Preface

About 30 years ago, the field of 3-dimensional topology was revo-

lutionized by Thurston’s Geometrization Theorem and by the unex-

pected appearance of hyperbolic geometry in purely topological prob-

lems. This book aims at introducing undergraduate students to some

of these striking developments. It grew out of notes prepared by the

author for a three-week course for undergraduates that he taught at

the Park City Mathematical Institute in June–July 2006. It covers

much more material than these lectures, but the written version in-

tends to preserve the overall spirit of the course. The ultimate goal,

attained in the last chapter, is to bring the students to a level where

they can understand the statements of Thurston’s Geometrization

Theorem for knot complements and, more generally, of the general

Geometrization Theorem for 3-dimensional manifolds recently proved

by G. Perelman. Another leading theme is the intrinsic beauty of

some of the mathematical objects involved, not just mathematically

but visually as well.

The first two-thirds of the book are devoted to 2-dimensional

geometry. After a brief discussion of the geometry of the euclidean

plane R2, the hyperbolic plane H2, and the sphere S2, we discuss the

construction of locally homogeneous spaces by gluing the sides of a

polygon. This leads to the investigation of the tessellations that are

associated to such constructions, with a special focus on one of the

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