  We start to study the problem of classifying smooth proper varieties over a
field k from the standpoint of A^1-homotopy theory. Motivated by the
topological theory of surgery, we discuss the problem of classifying up to
isomorphism all smooth proper varieties having a specified A^1-homotopy type.
Arithmetic considerations involving the sheaf of A^1-connected components lead
us to introduce several different notions of connectedness in A^1-homotopy
theory. We provide concrete links between these notions, connectedness of
points by chains of affine lines, and various rationality properties of
algebraic varieties (e.g., rational connectedness).
  We introduce the notion of an A^1-h-cobordism, an algebro-geometric analog of
the topological notion of h-cobordism, and use it as a tool to produce
non-trivial A^1-weak equivalences of smooth proper varieties. Also, we give
explicit computations of refined A^1-homotopy invariants, such as the
A^1-fundamental sheaf of groups, for some A^1-connected varieties. We observe
that the A^1-fundamental sheaf of groups plays a central yet mysterious role in
the structure of A^1-h-cobordisms. As a consequence of these observations, we
completely solve the classification problem for rational smooth proper surfaces
over an algebraically closed field: while there exist arbitrary dimensional
moduli of such surfaces, there are only countably many A^1-homotopy types, each
uniquely determined by the isomorphism class of its A^1-fundamental sheaf of
groups.
