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  "body": "\n## Summary\n\nThe Teichmüller space $T_g$ parametrises the complex structures on a fixed closed surface of genus $g$.\nEach point also carries a marking, which remembers the topology up to isotopy.\nFor $g \\ge 2$ it is a complex manifold of dimension $3g - 3$ and is homeomorphic to $\\R^{6g - 6}$.\nThe [[mapping class group|JV5NYG913V]] acts on it properly discontinuously, and the quotient is the [[moduli space of curves|NK9ATH4WB6]] of genus $g$.\n\n## Explanations\n\n### L1 human\n\nA closed surface of genus $g$, like a doughnut with $g$ holes, can be given many different shapes as a [[Riemann surface|5K0N8QMV6Q]].\nThe moduli space lists these shapes up to isomorphism, but surfaces with extra symmetries make it an orbifold rather than a manifold.\nTeichmüller space adds a marking, a way of naming the loops on the surface, which removes this problem.\nFor genus $1$ a marked torus is $\\C / (\\Z + \\tau \\Z)$ with $\\tau$ in the [[upper half-plane|X6XGGH5FPW]], and $T_1$ is the upper half-plane itself.\n",
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