## Geometry & Topology

### 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 $A1$–homotopy theory from part I of this paper. We then prove some $A1$–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 $A1$–homotopy theory.

#### Article information

Source
Geom. Topol., Volume 22, Number 2 (2018), 1181-1225.

Dates
Revised: 25 April 2017
Accepted: 24 May 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.gt/1517454118

Digital Object Identifier
doi:10.2140/gt.2018.22.1181

Mathematical Reviews number (MathSciNet)
MR3748687

Zentralblatt MATH identifier
06828607

#### Citation

Asok, Aravind; Hoyois, Marc; Wendt, Matthias. Affine representability results in $\mathbb{A}^1$–homotopy theory, II: Principal bundles and homogeneous spaces. Geom. Topol. 22 (2018), no. 2, 1181--1225. doi:10.2140/gt.2018.22.1181. https://projecteuclid.org/euclid.gt/1517454118