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

Abstract

The classifying space $BG$ of a topological group $G$ can be filtered by a sequence of subspaces $B\left(q,G\right)$, $q\ge 2$, 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 $B{\left(q,G\right)}_p\subset B\left(q,G\right)$ defined for a fixed prime $p$. We show that $B\left(q,G\right)$ is stably homotopy equivalent to a wedge of $B{\left(q,G\right)}_p$ 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 of order $2^{2n+1}$, $n\ge 2$, $B\left(2,G\right)$ does not have the homotopy type of a $K\left(\pi ,1\right)$ space, thus answering in a negative way a question posed by Adem. For a finite group $G$, we compute the complex $K$–theory of $B\left(2,G\right)$ modulo torsion.