Pontryagin-Thom, $π_0$ of the motivic sphere, and principal quadratic divisors

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.