Abstract

Morel–Voevodsky's A 1 -homotopy theory transports tools from algebraic topology into arithmetic and algebraic geometry, allowing us to draw arithmetic conclusions from topological arguments. Comparison results between classical and A 1 -homotopy theories can also be used in the reverse direction, allowing us to infer topological results from algebraic calculations. For example, see the article by Isaksen and Østvær on motivic stable homotopy groups in this volume. The present article will introduce unstable A 1 -homotopy theory and give several applications.