Algebraic vector bundles on spheres

Abstract

We determine the first non-stable ${\mathbb{A}}^1$-homotopy sheaf of $S{L}_n$. Using techniques of obstruction theory involving the ${\mathbb{A}}^1$-Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank $\ge d-1$ 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 ${\mathbb{A}}^1$-homotopy sheaves with real and complex realization.

35 pages; final version (before page proofs) to appear J. Top. Significantly reorganized and incorporates some material from http://arxiv.org/abs/1204.0770 (which will also soon be replaced)

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• Chow-Witt rings of classifying spaces for symplectic and special linear groups
• Generically split octonion algebras and A^1-homotopy theory
• The cancellation of projective modules of rank 2 with a trivial determinant
• Enumerating Non-Stable Vector Bundles
• Motivic stable cohomotopy and unimodular rows
• Two lives: Compositions of unimodular rows