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PreprintCC BY 4.0arXiv

Pontryagin-Thom, π0 of the motivic sphere, and principal quadratic divisors

  • Kyle Ormsby
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Abstract

Segal computed π0 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 Au(Γk) of finite étale k-schemes decorated with a unit, the marks by the Dress map d(L,u)=trL/k⟨ u⟩ into GW(k), and a Pontryagin-Thom map Au(Γk)→π0(𝕊k)(k) is shown to be surjective with kernel ker(d). This gives π0(𝕊k)(k)≅GW(k) for infinite perfect fields of characteristic ≠ 2 without Morel's computation. We then identify ker(d), by a reciprocity law on ℙ1, as the group of principal quadratic divisors of rational functions, so that π0 of the motivic sphere is the unit-decorated Burnside ring of Γk modulo principal quadratic divisors. The transversality step is proved by an elementary "re-topping" argument.

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Cite

BibTeXRIS
BibTeX
@misc{ormsby2026pontryaginthom,
  title = {{Pontryagin-Thom, $π_0$ of the motivic sphere, and principal quadratic divisors}},
  author = {Kyle Ormsby},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29065},
  eprint = {2610.29065},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29065},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Pontryagin-Thom, π_0 of the motivic sphere, and principal quadratic divisors
AU  - Kyle Ormsby
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29065
UR  - https://arxiv.org/abs/2610.29065
AB  - 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)=tr_L/k⟨ u⟩ into GW(k), and a Pontryagin-Thom map A^u(Γ_k)→π_0(𝕊_k)(k) is shown to be surjective with kernel ker(d). This gives π_0(𝕊_k)(k)≅GW(k) for infinite perfect fields of characteristic ≠ 2 without Morel's computation. We then identify ker(d), by a reciprocity law on 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.
ER  -