{
  "abstract": "\"Skeletal homology\" $K_{n}^{\\varepsilon}(X)$ refers to the homology of the chain complex $S_{n}^{\\varepsilon}(X)$ generated by skeletal $(n,\\varepsilon)$-simplices, i.e. functions from the $0$-skeleton of the standard simplex into a metric space $X$, with image diameter less than $\\varepsilon>0$. This homology was previously defined by Goldfarb, who showed that for finite metric spaces, it is isomorphic to the simplicial homology $H_{n}^Δ(VR_{\\varepsilon}(X))$ of the VR complex. We prove an isomorphism for arbitrary metric spaces, and introduce new methods to understand homology at scale. We define an invariant metric on $S_{n}^{\\varepsilon}(X)$, called the ultradiamond metric, that extends the uniform metric on skeletal simplices. With this metric we prove that \"close cycles are homologous\", which quickly leads to a host of stability results. We modify methods from singular homology to prove a strong generalization of Hausmann's Theorem, one of the two main justifications to use $H_{n}^Δ(VR_{\\varepsilon}(X))$ as a proxy for homology in discrete metric spaces. The second justification is Latchev's Theorem, for which we also prove a strong generalization. We define a homomorphism $ρ_{\\varepsilon}:H_{n}(X)\\rightarrow K_{n}^{\\varepsilon}(X)$ induced by repeated barycentric subdivision and restriction, the image of which we call \"real homology\" $H_{n}^{\\varepsilon}(X)$ at scale. We argue that $H_{n}^{\\varepsilon}(X)$ better represents bona fide homology at scale than $H_{n}^{Δ}(VR_{\\varepsilon}(X))$. To distinguish them, we define \"phantom homology\" to be $P_{n}^{\\varepsilon}(X)=K_{n}^{\\varepsilon}(X)/H_{n}^{\\varepsilon}(X)$, and use the stability of $K_{n}^{\\varepsilon}(X)$ to show that in collapse of Riemannian manifolds (e.g. the Berger Spheres), phantom homology can anticipate the abrupt drop in dimension that occurs in the limit.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.16009",
  "author_total": 2,
  "authors": [
    {
      "name": "Ivy Dey"
    },
    {
      "name": "Conrad Plaut"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30wtjqnm8g596.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.16009",
  "fulltext_url": null,
  "id": "30wtjqnm8g596",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.16009",
  "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.DG"
    ],
    [
      "openalex",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7169796469",
  "pdf_url": "https://arxiv.org/pdf/2607.16009",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30wtjqnm8g596.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2607.16009",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.16009"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Skeletal Homology",
  "title_text": "Skeletal Homology",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/30wtjqnm8g596",
  "venue": "arXiv",
  "versions": []
}