Nullification and cellularization of classifying spaces of finite groups

Abstract

Abstract. In this note we discuss the effect of the BZ/p-nullification P BZ/p and the BZ/p-cellularization CW BZ/p over classifying spaces of finite groups, and we relate them with the corresponding functors with respect to Moore spaces that have been intensively studied in the last years. We describe P BZ/p BG by means of a covering fibration, and we classify all finite groups G for which BG is BZ/p-cellular. We also carefully study the analogous functors in the category of groups, and their relationship with the fundamental groups of P BZ/p BG and CW BZ/p BG.

Cellular covers of groups

“…A similar construction is often denoted in topological contexts C W A X since it is inspired by the classical CW-approximation given by Whitehead, see [18,9,7]. Our description relies on [18], and we give only a rough sketch.…”

Cellular approximations of p‐local compact groups

“…The first objects studied in this context were classifying spaces of finite groups. In , the second author computed explicitly the ‐cellularization of classifying spaces of finite ‐groups. Then, while obtaining more information in the finite case , the case of connected compact Lie groups was undertaken in ; there more qualitative results were described, as a dichotomy result on the homotopy groups of , and also concrete examples for where , and .…”

The cellular structure of the classifying spaces of finite groups

“…A line of research of great relevance in the last years, and very related to our work, is cellular approximation in the category of groups ( [RS01], [Flo07], [DGS07], [DGS08]). …”

“…(The combined classification is rendered in a slightly simplified form as Theorem 5.1 of Section 5.) The p odd classification was a crucial ingredient we lacked in order to finish the characterization of CW BZ/p BG for all finite groups G (the other was the role of the subgroup O A (G), see below), solving a problem that was posed by Dror-Farjoun [Far95,3.C] in the case G = Z/p r , and partially solved in [Flo07] and [FS07] (see Section 2 below for an analysis of the previous cases). The latter paper showed the relationship between the upper homotopy of CW BZ/p BG and a specific strongly closed subgroup A 1 (S) of G. More precisely, for S a Sylow p-subgroup of G, A 1 (S) is the unique minimal strongly closed subgroup of S that contains all elements of order p in S. The importance of the subgroup A 1 (S) comes from the fact that it determines a great part of the structure of Hom(Z/p, G), and then of map * (BZ/p, BG).…”