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  "body": "\n## Definition\n\nLet $S$ be a closed, connected, oriented topological surface of [[genus|NJ787GZS1N]] $g \\ge 0$.\nA *marked Riemann surface* of type $S$ is a pair $(X, f)$, where $X$ is a [[Riemann surface|5K0N8QMV6Q]] and $f \\colon S \\to X$ is an orientation-preserving [[homeomorphism|05GZ11PWRY]].\nTwo marked Riemann surfaces $(X, f)$ and $(Y, h)$ are *equivalent* if there is a [[biholomorphism|8AJEMCZ29B]] $\\varphi \\colon X \\to Y$ such that $\\varphi \\circ f$ is isotopic to $h$.\nThe *Teichmüller space* $T(S)$ is the set of equivalence classes of marked Riemann surfaces of type $S$.\nIt depends up to natural bijection only on $g$ and is written $T_g$.\n\n## Remarks\n\nRiemann surfaces carry their orientation from the complex structure, which is why the marking is required to preserve orientation.\n$T_g$ carries a natural complete metric, the Teichmüller metric, defined from the least dilatation of [[quasiconformal maps|SB8KG46NHP]] between marked surfaces.\nFor $g \\ge 2$, $T_g$ is a complex manifold of dimension $3g - 3$, homeomorphic to $\\R^{6g - 6}$; $T_0$ is a point.\nThe mapping class group of $S$ acts on $T(S)$ by changing the marking, and the quotient is the moduli space $M_g$.\n\nExample: $g = 1$.\nFor $\\tau$ in the upper half-plane let $X_\\tau = \\C / (\\Z + \\tau \\Z)$.\nMark it by the homeomorphism $f_\\tau$ from the standard torus $\\R^2/\\Z^2$ induced by the real linear map sending $(1, 0)$ to $1$ and $(0, 1)$ to $\\tau$.\nEvery marked torus is equivalent to exactly one such $(X_\\tau, f_\\tau)$, so $T_1$ is identified with the upper half-plane.\n\nNon-example: a pair $(X, f)$ with $f$ orientation-reversing is not a marked Riemann surface, so it is not a point of $T(S)$.\n",
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