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  "body": "\n## Definition\n\nLet $S$ be a connected oriented [[surface|VGJJVBZ1MQ]], possibly with boundary $\\partial S$.\nLet $\\mathrm{Homeo}^+(S, \\partial S)$ be the group, under composition, of orientation-preserving [[homeomorphisms|0QT00PB5G2]] $f \\colon S \\to S$ with $f(x) = x$ for all $x \\in \\partial S$.\nTwo such homeomorphisms $f$ and $g$ are *isotopic* if there is a continuous map $H \\colon S \\times [0, 1] \\to S$ such that:\n\n1. each $H_t = H(\\cdot, t)$ lies in $\\mathrm{Homeo}^+(S, \\partial S)$;\n2. $H_0 = f$ and $H_1 = g$.\n\nThe *mapping class group* of $S$ is the quotient group\n$$\\mathrm{Mod}(S) = \\mathrm{Homeo}^+(S, \\partial S) / \\mathrm{Homeo}_0(S, \\partial S),$$\nwhere $\\mathrm{Homeo}_0(S, \\partial S)$ is the normal subgroup of homeomorphisms isotopic to the identity.\nIts elements, the isotopy classes, are *mapping classes*.\n\n## Remarks\n\nFor a surface with marked points or punctures, the homeomorphisms may permute the marked points, and the isotopies must keep them fixed.\nSome authors also allow orientation-reversing homeomorphisms; this gives the extended mapping class group, which contains $\\mathrm{Mod}(S)$ with index $2$ when $S$ admits an orientation-reversing homeomorphism fixing $\\partial S$.\n\nExample: for the torus $T^2 = \\R^2 / \\Z^2$, the map $\\mathrm{Mod}(T^2) \\to SL(2, \\Z)$ that records the action on the first homology group $H_1(T^2; \\Z) \\cong \\Z^2$ is an isomorphism.\nA matrix $A \\in SL(2, \\Z)$ maps $\\Z^2$ onto itself, so it induces a homeomorphism of $T^2$, and since $\\det A = 1$ it preserves orientation.\n\nNon-example: let $A = S^1 \\times [0, 1]$ be the annulus, with points written $(\\theta, t)$, and let $T(\\theta, t) = (\\theta + 2\\pi t, t)$, a [[Dehn twist|W890NRHGS4]].\nThe maps $H_s(\\theta, t) = (\\theta + 2\\pi s t, t)$ for $s \\in [0, 1]$ deform the identity into $T$, but for $0 \u003c s \u003c 1$ they rotate the boundary circle $S^1 \\times \\set{1}$, so they fail clause 1.\nIn fact $T$ is not isotopic to the identity in the sense above, and $\\mathrm{Mod}(A) \\cong \\Z$, generated by the class of $T$.\nFor the closed disk, the Alexander lemma gives $\\mathrm{Mod}(D^2) = 1$.\n",
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