Includes bibliographical references (p. 255-258) and index.
Contents:
Introduction
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.
Summary:
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.