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


Authors: Aravind Asok and Jean Fasel
Journal: J. Amer. Math. Soc. 28 (2015), 1031-1062
MSC (2010): Primary 14F42, 55S35, 13C10; Secondary 19A13, 19D45
Published electronically: August 7, 2014
MathSciNet review: 3369908
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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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Additional Information

Aravind Asok
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532
Email: asok@usc.edu

Jean Fasel
Affiliation: Fakultät Mathematik, Universität Duisburg-Essen, Campus Essen, Thea-Leymann-Strasse 9, D-45127 Essen, Germany
Email: jean.fasel@gmail.com

DOI: https://doi.org/10.1090/S0894-0347-2014-00818-3
Received by editor(s): June 11, 2013
Received by editor(s) in revised form: February 18, 2014, April 15, 2014, June 3, 2014, and June 10, 2014
Published electronically: August 7, 2014
Additional Notes: The first author was supported in part by NSF Awards DMS-0900813 and DMS-1966589.
The second author was supported by DFG Grant SFB Transregio 45.
Article copyright: © Copyright 2014 American Mathematical Society