Transfers on $\mathbb A^1$-connected components of quasi-split groups and the norm principle

TL;DR

The paper proves that for quasi-split groups over perfect fields, the sheaf of ${\mathbb A^1}$-connected components $\pi_0^{\mathbb A^1}(G)$ is strictly ${\mathbb A^1}$-invariant with transfers, which implies Merkurjev's norm principle for such groups. The authors establish $\mathbb A^1$-connectedness of the flag variety $G/B$ and develop transfers on the cellular ${\mathbb A^1}$-homology, leveraging these to transfer structural information to $\pi_0^{\mathbb A^1}(G)$. A key step is showing that the $\mathbb A^1$-fibration $T\to G\to G/T$ and the ${\mathbb A^1}$-connectedness of $G/T$ ensure the transfer-stability of the image of $\pi_1^{\mathbb A^1}(G/T)$. Consequently, the norm principle holds for all quasi-split groups over perfect fields, with a classification-free, ${\mathbb A^1}$-homotopy-theoretic approach.

Abstract

We show that the sheaf of $\mathbb A^1$-connected components of a quasi-split group over a perfect field is a strictly $\mathbb A^1$-invariant sheaf with (Voevodsky) transfers. As a consequence, we show that the norm principle holds for any quasi-split group over a perfect field.

Theorems & Definitions (33)

• Theorem 1.1
• Corollary 1.2
• Conjecture 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Lemma 2.5