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.
Chapter
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)
5.6 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
6.6 The Johnson Homomorphism
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
8. The Dehn–Nielsen–Baer Theorem
8.1 Statement of the Theorem
8.2 The Quasi-isometry Proof
9.1 The Braid Group: Three Perspectives
9.2 Basic Algebraic Structure of the Braid Group
9.4 Braid Groups and Symmetric Mapping Class Groups
PART 2. TEICHMÜLLER SPACE AND MODULI 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
11.1 Quasiconformal Maps and an Extremal Problem
11.3 Holomorphic Quadratic Differentials
11.4 Teichmüller Maps and Teichmüller's Theorems
11.6 Proof of Teichmüller's Uniqueness Theorem
11.7 Proof of Teichmüller's Existence Theorem
11.8 The Teichmüller Metric
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
13.6 Proof of the Classification Theorem
14.2 Pseudo-Anosov Stretch Factors
14.3 Properties of the Stable and Unstable Foliations
14.4 The Orbits of a Pseudo-Anosov Homeomorphism
14.5 Lengths and Intersection Numbers under Iteration
15.1 A Fundamental Example
15.2 A Sketch of the General Theory
15.4 Other Points of View
A Primer on Mapping Class Groups (PMS-49)
：A Primer on Mapping Class Groups (PMS-49)
( Princeton Mathematical Series )
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.
