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.

Why take the trouble to prove something Morel discovered* in 2004? Well, Morel's proof has always mystified me. It's beautiful, but it doesn't hew to the Pontryagin-Thom-type structure familiar to homotopy theorists, and I've always wondered... *invented

... 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.

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.