{
  "abstract": "Let $f:(\\C^n,0)\\to(\\C,0)$ be a holomorphic germ with an isolated critical point at the origin, and let $φ_f$ be the negative gradient flow of $\\re f$ with respect to a Riemannian metric. We show that the origin is an isolated invariant set of $φ_f$ and that its Conley index is the suspension of the Milnor fiber, \\[ h(φ_f,\\{0\\})\\;\\simeq\\;ΣF_f\\;\\simeq\\;\\bigvee_{μ(f)}S^n , \\] so that $μ(f)=\\rank \\CH_n(φ_f,\\{0\\};\\Z)$: the Milnor number is the rank of the homological Conley index of the flow in degree $n$. The proof uses continuation to a holomorphic morsification of $f$. Since $\\im f$ is a first integral of the flow, after a rotation of the target through an angle avoiding finitely many values there are no connecting orbits between the $μ(f)$ nondegenerate rest points that result, and the sum property of the Conley index applies. McCord's theorem then recovers the Poincaré--Hopf index of $-\\nabla\\re f$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10896",
  "author_total": 2,
  "authors": [
    {
      "name": "Jean-Paul Brasselet"
    },
    {
      "name": "Nivaldo Grulha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0y0710z1rw9kb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10896",
  "fulltext_url": null,
  "id": "0y0710z1rw9kb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10896",
  "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.DS"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7221152099",
  "pdf_url": "https://arxiv.org/pdf/2610.10896",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0y0710z1rw9kb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10896",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10896"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Milnor Fiber as a Local Conley Index",
  "title_text": "The Milnor Fiber as a Local Conley Index",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/0y0710z1rw9kb",
  "venue": "arXiv",
  "versions": []
}