{
  "abstract": "Let $\\widetilde{S}\\to S$ be an unbranched regular $p$-fold cyclic cover of a closed orientable surface $S$ of genus $g$. Two natural groups are associated to this cover. The first is the centralizer $\\mathrm{Mod}(\\widetilde{S},σ)$ of a chosen generator $σ$ of the deck transformation group in $\\mathrm{Mod}(\\widetilde{S})$. The second is the finite-index subgroup $\\mathrm{Mod}(S,[β])$ of $\\mathrm{Mod}(S)$ consisting of mapping classes that fix the nonzero class $[β]\\in H_1(S;\\mathbb Z/p\\mathbb Z)$ corresponding to the cover. For $p=2$, Sato computed the abelianizations of these groups. We compute their abelianizations for every odd prime $p$ and show that they exhibit a splitting phenomenon different from the case $p=2$. We also construct explicit abelianization maps using Morita's crossed homomorphism and the Prym representation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.24271",
  "author_total": 1,
  "authors": [
    {
      "name": "Xiyan Zhong"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/67q7spr53sgaa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.24271",
  "fulltext_url": null,
  "id": "67q7spr53sgaa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.24271",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7171567019",
  "pdf_url": "https://arxiv.org/pdf/2607.24271",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/67q7spr53sgaa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.24271",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.24271"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Abelianization of Symmetric Mapping Class Groups",
  "title_text": "Abelianization of Symmetric Mapping Class Groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/67q7spr53sgaa",
  "venue": "arXiv",
  "versions": []
}