  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.
