The Hurewicz map in motivic homotopy theory

Abstract

For an $\A^1$-connected pointed simplicial sheaf $\sX$ over a perfect field $k$, we prove that the Hurewicz map $π_1^{\A^1}(\sX) \to H_1^{\A^1}(\sX)$ is surjective. We also observe that the Hurewicz map for ${\P }_k^1$ is the abelianisation map. In the course of proving this result, we also show that for any morphism $\varphi$ of strongly $\A^1$-invariant sheaves of groups, the image and kernel of $\varphi$ are also strongly $\A^1$-invariant.

9 pages

Cited by in corpus (1)

• On the Whitehead theorem for nilpotent motivic spaces