An explicit KO‐degree map and applications

Abstract

The goal of this note is to study the analog in unstable A 1 -homotopy theory of the unit map from the motivic sphere spectrum to the Hermitian K-theory spectrum, i.e., the degree map in Hermitian K-theory. We show that "Suslin matrices", which are explicit maps from odd dimensional split smooth affine quadrics to geometric models of the spaces appearing in Bott periodicity in Hermitian K-theory, stabilize in a suitable sense to the unit map. As applications, we deduce that, which can be thought of as an extension of Matsumoto's celebrated theorem describing K 2 of a field. These results provide the first step in a program aimed at computing the sheaf π A 1 n (A n \ 0) for n ≥ 4.

The first stable homotopy groups of motivic spheres

“…Here f 0 (KQ) is the effective cover of hermitian K-theory arising in the slice filtration of the motivic stable homotopy category of F (this does not affect homotopy groups of nonnegative weight). The rightmost map in (1.2) is surjective for n ≥ −4 (compare with [4,Corollary 6]). The exact sequence (1.2) vastly generalizes computations in [51] for fields of cohomological dimension at most two, and in [13] and [19] for the real numbers.…”

Unstable motivic homotopy theory

“…The conjecture is proved for n = 4, again by analysis of a group in the 1-stem: π A 1 3 (A 3 − {0}). In spite of considerable attention, [AF13], [AF17], [AWW17], the groups π A 1 n (A n − {0}) have not been calculated for n ≥ 4, and the problem remains open.…”

A generalized Vaserstein symbol

“…As a consequence of all this, there is an isomorphism [X, GL/Sp] A 1 = GW 3 1 (R). It is argued in [AF2] that the morphisms of schemes GL 2n → A 2n , M → M t ψ 2n M induce an isomorphism GL/Sp ∼ = A of Nisnevich sheaves, where A = colim n A 2n (the transition maps are given by adding ψ 2 ). Altogether, we obtain an isomorphism [X, A] A 1 = GW 3 1 (R) and [X, A] A 1 is precisely A(R)/ ∼= W ′ E (R).…”

“…Following [AF2], we refer to this action as the conjugation action of R × on GW 3 1 (R) ∼ = V (R). Recall the we have already defined an action of R × on V (R) for any ring R with 2 ∈ R × in section 3.2.…”

“…which induces an action of R × = G m (R) on GW 3 1 (R) via the product map mentioned above. Again following [AF2], we refer to this action as the multiplicative action of R × = G m (R) on GW 3 1 (R) ∼ = V (R). It follows from the proof of [AF1, Proposition 3.5.1] that the multiplicative action coincides with the conjugation action.…”