Includes bibliographical references (p. 255-258) and index.
Unramified sheaves and strongly A¹-invariant sheaves
Unramified Milnor-Witt K-theories
Geometric versus canonical transfers
The Rost-Schmid complex of a strongly A¹-invariant sheaf
A¹-homotopy sheaves and A¹-homology sheaves
A¹-coverings, [Pi]A¹1 (Pn) and [Pi]A¹1 (SLn)
A¹-homotopy and algebraic vector bundles
The affine B.G. property for the linear groups and the Grassmanian
The (Affine) B.G. property for simplicial sheaves
Recollection on obstruction theory.
This text deals with A¹-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 A¹-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 A¹-homotopy sheaves, A¹-homotogy sheaves, and sheaves with generalized transfers, as well as to algebraic vector bundles over affine smooth varieties.