Editing Low-dimensional topology (section)

===Poincaré conjecture and geometrization===
{{Main|Geometrization conjecture}}

[[Thurston's geometrization conjecture]] states that certain three-dimensional [[topological space]]s each have a unique geometric structure that can be associated with them. It is an analogue of the [[uniformization theorem]] for two-dimensional [[surface (topology)|surface]]s, which states that every [[simply connected|simply-connected]] [[Riemann surface]] can be given one of three geometries ([[Euclidean geometry|Euclidean]], [[Spherical geometry|spherical]], or [[hyperbolic geometry|hyperbolic]]).
In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed [[3-manifold]] can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.  The conjecture was proposed by {{harvs|txt|authorlink=William Thurston|first=William|last= Thurston|year= 1982}}, and implies several other conjectures, such as the [[Poincaré conjecture]] and Thurston's [[elliptization conjecture]].<ref>{{citation
 | last = Thurston | first = William P. | authorlink = William Thurston
 | doi = 10.1090/S0273-0979-1982-15003-0
 | issue = 3
 | journal = [[Bulletin of the American Mathematical Society]]
 | mr = 648524
 | pages = 357–381
 | series = New Series
 | title = Three-dimensional manifolds, Kleinian groups and hyperbolic geometry
 | volume = 6
 | year = 1982| doi-access = free
 }}.</ref>