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  "body": "\n## Summary\n\nThe mapping class group of a surface is the [[group|G5S1M9]] of its orientation-preserving [[homeomorphisms|0QT00PB5G2]] up to deformation.\nTwo homeomorphisms give the same element when one can be continuously deformed into the other.\nIt records the symmetries of the surface that cannot be undone by a deformation.\nThe mapping class group of the torus is $SL(2, \\Z)$, and the [[braid groups|BNR4H5GCTP]] are mapping class groups of punctured disks.\nFor closed orientable surfaces it is generated by finitely many [[Dehn twists|W890NRHGS4]], by the [[Dehn–Lickorish theorem|BG3EH18KRE]], and it acts on Teichmüller space.\n\n## Explanations\n\n### L1 human\n\nTake a surface, such as the surface of a doughnut, and map it to itself without tearing, in a way that keeps its orientation.\nMany such maps are just small wiggles that can be undone continuously; these are ignored.\nWhat remains are the genuinely different ways to map the surface to itself.\nOn 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.\nThe twists along two such circles, one around the tube and one along it, generate the group.\nThe mapping class group of the torus is the group $SL(2, \\Z)$ of integer $2 \\times 2$ matrices with determinant $1$.\n",
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