A1-Algebraic Topology over a Field

This text deals with A ¹ -homotopy theory over a base field, i.e., with the natural homotopy theory associated to the category of smooth varieties over a field in which the affine line is imposed to be contractible. It is a natural sequel to the foundational paper on A ¹ -homotopy theory written together with V. Voevodsky. Inspired by classical results in algebraic topology, we present new techniques, new results and applications related to the properties and computations of A ¹ -homotopy sheaves, A ¹ -homology sheaves, and sheaves with generalized transfers, as well as to algebraic vector bundles over affine smooth varieties.