Algebraic vector bundles on spheres
Résumé
We determine the first non-stable A1-homotopy sheaf of SLn. 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.