Algebraic & Geometric Topology

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

Cihan Okay

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The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G), q2, 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)pB(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, n2, 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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R10: Fiber bundles
Secondary: 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 55Q52: Homotopy groups of special spaces

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


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.

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