Thurston's Work on Surfaces
1st Edition
Published on 20. July 2022
272 pages
978-1-4008-3903-2 (ISBN)
Description
This book provides a detailed exposition of William Thurston's work on surface homeomorphisms, available here for the first time in English. Based on material of Thurston presented at a seminar in Orsay from 1976 to 1977, it covers topics such as the space of measured foliations on a surface, the Thurston compactification of Teichmüller space, the Nielsen-Thurston classification of surface homeomorphisms, and dynamical properties of pseudo-Anosov diffeomorphisms. Thurston never published the complete proofs, so this text is the only resource for many aspects of the theory.
Thurston was awarded the prestigious Fields Medal in 1982 as well as many other prizes and honors, and is widely regarded to be one of the major mathematical figures of our time. Today, his important and influential work on surface homeomorphisms is enjoying continued interest in areas ranging from the Poincaré conjecture to topological dynamics and low-dimensional topology.
Conveying the extraordinary richness of Thurston's mathematical insight, this elegant and faithful translation from the original French will be an invaluable resource for the next generation of researchers and students.
Thurston was awarded the prestigious Fields Medal in 1982 as well as many other prizes and honors, and is widely regarded to be one of the major mathematical figures of our time. Today, his important and influential work on surface homeomorphisms is enjoying continued interest in areas ranging from the Poincaré conjecture to topological dynamics and low-dimensional topology.
Conveying the extraordinary richness of Thurston's mathematical insight, this elegant and faithful translation from the original French will be an invaluable resource for the next generation of researchers and students.
More details
Other editions
Additional editions
Albert Fathi | Francois Laudenbach | Valentin Poenaru
Thurston's Work on Surfaces
Book
04/2012
Princeton University Press
€68.50
Shipment within 10-20 days
Persons
Albert Fathi is professor at the École Normale Supérieure de Lyon. François Laudenbach is professor emeritus at the University of Nantes. Valentin Poénaru is professor emeritus at the Université Paris-Sud, Orsay. Djun Kim is a Skylight research associate in mathematics at the University of British Columbia. Dan Margalit is assistant professor of mathematics at Georgia Institute of Technology. He is the coauthor of A Primer on Mapping Class Groups (Princeton).
Content
- Cover Page
- Half-title Page
- Title Page
- Copyright Page
- Contents
- Preface
- Foreword to the First Edition
- Foreword to the Second Edition
- Translators' Notes
- Acknowledgments
- Abstract
- 1. An Overview of Thurston's Theorems on Surfaces
- 1.1 Introduction
- 1.2 The Space of Simple Closed Curves
- 1.3 Measured Foliations
- 1.4 Teichm¨uller Space
- 1.5 Pseudo-Anosov Diffeomorphisms
- 1.6 The Case of the Torus
- 2. Some Reminders about the Theory of Surface Diffeomorphisms
- 2.1 The Space of Homotopy Equivalences of a Surface
- 2.2 The Braid Groups
- 2.3 Diffeomorphisms of the Pair of Pants
- 3. Review of Hyperbolic Geometry in Dimension 2
- 3.1 A Little Hyperbolic Geometry
- 3.2 The Teichm¨uller Space of the Pair of Pants
- 3.3 Generalities on the Geometric Intersection of Simple Closed Curves
- 3.4 Systems of Simple Closed Curves and Hyperbolic Isometries
- 4. The Space of Simple Closed Curves in a Surface
- 4.1 The Weak Topology on the Space of Simple Closed Curves
- 4.2 The Space of Multicurves
- 4.3 An Explicit Parametrization of the Space of Multicurves
- A. Pair of Pants Decompositions of a Surface
- 5. Measured Foliations
- 5.1 Measured Foliations and the Euler-Poincar´e Formula
- 5.2 Poincar´e Recurrence and the Stability Lemma
- 5.3 Measured Foliations and Simple Closed Curves
- 5.4 Curves as Measured Foliations
- B. Spines of Surfaces
- 6 The Classification of Measured Foliations
- 6.1 Foliations of the Annulus
- 6.2 Foliations of the Pair of Pants
- 6.3 The Pants Seam
- 6.4 The Normal Form of a Foliation
- 6.5 Classification of Measured Foliations
- 6.6 Enlarged Curves as Functionals
- 6.7 Minimality of the Action of the Mapping Class Group on PMF
- 6.8 Complementary Measured Foliations
- C. Explicit Formulas for Measured Foliations
- 7. Teichm¨uller Space
- 8. The Thurston Compactification of Teichm¨uller Space
- 8.1 Preliminaries
- 8.2 The Fundamental Lemma
- 8.3 The Manifold
- D. Estimates of Hyperbolic Distances
- D.1 The Hyperbolic Distance from i to a Point z0
- D.2 Relations between the Sides of a Right Hyperbolic Hexagon
- D.3 Bounding Distances in Pairs of Pants
- 9. The Classification of Surface Diffeomorphisms
- 9.1 Preliminaries
- 9.2 Rational Foliations (the Reducible Case)
- 9.3 Arational Measured Foliations
- 9.4 Arational Foliations with ? = 1 (the Finite Order Case)
- 9.5 Arational Foliations with ? ? 1 (the Pseudo-Anosov Case)
- 9.6 Some Properties of Pseudo-Anosov Diffeomorphisms
- 10. Some Dynamics of Pseudo-Anosov Diffeomorphisms
- 10.1 Topological Entropy
- 10.2 The Fundamental Group and Entropy
- 10.3 Subshifts of Finite Type
- 10.4 The Entropy of Pseudo-Anosov Diffeomorphisms
- 10.5 Constructing Markov Partitions for Pseudo-Anosov Diffeomorphisms
- 10.6 Pseudo-Anosov Diffeomorphisms are Bernoulli
- 11. Thurston's Theory for Surfaces with Boundary
- 11.1 The Spaces of Curves and Measured Foliations
- 11.2 Teichm¨uller Space and Its Compactification
- 11.3 A Sketch of the Classification of Diffeomorphisms
- 11.4 Thurston's Classification and Nielsen's Theorem
- 11.5 The Spectral Theorem
- 12. Uniqueness Theorems for Pseudo-Anosov Diffeomorphisms
- 12.1 Statement of Results
- 12.2 The Perron-Frobenius Theorem and Markov Partitions
- 12.3 Unique Ergodicity
- 12.4 The Action of Pseudo-Anosovs on PMF
- 12.5 Uniqueness of Pseudo-Anosov Maps
- 13. Constructing Pseudo-Anosov Diffeomorphisms
- Index
- 13.1 Generalized Pseudo-Anosov Diffeomorphisms
- 13.2 A Construction by Ramified Covers
- 13.3 A Construction by Dehn Twists
- 14. Fibrations over S¹ with Pseudo-Anosov Monodromy
- 14.1 The Thurston Norm
- 14.2 The Cone C of Nonsingular Classes
- 14.3 Cross Sections to Flows
- 15. Presentation of the Mapping Class Group
- 15.1 Preliminaries
- 15.2 A Method for Presenting the Mapping Class Group
- 15.3 The Cell Complex of Marked Functions
- 15.4 The Marking Complex
- 15.5 The Case of the Torus
- Bibliography
