Low-Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots
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...
Search in google:
Pre-order your copy today and receive a special introductory price of US$39.00! Offer ends July 30, 2009 and applies to individuals only. 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.
IAS/Park City Mathematics InstituteCh. 1 The euclidean plane 1Ch. 2 The hyperbolic plane 11Ch. 3 The 2-dimensional sphere 47Ch. 4 Gluing constructions 55Ch. 5 Gluing examples 89Ch. 6 Tessellations 133Ch. 7 Group actions and fundamental domains 185Ch. 8 The Farey tessellation and circle packing 207Ch. 9 The 3-dimensional hyperbolic space 227Ch. 10 Kleinian groups 241Ch. 11 The figure-eight knot complement 293Ch. 12 Geometrization theorems in dimension 3 315Appendix Tool Kit 355Supplemental bibliography and references 365Index 377
