## Algebraic & Geometric Topology

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

Cihan Okay

#### Abstract

The classifying space $BG$ of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, $q≥2$, 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 $B(q,G)p⊂B(q,G)$ 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 of order $22n+1$, $n≥2$, $B(2,G)$ does not have the homotopy type of a $K(π,1)$ space, thus answering in a negative way a question posed by Adem. For a finite group $G$, we compute the complex $K$–theory of $B(2,G)$ modulo torsion.

#### Article information

Source
Algebr. Geom. Topol., Volume 14, Number 4 (2014), 2223-2257.

Dates
Revised: 12 September 2013
Accepted: 18 September 2013
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715963

Digital Object Identifier
doi:10.2140/agt.2014.14.2223

Mathematical Reviews number (MathSciNet)
MR3331614

Zentralblatt MATH identifier
1305.55008

#### Citation

Okay, Cihan. Homotopy colimits of classifying spaces of abelian subgroups of a finite group. Algebr. Geom. Topol. 14 (2014), no. 4, 2223--2257. doi:10.2140/agt.2014.14.2223. https://projecteuclid.org/euclid.agt/1513715963

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