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Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots
Francis Bonahon, University of Southern California, Los Angeles, CA
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Student Mathematical Library
2009; 384 pp; softcover
Volume: 49
ISBN-10: 0-8218-4816-X
ISBN-13: 978-0-8218-4816-6
List Price: US$54
Institutional Members: US$43.20
All Individuals: US$43.20
Order Code: STML/49
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See also:

Hamilton's Ricci Flow - Bennett Chow, Peng Lu and Lei Ni

Low Dimensional Topology - Tomasz S Mrowka and Peter S Ozsvath

Lectures on Surfaces: (Almost) Everything You Wanted to Know about Them - Anatole Katok and Vaughn Climenhaga

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

"...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

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