The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,... more 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. Then 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 2n+1 , n ≥ 2, B(2, G) does not have the homotopy type of a K(π, 1) space. For a finite group G, we compute the complex K-theory of B(2, G) modulo torsion.
The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,... more 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. Then 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 2n+1 , n ≥ 2, B(2, G) does not have the homotopy type of a K(π, 1) space. For a finite group G, we compute the complex K-theory of B(2, G) modulo torsion.
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Papers by Cihan Okay