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  "body": "\n## Statement\n\nLet $g \\ge 1$ and let $S_g$ be a closed orientable [[surface|VGJJVBZ1MQ]] of [[genus|NJ787GZS1N]] $g$.\nLet $\\operatorname{Mod}(S_g)$ be its [[mapping class group|JV5NYG913V]], the group of isotopy classes of orientation-preserving homeomorphisms of $S_g$.\nThen $\\operatorname{Mod}(S_g)$ is generated by finitely many [[Dehn twists|W890NRHGS4]] about non-separating simple closed curves.\nMore precisely, Dehn twists about $3g - 1$ suitable curves suffice.\n\n## Remarks\n\nDehn proved that finitely many twists generate the group, and Lickorish later gave the explicit set of $3g - 1$ curves.\nHumphries reduced the number to $2g + 1$, which is the smallest possible for $g \\ge 2$.\nIdea of the proof: argue by induction on the genus with the curve complex or a connected graph of non-separating curves.\nTwo non-separating curves can be joined by a chain of curves, each meeting the next in one point.\nTwists along such chains move any non-separating curve to a fixed one.\nThis reduces the problem to the stabiliser of one curve, which is related to the mapping class group of a surface of smaller complexity.\n\nThe theorem at work: for $g = 1$ the mapping class group of the torus is $SL(2, \\Z)$, acting on $H_1(T^2; \\Z) = \\Z^2$.\nThe twists about the two standard curves act by $\\begin{pmatrix} 1 \u0026 1 \\\\ 0 \u0026 1 \\end{pmatrix}$ and $\\begin{pmatrix} 1 \u0026 0 \\\\ -1 \u0026 1 \\end{pmatrix}$, and these two matrices generate $SL(2, \\Z)$.\nFor $g = 0$ the mapping class group of the sphere is trivial.\n\n## Explanations\n\n### L1 human\n\nThe ways to deform a closed surface onto itself, counted up to continuous wiggling, form a group.\nThe theorem says that every such deformation is a composition of a few basic moves, twists along loops.\nFor a surface with $g$ holes, $3g - 1$ fixed loops are enough.\n\n### L2 human\n\nFinite generation by twists is the starting point of most work on mapping class groups.\nIt is the starting point for finite presentations of the group, and it lets one compute its abelianization, which is trivial for $g \\ge 3$.\nIt is the key step in the proof of the [[Lickorish–Wallace theorem|1YP4RF1QXV]].\nThere a Heegaard gluing map is written as a product of twists, and each twist is realised by surgery on a curve.\nIt is also used to show that the action on homology maps $\\operatorname{Mod}(S_g)$ onto the symplectic group $Sp(2g, \\Z)$.\n",
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