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.
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