Editing Low-dimensional topology (section)

===Teichmüller space===
{{Main|Teichmüller space}}

In [[mathematics]], the '''Teichmüller space''' ''T<sub>X</sub>'' of a (real) topological surface ''X'', is a space that parameterizes [[complex manifold|complex structures]] on ''X'' up to the action of [[homeomorphism]]s that are [[Homotopy#Isotopy|isotopic]] to the [[identity function|identity homeomorphism]]. Each point in ''T<sub>X</sub>'' may be regarded as an isomorphism class of 'marked' [[Riemann surface]]s where a 'marking' is an isotopy class of homeomorphisms from ''X'' to ''X''. 
The Teichmüller space is the [[orbifold|universal covering orbifold]] of the (Riemann) moduli space.

Teichmüller space has a canonical [[complex number|complex]] [[manifold]] structure and a wealth of natural metrics. The underlying topological space of Teichmüller space was studied by Fricke, and the Teichmüller metric on it  was introduced by {{harvs|txt|authorlink=Oswald Teichmüller|first=Oswald |last=Teichmüller|year=1940}}.<ref>{{citation
 | last = Teichmüller | first = Oswald | authorlink = Oswald Teichmüller
 | issue = 22
 | journal = Abh. Preuss. Akad. Wiss. Math.-Nat. Kl.
 | mr = 0003242
 | page = 197
 | title = Extremale quasikonforme Abbildungen und quadratische Differentiale
 | volume = 1939
 | year = 1940}}.</ref>