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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

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.

homotopy colimit, classifying space, $K$–theory, descending central series
Mathematical Subject Classification 2010
Primary: 55R10
Secondary: 55N15, 55Q52
Received: 18 July 2013
Revised: 12 September 2013
Accepted: 18 September 2013
Published: 28 August 2014
Cihan Okay
Department of Mathematics
The University of British Columbia
Vancouver BC V6T 1Z2