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Encyclopedia > Teichmuller 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 structures of Riemann surfaces whose underlying topological structure is the same as that of X. It is named after the German mathematician Oswald Teichmüller. For other meanings of mathematics or math, see mathematics (disambiguation). ... Riemann surface for the function f(z) = sqrt(z) In mathematics, particularly in complex analysis, a Riemann surface, named after Bernhard Riemann, is a one-dimensional complex manifold. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... Oswald Teichmüller (June 18, 1913 - September 11, 1943) was a German mathematician who introduced quasi-conformal mappings and differential geometric methods into complex analysis. ...

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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. In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ... The word isotopic has a number of different meanings, including: In the physical sciences, to do with chemical isotopes; In mathematics, to do with a relation called isotopism. ... In mathematics, in the sub-field of geometric topology, the mapping class group is an important algebraic invariant of a topological space. ...


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 3g-3+n. These are the cases of "finite type".


Note that, even though a compact surface with a point removed and the same surface with a disc removed are topologically the same, a complex structure on the surface behaves very differently around a point and around a removed disc. In particular, the boundary of the removed disc becomes an "ideal boundary" for the Riemann surface, and isomorphisms between surfaces with non-empty ideal boundary must take this ideal boundary into account. Varying the structure quasiconformally along the ideal boundary shows that the Teichmüller space of a Riemann surface with nonempty ideal boundary must be infinite-dimensional.


Teichmüller metric

There is, in general, no isomorphism from one Riemann surface to another of the same topological type that is isotopic to the identity. There is, however, always a quasiconformal map from one to the other that is isotopic to the identity, and the measure of how far such a map is from being conformal (i.e., holomorphic) gives a metric on TX, called the Teichmüller metric. In mathematics, a mapping w = f(z) is angle-preserving or (more usually) conformal at a point z0, if it preserves oriented angles between curves through z0, as well as their orientation, i. ... Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point. ...


Examples of Teichmüller spaces

A sphere with four points removed and a torus both have Teichmüller spaces of complex dimension 1.


  Results from FactBites:
 
PlanetMath: Teichmüller space (108 words)
quasiconformal maps to the category of Teichmüller spaces (as a subcategory of metric spaces).
Cross-references: metric spaces, subcategory, maps, quasiconformal, category, contravariant functor, isometric mapping, isometry, mapping, maximal dilatation, distance, equivalence relation, initial point, relation, equivalence classes, conformal mapping, homotopic, onto, quasiconformal mapping, sense-preserving, Riemann surface
This is version 5 of Teichmüller space, born on 2004-04-23, modified 2005-03-07.
Howard Masur - Papers (694 words)
It was introduced by Harvey as an analogy, in the context of Teichmuller space, for Tits buildings for symmetric spaces, and has been studied by Harer and Ivanov as a tool for understanding mapping class groups of surfaces.
In a certain sense this hyperbolicity is an explanation of why the Teichmuller space has some negative-curvature properties in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space fails most obviously in the regions corresponding to surfaces where some curve is extremely short.
The complex of curves exactly encodes the intersection patterns of this family of regions (it is the "nerve" of the family), and we show that its hyperbolicity means that the Teichmuller space is "relatively hyperbolic" with respect to this family.
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