Jean Fasel

Algebraic vector bundles on spheres

We determine the first non-stable A1-homotopy sheaf of SL n. Using techniques of obstruction theory involving the A1-Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank at least d‐1 on split smooth affine quadrics of dimension 2d‐1. These computations allow us to answer a question posed by Nori, which gives a criterion for completability of certain unimodular rows. Furthermore, we study compatibility of our computations of A1-homotopy sheaves with real and complex realization.

A cohomological classification of vector bundles on smooth affine threefolds

We give a cohomological classification of vector bundles of rank

Some remarks on orbit sets of unimodular rows

2

On Stably Free Modules over Affine Algebras

on a smooth affine threefold over an algebraically closed field having characteristic unequal to

Stably free modules over smooth affine threefolds

2

Mennicke symbols, K -cohomology and a Bass-Kubota theorem

. As a consequence we deduce that cancellation holds for rank

Families of \mathbb{A}^1-contractible affine threefolds

2

Trivial Witt groups of flag varieties

vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first non-stable

Groupes de Chow-Witt

{\mathbb A}^1

The Chow-Witt ring.

-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morels