{
  "abstract": "We prove that every mod $k$ integral cycle $T$ of sufficiently high codimension in an oriented Riemannian manifold $\\mathcal{M}$ can be approximated in flat norm by a cycle which is a $\\mathbb{Z}_k$-smooth submanifold $Σ$ of nearly the same area, up to a singular set of codimension 5. If in addition $T$ belongs to a mod $k$ homology class admitting an embedded $\\mathbb{Z}_k$-smooth manifold representative, then $Σ$ can be taken fully regular, up to the codimension one stratum of classical singularities. This extends the smooth approximation theory to cycles with coefficients in $\\mathbb{Z}/k\\mathbb{Z}$ in stable regimes. Moreover, we provide examples showing the impossibility of locally approximating minimizing cycles by smooth or $\\mathbb{Z}_k$-smooth submanifolds, even in homology classes with smooth or $\\mathbb{Z}_k$-smooth representatives.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20499",
  "author_total": 2,
  "authors": [
    {
      "name": "Gianmarco Caldini"
    },
    {
      "name": "Anna Skorobogatova"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fd75geddxsft9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20499",
  "fulltext_url": null,
  "id": "fd75geddxsft9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20499",
  "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.AT"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7222216773",
  "pdf_url": "https://arxiv.org/pdf/2610.20499",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fd75geddxsft9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20499",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20499"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "$\\mathbb{Z}_k$-smooth approximation of mod $k$ integral cycles",
  "title_text": "ℤ_k-smooth approximation of mod k integral cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/fd75geddxsft9",
  "venue": "arXiv",
  "versions": []
}