An explicit KO ‐degree map and applications

The 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 KiMW(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 πnA1(An∖0) for n⩾4 .

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