On periodic homotopy and homology equivalences of spaces

Abstract

There are at least two ways to approach the homotopy theory of spaces `at chromatic height $n$': one may localize with respect to $T(n)$-homology or with respect to $v_n$-periodic homotopy groups. It was already observed by Bousfield that these two options yield rather different results. We build on his work to prove precise comparison results between the two notions. A crucial concept is a more robust notion of $T(n)$-equivalence that we call `parametric $T(n)$-equivalence': this is a map of spaces that induces an equivalence on $\infty$-categories of local systems valued in $T(n)$-local spectra. Our results are sharpest in the case of infinite loop spaces, where amongst other things we prove a $T(n)$-local version of a result of Kuhn on the Morava $K$-theory of the Whitehead tower. As a corollary of our results we also produce a formula for the $L_n^f$-localization of an infinite loop space $\Omega^\infty E$ of a spectrum satisfying $L_{n-1}^f E \simeq 0$.