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

Okay, Cihan¹

¹Department of Mathematics, The University of British Columbia, Vancouver BC V6T 1Z2, Canada

Algebraic and Geometric Topology, Tome 14 (2014) no. 4, pp. 2223-2257

Résumé

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.

Détail

DOI : 10.2140/agt.2014.14.2223

Classification : 55R10, 55N15, 55Q52
Keywords: homotopy colimit, classifying space, $K$–theory, descending central series

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