{
  "abstract": "We discuss the question of when a homology or cohomology class with twisted integer coefficients of a manifold $X$ is realised by a submanifold. While this question is classical in nature, providing an answer requires relatively modern techniques from parametrised homotopy theory. More specifically, we introduce cobordism classes twisted by a coefficient system and then define a twisted Thom space $\\operatorname{M^\\mathrm{tw}O}(n)$ over $\\operatorname{BO}(1)$, which serves as the classifying object for this cobordism theory under a twisted Pontryagin-Thom construction. As a result, a twisted homology class is realisable if and only if its Poincaré dual is the image of the twisted Thom class in $\\operatorname{M^\\mathrm{tw}O}(n)$ under a parametrised map $X \\to \\operatorname{M^\\mathrm{tw}O}(n)$ over $\\operatorname{BO}(1)$. By showing that $\\operatorname{M^\\mathrm{tw}O}(n)$ is parametrised homotopy equivalent to the unbased homotopy orbits of $\\operatorname{MSO}(n)$ under the orientation-reversing involution, we are able to construct its parametrised Postnikov tower to derive obstructions to realisability. We conclude by giving the first known examples of non-realisable integer homology classes in non-orientable manifolds.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.24462",
  "author_total": 3,
  "authors": [
    {
      "name": "Mark Grant"
    },
    {
      "name": "Michael Jung"
    },
    {
      "name": "Baylee Schutte"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5kfz9df4bdz0z.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.24462",
  "fulltext_url": null,
  "id": "5kfz9df4bdz0z",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.24462",
  "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.GT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7171511262",
  "pdf_url": "https://arxiv.org/pdf/2607.24462",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5kfz9df4bdz0z.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2607.24462",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.24462"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On Realisability of Twisted Homology",
  "title_text": "On Realisability of Twisted Homology",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:25Z",
  "url": "https://research.draconitic.com/paper/5kfz9df4bdz0z",
  "venue": "arXiv",
  "versions": []
}