A1-Algebraic Topology over a Field

A1-Algebraic Topology over a Field


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.

Table of contents

Front Matter....Pages i-x
Introduction....Pages 1-13
Unramified Sheaves and Strongly $${\mathbb{A}}^{1}$$ -Invariant Sheaves....Pages 15-48
Unramified MilnorWitt K-Theories....Pages 49-80
Geometric Versus Canonical Transfers....Pages 81-112
The RostSchmid Complex of a Strongly $${\mathbb{A}}^{1}$$ -Invariant Sheaf....Pages 113-148
$${\mathbb{A}}^{1}$$ -Homotopy Sheaves and $${\mathbb{A}}^{1}$$ -Homology Sheaves....Pages 149-175
$${\mathbb{A}}^{1}$$ -Coverings, $${\pi }_{1}^{{\mathbb{A}}^{1} }({\mathbb{P}}^{n})$$ and $${\pi }_{1}^{{\mathbb{A}}^{1} }(S{L}_{n})$$ ....Pages 177-197
$${\mathbb{A}}^{1}$$ -Homotopy and Algebraic Vector Bundles....Pages 199-207
The Affine B.G. Property for the Linear Groups and the Grassmanian....Pages 209-226
Back Matter....Pages 227-259


  • Author: Fabien Morel (auth.)
  • Edition: 1
  • Publication Date: 2012
  • Publisher: Springer-Verlag Berlin Heidelberg
  • ISBN-10: 3642295134
  • ISBN-13: 9783642295133
  • Pages: 270
  • Format: pdf
  • Size: 3.3M
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