On the comparison of stable and unstable $p$-completion

Abstract

In this note we show that a $p$-complete nilpotent space $X$ has a $p$-complete suspension spectrum if and only if its homotopy groups ${\pi }_{\ast }X$ are bounded $p$-torsion. In contrast, if ${\pi }_{\ast }X$ is not all bounded $p$-torsion, we locate uncountable rational vector spaces in the integral homology and in the stable homotopy groups of $X$. To prove this, we establish a homological criterion for $p$-completeness of connective spectra. Moreover, we illustrate our results by studying the stable homotopy groups of $K({\mathbb{Z}}_p,n)$ via Goodwillie calculus.

All comments welcome