Motivic spheres and the image of the Suslin--Hurewicz map
Aravind Asok, Jean Fasel, Ben Williams

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 to Milnor K-theory in degree admits an interpretation in terms of unstable -homotopy sheaves of the general linear group. Using this identification, we establish Suslin's conjecture in degree for infinite fields having characteristic unequal to or . We do this by linking the relevant unstable -homotopy sheaf of the general linear group to the stable -homotopy of motivic spheres.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models
