Pontryagin-Thom, ${\pi }_0$ of the motivic sphere, and principal quadratic divisors

Segal computed ${\pi }_0$ 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 $A^u({\mathrm{\Gamma }}_k)$ of finite étale $k$-schemes decorated with a unit, the marks by the Dress map $d(L,u)={\mathrm{t}\mathrm{r}}_{L\mathrm{/}k}⟨u⟩$ into $\mathrm{G}\mathrm{W}(k)$, and a Pontryagin-Thom map $A^u({\mathrm{\Gamma }}_k)\to {\pi }_0({\mathbb{S}}_k)(k)$ is shown to be surjective with kernel $\ker (d)$. This gives ${\pi }_0({\mathbb{S}}_k)(k)\cong \mathrm{G}\mathrm{W}(k)$ for infinite perfect fields of characteristic $\mathrm{\ne }2$ 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 ${\pi }_0$ of the motivic sphere is the unit-decorated Burnside ring of ${\mathrm{\Gamma }}_k$ modulo principal quadratic divisors. The transversality step is proved by an elementary “re-topping” argument.