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Difference between revisions of "Poincar Conjecture"
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The '''Poincaré conjecture''' is one of the seven [[Millenium Problems]], and is the only one that has been solved, in 2003 by [[Grigori Perelman]].
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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]].
  
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==The Conjecture States...==
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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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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."
 

Latest revision as of 14:12, 31 March 2017

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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