Affine representability results in ${\mathbb A}^{1}$-homotopy theory III: Finite fields and complements

Abstract

We give a streamlined proof of A(1)-representability for G-torsors under isotropic reductive groups, extending previous results in this sequence of papers to finite fields. We then analyze a collection of group homomorphisms that yield fiber sequences in A(1)-homotopy theory, and identify the final examples of motivic spheres that arise as homogeneous spaces for reductive groups.