A1-Algebraic Topology over a Field

1. Front Matter

   Pages i-x

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2. Introduction

   • Fabien Morel

   Pages 1-13

3. Unramified Sheaves and Strongly \({\mathbb{A}}^{1}\)-Invariant Sheaves

   • Fabien Morel

   Pages 15-48

4. Unramified Milnor–Witt K-Theories

   • Fabien Morel

   Pages 49-80

5. Geometric Versus Canonical Transfers

   • Fabien Morel

   Pages 81-112

6. The Rost–Schmid Complex of a Strongly \({\mathbb{A}}^{1}\)-Invariant Sheaf

   • Fabien Morel

   Pages 113-148

7. \({\mathbb{A}}^{1}\)-Homotopy Sheaves and \({\mathbb{A}}^{1}\)-Homology Sheaves

   • Fabien Morel

   Pages 149-175

8. \({\mathbb{A}}^{1}\)-Coverings, \({\pi }_{1}^{{\mathbb{A}}^{1} }({\mathbb{P}}^{n})\) and \({\pi }_{1}^{{\mathbb{A}}^{1} }(S{L}_{n})\)

   • Fabien Morel

   Pages 177-197

9. \({\mathbb{A}}^{1}\)-Homotopy and Algebraic Vector Bundles

   • Fabien Morel

   Pages 199-207

10. The Affine B.G. Property for the Linear Groups and the Grassmanian

    • Fabien Morel

    Pages 209-226

11. Back Matter

    Pages 227-259

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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.

• 14-XX; 19-XX; 55-XX
• Algebraic vector bundles
• Milnor-Witt K-theory
• Motivic homotopy theory
• Sheaves with transfers

• Institut für Mathematik, Universität München, München, Germany

  Fabien Morel

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