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

TL;DR

This paper interprets Suslin's conjecture on the image of the Hurewicz map in algebraic K-theory using unstable ${ m f A}^1$-homotopy sheaves, proving it in degree 5 for certain fields by connecting to motivic spheres.

Contribution

It provides a new interpretation of Suslin's conjecture in terms of unstable ${ m f A}^1$-homotopy sheaves and proves the conjecture in degree 5 for specific fields.

Findings

Confirmed Suslin's conjecture in degree 5 for fields with characteristic not 2 or 3.

Linked unstable ${ m f A}^1$-homotopy sheaves to motivic spheres.

Established a new interpretation of the conjecture via ${ m f A}^1$-homotopy theory.

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 $2$ or $3$. We do this by linking the relevant unstable ${\mathbb A}^1$-homotopy sheaf of the general linear group to the stable ${\mathbb A}^1$-homotopy of motivic spheres.