{
  "abstract": "We study torus-exterior replacement in four-manifolds, with the construction controlled by the peripheral homomorphism. Using torus exteriors arising from the work of Boyle and Kanenobu-Kazama, with explicitly controlled fundamental groups and peripheral systems, we construct smooth homotopy 4-spheres and manifolds homeomorphic to S^2 x S^2 and \\#_3(S^2 x S^2). We also study direct gluings of two torus exteriors. A swap gluing of two one-null exteriors kills both meridians and gives a simply connected manifold with Euler characteristic \\(4\\) and signature \\(0\\), while a meridian-preserving gluing of doubly-null exteriors retains the complement groups as an amalgam over the meridian subgroup. Twisted peripheral gluings give cyclic and explicit finite nonabelian fundamental groups. We also consider the full Kanenobu-Kazama \\(1\\)-handle family and Litherland tori with full-rank peripheral subgroup; the latter have no null primitive boundary slope and lead naturally to amalgamated-product fundamental groups.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25080",
  "author_total": 2,
  "authors": [
    {
      "name": "Anar Akhmedov"
    },
    {
      "name": "Azer Akhmedov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9segnzgq6yp7f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25080",
  "fulltext_url": null,
  "id": "9segnzgq6yp7f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25080",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AT"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Geometric and Algebraic Topology"
    ]
  ],
  "openalex": "W7214102666",
  "pdf_url": "https://arxiv.org/pdf/2609.25080",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9segnzgq6yp7f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25080",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25080"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Nonabelian Torus Exteriors in $\\mathbb S^4$ and Torus Replacement in Four-Manifolds",
  "title_text": "Nonabelian Torus Exteriors in 𝕊^4 and Torus Replacement in Four-Manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/9segnzgq6yp7f",
  "venue": "arXiv",
  "versions": []
}