A compact 2dimensional surface without boundary is topologically homeomorphic to a 2sphere if every loop can be continuously tightened to a point. The Poincaré conjecture asserts that the same is true for 3dimensional spaces.


Field  Geometric topology 

Conjectured by  Henri Poincaré 
Conjectured in  1904 
First proof by  Grigori Perelman 
First proof in  2006 
Implied by 

Equivalent to 

Generalizations  Generalized Poincaré conjecture 
The Poincaré Conjecture is a question about spheres in mathematics. It is named after Henri Poincaré, the French mathematician and physicist who formulated it in 1904.
The sphere (also called the 2sphere, as it is a 2dimensional surface, although it is usually seen as inside a three dimensional space) has the property that any loop on it can be contracted to a point (if a rubber band is wrapped around the sphere, it is possible to slide it down to a point). Mathematicians say that the 2sphere is simply connected. Other spaces do not have this property, for example the donut: a rubber band that goes around the whole donut once cannot be slid down to a point without it leaving the surface.
Mathematicians knew that this property was unique to the 2sphere, in the sense that any other simply connected space that does not have edges and is small enough (in mathematician terms, that is compact) is in fact the 2sphere. It is no longer true if we remove the idea of smallness however, as an infinitely large plane is also simply connected. Also, a regular disk (a circle and its interior) is simply connected, but it has an edge (the bounding circle).
The conjecture asks whether the same is true for the 3sphere, which is an object living naturally in four dimensions. This question motivated much of modern mathematics, especially in the field of topology. The question was finally settled in 2002 by Grigori Perelman, a Russian mathematician, with methods from geometry, showing that it is indeed true. He was awarded a Fields medal for his work, which he declined.
The Poincaré conjecture can also be extended to higher dimensions: this is the generalised Poincaré conjecture. Surprisingly, it was easier to prove the fact for higherdimensional spheres: In 1960, Smale proved it to be true for the 5sphere, 6sphere and higher, and in 1982, Freedman proved that it was also true for the 4sphere, for which he was awarded a Fields medal.