A primer on mapping class groups (Princeton mathematical series)
The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time givi...
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| Hlavní autoři: | , |
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| Médium: | E-kniha Kniha |
| Jazyk: | angličtina |
| Vydáno: |
Princeton, N.J
Princeton University Press
2011
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| Vydání: | 1 |
| Edice: | Princeton Mathematical Series |
| Témata: | |
| ISBN: | 0691147949, 9780691147949, 1400839041, 9781400839049 |
| On-line přístup: | Získat plný text |
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- A primer on mapping class groups (Princeton mathematical series) -- Contents -- Preface -- Acknowledgments -- Overview -- Part 1: Mapping Class Groups -- Chapter One: Curves, Surfaces, and Hyperbolic Geometry -- Chapter Two: Mapping Class Group Basics -- Chapter Three: Dehn Twists -- Chapter Four: Generating the Mapping Class Group -- Chapter Five: Presentations and Low-dimensional Homology -- Chapter Six: The Symplectic Representation and the Torelli Group -- Chapter Seven: Torsion -- Chapter Eight: The Dehn-Nielsen-Baer Theorem -- Chapter Nine: Braid Groups -- Part 2: Teichmüller Space and Moduli Space -- Chapter Ten: Teichmüller Space -- Chapter Eleven: Teichmüller Geometry -- Chapter Twelve: Moduli Space -- Part 3: The Classification and Pseudo-Anosov Theory -- Chapter Thirteen: The Nielsen-Thurston Classification -- Chapter Fourteen: Pseudo-Anosov Theory -- Chapter Fifteen: Thurston's Proof -- Bibliography -- Index
- Front Matter Table of Contents Preface Acknowledgments Overview Chapter One: Curves, Surfaces, and Hyperbolic Geometry Chapter Two: Mapping Class Group Basics Chapter Three: Dehn Twists Chapter Four: Generating the Mapping Class Group Chapter Five: Presentations and Low-dimensional Homology Chapter Six: The Symplectic Representation and the Torelli Group Chapter Seven: Torsion Chapter Eight: The Dehn—Nielsen—Baer Theorem Chapter Nine: Braid Groups Chapter Ten: Teichmüller Space Chapter Eleven: Teichmüller Geometry Chapter Twelve: Moduli Space Chapter Thirteen: The Nielsen—Thurston Classification Chapter Fourteen: Pseudo-Anosov Theory Chapter Fifteen: Thurston’s Proof Bibliography Index
- 8. The Dehn-Nielsen-Baer Theorem -- 8.1 Statement of the Theorem -- 8.2 The Quasi-isometry Proof -- 8.3 Two Other Viewpoints -- 9. Braid Groups -- 9.1 The Braid Group: Three Perspectives -- 9.2 Basic Algebraic Structure of the Braid Group -- 9.3 The Pure Braid Group -- 9.4 Braid Groups and Symmetric Mapping Class Groups -- PART 2. TEICHMÜLLER SPACE AND MODULI SPACE -- 10. Teichmüller Space -- 10.1 Definition of Teichmüller Space -- 10.2 Teichmüller Space of the Torus -- 10.3 The Algebraic Topology -- 10.4 Two Dimension Counts -- 10.5 The Teichmüller Space of a Pair of Pants -- 10.6 Fenchel-Nielsen Coordinates -- 10.7 The 9g - 9 Theorem -- 11. Teichmüller Geometry -- 11.1 Quasiconformal Maps and an Extremal Problem -- 11.2 Measured Foliations -- 11.3 Holomorphic Quadratic Differentials -- 11.4 Teichmüller Maps and Teichmüller's Theorems -- 11.5 Grötzsch's Problem -- 11.6 Proof of Teichmüller's Uniqueness Theorem -- 11.7 Proof of Teichmüller's Existence Theorem -- 11.8 The Teichmüller Metric -- 12. Moduli Space -- 12.1 Moduli Space as the Quotient of Teichmüller Space -- 12.2 Moduli Space of the Torus -- 12.3 Proper Discontinuity -- 12.4 Mumford's Compactness Criterion -- 12.5 The Topology at Infinity of Moduli Space -- 12.6 Moduli Space as a Classifying Space -- PART 3. THE CLASSIFICATION AND PSEUDO-ANOSOV THEORY -- 13. The Nielsen-Thurston Classi.cation -- 13.1 The Classi.cation for the Torus -- 13.2 The Three Types of Mapping Classes -- 13.3 Statement of the Nielsen-Thurston Classification -- 13.4 Thurston's Geometric Classification of Mapping Tori -- 13.5 The Collar Lemma -- 13.6 Proof of the Classification Theorem -- 14. Pseudo-Anosov Theory -- 14.1 Five Constructions -- 14.2 Pseudo-Anosov Stretch Factors -- 14.3 Properties of the Stable and Unstable Foliations -- 14.4 The Orbits of a Pseudo-Anosov Homeomorphism
- Cover -- Contents -- Preface -- Acknowledgments -- Overview -- PART 1. MAPPING CLASS GROUPS -- 1. Curves, Surfaces, and Hyperbolic Geometry -- 1.1 Surfaces and Hyperbolic Geometry -- 1.2 Simple Closed Curves -- 1.3 The Change of Coordinates Principle -- 1.4 Three Facts about Homeomorphisms -- 2. Mapping Class Group Basics -- 2.1 Definition and First Examples -- 2.2 Computations of the Simplest Mapping Class Groups -- 2.3 The Alexander Method -- 3. Dehn Twists -- 3.1 Definition and Nontriviality -- 3.2 Dehn Twists and Intersection Numbers -- 3.3 Basic Facts about Dehn Twists -- 3.4 The Center of the Mapping Class Group -- 3.5 Relations between Two Dehn Twists -- 3.6 Cutting, Capping, and Including -- 4. Generating the Mapping Class Group -- 4.1 The Complex of Curves -- 4.2 The Birman Exact Sequence -- 4.3 Proof of Finite Generation -- 4.4 Explicit Sets of Generators -- 5. Presentations and Low-dimensional Homology -- 5.1 The Lantern Relation and H[sub(1)] (Mod(S) -- Z) -- 5.2 Presentations for the Mapping Class Group -- 5.3 Proof of Finite Presentability -- 5.4 Hopf's Formula and H[sub(2)] (Mod(S) -- Z) -- 5.5 The Euler Class -- 5.6 Surface Bundles and the Meyer Signature Cocycle -- 6. The Symplectic Representation and the Torelli Group -- 6.1 Algebraic Intersection Number as a Symplectic Form -- 6.2 The Euclidean Algorithm for Simple Closed Curves -- 6.3 Mapping Classes as Symplectic Automorphisms -- 6.4 Congruence Subgroups, Torsion-free Subgroups, and Residual Finiteness -- 6.5 The Torelli Group -- 6.6 The Johnson Homomorphism -- 7. Torsion -- 7.1 Finite-order Mapping Classes versus Finite-order Homeomorphisms -- 7.2 Orbifolds, the 84(g - 1) Theorem, and the 4g + 2 Theorem -- 7.3 Realizing Finite Groups as Isometry Groups -- 7.4 Conjugacy Classes of Finite Subgroups -- 7.5 Generating the Mapping Class Group with Torsion
- 14.5 Lengths and Intersection Numbers under Iteration -- 15. Thurston's Proof -- 15.1 A Fundamental Example -- 15.2 A Sketch of the General Theory -- 15.3 Markov Partitions -- 15.4 Other Points of View -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z
- Part 1. Mapping Class Groups --
- Part 3. The Classification and Pseudo-Anosov Theory --
- Chapter Nine. Braid Groups
- Chapter One. Curves, Surfaces, and Hyperbolic Geometry
- Index
- Chapter Four. Generating The Mapping Class Group
- Acknowledgments
- Chapter Eleven. Teichmüller Geometry
- Chapter Five. Presentations And Low-Dimensional Homology
- Chapter Eight. The Dehn–Nielsen–Baer Theorem
- Preface
- Part 2. Teichmüller Space and Moduli Space --
- Chapter Six. The Symplectic Representation and the Torelli Group
- Overview
- Chapter Seven. Torsion
- Chapter Three. Dehn Twists
- Chapter Fifteen. Thurston’S Proof
- -
- Chapter Thirteen. The Nielsen–Thurston Classification
- Chapter Twelve. Moduli Space
- Chapter Fourteen. Pseudo-Anosov Theory
- /
- Contents
- Frontmatter --
- Chapter Two. Mapping Class Group Basics
- Chapter Ten. Teichmüller Space
- Bibliography