{
  "id": "1307.2950",
  "version": "v3",
  "published": "2013-07-11T00:05:04.000Z",
  "updated": "2014-12-14T22:39:18.000Z",
  "title": "Homotopy colimits of classifying spaces of abelian subgroups of a finite group",
  "authors": [
    "Cihan Okay"
  ],
  "journal": "Algebr. Geom. Topol. 14 (2014) 2223-2257",
  "doi": "10.2140/agt.2014.14.2223",
  "categories": [
    "math.AT",
    "math.GR",
    "math.KT"
  ],
  "abstract": "The classifying space BG of a topological group $G$ can be filtered by a sequence of subspaces $B(q,G)$, 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 $B(q,G)_p$ of $B(q,G)$ 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, $B(2,G)$ does not have the homotopy type of a $K(\\pi,1)$ space. For a finite group $G$, we compute the complex K-theory of $B(2,G)$ modulo torsion.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-09-19T00:30:44.000Z",
      "abstract": "The classifying space BG of a topological group G can be filtered by a sequence of subspaces B(q,G), 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 of 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, B(2,G) does not have the homotopy type of a K(\\pi,1) space. For a finite group G, we compute the complex K-theory of B(2,G) modulo torsion.",
      "comment": "Preliminary version",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2014-12-14T22:39:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "homotopy colimits",
      "finite group",
      "abelian subgroups",
      "homotopy type",
      "abelian groups play"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1307.2950O"
    }
  }
}