${\mathbb{A}}^1$-algebraic topology

Abstract

We present some recent results in A1-algebraic topology, which means both in A1-homotopy theory of schemes and its relationship with algebraic geometry. This refers to the classical relationship between homotopy theory and (differential) topology. We explain several examples of “motivic” versions of classical results: the theory of the Brouwer degree, the classification of A1-coverings through the A1-fundamental group, the Hurewicz Theorem and the A1-homotopy of algebraic spheres, and the A1-homotopy classification of vector bundles. We also give some applications and perspectives.