{
  "id": "1907.10212",
  "version": "v1",
  "published": "2019-07-24T02:36:44.000Z",
  "updated": "2019-07-24T02:36:44.000Z",
  "title": "Rewriting systems and the multiplicative structure in the cohomology of $π_1(Σ_{g_1}\\times\\cdots\\timesΣ_{g_n})$ with twisted $\\mathbb{Z}$-coefficients",
  "authors": [
    "Natalia Cadavid-Aguilar",
    "Jesús González"
  ],
  "comment": "32 pages",
  "categories": [
    "math.AT"
  ],
  "abstract": "Let $\\pi_g$ stand for the fundamental group of the closed orientable surface $\\Sigma_g$ of genus $g$. We use a finite complete rewriting system for $\\pi_g$ in order to produce an explicit contracting homotopy for the standard minimal $\\pi_g$-free resolution $M_*^g$ of the trivial $\\pi_g$-module $\\mathbb{Z}$. This allows us to construct an explicit diagonal approximation for any $\\mathbb{Z}$-tensor product $M_*^{g_1}\\otimes\\cdots\\otimes M_*^{g_n}$ which, in turn, yields an efficient method to compute the structure of the cohomology product maps $H^*(\\pi_{g_1}\\times\\cdots\\times\\pi_{g_n};M)\\otimes H^*(\\pi_{g_1}\\times\\cdots\\times\\pi_{g_n};M')\\to H^*(\\pi_{g_1}\\times\\cdots\\times\\pi_{g_n};M\\otimes M').$ Details and explicit examples are spelled out for $n=1$ and $n=2$ when the abelian group structure underlying both $M$ and $M'$ is $\\mathbb{Z}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-24T02:36:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20J06",
      "20F10",
      "68Q42"
    ],
    "keywords": [
      "multiplicative structure",
      "coefficients",
      "cohomology product maps",
      "finite complete rewriting system",
      "abelian group structure"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}