Difference between revisions of "Poincaré Conjecture"

Revision as of 21:44, 29 September 2024

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-The '''Poincaré Conjecture''' which was originally a [[conjecture]], was solved in 2003 and is now a [[theorem]]. It states that every closed topological three-dimensional [[manifold]] is [[homeomorphism|homeomorphic]] to a [[hypersphere|3-sphere]].
+#REDIRECT[[Poincaré Conjecture]]
-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 <math>n=1</math> is trivial, the case <math>n=2</math> is a classic problem, and the truth of the statement for <math>n\ge 4</math> 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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