Splitting vector bundles outside the stable range and A^1-homotopy sheaves of punctured affine spaces

Abstract

We discuss the relationship between the ${\mathbb{A}}^1$-homotopy sheaves of ${\mathbb{A}}^n\setminus 0$ and the problem of splitting off a trivial rank $1$ summand from a rank $n$-vector bundle. We begin by computing ${\pi }_3^{{\mathbb{A}}^1}({\mathbb{A}}^3\setminus 0)$, and providing a host of related computations of "non-stable" ${\mathbb{A}}^1$-homotopy sheaves. We then use our computation to deduce that a rank $3$ vector bundle on a smooth affine $4$-fold over an algebraically closed field having characteristic unequal to $2$ splits off a trivial rank $1$ summand if and only if its third Chern class (in Chow theory) is trivial. This result provides a positive answer to a case of a conjecture of M.P. Murthy.

33 pages; v3: Completely rewritten and reorganized (along with http://arxiv.org/abs/1204.0770 and http://arxiv.org/abs/1204.4538). v4: final version (before page proofs), accepted for publication in Journal of the AMS

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