In mathematics, given a Riemann surface *X*, the **Teichmüller space** of *X*, notated *T*_{X} 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 = âˆ’1. ...
On a sphere, the sum of the angles of a triangle is not equal to 180Â°. 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 TeichmÃ¼ller (June 18, 1913 - September 11, 1943) was a German mathematician who introduced quasi-conformal mappings and differential geometric methods into complex analysis. ...
## 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 *T*_{X}: whereas moduli space identifies all surfaces which are isomorphic, *T*_{X} 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 *T*_{X}. 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 *T*_{X} 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 *T*_{X} 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 *T*_{X} is 3*g*-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 *T*_{X}, 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. |