In mathematics, the Teichmüller space T ( S ) {\displaystyle T(S)} of a (real) topological (or differential) surface S {\displaystyle S} is a space that parametrizes complex structures on S {\displaystyle S} up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller. Each point in a Teichmüller space T ( S ) {\displaystyle T(S)} may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S {\displaystyle S} to itself. It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension 6 g − 6 {\displaystyle 6g-6} for a surface of genus g ≥ 2 {\displaystyle g\geq 2} . In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space. The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of research. The sub-field of mathematics that studies the Teichmüller space is called Teichmüller theory.
History Moduli spaces for Riemann surfaces and related Fuchsian groups have been studied since the work of Bernhard Riemann (1826–1866), who knew that 6 g − 6 {\displaystyle 6g-6} real parameters were needed to describe the variations of complex structures on a surface of genus g ≥ 2 {\displaystyle g\geq 2} . The early study of Teichmüller space, in the late nineteenth–early twentieth century, was geometric and founded on the interpretation of Riemann surfaces as hyperbolic surfaces. Among the main contributors were Felix Klein, Henri Poincaré, Paul Koebe, Jakob Nielsen, Robert Fricke and Werner Fenchel. The main contribution of Teichmüller to the study of moduli was the introduction of quasiconformal mappings to the subject. They allow us to give much more depth to the study of moduli spaces by endowing them with additional features that were not present in the previous, more elementary works. After World War II the subject was developed further in this analytic vein, in particular by Lars Ahlfors and Lipman Bers. The theory continues to be active, with numerous studies of the complex structure of Teichmüller space (introduced by Bers). The geometric vein in the study of Teichmüller space was revived following the work of William Thurston in the late 1970s, who introduced a geometric compactification which he used in his study of the mapping class group of a surface. Other more combinatorial objects associated to this group (in particular the curve complex) have also been related to Teichmüller space, and this is a very active subject of research in geometric group theory.
Definitions
Teichmüller space from complex structures Let S {\displaystyle S} be an orientable smooth surface (a differentiable manifold of dimension 2). Informally the Teichmüller space T ( S ) {\displaystyle T(S)} of S {\displaystyle S} is the space of Riemann surface structures on S {\displaystyle S} up to isotopy. Formally it can be defined as follows. Two complex structures X , Y {\displaystyle X,Y} on S {\displaystyle S} are said to be equivalent if there is a diffeomorphism f ∈ Diff ( S ) {\displaystyle f\in \operatorname {Diff} (S)} such that:
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