Mapping Class Group

The mapping class group of an oriented surface is the group of orientation-preserving homeomorphisms modulo isotopy, with composition as the group operation. For a closed connected orientable surface of genus , it is denoted (Guo and Sun 2026).

The involution rank measures how many group involutions are needed to generate this group. Guo and Sun (2026) proved that .

See also

Group Generators, Group Involution, Homeomorphism, Involution Rank, Isotopy, Surface

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References

Guo, Q. and Sun, D. "The Involution Rank of

Is Four." 16 Oct 2026. https://arxiv.org/abs/2610.22774.

Cite this as:

Weisstein, Eric W. "Mapping Class Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MappingClassGroup.html

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