Abstract
We determine the first non-stable A 1 -homotopy sheaf of SL n . Using techniques of obstruction theory involving the A 1 -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 2 d - 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 A 1 -homotopy sheaves with real and complex realization.
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