Algebraic vector bundles on spheres

Aravind Asok; Jean Fasel

doi:10.1112/jtopol/jtt046

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arXiv · 1204.4538

Algebraic vector bundles on spheres

Aravind Asok·Jean Fasel

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Abstract

We determine the first non-stable ${\mathbb A}^1$-homotopy sheaf of $SL_n$. 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 $\geq d-1$ 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.

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BibTeX RIS

Aravind Asok, Jean Fasel. 2012-04-20. Algebraic vector bundles on spheres. https://doi.org/10.1112/jtopol%2Fjtt046

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