Homotopy colimits of classifying spaces of abelian subgroups of a finite group

TL;DR

This paper studies the homotopy types of classifying spaces of finite groups filtered by central series, introducing prime-specific subspaces and analyzing their properties, especially for extraspecial 2-groups, with implications for K-theory.

Contribution

It introduces prime-specific subspaces of classifying spaces and analyzes their homotopy equivalences, especially for extraspecial 2-groups, advancing understanding of their topological and algebraic structures.

Findings

$B(q,G)$ is stably homotopy equivalent to a wedge of $B(q,G)_p$ spaces.

Colimits of abelian groups in these spaces are finite for extraspecial 2-groups.

$B(2,G)$ for extraspecial 2-groups is not a $K(\pi,1)$ space.

Abstract

The classifying space BG of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, using the descending central series of free groups. If $G$ is finite, describing them as homotopy colimits is convenient when applying homotopy theoretic methods. In this paper we introduce natural subspaces $B(q,G)_p$ of $B(q,G)$ defined for a fixed prime $p$. We show that $B(q,G)$ is stably homotopy equivalent to a wedge of $B(q,G)_p$ as $p$ runs over the primes dividing the order of $G$. Colimits of abelian groups play an important role in understanding the homotopy type of these spaces. Extraspecial $2$-groups are key examples, for which these colimits turn out to be finite. We prove that for extraspecial 2-groups, $B(2,G)$ does not have the homotopy type of a $K(\pi,1)$ space. For a finite group $G$, we compute the complex K-theory of $B(2,G)$ modulo torsion.