On periodic homotopy and homology equivalences of spaces

TL;DR

This paper compares two approaches to the homotopy theory of spaces at chromatic height n, introducing a robust notion of T(n)-equivalence and analyzing their differences, especially for infinite loop spaces.

Contribution

It establishes precise comparison results between T(n)-localization and v_n-periodic homotopy groups, introducing parametric T(n)-equivalence and extending results to infinite loop spaces.

Findings

Sharp comparison results between T(n)-localization and v_n-periodic homotopy groups.

Introduction of parametric T(n)-equivalence as a more robust notion.

A formula for the L_n^f-localization of infinite loop 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 $L_{n-1}^f E \simeq 0$.