R-equivalence and A^1-connectedness in anisotropic groups

TL;DR

This paper establishes a deep connection between R-equivalence and A^1-equivalence in anisotropic semisimple algebraic groups, revealing structural properties of their A^1-connected components over fields.

Contribution

It proves that R-equivalence coincides with A^1-equivalence for anisotropic, simply connected groups, and shows Sing_*(G) is not A^1-local, leading to abelian sheaf structures.

Findings

R-equivalence and A^1-equivalence are equivalent in the specified groups

Sing_*(G) is not A^1-local for these groups

A^1-connected components form an abelian sheaf

Abstract

We show that if G is an anisotropic, semisimple, absolutely almost simple, simply connected group over a field k, then two elements of G over any field extension of k are R-equivalent if and only if they are A^1-equivalent. As a consequence, we see that Sing_*(G) cannot be A^1-local for such groups. This implies that the A^1-connected components of a semisimple, absolutely almost simple, simply connected group over a field k form a sheaf of abelian groups.