Editing Conjecture (section)

===Poincaré conjecture===
{{main|Poincaré conjecture}} 
In [[mathematics]], the [[Poincaré conjecture]] is a [[theorem]] about the [[Characterization (mathematics)|characterization]] of the [[3-sphere]], which is the hypersphere that bounds the [[unit ball]] in four-dimensional space. The conjecture states that: {{Blockquote|Every [[simply connected]], [[closed manifold|closed]] 3-[[manifold]] is [[homeomorphic]] to the 3-sphere.|sign=|source=}} An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called [[homotopy equivalence]]: if a 3-manifold is ''homotopy equivalent'' to the 3-sphere, then it is necessarily ''homeomorphic'' to it.

Originally conjectured by [[Henri Poincaré]] in 1904, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a [[Closed manifold|closed]] [[3-manifold]]). The Poincaré conjecture claims that if such a space has the additional property that each [[path (topology)|loop]] in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An [[generalized Poincaré conjecture|analogous result]] has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, [[Grigori Perelman]] presented a proof of the conjecture in three papers made available in 2002 and 2003 on [[arXiv]]. The proof followed on from the program of [[Richard S. Hamilton]] to use the [[Ricci flow]] to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricci flow, called ''Ricci flow with surgery'' to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method "converged" in three dimensions.<ref>{{cite journal | last = Hamilton | first = Richard S. | author-link = Richard S. Hamilton | title = Four-manifolds with positive isotropic curvature | journal = Communications in Analysis and Geometry | volume = 5 | issue = 1 | pages = 1&ndash;92 | year = 1997 | doi =  10.4310/CAG.1997.v5.n1.a1| mr = 1456308 | zbl = 0892.53018| doi-access = free }}</ref>  Perelman completed this portion of the proof. Several teams of mathematicians have verified that Perelman's proof is correct.

The Poincaré conjecture, before being proven, was one of the most important open questions in [[topology]].