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Teichmüller space

In [mathematics] , given a [Riemann surface] X , the Teichmüller space of X , notated TX or Teich( X ), is a [complex] [manifold] whose points represent all [complex structure] s of Riemann surfaces whose underlying [topological structure] is the same as that of X . It is named after the German mathematician [Oswald Teichmüller] .

Relation to moduli space

The Teichmüller space of a surface is related to its [moduli space] , but preserves more information about the surface. More precisely, the surface X (or its underlying topological structure) provides a marking X → Y of each Riemann surface Y represented in TX : whereas moduli space identifies all surfaces which are isomorphic, TX only identifies those surfaces which are isomorphic via a [biholomorphic] map f that is [isotopic] to the identity (with respect to the marking, hence its need). The automorphisms of X , up to isotopy, form a discrete group (the Teichmüller modular group , or [mapping class group] of X ) that acts on TX . The action is as follows: if [ g ] is an element of the mapping class group of X , then [ g ] sends the point represented by the marking h: X → Y to the point with the marking hg : X → X → Y . The quotient of TX by this action is precisely the moduli space of X .

Properties of TX

The Teichmüller space of X is a complex manifold. Its complex dimension depends on topological properties of X . If X is obtained from a compact surface of genus g (take g greater than 1) by removing n points, then the dimension of TX is 3 g − 3 n . These are the cases of "finite type".

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