{
  "abstract": "For a finite simple graph $G$, the matching complex $M(G)$ is the simplicial complex whose vertex set is the edge set of $G$ and whose simplices are all the matchings in $G$. The topology of the matching complex of the $m\\times n$ grid graph $G_{m\\times n}$ is known only for $m = 1,2$, in which cases it is homotopy equivalent to a wedge of spheres. In this article, we study the matching complex $M(G_{3 \\times n})$. We prove that for $n\\ge2$, its reduced homology vanishes in dimensions $i \\leq n-2$ and in top dimension, while $\\tilde{H}_{n-1}(M(G_{3\\times n}))\\neq 0$. We also show that $M(G_{3 \\times n})$ is simply connected for $n \\geq 3$. Consequently, the topological connectivity of $M(G_{3\\times n})$ is $n-2$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27366",
  "author_total": 3,
  "authors": [
    {
      "name": "Pratiksha Chauhan"
    },
    {
      "name": "Anchal Sharma"
    },
    {
      "name": "Samir Shukla"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5rjh9y4j49kmv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27366",
  "fulltext_url": null,
  "id": "5rjh9y4j49kmv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27366",
  "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"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7214244171",
  "pdf_url": "https://arxiv.org/pdf/2609.27366",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5rjh9y4j49kmv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27366",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27366"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Homology of matching complexes of $3\\times n$ grid graphs",
  "title_text": "Homology of matching complexes of 3× n grid graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/5rjh9y4j49kmv",
  "venue": "arXiv",
  "versions": []
}