Post

New paper day! 🥳 🧮 Pontryagin-Thom, π₀ of the motivic sphere spectrum, and principal quadratic forms arxiv.org/abs/2610.29065 wherein I provide a novel proof of Morel's theorem identifying π₀(𝕊ₖ) with the Grothendieck-Witt ring GW(k) of quadratic forms.

The paper introduces a motivic Pontryagin-Thom construction PT taking a pair (L,u) — L a finite étale k-algebra, u a unit in L — to a framed correspondence e(L,u). Think of e(L,u) as the points Spec(L) in 𝔸¹ with framing (tangent data) u.

The source of PT is Aᵘ(Γₖ), the "unit decorated Burnside ring", which also admits a surjective map d to GW(k) taking (L,u) — L/k now a finite separable field extension; these generate Aᵘ(Γₖ) — to the corresponding trace form.

The main contributions here are to 1. define Aᵘ(Γₖ) and recognize that it's the correct domain for PT, 2. prove this via a Position Lemma (an elementary "re-topping" method) that allows us to avoid non-reduced points,

3. use the Scheja-Storch form w and a theorem of Cazanave on naïve 𝔸¹-homotopies to prove that w∘PT surjects onto GW(k) with kernel ker(d). By Voevodsky's theory of framed correspondences, this proves π₀(𝕊ₖ)≅GW(k).

Beyond this (section 6 of the paper), I demonstrate that the kernel of PT is generated by what I call 'principal quadratic divisors', with the proof passing through an elaboration of Scharlau's quadratic reciprocity law on ℙ¹ — the quadratic version of principal divisors on ℙ¹ having degree 0.

... if we can determine π₀(𝕊ₖ) without recourse to graded endomorphisms and Milnor-Witt K-theory. This paper demonstrates that we can, and I hope that it opens the door to theorems where Morel's pattern can't be followed, e.g. in equivariant stable motivic homotopy theory.

(Aside: Morel’s proof runs through the heart of the homotopy t-structure on SH(k), where π₀(𝕊ₖ)_* is the initial homotopy module and must be identified in all weights at once. Pontryagin–Thom methods work in weight zero alone and need nothing from the heart, which is understood only over fields.)

See the paper arxiv.org/abs/2610.29065 or my slides e-infinity.space/files/taters... for more info. I'll speak about this paper at the AMS Western Sectional meeting at ASU on Nov 7.