An explicit KO-degree map and applications

Abstract

The goal of this note is to study the analog in unstable ${\mathbb{A}}^1$-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 $K_i^{MW}(F)=G{W}_i^i(F)$ for $i\le 3$, 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 ${\pi }_n^{{\mathbb{A}}^1}({\mathbb{A}}^n\setminus 0)$ for $n\ge 4$.

36 Pages, Final version, to appear Journal of Topology

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