{
  "abstract": "Segal computed $π_0$ of the equivariant sphere spectrum as the Burnside ring by a Pontryagin-Thom construction, transversality, and marks. We carry out the same argument in the motivic setting over a field $k$: the Burnside ring is replaced by the Grothendieck ring $A^u(Γ_k)$ of finite étale $k$-schemes decorated with a unit, the marks by the Dress map $d(L,u)=\\mathrm{tr}_{L/k}\\langle u\\rangle$ into $\\mathrm{GW}(k)$, and a Pontryagin-Thom map $A^u(Γ_k)\\toπ_0(\\mathbb{S}_k)(k)$ is shown to be surjective with kernel $\\ker(d)$. This gives $π_0(\\mathbb{S}_k)(k)\\cong\\mathrm{GW}(k)$ for infinite perfect fields of characteristic $\\ne 2$ without Morel's computation. We then identify $\\ker(d)$, by a reciprocity law on $\\mathbb{P}^1$, as the group of principal quadratic divisors of rational functions, so that $π_0$ of the motivic sphere is the unit-decorated Burnside ring of $Γ_k$ modulo principal quadratic divisors. The transversality step is proved by an elementary \"re-topping\" argument.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.29065",
  "author_total": 1,
  "authors": [
    {
      "name": "Kyle Ormsby"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45375ffe2fasz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.29065",
  "fulltext_url": null,
  "id": "45375ffe2fasz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.29065",
  "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.AG"
    ],
    [
      "arxiv",
      "math.AT"
    ],
    [
      "openalex",
      "2610|Homotopy and Cohomology in Algebraic Topology"
    ]
  ],
  "openalex": "W7223240951",
  "pdf_url": "https://arxiv.org/pdf/2610.29065",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45375ffe2fasz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2610.29065",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.29065"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Pontryagin-Thom, $π_0$ of the motivic sphere, and principal quadratic divisors",
  "title_text": "Pontryagin-Thom, π_0 of the motivic sphere, and principal quadratic divisors",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/45375ffe2fasz",
  "venue": "arXiv",
  "versions": []
}