad otherwise. General examples of ${F_p}$-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 ${F_p}$-bad. This provides the first example of an ${F_p}$-bad space of finite type and implies that the $p$-profinite completion of a free group on two generators must have nontrivial higher $\bmod \text {-}p$ homology as a discrete group. A major part of the paper is devoted to showing that the desirable properties of nilpotent spaces under the $p$-adic completion can be extended to the wider class of $p$-seminilpotent spaces.