On the ${\mathbb{A}}^1$-invariance of ${\mathrm{K}}_2$ of Chevalley groups of simply-laced type

In this paper we study the ${\mathbb{A}}^1$-invariance of the unstable functor ${\mathrm{K}}_2(\mathrm{\Phi },R)$ in the case when $\mathrm{\Phi }$ is an irreducible root system of type $\mathsf{A}\mathsf{D}\mathsf{E}$ containing ${\mathsf{A}}_4$ and not of type ${\mathsf{E}}_8$. We show that in the geometric case, i. e. when $R$ is a regular ring containing a field $k$ one has ${\mathrm{K}}_2(\mathrm{\Phi },R[t])={\mathrm{K}}_2(\mathrm{\Phi },R)$, which allows one to interpret the unstable ${\mathrm{K}}_2$ groups as ${\mathbb{A}}^1$-fundamental groups of Chevalley–Demazure group schemes in the ${\mathbb{A}}^1$-homotopy category. We also prove a variant of “early stability” theorem which allows one to find a generating set of ${\mathrm{K}}_2(\mathrm{\Phi },A[X_1,\dots X_n])$ in the case when $A$ is a Dedekind domain.