A1 Algebraic Topology over a Field 1 Introduction 2 Unramified sheaves and strongly A1 invariant sheaves 3 Unramified Milnor Witt K theories 4 Geometric versus canonical transfers 5 The Rost Schmid complex of a strongly A1 invariant sheaf 6 A1 homotopy sheaves and A1 homology sheaves 7 A1 coverings 8 A1 homotopy and algebraic vector bundles 9 The affine B.G. property for the linear groups and the Grassmanian 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 homology sheaves and sheaves with generalized transfers as well as to algebraic vector bundles over affine smooth varieties