An explicit KO-degree map and applications

Abstract

International audienceThe goal of this note is to study the analog in unstable A1-homotopy theory of the unit map from the motivic sphere spectrum to the Hermitian K-theory spectrum, that is, 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 KMWi(F)=GWii(F) for i⩽3, which can be thought of as an extension of Matsumoto's celebrated theorem describing K2 of a field. These results provide the first step in a program aimed at computing the sheaf πA1n(An∖0) for n⩾4