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Poincaré conjecture

 For compact 2-dimensional surfaces without boundary, if every loop can be continuously tightened to a point, then the surface is topologically homeomorphic to a 2-sphere, usually just called a sphere. The Poincaré conjecture asserts that the same is true for 3-dimensional surfaces.
In [mathematics] , the Poincaré conjecture (French, ) is a [theorem] about the [characterization] of the [three-dimensional sphere] among [three-dimensional manifolds] . Originally conjectured by [Henri Poincaré] , the claim concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a [closed] [3-manifold] ). The Poincaré conjecture claims that if such a space has the additional property that each [loop] in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An [analogous result] has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, [Grigori Perelman] sketched a proof of the conjecture in a series of papers made available in 2002 and 2003. The proof followed the program of [Richard Hamilton] . Several high-profile teams of mathematicians have since verified the correctness of Perelman's proof.

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