{
  "id": "1309.4192",
  "version": "v1",
  "published": "2013-09-17T06:12:26.000Z",
  "updated": "2013-09-17T06:12:26.000Z",
  "title": "New lower bounds for the topological complexity of aspherical spaces",
  "authors": [
    "Mark Grant",
    "Gregory Lupton",
    "John Oprea"
  ],
  "comment": "15 pages, 4 figures",
  "categories": [
    "math.AT"
  ],
  "abstract": "We show that the topological complexity of an aspherical space $X$ is bounded below by the cohomological dimension of the direct product $A\\times B$, whenever $A$ and $B$ are subgroups of $\\pi_1(X)$ whose conjugates intersect trivially. For instance, this assumption is satisfied whenever $A$ and $B$ are complementary subgroups of $\\pi_1(X)$. This gives computable lower bounds for the topological complexity of many groups of interest (including semidirect products, pure braid groups, certain link groups, and Higman's acyclic four-generator group), which in some cases improve upon the standard lower bounds in terms of zero-divisors cup-length. Our results illustrate an intimate relationship between the topological complexity of an aspherical space and the subgroup structure of its fundamental group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-17T06:12:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55M99",
      "55P20",
      "55M30",
      "20J06",
      "68T40"
    ],
    "keywords": [
      "topological complexity",
      "aspherical space",
      "higmans acyclic four-generator group",
      "standard lower bounds",
      "pure braid groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.4192G"
    }
  }
}