Dehn–Nielsen theorem

In mathematics, and more precisely in topology, the mapping class group of a surface, sometimes called the modular group or Teichmüller modular group, is the group of homeomorphisms of the surface viewed up to continuous (in the compact-open topology) deformation. It is of fundamental importance for the study of 3-manifolds via their embedded surfaces and is also studied in algebraic geometry in relation to moduli problems for curves.

The mapping class group can be defined for arbitrary manifolds (indeed, for arbitrary topological spaces) but the 2-dimensional setting is the most studied in group theory.

The mapping class group of surfaces are related to various other groups, in particular braid groups and outer automorphism groups.

The mapping class group appeared in the first half of the twentieth century. Its origins lie in the study of the topology of hyperbolic surfaces, and especially in the study of the intersections of closed curves on these surfaces. The earliest contributors were Max Dehn and Jakob Nielsen: Dehn proved finite generation of the group, and Nielsen gave a classification of mapping classes and proved that all automorphisms of the fundamental group of a surface can be represented by homeomorphisms (the Dehn–Nielsen–Baer theorem).

The Dehn–Nielsen theory was reinterpreted in the mid-seventies by Thurston who gave the subject a more geometric flavour and used this work to great effect in his program for the study of three-manifolds.

More recently the mapping class group has been by itself a central topic in geometric group theory, where it provides a testing ground for various conjectures and techniques.

Let ${\displaystyle S}$ be a connected, closed, orientable surface and ${\displaystyle \mathrm {Homeo} ^{+}(S)}$ the group of orientation-preserving, or positive, homeomorphisms of ${\displaystyle S}$. This group has a natural topology, the compact-open topology. It can be defined easily by a distance function: if we are given a metric ${\displaystyle d}$ on ${\displaystyle S}$ inducing its topology then the function defined by

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