Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects: LMS-CMI Research School, London, July 2018 (Lecture Notes in Mathematics, 2292)

This book provides an introduction to state-of-the-art applications of homotopy theory to arithmetic geometry. The contributions to this volume are based on original lectures by leading researchers at the LMS-CMI Research School on ‘Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects’ and the Nelder Fellow Lecturer Series, which both took place at Imperial College London in the summer of 2018. The contribution by Brazelton, based on the lectures by Wickelgren, provides an introduction to arithmetic enumerative geometry, the notes of Cisinski present motivic sheaves and new cohomological methods for intersection theory, and Schlank’s contribution gives an overview of the use of étale homotopy theory for obstructions to the existence of rational points on algebraic varieties. Finally, the article by Asok and Østvær, based in part on the Nelder Fellow lecture series by Østvær, gives a survey of the interplay between motivic homotopy theory and affine algebraic geometry, with a focus on contractible algebraic varieties. Now a major trend in arithmetic geometry, this volume offers a detailed guide to the fascinating circle of recent applications of homotopy theory to number theory. It will be invaluable to research students entering the field, as well as postdoctoral and more established researchers. 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 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 ζ-Functions 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 5.2.5 A Snapshot of the Stable Motivic Homotopy Category Stable Representablity of Algebraic K-theory Milnor–Witt K-theory 5.3 Concrete A1-weak Equivalences 5.3.1 Constructing A1-weak Equivalences of Smooth Schemes 5.3.2 A1-weak Equivalences vs. Weak Equivalences 5.3.3 Cancellation Questions and A1-weak Equivalences 5.3.4 Danielewski Surfaces and Generalizations 5.3.5 Building Quasi-Affine A1-contractible Varieties Unipotent Quotients Other Quasi-Affine A1-contractible Varieties 5.4 Further Computations in A1-homotopy Theory 5.4.1 A1-homotopy Sheaves Basic Definitions A1-rigid Varieties Embed into H(k) 5.4.2 A1-connectedness and Geometry A1-connectedness and Rationality Properties 5.4.3 A1-homotopy Sheaves Spheres and Brouwer Degree 5.4.4 A1-homotopy at Infinity One-point Compactifications Stable End Spaces 5.5 Cancellation Questions and A1-contractibility 5.5.1 The Biregular Cancellation Problem 5.5.2 A1-contractibility vs Topological Contractibility Affine Lines on Topologically Contractible Surfaces Chow Groups and Vector Bundles on Topologically Contractible Surfaces 5.5.3 Cancellation Problems and the Russell Cubic The Russell Cubic and Equivariant K-theory Higher Chow Groups and Stable A1-contractibility 5.5.4 A1-contractibility of the Koras–Russell Threefold 5.5.5 Koras–Russell Fiber Bundles References Index This Book Provides An Introduction To State-of-the-art Applications Of Homotopy Theory To Arithmetic Geometry. The Contributions To This Volume Are Based On Original Lectures By Leading Researchers At The Lms-cmi Research School On ‘homotopy Theory And Arithmetic Geometry - Motivic And Diophantine Aspects’ And The Nelder Fellow Lecturer Series, Which Both Took Place At Imperial College London In The Summer Of 2018. The Contribution By Brazelton, Based On The Lectures By Wickelgren, Provides An Introduction To Arithmetic Enumerative Geometry, The Notes Of Cisinski Present Motivic Sheaves And New Cohomological Methods For Intersection Theory, And Schlank’s Contribution Gives An Overview On The Use Of étale Homotopy Theory For Obstructions On The Existence Of Rational Points For Algebraic Varieties. Finally, The Article By Asok And Østvær, Based In Part On The Nelder Fellow Lecture Series By Østvær, Gives A Survey Of The Interplay Between Motivic Homotopy Theory And Affine Algebraic Geometry, With A Focus On Contractible Algebraic Varieties. Now A Major Trend In Arithmetic Geometry, This Volume Offers A Detailed Guide To The Fascinating Circle Of Recent Applications Of Homotopy Theory To Number Theory. It Will Be Invaluable To Research Students Entering The Field, As Well As Postdoctoral And More Established Researchers. This book provides an introduction to state-of-the-art applications of homotopy theory to arithmetic geometry. The contributions to this volume are based on original lectures by leading researchers at the LMS-CMI Research School on Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects and the Nelder Fellow Lecturer Series, which both took place at Imperial College London in the summer of 2018. The contribution by Brazelton, based on the lectures by Wickelgren, provides an introduction to arithmetic enumerative geometry, the notes of Cisinski present motivic sheaves and new cohomological methods for intersection theory, and Schlanks contribution gives an overview of the use of etale homotopy theory for obstructions to the existence of rational points on algebraic varieties. Finally, the article by Asok and stvr, based in part on the Nelder Fellow lecture series by stvr, gives a survey of the interplay between motivic homotopy theory and affine algebraic geometry, with a focus on contractible algebraic varieties. Now a major trend in arithmetic geometry, this volume offers a detailed guide to the fascinating circle of recent applications of homotopy theory to number theory. It will be invaluable to research students entering the field, as well as postdoctoral and more established researchers