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

Okay, Cihan

doi:10.2140/agt.2014.14.2223

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

Cihan Okay

Algebraic & Geometric Topology 14 (2014) 2223–2257

DOI: 10.2140/agt.2014.14.2223

Abstract

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The classifying space $B G$ of a topological group $G$ can be filtered by a sequence of subspaces $B (q, G)$ , $q \geq 2$ , 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 $B {(q, G)}_{p} \subset B (q, G)$ 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 $2^{2 n + 1}$ , $n \geq 2$ , $B (2, G)$ 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.

Keywords

homotopy colimit, classifying space, $K$–theory, descending central series

Mathematical Subject Classification 2010

Primary: 55R10

Secondary: 55N15, 55Q52

References

Publication

Received: 18 July 2013

Revised: 12 September 2013

Accepted: 18 September 2013

Published: 28 August 2014

Authors

Cihan Okay
Department of Mathematics The University of British Columbia Vancouver BC V6T 1Z2 Canada
http://www.math.ubc.ca/~okay/

Volume 14, issue 4 (2014)