A primer on mapping class groups

Given a compact connected orientable surface S there are two fundamental objects attached: a group and a space. The group is the mapping class group of S, denoted by Mod(S). This group is defined by the isotopy classes of orientation-preserving homeomorphism from S to itself. Equivalently, Mod(S) may be defined using diffeomorphisms instead of homeomorphisms or homotopy classes instead of isotopy classes. The space is the Teichmüller space of S, Teich(S). Teichmüller space and moduli space are fundamental objects in fields like low-dimensional topology, algebraic geometry and mathematical physics. If X (S) < 0, the Teichmüller space can be thought of as the set of homotopy classes of hyperbolic structures of S or, equivalently, as the set of isotopy classes of hyperbolic metrics on S, HypMet(S). The group and the space are connected through the moduli space in the following way. The group of orientation-preserving diffeomorphisms of S, Diff+(S) acts on HypMet(S) and this action descends to an action of Mod(S) on Teich(S) which is properly discontinuous. The quotient space,