Editing Diffeomorphism (section)

===Connectedness===
For manifolds, the diffeomorphism group is usually not connected. Its component group is called the [[mapping class group]]. In dimension 2 (i.e. [[Surface (topology)|surface]]s), the mapping class group is a [[finitely presented group]] generated by [[Dehn twist]]s; this has been proved by [[Max Dehn]], [[W. B. R. Lickorish]], and [[Allen Hatcher]]).{{Citation needed|date=December 2009}} Max Dehn and [[Jakob Nielsen (mathematician)|Jakob Nielsen]] showed that it can be identified with the [[outer automorphism group]] of the [[fundamental group]] of the surface.

[[William Thurston]] refined this analysis by [[Nielsen-Thurston classification|classifying elements of the mapping class group]] into three types: those equivalent to a [[Periodic function#Periodic mapping|periodic]] diffeomorphism; those equivalent to a diffeomorphism leaving a simple closed curve invariant; and those equivalent to [[Pseudo-Anosov map|pseudo-Anosov diffeomorphisms]]. In the case of the [[torus]] <math>S^1\times S^1=\R^2/\Z^2</math>, the mapping class group is simply the [[modular group]] <math>\text{SL}(2,\Z)</math> and the classification becomes classical in terms of [[Möbius transformation#Elliptic transforms|elliptic]], [[Möbius transformation#Parabolic transforms|parabolic]] and [[Möbius transformation#Hyperbolic transforms|hyperbolic]] matrices. Thurston accomplished his classification by observing that the mapping class group acted naturally on a [[Compactification (mathematics)|compactification]] of [[Teichmüller space]]; as this enlarged space was homeomorphic to a closed ball, the [[Brouwer fixed-point theorem]] became applicable. Smale [[conjecture]]d that if <math>M</math> is an [[Orientability#Orientability_of_manifolds|oriented]] smooth closed manifold, the [[identity component]] of the group of orientation-preserving diffeomorphisms is [[Simple group|simple]]. This had first been proved for a product of circles by [[Michel Herman]]; it was proved in full generality by Thurston.