Overview

This book deals with two fundamental objects attached to a surface S and how they relate to each other: a group and a space. The group is the mapping class group of S, denoted by Mod(S). It is defined as the group of isotopy classes of orientation-preserving diffeomorphisms of S. The space is the Teichmüller space of S, a metric space homeomorphic to an open ball. The book considers the relations between the algebraic structure of Mod(S), the geometry of Teich(S), and the topology of M(S). Underlying these connections is the combinatorial topology of the surface S. The Nielsen–Thurston classification theorem, which gives a particularly nice representative for each element of Mod(S), is also discussed.