Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots
About this Title
Francis Bonahon, University of Southern California, Los Angeles, CA
Publication: The Student Mathematical Library
Publication Year 2009: Volume 49
ISBNs: 978-0-8218-4816-6 (print); 978-1-4704-1634-8 (online)
MathSciNet review: MR2522946
MSC: Primary 57M50; Secondary 57-01, 57M25
The study of 3-dimensional spaces brings together elements from several areas of mathematics. The most notable are topology and geometry, but elements of number theory and analysis also make appearances. In the past 30 years, there have been striking developments in the mathematics of 3-dimensional manifolds. This book aims to introduce undergraduate students to some of these important developments.
Low-Dimensional Geometry starts at a relatively elementary level, and its early chapters can be used as a brief introduction to hyperbolic geometry. However, the ultimate goal is to describe the very recently completed geometrization program for 3-dimensional manifolds. The journey to reach this goal emphasizes examples and concrete constructions as an introduction to more general statements. This includes the tessellations associated to the process of gluing together the sides of a polygon. Bending some of these tessellations provides a natural introduction to 3-dimensional hyperbolic geometry and to the theory of kleinian groups, and it eventually leads to a discussion of the geometrization theorems for knot complements and 3-dimensional manifolds.
This book is illustrated with many pictures, as the author intended to share his own enthusiasm for the beauty of some of the mathematical objects involved. However, it also emphasizes mathematical rigor and, with the exception of the most recent research breakthroughs, its constructions and statements are carefully justified.
Undergraduate students interested in topology and/or geometry of low-dimensional manifolds, particularly 3-manifolds.
Table of Contents
- Chapter 1. The euclidean plane
- Chapter 2. The hyperbolic plane
- Chapter 3. The 2-dimensional sphere
- Chapter 4. Gluing constructions
- Chapter 5. Gluing examples
- Chapter 6. Tessellations
- Chapter 7. Group actions and fundamental domains
- Chapter 8. The Farey tessellation and circle packing
- Chapter 9. The 3-dimensional hyperbolic space
- Chapter 10. Kleinian groups
- Chapter 11. The figure-eight knot complement
- Chapter 12. Geometrization theorems in dimension 3
- Appendix. Tool kit