{
  "id": "1202.3222",
  "version": "v1",
  "published": "2012-02-15T07:33:12.000Z",
  "updated": "2012-02-15T07:33:12.000Z",
  "title": "Pillar switchings and acyclic embedding in mapping class group",
  "authors": [
    "Chan-Seok Jeong",
    "Yongjin Song"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1005.1300 by other authors",
  "journal": "Int. J. Math. 24, 1350103 (2013), 21 pages",
  "doi": "10.1142/S0129167X13501036",
  "categories": [
    "math.AT"
  ],
  "abstract": "The braid group $B_g$ is embedded in the ribbon braid group that is defined to be the mapping class group $\\Gamma_{0,(g),1}$. By gluing two copies of surface $S_{0,g+2}$ along $g+1$ holes, we get surface $S_{g,1}$. A pillar switching is a self-homeomorphism of $S_{g,1}$ which switches two pillars of surfaces by $180{}^\\circ$ horizontal rotation. We analyze the actions of pillar switchings on $\\pi_1 S_{g,1}$ and then give concrete expressions of pillar switchings in terms of standard Dehn twists. The map $\\psi: B_g \\rightarrow \\Gamma_{g,1}$ sending the generators of $B_g$ to pillar switchings on $S_{g,1}$ is defined by extending the embedding $B_g \\hookrightarrow \\Gamma_{0,(g+1),1}$. We show that this map is injective by analyzing the actions of pillar switchings on $\\pi_1 S_{g,1}$. The second part of this paper is to prove that this map induces a trivial homology homomorphism in the stable range. For the proof we use the categorical delooping. We construct a suitable monoidal 2-functor from tile category to surface category and show that this functor thus induces a map of double loop spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-15T07:33:12.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P48",
      "55R37",
      "57M50"
    ],
    "keywords": [
      "pillar switching",
      "mapping class group",
      "acyclic embedding",
      "ribbon braid group",
      "standard dehn twists"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.3222J"
    }
  }
}