# <strong>A</strong><sup>1</sup>-algebraic topology

### Fabien Morel

Universität München, Germany

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## Abstract

We present some recent results in **A**1-algebraic topology, which means both in **A**1-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 **A**1-coverings through the **A**1-fundamental group, the Hurewicz Theorem and the **A**1-homotopy of algebraic spheres, and the **A**1-homotopy classification of vector bundles. We also give some applications and perspectives.