{
  "id": "math/0410539",
  "version": "v3",
  "published": "2004-10-25T22:30:20.000Z",
  "updated": "2005-09-04T11:54:29.000Z",
  "title": "Discrete Morse theory and graph braid groups",
  "authors": [
    "Daniel Farley",
    "Lucas Sabalka"
  ],
  "comment": "Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-44.abs.html",
  "journal": "Algebr. Geom. Topol. 5 (2005) 1075-1109",
  "categories": [
    "math.GR",
    "math.AT",
    "math.GT"
  ],
  "abstract": "If Gamma is any finite graph, then the unlabelled configuration space of n points on Gamma, denoted UC^n(Gamma), is the space of n-element subsets of Gamma. The braid group of Gamma on n strands is the fundamental group of UC^n(Gamma). We apply a discrete version of Morse theory to these UC^n(Gamma), for any n and any Gamma, and provide a clear description of the critical cells in every case. As a result, we can calculate a presentation for the braid group of any tree, for any number of strands. We also give a simple proof of a theorem due to Ghrist: the space UC^n(Gamma) strong deformation retracts onto a CW complex of dimension at most k, where k is the number of vertices in Gamma of degree at least 3 (and k is thus independent of n).",
  "revisions": [
    {
      "version": "v3",
      "updated": "2005-09-04T11:54:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20F65",
      "20F36",
      "57M15",
      "57Q05",
      "55R80"
    ],
    "keywords": [
      "graph braid groups",
      "discrete morse theory",
      "strong deformation retracts",
      "unlabelled configuration space",
      "fundamental group"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math.....10539F"
    }
  }
}