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Table of Contents
2.3.2 Sketch of Proof for Theorem 4
2.3.3 A1-Milnor Numbers
2.3.4 An Arithmetic Count of the Lines on a Smooth Cubic Surface
2.3.5 An Arithmetic Count of the Lines Meeting 4Lines in Space
Notation Guide
References
3 Cohomological Methods in Intersection Theory
3.1 Introduction
3.2 Étale Motives
3.2.1 The h-topology
3.2.2 Construction of Motives, After Voevodsky
3.2.3 Functoriality
3.2.4 Representability Theorems
3.3 Finiteness and Euler Characteristic
3.3.1 Locally Constructible Motives
3.3.2 Integrality of Traces and Rationality of -Functions
Intro
Preface
Contents
1 Homotopy Theory and Arithmetic Geometry-Motivic and Diophantine Aspects: An Introduction
1.1 Overview of Themes
1.2 Summaries of Individual Contributions
References
2 An Introduction to A1-Enumerative Geometry
2.1 Introduction
2.2 Preliminaries
2.2.1 Enriching the Topological Degree
2.2.2 The Grothendieck-Witt Ring
2.2.3 Lannes' Formula
2.2.4 The Unstable Motivic Homotopy Category
2.2.5 Colimits
2.2.6 Purity
2.3 A1-enumerative Geometry
2.3.1 The Eisenbud-Khimshiashvili-Levine Signature Formula
3.3.3 Grothendieck-Verdier Duality
3.3.4 Generic Base Change: A Motivic Variation on Deligne's Proof
3.4 Characteristic Classes
3.4.1 Künneth Formula
3.4.2 Grothendieck-Lefschetz Formula
References
4 Étale Homotopy and Obstructions to Rational Points
4.1 Introduction
4.2 ∞-Categories
4.2.1 Motivation
4.2.2 Quasi-Categories
4.2.3 ∞-Groupoids and the Homotopy Hypothesis
4.2.4 Quasi-Categories from Topological Categories
4.2.5 ∞-Category Theory
4.2.6 The Homotopy Category
4.2.7 ∞-Categories and Homological Algebra
4.2.8 Stable ∞-Categories
4.2.9 Localization
4.3 ∞-Topoi
4.3.1 Definitions
4.3.2 The Shape of an ∞-Topos
4.4 Obstruction Theory
4.4.1 Obstruction Theory for Homotopy Types
4.4.2 For ∞-Topoi and Linear(ized) Versions
4.5 Étale Homotopy and Rational Points
4.5.1 The étale ∞-Topos
4.5.2 Rational Points
4.5.3 The Local-to-Global Principle
4.6 Galois Theory and Embedding Problems
4.6.1 Topoi and Embedding Problems
References
5 A1-homotopy Theory and Contractible Varieties: A Survey
5.1 Introduction: Topological and Algebro-Geometric Motivations
5.1.1 Open Contractible Manifolds
5.1.2 Contractible Algebraic Varieties
5.2 A User's Guide to A1-homotopy Theory
5.2.1 Brief Topological Motivation
5.2.2 Homotopy Functors in Algebraic Geometry
5.2.3 The Unstable A1-homotopy Category: Construction
Spaces
Nisnevich and cdh Distinguished Squares
Localization
5.2.4 The Unstable A1-homotopy Category: Basic Properties
Motivic Spheres
Representability Statements
Representability of Chow Groups
The Purity Isomorphism
Comparison of Nisnevich and cdh-local A1-weak Equivalences
2.3.3 A1-Milnor Numbers
2.3.4 An Arithmetic Count of the Lines on a Smooth Cubic Surface
2.3.5 An Arithmetic Count of the Lines Meeting 4Lines in Space
Notation Guide
References
3 Cohomological Methods in Intersection Theory
3.1 Introduction
3.2 Étale Motives
3.2.1 The h-topology
3.2.2 Construction of Motives, After Voevodsky
3.2.3 Functoriality
3.2.4 Representability Theorems
3.3 Finiteness and Euler Characteristic
3.3.1 Locally Constructible Motives
3.3.2 Integrality of Traces and Rationality of -Functions
Intro
Preface
Contents
1 Homotopy Theory and Arithmetic Geometry-Motivic and Diophantine Aspects: An Introduction
1.1 Overview of Themes
1.2 Summaries of Individual Contributions
References
2 An Introduction to A1-Enumerative Geometry
2.1 Introduction
2.2 Preliminaries
2.2.1 Enriching the Topological Degree
2.2.2 The Grothendieck-Witt Ring
2.2.3 Lannes' Formula
2.2.4 The Unstable Motivic Homotopy Category
2.2.5 Colimits
2.2.6 Purity
2.3 A1-enumerative Geometry
2.3.1 The Eisenbud-Khimshiashvili-Levine Signature Formula
3.3.3 Grothendieck-Verdier Duality
3.3.4 Generic Base Change: A Motivic Variation on Deligne's Proof
3.4 Characteristic Classes
3.4.1 Künneth Formula
3.4.2 Grothendieck-Lefschetz Formula
References
4 Étale Homotopy and Obstructions to Rational Points
4.1 Introduction
4.2 ∞-Categories
4.2.1 Motivation
4.2.2 Quasi-Categories
4.2.3 ∞-Groupoids and the Homotopy Hypothesis
4.2.4 Quasi-Categories from Topological Categories
4.2.5 ∞-Category Theory
4.2.6 The Homotopy Category
4.2.7 ∞-Categories and Homological Algebra
4.2.8 Stable ∞-Categories
4.2.9 Localization
4.3 ∞-Topoi
4.3.1 Definitions
4.3.2 The Shape of an ∞-Topos
4.4 Obstruction Theory
4.4.1 Obstruction Theory for Homotopy Types
4.4.2 For ∞-Topoi and Linear(ized) Versions
4.5 Étale Homotopy and Rational Points
4.5.1 The étale ∞-Topos
4.5.2 Rational Points
4.5.3 The Local-to-Global Principle
4.6 Galois Theory and Embedding Problems
4.6.1 Topoi and Embedding Problems
References
5 A1-homotopy Theory and Contractible Varieties: A Survey
5.1 Introduction: Topological and Algebro-Geometric Motivations
5.1.1 Open Contractible Manifolds
5.1.2 Contractible Algebraic Varieties
5.2 A User's Guide to A1-homotopy Theory
5.2.1 Brief Topological Motivation
5.2.2 Homotopy Functors in Algebraic Geometry
5.2.3 The Unstable A1-homotopy Category: Construction
Spaces
Nisnevich and cdh Distinguished Squares
Localization
5.2.4 The Unstable A1-homotopy Category: Basic Properties
Motivic Spheres
Representability Statements
Representability of Chow Groups
The Purity Isomorphism
Comparison of Nisnevich and cdh-local A1-weak Equivalences