A^1-algebraic topology over a field, by Fabien Morel

This text is a revised and extended version of preprint* 794*: July 26, 2006, On the structure of A^1-homotopy sheaves, I 0794. In this work we prove some basic results in the context of A^1-homotopy theory of smooth schemes over a field k : the analogue of the Brouwer degree, the Hurewicz theorem, the theory of A^1-coverings and its relation to the fundamental A^1-homotopy sheaf, and some fundamental computations involving unramified Milnor-Witt K-theory like the fundamental A^1-homotopy sheaves of P^n and SL_n .

Fabien Morel <morel@mathematik.uni-muenchen.de>