{
  "id": "2403.16322",
  "version": "v1",
  "published": "2024-03-24T23:06:50.000Z",
  "updated": "2024-03-24T23:06:50.000Z",
  "title": "On the mapping class group action on the homology of surface covers",
  "authors": [
    "Igor Spiridonov"
  ],
  "comment": "11 pages, 2 figures",
  "categories": [
    "math.GT"
  ],
  "abstract": "Let $\\phi \\in {\\rm Mod}(\\Sigma)$ be an arbitrary element of the mapping class group of a closed orientable surface $\\Sigma$ of genus at least $2$. For any characteristic cover $\\widetilde{\\Sigma} \\to \\Sigma$ one can consider the linear subspace ${\\rm H}_1^{f.o.}(\\widetilde{\\Sigma}, \\mathbb{Q})^\\phi \\subseteq {\\rm H}_1(\\widetilde{\\Sigma}, \\mathbb{Q})$ consisting of all homology classes with finite $\\phi$-orbit. We prove that $\\dim {\\rm H}_1^{f.o.}(\\widetilde{\\Sigma}, \\mathbb{Q})^\\phi$ can be arbitrary large for any fixed $\\phi \\in {\\rm Mod}(\\Sigma)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-24T23:06:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "57K20",
      "57M10",
      "57M12"
    ],
    "keywords": [
      "mapping class group action",
      "surface covers",
      "arbitrary element",
      "characteristic cover",
      "linear subspace"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}