Three-dimensional Geometry and Topology, Volume 1

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Princeton University Press, 1997 - Mathematics - 311 pages
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This book develops some of the extraordinary richness, beauty, and power of geometry in two and three dimensions, and the strong connection of geometry with topology. Hyperbolic geometry is the star. A strong effort has been made to convey not just denatured formal reasoning (definitions, theorems, and proofs), but a living feeling for the subject. There are many figures, examples, and exercises of varying difficulty.

This book was the origin of a grand scheme developed by Thurston that is now coming to fruition. In the 1920s and 1930s the mathematics of two-dimensional spaces was formalized. It was Thurston's goal to do the same for three-dimensional spaces. To do this, he had to establish the strong connection of geometry to topology--the study of qualitative questions about geometrical structures. The author created a new set of concepts, and the expression "Thurston-type geometry" has become a commonplace.

Three-Dimensional Geometry and Topology had its origins in the form of notes for a graduate course the author taught at Princeton University between 1978 and 1980. Thurston shared his notes, duplicating and sending them to whoever requested them. Eventually, the mailing list grew to more than one thousand names. The book is the culmination of two decades of research and has become the most important and influential text in the field. Its content also provided the methods needed to solve one of mathematics' oldest unsolved problems--the Poincaré Conjecture.

In 2005 Thurston won the first AMS Book Prize, for Three-dimensional Geometry and Topology. The prize recognizes an outstanding research book that makes a seminal contribution to the research literature. Thurston received the Fields Medal, the mathematical equivalent of the Nobel Prize, in 1982 for the depth and originality of his contributions to mathematics. In 1979 he was awarded the Alan T. Waterman Award, which recognizes an outstanding young researcher in any field of science or engineering supported by the National Science Foundation.

 

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Contents

What Is a Manifold?
3
11 Polygons and Surfaces
4
12 Hyperbolic Surfaces
7
13 The Totality of Surfaces
17
14 Some ThreeManifolds
31
Hyperbolic Geometry and Its Friends
43
21 Negatively Curved Surfaces in Space
45
22 The Inversive Models
53
35 Discrete Groups
153
36 Bundles and Connections
158
37 Contact Structures
168
38 The Eight Model Geometries
179
39 Piecewise Linear Manifolds
190
310 Smoothings
193
The Structure of Discrete Groups
209
42 Euclidean Manifolds and Crystallographic Groups
221

23 The Hyperboloid Model and the Klein Model
64
24 Some Computations in Hyperbolic Space
74
25 Hyperbolic Isometries
86
26 Complex Coordinates for Hyperbolic ThreeSpace
98
27 The Geometry of the ThreeSphere
103
Geometric Manifolds
109
32 Triangulations and Gluings
118
33 Geometric Structures on Manifolds
125
34 The Developing Map and Completeness
139
43 ThreeDimensional Euclidean Manifolds
231
44 Elliptic ThreeManifolds
242
45 The ThickThin Decomposition
253
46 Teichmuller Space
258
47 ThreeManifolds Modeled on Fibered Geometries
277
Glossary
289
Bibliography
295
Index
301
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Page 297 - Hatcher, A. (1983): Hyperbolic structures of arithmetic type on some link complements. J. London Math. Soc. 27, 345-355 Heath-Brown, DR, Patterson, SJ (1979): The distribution of Kummer sums at prime arguments.

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