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

By A. Asok, Jean Fasel. 2012

57 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4f57d4d873f568aa3b102863d1fcf24e8d0bad5b

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Geometric models for higher Grothendieck-Witt groups in A1-homotopy theory (2013, methodology): “Our result has already been used in the work of [1,18] and opens the door to a classification of unstable operations in Grothendieck–Witt theory as done in [11] for K -theory.”