{
  "abstract": "We compute the first homology of the unordered configuration space of $n$ open unit-squares in the $w$ by $h$ rectangle for all positive integers $n$, $w$, and $h$. Letting $w$ and $h$ vary, these square configuration spaces form a bi-filtration of the unordered configuration space of $n$ points in the plane, so their fundamental groups approximate the $n$-strand braid group. Using a pair of spectral sequences, we prove that these square and point configuration spaces have naturally isomorphic first homology groups if $w,h\\ge 3$ and $n\\le wh-3$. In the $h=2$ case, we show that the first homology groups of these square configuration spaces are free of rank $n-1$ for $n\\le 2w-2$, whereas they are free of rank $w-1$ for $n=2w-1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12633",
  "author_total": 3,
  "authors": [
    {
      "name": "Minh Pham"
    },
    {
      "name": "Nicholas Wawrykow"
    },
    {
      "name": "Jacob Zweiback"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/78q35bqym685d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12633",
  "fulltext_url": null,
  "id": "78q35bqym685d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12633",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7221568407",
  "pdf_url": "https://arxiv.org/pdf/2610.12633",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/78q35bqym685d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12633",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12633"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The first homology of square-section braid groups",
  "title_text": "The first homology of square-section braid groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/78q35bqym685d",
  "venue": "arXiv",
  "versions": []
}