Strong ${\mathbb{A}}^1$-invariance of ${\mathbb{A}}^1$-connected components of reductive algebraic groups

Abstract

We show that the sheaf of ${\mathbb{A}}^1$-connected components of a reductive algebraic group over a perfect field is strongly ${\mathbb{A}}^1$-invariant. As a consequence, torsors under such groups give rise to ${\mathbb{A}}^1$-fiber sequences. We also show that sections of ${\mathbb{A}}^1$-connected components of anisotropic, semisimple, simply connected algebraic groups over an arbitrary field agree with their $R$-equivalence classes, thereby removing the perfectness assumption in the previously known results about the characterization of isotropy in terms of affine homotopy invariance of Nisnevich locally trivial torsors.

15 pages, comments are welcome, v4: final version before page proofs, accepted for publication by the Journal of Topology