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arXiv · 2101.01489

The Hurewicz map in motivic homotopy theory

Utsav Choudhury·Amit Hogadi
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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.

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Utsav Choudhury, Amit Hogadi. 2021-01-05. The Hurewicz map in motivic homotopy theory. https://doi.org/10.2140/akt.2022.7.179

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