A1-algebraic topology over a field

Description

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. Read more... Introduction -- Background -- Materials and Methods -- Results and Discussions -- Conclusion. Unramified Sheaves and Strongly \mathbbA1A1-Invariant Sheaves -- Unramified Milnor-Witt K-Theories -- Geometric Versus Canonical Transfers -- The Rost-Schmid Complex of a Strongly \mathbbA1A1-Invariant Sheaf -- \mathbbA1A1-Homotopy Sheaves and \mathbbA1A1-Homology Sheaves -- \mathbbA1A1-Coverings, p1\mathbbA1 (\mathbbPn)1A1(Pn) and p1\mathbbA1 (SLn)1A1(SLn) -- \mathbbA1A1-Homotopy and Algebraic Vector Bundles -- The Affine B.G. Property for the Linear Groups and the Grassmanian