{
  "abstract": "For $n\\geq3$, let $\\overline B_n=B_n/[P_n,P_n]$ be the crystallographic quotient of the Artin braid group, with translation subgroup $\\overline P_n=P_n/[P_n,P_n]\\cong\\mathbb Z^N$, where $N=\\binom n2$. We prove that every compact connected flat $3$-manifold $M$, considered up to affine equivalence, is the base of a geometric torus bundle whose total space is a compact flat manifold with fundamental group a Bieberbach subgroup of some $\\overline B_n$. More precisely, for every $n\\geq n_M$, where $n_M\\in\\{3,4,5\\}$ is an explicit optimal threshold depending on $M$, we construct a Bieberbach subgroup $Γ_n(M)\\leq\\overline B_n$ of dimension $N$ and a complete normal subgroup $K_n(M)\\cong\\mathbb Z^{N-3}$ consisting of translations such that $$ Γ_n(M)/K_n(M)\\congπ_1(M). $$ We also determine which of the associated torus-bundle group extensions split: for the realizations constructed here, splitting occurs for exactly four of the ten affine-equivalence classes of compact flat $3$-manifolds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20478",
  "author_total": 1,
  "authors": [
    {
      "name": "Oscar Ocampo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2np746j77ba2x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20478",
  "fulltext_url": null,
  "id": "2np746j77ba2x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20478",
  "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.GR"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7222202387",
  "pdf_url": "https://arxiv.org/pdf/2610.20478",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2np746j77ba2x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20478",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20478"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds",
  "title_text": "Crystallographic quotients of Artin braid groups and torus bundles over flat $3$-manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/2np746j77ba2x",
  "venue": "arXiv",
  "versions": []
}