${\mathbb{A}}^1$-connected components of classifying spaces and purity for torsors

In this paper, we study the Nisnevich sheafification ${\mathcal{H}}_{\acute{e}t}^1(G)$ of the presheaf associating to a smooth scheme the set of isomorphism classes of $G$-torsors, for a reductive group $G$. We show that if $G$-torsors on affine lines are extended, then ${\mathcal{H}}_{\acute{e}t}^1(G)$ is homotopy invariant and show that the sheaf is unramified if and only if Nisnevich-local purity holds for $G$-torsors. We also identify the sheaf ${\mathcal{H}}_{\acute{e}t}^1(G)$ with the sheaf of ${\mathbb{A}}^1$-connected components of the classifying space ${\rm B}_{\acute{e}t}G$. This establishes the homotopy invariance of the sheaves of components as conjectured by Morel. It moreover provides a computation of the sheaf of ${\mathbb{A}}^1$-connected components in terms of unramified $G$-torsors over function fields whenever Nisnevich-local purity holds for $G$-torsors.