  We study the problem of classifying projectivizations of rank-two vector
bundles over ${\mathbb P}^2$ up to various notions of equivalence that arise
naturally in ${\mathbb A}^1$-homotopy theory, namely ${\mathbb A}^1$-weak
equivalence and ${\mathbb A}^1$-$h$-cobordism.
  First, we classify such varieties up to ${\mathbb A}^1$-weak equivalence:
over algebraically closed fields having characteristic unequal to two the
classification can be given in terms of characteristic classes of the
underlying vector bundle. When the base field is ${\mathbb C}$, this
classification result can be compared to a corresponding topological result and
we find that the algebraic and topological homotopy classifications agree.
  Second, we study the problem of classifying such varieties up to ${\mathbb
A}^1$-$h$-cobordism using techniques of deformation theory. To this end, we
establish a deformation rigidity result for ${\mathbb P}^1$-bundles over
${\mathbb P}^2$ which links ${\mathbb A}^1$-$h$-cobordisms to deformations of
the underlying vector bundles. Using results from the deformation theory of
vector bundles we show that if $X$ is a ${\mathbb P}^1$-bundle over ${\mathbb
P}^2$ and $Y$ is the projectivization of a direct sum of line bundles on
${\mathbb P}^2$, then if $X$ is ${\mathbb A}^1$-weakly equivalent to $Y$, $X$
is also ${\mathbb A}^1$-$h$-cobordant to $Y$.
  Finally, we discuss some subtleties inherent in the definition of ${\mathbb
A}^1$-$h$-cobordism. We show, for instance, that direct ${\mathbb
A}^1$-$h$-cobordism fails to be an equivalence relation.
