The Hurewicz map in motivic homotopy theory

Abstract

. For an A 1 -connected pointed simplicial sheaf X over a perfect field k , we prove that the Hurewicz map π A 1 1 ( X ) → H A 1 1 ( X ) is surjective. We also observe that the Hurewicz map for P 1 k is the abelianisation map. In the course of proving this result, we also show that for any morphism φ of strongly A 1 -invariant sheaves of groups, the image and kernel of φ are also strongly A 1 -invariant.