Poincare conjecture

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The Poincaré Conjecture is widely considered the most important unsolved problem in topology. It was first formulated by Henri Poincaré in 1904. Recently, a $1,000,000 prize was offered for its solution.

The conjecture is that every simply-connected closed 3-manifold is homeomorphic to a 3-sphere. (See the fundamental group article for a definition of "simply-connected".)

Analogues of the Poincaré Conjecture in dimensions other than 3 can also be formulated. The difficulty of low-dimensional topology is highlighted by the fact these analogues have now all been proved, while the original 3-dimensional version of Poincaré's conjecture remains unsolved. Its solution is central to the problem of classifying 3-manifolds.

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