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We start to study the problem of classifying smooth proper varieties over afield k from the standpoint of A^1-homotopy theory. Motivated by thetopological theory of surgery, we discuss the problem of classifying up toisomorphism all smooth proper varieties having a specified A^1-homotopy type.Arithmetic considerations involving the sheaf of A^1-connected components leadus to introduce several different notions of connectedness in A^1-homotopytheory. We provide concrete links between these notions, connectedness ofpoints by chains of affine lines, and various rationality properties ofalgebraic varieties (e.g., rational connectedness).  We introduce the notion of an A^1-h-cobordism, an algebro-geometric analog ofthe topological notion of h-cobordism, and use it as a tool to producenon-trivial A^1-weak equivalences of smooth proper varieties. Also, we giveexplicit computations of refined A^1-homotopy invariants, such as theA^1-fundamental sheaf of groups, for some A^1-connected varieties. We observethat the A^1-fundamental sheaf of groups plays a central yet mysterious role inthe structure of A^1-h-cobordisms. As a consequence of these observations, wecompletely solve the classification problem for rational smooth proper surfacesover an algebraically closed field: while there exist arbitrary dimensionalmoduli of such surfaces, there are only countably many A^1-homotopy types, eachuniquely determined by the isomorphism class of its A^1-fundamental sheaf ofgroups.

Smooth varieties up to A^1-homotopy and algebraic h-cobordisms