An Introduction to Geometric Topology

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.

486 pages, more than 200 colour figures. Second version of April 2022, with minor corrections

Cited by in corpus (17)

• Currents, Systoles, and Compactifications of Character Varieties
• Minimal Entropy of $3$-manifolds
• Stable integral simplicial volume of 3-manifolds
• The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds
• Hierarchies for Relatively Hyperbolic Virtually Special Groups
• Mirzakhani's work on earthquake flow
• Well-posedness of Hersch-Szegő's center of mass by hyperbolic energy minimization
• Cabling Legendrian and transverse knots
• On the diffeomorphism type of Seifert fibered spherical 3-orbifolds
• A small closed convex projective 4-manifold via Dehn filling
• Indefinite Stein fillings and Pin(2)-monopole Floer homology
• Immersions of surfaces into SL(2,C) and into the space of geodesics of Hyperbolic space
• On the IRS compactification of moduli space
• Topological obstructions to nonnegative scalar curvature and mean convex boundary
• Finiteness Theorems for Gromov-Hyperbolic Spaces and Groups
• Invariants of PSL(n,R)-Fuchsian representations and a slice of Hitchin components
• Monopole Floer homology and SOLV geometry