Editing Geometric topology (section)

===Low-dimensional topology===
{{main|Low-dimensional topology}}
[[Low-dimensional topology]] includes:
* [[Surface (topology)|Surfaces]] (2-manifolds)
* [[3-manifold]]s
* [[4-manifold]]s
each have their own theory, where there are some connections.

Low-dimensional topology is strongly geometric, as reflected in the [[uniformization theorem]] in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, negative curvature/hyperbolic – and the [[geometrization conjecture]] (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of 8 possible geometries.

2-dimensional topology can be studied as [[complex geometry]] in one variable ([[Riemann surface]]s are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can be studied from the point of view of complex geometry in two variables (complex surfaces), though not every 4-manifold admits a complex structure.