#### Volume 22, issue 2 (2018)

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

### Aravind Asok, Marc Hoyois and Matthias Wendt

Geometry & Topology 22 (2018) 1181–1225
##### Abstract

We establish a relative version of the abstract “affine representability” theorem in ${\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1}$–homotopy theory from part I of this paper. We then prove some ${\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1}$–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 ${\mathbb{A}}^{\phantom{\rule{0.3em}{0ex}}1}$–homotopy theory.

##### Keywords
motivic homotopy theory, principal bundles
##### Mathematical Subject Classification 2010
Primary: 14F42, 14L10, 20G15, 55R15
##### Publication
Received: 13 July 2016
Revised: 25 April 2017
Accepted: 24 May 2017
Published: 16 January 2018
Proposed: Haynes R Miller
Seconded: Mark Behrens, Stefan Schwede
##### Authors
 Aravind Asok Department of Mathematics University of Southern California Los Angeles, CA United States Marc Hoyois Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States Matthias Wendt Fakultät Für Mathematik Universität Duisburg-Essen Essen Germany