{
  "id": "2209.08148",
  "version": "v1",
  "published": "2022-09-16T19:44:19.000Z",
  "updated": "2022-09-16T19:44:19.000Z",
  "title": "Computations in the unstable homology of moduli spaces of Riemann surfaces",
  "authors": [
    "Carl-Friedrich Bödigheimer",
    "Felix Boes",
    "Florian Kranhold"
  ],
  "comment": "38 pages, 18 figures, 8 tables",
  "categories": [
    "math.AT",
    "math.GT"
  ],
  "abstract": "In this article we give a survey of homology computations for moduli spaces $\\mathfrak{M}_{g,1}^m$ of Riemann surfaces with genus $g\\geqslant 0$, one boundary curve, and $m\\geqslant 0$ punctures. While rationally and stably this question has a satisfying answer by the Madsen-Weiss theorem, the unstable homology remains notoriously complicated. We discuss calculations with integral, mod-2, and rational coefficients. Furthermore, we determine, in most cases, explicit generators using homology operations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-16T19:44:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32G15",
      "55R40",
      "57K20",
      "58D05"
    ],
    "keywords": [
      "moduli spaces",
      "riemann surfaces",
      "homology operations",
      "madsen-weiss theorem",
      "unstable homology remains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 38,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}