Motivic spheres and the image of the Suslin--Hurewicz map

Abstract

We show that an old conjecture of A.A. Suslin characterizing the image of a Hurewicz map from Quillen K-theory in degree $n$ to Milnor K-theory in degree $n$ admits an interpretation in terms of unstable ${\mathbb A}^1$-homotopy sheaves of the general linear group. Using this identification, we establish Suslin's conjecture in degree $5$ for infinite fields having characteristic unequal to $2$