A Primer on Mapping Class Groups (PMS-49) / Dan Margalit, Benson Farb.
- Course Book
- Princeton, NJ : Princeton University Press, 
- Princeton mathematical series ; 49.
Princeton Mathematical Series ; 49
1 online resource (489 p.)
- Mappings (Mathematics)
Class groups (Mathematics)
- Electronic books.
- 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.
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
- Description based upon print version of record.
Includes bibliographical references and index.
Description based on online resource; title from PDF title page (publisher's Web site, viewed 08. Jul 2019)
- Margalit, Dan, author.
- Publisher Number:
- 10.1515/9781400839049 doi
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