{
  "id": "1311.5705",
  "version": "v4",
  "published": "2013-11-22T10:40:36.000Z",
  "updated": "2014-02-05T11:26:23.000Z",
  "title": "The 2-torsion in the second homology of the genus $3$ mapping class group",
  "authors": [
    "Wolfgang Pitsch"
  ],
  "comment": "4 pages, 2 figures, some typos corrected, v3. Reference to previous work by Sakasai proving the result by different methods added, See the Abstract",
  "categories": [
    "math.AT",
    "math.GR",
    "math.KT"
  ],
  "abstract": "This work is NOT to be used as reference. First, because as C.F.~B\\\"odigheimer and M.~Korkmaz pointed to us the computation of the $\\mathbf{Z}_2$ factor that remained undecided in M.~Korkmaz and A. Stipsicz, {\\em The second homology groups of mapping class groups of orientable surfaces.} Math. Proc. Camb. Phil. Soc., was shown to exist by Skasai, see hi Theorem 4.9 and Corollary 4.10 in {\\em Lagrangian mapping class groups from a group homological point of view.} Algebr. Geom. Topol. 12 (2012), no. 1, 267--291. Second, because one could obtain this result by gathering old results in the literature, first by noticing as Korkmaz kindly reminded me, that D.~Johnson, in \\emph{Homeomorphisms of a surface which act trivially on homology} Porc. AMS Volume 75, Number 1, 1979. proved that the quotient of the Torelli group $\\mathcal{T}_g/[\\mathcal{T}_g,\\mathcal{M}_g]$ is trivial for $g\\geq 3$, the five term exact sequence then implies that the $\\mathbf{Z}_2$ factor in Stein's computation of $H_2(Sp(6,\\mathbf{Z});\\mathbf{Z}) = \\mathbf{Z}\\oplus\\mathbf{Z}_2$ (see his {\\em The Schur Multipliers of $Sp_6(\\mathbf{Z}), Spin_8(\\mathbf{Z}), Spin_7(\\mathbf{Z}),$ and $F_4(\\mathbf{Z})$.} Math. Ann. 215 (1975), 173--193. ), detects the undecided $\\mathbf{Z}_2$ factor in $H_2(\\mathbf{M}_3;\\mathbf{Z})$.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2014-02-05T11:26:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J06",
      "57M60"
    ],
    "keywords": [
      "second homology groups",
      "lagrangian mapping class groups",
      "term exact sequence",
      "hi theorem",
      "computation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 4,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1311.5705P"
    }
  }
}