Basic structure of A^1-homotopy sheaves as unramified sheaves on the category of smooth $k$-schemes. A^1-motives and A^1-homology of spaces. The Hurewicz theorem. Description of the H_0 of smash powers of G_m's in terms of Milnor-Witt K-theory. Consequences and examples of computations. The first non-trivial A^1-homotopy sheaf of algebraic spheres. The Brouwer degree.

A^1-homotopy classification of rank n vector bundles. Application to the Euler class for rank n vector bundles over dim n affine smooth n-dimensional schemes and to stably free vector bundles.

The theory of A^1-coverings, A^1-universal coverings and A^1-fundamental group. Examples of computations (for surfaces for instance). Some perspectives towards a ``surgery classification" of smooth projective A^1-connected varieties and more.

References:

F. Morel. An introduction to A^1-homotopy theory <http://www.mathematik.uni-muenchen.de/%7Emorel/lectureTrieste.ps>, In Contemporary Developments in Algebraic K-theory, ICTP Lecture notes, 15 (2003), pp. 357-441, M. Karoubi, A.O. Kuku , C. Pedrini (ed.).

F. Morel. A^1-algebraic topology over a field <http://www.mathematik.uni-muenchen.de/~morel/A1homotopy.pdf>,
also available as preprint *806 *on the K-theory server, at <http://www.math.uiuc.edu/K-theory/0806/A1homotopy.pdf>

F. Morel. A^1-homotopy classification of vector bundles over smooth affine schemes <http://www.mathematik.uni-muenchen.de/%7Emorel/bgln.pdf>, Preprint.

F. Morel and V. Voevodsky, A^1-homotopy theory of schemes, Publications Mathématiques de l'I.H.E.S, volume 90.