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A Primer on Mapping Class Groups (PMS-49)

Book Description

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 giving full details and keeping the text nearly self-contained. The book is suitable for graduate students.

A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.

Table of Contents

  1. Cover
  2. Half title
  3. Title
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Acknowledgments
  9. Overview
  10. Part 1. Mapping Class Groups
    1. 1. Curves, Surfaces, and Hyperbolic Geometry
      1. 1.1 Surfaces and Hyperbolic Geometry
      2. 1.2 Simple Closed Curves
      3. 1.3 The Change of Coordinates Principle
      4. 1.4 Three Facts about Homeomorphisms
    2. 2. Mapping Class Group Basics
      1. 2.1 Definition and First Examples
      2. 2.2 Computations of the Simplest Mapping Class Groups
      3. 2.3 The Alexander Method
    3. 3. Dehn Twists
      1. 3.1 Definition and Nontriviality
      2. 3.2 Dehn Twists and Intersection Numbers
      3. 3.3 Basic Facts about Dehn Twists
      4. 3.4 The Center of the Mapping Class Group
      5. 3.5 Relations between Two Dehn Twists
      6. 3.6 Cutting, Capping, and Including
    4. 4. Generating the Mapping Class Group
      1. 4.1 The Complex of Curves
      2. 4.2 The Birman Exact Sequence
      3. 4.3 Proof of Finite Generation
      4. 4.4 Explicit Sets of Generators
    5. 5. Presentations and Low-dimensional Homology
      1. 5.1 The Lantern Relation and H1 (Mod(S); Z)
      2. 5.2 Presentations for the Mapping Class Group
      3. 5.3 Proof of Finite Presentability
      4. 5.4 Hopf’s Formula and H2(Mod(S); Z)
      5. 5.5 The Euler Class
      6. 5.6 Surface Bundles and the Meyer Signature Cocycle
    6. 6. The Symplectic Representation and the Torelli Group
      1. 6.1 Algebraic Intersection Number as a Symplectic Form
      2. 6.2 The Euclidean Algorithm for Simple Closed Curves
      3. 6.3 Mapping Classes as Symplectic Automorphisms
      4. 6.4 Congruence Subgroups, Torsion-free Subgroups, and Residual Finiteness
      5. 6.5 The Torelli Group
      6. 6.6 The Johnson Homomorphism
    7. 7. Torsion
      1. 7.1 Finite-order Mapping Classes versus Finite-order Homeomorphisms
      2. 7.2 Orbifolds, the 84(g − 1) Theorem, and the 4g + 2 Theorem
      3. 7.3 Realizing Finite Groups as Isometry Groups
      4. 7.4 Conjugacy Classes of Finite Subgroups
      5. 7.5 Generating the Mapping Class Group with Torsion
    8. 8. The Dehn–Nielsen–Baer Theorem
      1. 8.1 Statement of the Theorem
      2. 8.2 The Quasi-isometry Proof
      3. 8.3 Two Other Viewpoints
    9. 9. Braid Groups
      1. 9.1 The Braid Group: Three Perspectives
      2. 9.2 Basic Algebraic Structure of the Braid Group
      3. 9.3 The Pure Braid Group
      4. 9.4 Braid Groups and Symmetric Mapping Class Groups
  11. Part 2. Teichmüller Space And Moduli Space
    1. 10. Teichmüller Space
      1. 10.1 Definition of Teichmüller Space
      2. 10.2 Teichmüller Space of the Torus
      3. 10.3 The Algebraic Topology
      4. 10.4 Two Dimension Counts
      5. 10.5 The Teichmüller Space of a Pair of Pants
      6. 10.6 Fenchel–Nielsen Coordinates
      7. 10.7 The 9g − 9 Theorem
    2. 11. Teichmüller Geometry
      1. 11.1 Quasiconformal Maps and an Extremal Problem
      2. 11.2 Measured Foliations
      3. 11.3 Holomorphic Quadratic Differentials
      4. 11.4 Teichmüller Maps and Teichmüller’s Theorems
      5. 11.5 Grötzsch’s Problem
      6. 11.6 Proof of Teichmüller’s Uniqueness Theorem
      7. 11.7 Proof of Teichmüller’s Existence Theorem
      8. 11.8 The Teichmüller Metric
    3. 12. Moduli Space
      1. 12.1 Moduli Space as the Quotient of Teichmüller Space
      2. 12.2 Moduli Space of the Torus
      3. 12.3 Proper Discontinuity
      4. 12.4 Mumford’s Compactness Criterion
      5. 12.5 The Topology at Infinity of Moduli Space
      6. 12.6 Moduli Space as a Classifying Space
  12. Part 3. The Classification And Pseudo-Anosov Theory
    1. 13. The Nielsen–Thurston Classification
      1. 13.1 The Classification for the Torus
      2. 13.2 The Three Types of Mapping Classes
      3. 13.3 Statement of the Nielsen–Thurston Classification
      4. 13.4 Thurston’s Geometric Classification of Mapping Tori
      5. 13.5 The Collar Lemma
      6. 13.6 Proof of the Classification Theorem
    2. 14. Pseudo-Anosov Theory
      1. 14.1 Five Constructions
      2. 14.2 Pseudo-Anosov Stretch Factors
      3. 14.3 Properties of the Stable and Unstable Foliations
      4. 14.4 The Orbits of a Pseudo-Anosov Homeomorphism
      5. 14.5 Lengths and Intersection Numbers under Iteration
    3. 15. Thurston’s Proof
      1. 15.1 A Fundamental Example
      2. 15.2 A Sketch of the General Theory
      3. 15.3 Markov Partitions
      4. 15.4 Other Points of View
  13. Bibliography
  14. Index