{
  "id": "2112.10864",
  "version": "v2",
  "published": "2021-12-20T21:19:05.000Z",
  "updated": "2022-03-16T19:14:30.000Z",
  "title": "Moduli spaces of Riemann surfaces as Hurwitz spaces",
  "authors": [
    "Andrea Bianchi"
  ],
  "comment": "52 pages, 5 figures; the introduction has been changed",
  "categories": [
    "math.AT"
  ],
  "abstract": "We consider the moduli space $\\mathfrak{M}_{g,n}$ of Riemann surfaces of genus $g\\ge0$ with $n\\ge1$ ordered and directed marked points. For $d\\ge 2g+n-1$, we show that $\\mathfrak{M}_{g,n}$ is homotopy equivalent to a component of the generalised Hurwitz space $\\mathrm{Hur}^{\\Delta}(\\mathfrak{S}_d^{\\mathrm{geo}})$, associated with the partially multiplicative quandle $\\mathfrak{S}_d^{\\mathrm{geo}}$. As an application, we give a new proof of the Mumford conjecture on the stable rational cohomology of moduli spaces of Riemann surfaces. We also provide a combinatorial model for the infinite loop space $\\Omega^{\\infty-2}\\mathrm{MTSO}(2)$ of Hurwitz flavour.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-16T19:14:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55P35",
      "55P62",
      "55R80",
      "57T25"
    ],
    "keywords": [
      "moduli space",
      "riemann surfaces",
      "infinite loop space",
      "generalised hurwitz space",
      "homotopy equivalent"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}