  We discuss algebraic vector bundles on smooth k-schemes X contractible from
the standpoint of A^1-homotopy theory; when k = C, the smooth manifolds X(C)
are contractible as topological spaces. The integral algebraic K-theory and
integral motivic cohomology of such schemes are that of Spec k. One might hope
that furthermore, and in analogy with the classification of topological vector
bundles on manifolds, algebraic vector bundles on such schemes are all
isomorphic to trivial bundles; this is almost certainly true when the scheme is
affine. However, in the non-affine case this is false: we show that
(essentially) every smooth A^1-contractible strictly quasi-affine scheme that
admits a U-torsor whose total space is affine, for U a unipotent group,
possesses a non-trivial vector bundle. Indeed we produce explicit arbitrary
dimensional families of non-isomorphic such schemes, with each scheme in the
family equipped with "as many" (i.e., arbitrary dimensional moduli of)
non-isomorphic vector bundles, of every sufficiently large rank n, as one
desires; neither the schemes nor the vector bundles on them are distinguishable
by algebraic K-theory. We also discuss the triviality of vector bundles for
certain smooth complex affine varieties whose underlying complex manifolds are
contractible, but that are not necessarily A^1-contractible.
