---
tag: JV5NYG913V
kind: concept
name: Mapping class group
msc: ["57K20", "20F38"]
specializes: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Mapping_class_group
same_as:
  wikidata: Q3845212
  wikipedia: Mapping class group
---

## Summary

The mapping class group of a surface is the [[group|G5S1M9]] of its orientation-preserving [[homeomorphisms|0QT00PB5G2]] 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|BNR4H5GCTP]] are mapping class groups of punctured disks.
For closed orientable surfaces it is generated by finitely many [[Dehn twists|W890NRHGS4]], by the [[Dehn–Lickorish theorem|BG3EH18KRE]], and it acts on Teichmüller space.

## Explanations

### L1 human

Take a surface, such as the surface of a doughnut, and map it to itself without tearing, in a way that keeps its orientation.
Many such maps are just small wiggles that can be undone continuously; these are ignored.
What remains are the genuinely different ways to map the surface to itself.
On the torus, cutting along a circle around the tube, twisting one side by a full turn and gluing back gives a map that cannot be undone.
The twists along two such circles, one around the tube and one along it, generate the group.
The mapping class group of the torus is the group $SL(2, \Z)$ of integer $2 \times 2$ matrices with determinant $1$.