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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| Language: | English |
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Princeton, N.J
Princeton University Press
2011
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| Edition: | 1 |
| Series: | Princeton Mathematical Series |
| Subjects: | |
| ISBN: | 0691147949, 9780691147949, 1400839041, 9781400839049 |
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| Abstract | 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. |
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| AbstractList | 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. No detailed description available for "A Primer on Mapping Class Groups (PMS-49)". 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 keeping the text nearly self-contained. 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. 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 Groupsbegins 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. |
| Author | Farb, Benson Margalit, Dan |
| Author_xml | – sequence: 1 fullname: Farb, Benson – sequence: 2 fullname: Margalit, Dan |
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| Copyright | 2012 Princeton University Press |
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| Keywords | Hyperbolic geometry Homology (mathematics) Bijection Subset Rectangle Subgroup Conjugacy class Closed geodesic Vector space Mathematical induction Euler characteristic Eigenvalues and eigenvectors Pair of pants (mathematics) Geometric group theory Fundamental domain Corollary Permutation Simply connected space Intersection number (graph theory) Finitely presented Curve Disk (mathematics) Continuous function Topology Cohomology Homomorphism Riemannian manifold Riemann surface Jordan curve theorem Mapping class group Existential quantification Dehn twist Division by zero Infimum and supremum Coset Dimension (vector space) Finite group Big O notation Homeomorphism Homotopy Special case Quasi-isometry Foliation Summation Train track (mathematics) Orbifold Diffeomorphism Equivalence class Metric space Diagram (category theory) Coordinate system Fundamental group Upper and lower bounds Linear map Covering space Pointwise Moduli space Theorem Exact sequence Free group Upper half-plane Elementary matrix Automorphism Compact space Disjoint union Braid group Compactification (mathematics) Intersection (set theory) Transverse measure Quadratic differential |
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| Notes | Includes bibliographical references (p. [447]-463) and index |
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| Snippet | 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... No detailed description available for "A Primer on Mapping Class Groups (PMS-49)". |
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| SubjectTerms | Algebraic Automorphism Big O notation Bijection Braid group Class groups (Mathematics) Closed geodesic Cohomology Compact space Compactification (mathematics) Conjugacy class Continuous function Coordinate system Corollary Coset Covering space Curve Dehn twist Diagram (category theory) Diffeomorphism Dimension (vector space) Disjoint union Disk (mathematics) Division by zero Eigenvalues and eigenvectors Elementary matrix Equivalence class Euler characteristic Exact sequence Existential quantification Finite group Finitely presented Foliation Free group Fundamental domain Fundamental group Geometric group theory Geometry Homeomorphism Homology (mathematics) Homomorphism Homotopy Hyperbolic geometry Infimum and supremum Intersection (set theory) Intersection number (graph theory) Jordan curve theorem Linear map Mapping class group Mappings (Mathematics) Mathematical induction MATHEMATICS MATHEMATICS / Advanced bisacsh MATHEMATICS / Geometry / Algebraic MATHEMATICS / Geometry / Algebraic bisacsh MATHEMATICS / Geometry / General MATHEMATICS / Topology bisacsh Metric space Moduli space Orbifold Pair of pants (mathematics) PBMW PBPD Permutation Pointwise Quadratic differential Quasi-isometry Rectangle Riemann surface Riemannian manifold Simply connected space Special case Subgroup Subset Summation Theorem Topology Train track (mathematics) Transverse measure Upper and lower bounds Upper half-plane Vector space |
| SubjectTermsDisplay | Algebraic Geometry Mathematics PBMW PBPD Topology |
| TableOfContents | 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 |
| Title | A primer on mapping class groups (Princeton mathematical series) |
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