An Introduction to Geometric Topology

TL;DR

This book offers a comprehensive, self-contained introduction to the topology and geometry of surfaces and three-manifolds, emphasizing Thurston's geometrisation theorem and related foundational results in geometric topology.

Contribution

It provides complete proofs and a unified presentation of key concepts and theorems in geometric topology, including Thurston's geometrisation and hyperbolic Dehn filling.

Findings

Thurston's geometrisation of three-manifolds explained and proved

Complete proofs of Mostow's rigidity and the thick-thin decomposition

Classification of three-manifolds and surface diffeomorphisms

Abstract

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.