## Geometry & Topology

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

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

Aravind Asok, Marc Hoyois, and Matthias Wendt

#### 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.

#### Article information

**Source**

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

**Dates**

Received: 13 July 2016

Revised: 25 April 2017

Accepted: 24 May 2017

First available in Project Euclid: 1 February 2018

**Permanent link to this document**

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

**Subjects**

Primary: 14F42: Motivic cohomology; motivic homotopy theory [See also 19E15] 14L10: Group varieties 20G15: Linear algebraic groups over arbitrary fields 55R15: Classification

**Keywords**

motivic homotopy theory principal bundles

#### 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