Splitting vector bundles outside the stable range and 𝔸¹-homotopy sheaves of punctured affine spaces

Asok, Aravind; Fasel, Jean

doi:10.1090/S0894-0347-2014-00818-3

Splitting vector bundles outside the stable range and ${\mathbb A}^1$-homotopy sheaves of punctured affine spaces
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by Aravind Asok and Jean Fasel

J. Amer. Math. Soc. 28 (2015), 1031-1062

DOI: https://doi.org/10.1090/S0894-0347-2014-00818-3

Published electronically: August 7, 2014

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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 $\boldsymbol {\pi }_3^{{\mathbb A}^1}({\mathbb A}^3 {\setminus } 0)$ 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.

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