Mapping Class Groups

The mapping class group Mod S of an orientable surface S is defined as the group of isotopy classes of orientation-preserving diffeomorphisms S → S . In addition to being a central object of the topology of surfaces (cf. 2.1), these groups also play an important role in the theory of Teichmuller spaces and in algebraic geometry, where they are known underthe name Teichmuller modular groups or simply modular groups . Our notations are derived from the latter terminology. There are several closely related groups, which also deserve the name of the mapping classgroups (or Teichmuller modular groups). First of all, one may consider the extended mapping class group Mod ◊ S of S defined as the group of the isotopy classes of all diffeomorphisms S → S . The pure mapping class group PMod S of S is defined as the group of isotopy classes of all orientation-preserving diffeomorphisms S → S preserving setwise all boundary components of S . Finally, one may consider the group ℳ S of all (orientation-preserving) diffeomorphisms S → S fixed on the boundary ∂ S , considered up to isotopies fixed on the boundary. If ∂ S ≠ 0, then diffeomorphisms fixed on ∂ S are automatically orientation-preserving. If ∂ S = 0, then, of course, ℳ S = Mod S . All these groups could be also defined as the 0- th homotopy groups of suitable diffeomorphisms groups of S . For example, Mod S = π 0 (Diff( S )), where Diff( S ) is the group of all orientation-preserving diffeomorphisms of S considered with, for example, C ∞ -topology (or any other reasonable topology).