A Primer on Mapping Class Groups (PMS-49) / / Benson Farb, Dan Margalit.

• Superior document: Title is part of eBook package: De Gruyter Princeton Mathematical Series eBook Package
• VerfasserIn: Farb, Benson,; Margalit, Dan,
• Place / Publishing House: Princeton, NJ : : Princeton University Press, , [2011]; ©2012
• Year of Publication: 2011
• Edition: Course Book
• Language: English
• Series: Princeton Mathematical Series ; 49
• Physical Description: 1 online resource (512 p.) :; 115 line illus.

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

Frontmatter -- 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