Mapping class group

Summary

The mapping class group of a surface is the group of its orientation-preserving homeomorphisms up to deformation. Two homeomorphisms give the same element when one can be continuously deformed into the other. It records the symmetries of the surface that cannot be undone by a deformation. The mapping class group of the torus is $SL(2, \Z)$, and the braid groups are mapping class groups of punctured disks. For closed orientable surfaces it is generated by finitely many Dehn twists, by the Dehn–Lickorish theorem, and it acts on Teichmüller space.