Poincaré Conjecture

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The Poincaré conjecture is one of the seven Millennium Problems, and is the only one that has been solved, in 2003 by Grigori Perelman.

In elementary terms, the Poincaré conjecture states that the only three-manifold with no "holes" is the three-sphere. This would also show that the only n-manifold with no "holes" is the n-sphere; the case $n=1$ is trivial, the case $n=2$ is a classic problem, and the truth of the statement for $n\ge 4$ was verified by Stephen Smale in 1961. More rigorously, the conjecture is expressed as "Every simply connected, compact three-manifold (without boundary) is homeomorphic to the three-sphere."

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