Motivic spheres and the image of the Suslin-Hurewicz map

摘要

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 A1-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 or 3. We do this by linking the relevant unstable A1-homotopy sheaf of the general linear group to the stable A1-homotopy of motivic spheres.