An Introduction to Geometric Topology

This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002.The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds.It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem.The text should be suitable for a master or PhD student in mathematics interested in geometry, or for any curious mathematician with a standard background in topology and analysis.This is the third edition of the book, published in August 2023. With respect to the first edition, it contains some minor corrections, and all the interior figures are in colour. Read more