Affine representability results in $\mathbb{A}^1$–homotopy theory, II: Principal bundles and homogeneous spaces

Abstract

We establish a relative version of the abstract “affine representability” theorem in ${\mathbb{A}}^1$–homotopy theory from part I of this paper. We then prove some ${\mathbb{A}}^1$–invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass–Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in ${\mathbb{A}}^1$–homotopy theory.