The Hurewicz map in motivic homotopy theory

TL;DR

This paper investigates the properties of the Hurewicz map in motivic homotopy theory, establishing its surjectivity for certain sheaves and analyzing its behavior for projective lines, advancing understanding of algebraic topology over fields.

Contribution

It proves the surjectivity of the Hurewicz map for $\\A^1$-connected sheaves and shows that the image and kernel of morphisms between strongly $\\A^1$-invariant sheaves are also strongly invariant.

Findings

Hurewicz map $\\pi_1^{\A^1} \to H_1^{\A^1}$ is surjective for $\\A^1$-connected sheaves.

Hurewicz map for $\\P^1_k$ is the abelianisation map.

Image and kernel of morphisms of strongly $\\A^1$-invariant sheaves are also strongly $\\A^1$-invariant.

Abstract

For an $\A^1$-connected pointed simplicial sheaf $\sX$ over a perfect field $k$, we prove that the Hurewicz map $\pi_1^{\A^1}(\sX) \to H_1^{\A^1}(\sX)$ 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 $\phi$ of strongly $\A^1$-invariant sheaves of groups, the image and kernel of $\phi$ are also strongly $\A^1$-invariant.