Editing Low-dimensional topology (section)

==Two dimensions==
{{Main|surface (topology)}}

A [[Surface (topology)|surface]] is a [[two-dimensional]], [[topological manifold]]. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional [[Euclidean space]] '''R'''<sup>3</sup>—for example, the surface of a [[ball (mathematics)|ball]]. On the other hand, there are surfaces, such as the [[Klein bottle]], that cannot be [[embedding|embedded]] in three-dimensional Euclidean space without introducing [[singularity theory|singularities]] or self-intersections.

===Classification of surfaces===
The ''[[classification theorem]] of closed surfaces'' states that any [[connected (topology)|connected]] [[closed manifold|closed]] surface is homeomorphic to some member of one of these three families:

# the sphere;
# the [[connected sum]] of ''g'' [[torus|tori]], for <math>g \geq 1</math>;
# the connected sum of ''k'' [[real projective plane]]s, for <math>k \geq 1</math>.

The surfaces in the first two families are [[orientability|orientable]]. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number ''g'' of tori involved is called the ''genus'' of the surface. The sphere and the torus have [[Euler characteristic]]s 2 and 0, respectively, and in general the Euler characteristic of the connected sum of ''g'' tori is {{nowrap|2 &minus; 2''g''}}.

The surfaces in the third family are nonorientable. The Euler characteristic of the real projective plane is 1, and in general the Euler characteristic of the connected sum of ''k'' of them is {{nowrap|2 &minus; ''k''}}.

===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>

===Uniformization theorem===
{{Main|Uniformization theorem}}

In [[mathematics]], the '''uniformization theorem'''  says that every [[simply connected]] [[Riemann surface]] is [[Conformal equivalence|conformally equivalent]] to one of the three domains: the open [[unit disk]], the [[complex plane]], or the [[Riemann sphere]]. In particular it admits a [[Riemannian metric]] of [[constant curvature]]. This classifies Riemannian surfaces as elliptic (positively curved—rather, admitting a constant positively curved metric), parabolic (flat), and hyperbolic (negatively curved) according to their [[universal cover]].

The uniformization theorem is a generalization of the [[Riemann mapping theorem]] from proper simply connected [[open set|open]] [[subset]]s of the plane to arbitrary simply connected Riemann surfaces.