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

Abstract

We discuss the relationship between theA1{\mathbb A}^1-homotopy sheaves ofAn∖0{\mathbb A}^n {\setminus } 0and the problem of splitting off a trivial rank11summand from a ranknnvector bundle. We begin by computingπ3A1(A3∖0)\boldsymbol {\pi }_3^{{\mathbb A}^1}({\mathbb A}^3 {\setminus } 0)and providing a host of related computations of “non-stable”A1{\mathbb A}^1-homotopy sheaves. We then use our computation to deduce that a rank33vector bundle on a smooth affine44-fold over an algebraically closed field having characteristic unequal to22splits off a trivial rank11summand 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.