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Home    Orders: US  Orders: other countries   Shopping Cart Matrix Editions future books   Teichmuller Theory and Applications to Geometry, Topology, and Dynamics Volume 2: Four Theorems by William Thurston By John H. Hubbard with contributions by Adrien Douady, William Dunbar, and Roland Roeder, as well as Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska, and Sudeb Mitra Volume 1 set up the Teichmüller theory necessary for discussing Thurston's theorems. Volume 2 proves Thurston's four theorems: The classification of homeomorphisms of surfaces. The topological characterization of rational maps. The hyperbolization theorem for 3-manifolds that fiber over the circle. The hyperbolization theorem for Haken 3-manifolds. From the preface to volume 1: Not only are the theorems of extraordinary beauty in themselves, but the methods of proof that Thurston introduced were so novel and displayed such amazing geometric insight that to this day they have barely entered the accepted methods of mathematicians in the field. The results sound more or less unrelated, but they are linked by a common thread: each one goes from topology to geometry. Each says that either a topological problem has a natural geometry, or there is an understandable obstruction. The proofs are closely related: you use the topology to set up an analytic mapping from a Teichmüller space to itself; the geometry arises from a fixed point of this mapping. Thurston proceeds to show that if there is no fixed point, then some system of simple closed curves on the surface is an obstruction to finding a solution. Thus the proofs of the theorems are somehow similar, although the details and difficulty are very different....

Contents: Chapter 8 The Classification of Homeomorphisms of Surfaces Chapter 9 Rational Functions Chapter 10 Geometry of Hyperbolic Space Chapter 11 The Ahlfors Finiteness Theorem and Rigidity Theorems Chapter 12 A Hyperbolic Structure for Mapping Tori of Pseudo-Anosov Homeomorphisms Chapter 13 Some Topology of 3-manifolds: Hierarchies Chapter 14 Andreev's Theorem on Hyperbolic Polyhedra Chapter 15 The Skinning Lemma

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