Homotopy idempotent functors on classifying spaces

TL;DR

This paper explores how homotopy idempotent functors like $P_{B ext{Z}/p}$ and $CW_{B ext{Z}/p}$ affect the mod $p$ structure of classifying spaces, relating them to other localization and completion tools.

Contribution

It analyzes the impact of these functors on a broad class of spaces, including classifying spaces of Lie groups, and clarifies their relationship with other mod $p$ homotopy functors.

Findings

Homotopy functors significantly alter the mod $p$ structure of classifying spaces.

Relationships between $P_{B ext{Z}/p}$, $CW_{B ext{Z}/p}$, and other localization functors are established.

The study extends understanding of mod $p$ homotopy theory for a wide class of spaces.

Abstract

Fix a prime $p$. Since their definition in the context of Localization Theory, the homotopy functors $P_{B\Z/p}$ and $CW_{B\Z/p}$ have shown to be powerful tools to understand and describe the mod $p$ structure of a space. In this paper, we study the effect of these functors on a wide class of spaces which includes classifying spaces of compact Lie groups and their homotopical analogues. Moreover, we investigate their relationship in this context with other relevant functors in the analysis of the mod $p$ homotopy, such as Bousfield-Kan completion and Bousfield homological localization.