Student Mathematical Library
Volume: 49;
2009;
384 pp;
Softcover
MSC: Primary 51; 30; 57;
Print ISBN: 978-0-8218-4816-6
Product Code: STML/49
List Price: $54.00
Individual Price: $43.20
Electronic ISBN: 978-1-4704-1634-8
Product Code: STML/49.E
List Price: $54.00
Individual Price: $43.20
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Supplemental Materials
Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots
Share this pageFrancis Bonahon
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.
This book is published in cooperation with IAS/Park City Mathematics Institute
Readership
Undergraduate students interested in topology and/or geometry of low-dimensional manifolds, particularly 3-manifolds.
Reviews & Endorsements
...an essential book, if just because no other yet competes for this crucial niche, but also happily excellent in every respect—passionately told, expertly rendered, exquisitely organized, and sumptuously illustrated. ... Essential.
-- CHOICE Magazine
Table of Contents
Table of Contents
Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots
- Cover Cover11 free
- Title page iii5 free
- Table of contents vii9 free
- Subseries preface xi13 free
- Preface xiii15 free
- The euclidean plane 119 free
- The hyperbolic plane 1129
- The 2-dimensional sphere 4765
- Gluing constructions 5573
- Gluing examples 89107
- Tessellations 133151
- Group actions and fundamental domains 185203
- The Farey tessellation and circle packing 207225
- The 3-dimensional hyperbolic space 227245
- Kleinian groups 241259
- The figure-eight knot complement 293311
- Geometrization theorems in dimension 3 315333
- Tool kit 355373
- Bibliography and references 365383
- Index 377395 free
- Back Cover Back Cover1403