Non-stable $K_1$-functors and $R$-equivalence

Let $A$ be a commutative ring. The elementary subgroup $E_n(A)$ of $SL_n(A)$ is the subgroup generated by the elementary transvections $e+te_{ij}$, where $1\leqslant i,j\leqslant n$ are distinct and $t$ is any element of $A$. This notion generalizes to any reductive $A$-group scheme $G$ satisfying a suitable isotropy condition. Namely, one defines the elementary subgroup $E(A)$ of the group of $A$-points $G(A)$ as the subgroup generated by the $A$-points of unipotent radicals of parabolic subgroups of $G$. The functor $K_1^G(-)=G(-)/E(-)$ on the category of commutative $A$-algebras is called the non-stable $K_1$-functor associated to $G$. If $A=k$ is a field and $G$ is semisimple, $E(k)$ is nothing but the group $G(k)^+$ introduced by J. Tits; in this case $K_1^G(k)=W(k,G)$ is also called the Whitehead group of $G$, and its computation is the subject of the Kneser–Tits problem. In this context, it has been known for some time that if $G$ is simply connected, then $K_1^G(k)$ coincides with the $R$-equivalence class group $G(k)/R$ in the sense of Yu. Manin. We generalize this identification to reductive groups over rings other than fields and apply it to the study of birational properties of $K_1^G$ and $G(-)/R$. The talk is based on a joint work with P. Gille.