Mapping class group of a surface

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.

https://en.wikipedia.org/wiki/Mapping_class_group_of_a_surface