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| {\bf Conjecture} (Poincar\'e) {\em Every 3-manifold which is homotopy equivalent to the 3-sphere is in fact homeomorphic to it. Or, in a more elementary form, every simply-connected compact 3-manifold is homeomorphic to $S^3$.} |
{\bf Conjecture} (Poincar\'e) {\em Every 3-manifold which is homotopy equivalent to the 3-sphere is in fact homeomorphic to it. Or, in a more elementary form, every simply-connected compact 3-manifold is homeomorphic to $S^3$.} |
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| The first statement is known to be true when 3 is replaced by any other number, but has thus far resisted proof in the 3-dimensional case. |
The first statement is known to be true when 3 is replaced by any other number, but has thus far resisted proof in the 3-dimensional case. |
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The Poincar\'e conjecture is one of the Clay Institute Millenium Prize Problems. You can read more about the conjecture <a href="http://www.claymath.org/Millennium_Prize_Problems/Poincare_Conjecture/">here.</a> |
| The Poincar\'e conjecture is one of the Clay Institute Millenium Prize Problems. You can read more about the conjecture \PMlinkexternal{here.}{http://www.claymath.org/Millennium_Prize_Problems/Poincare_Conjecture/} |
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| In 2003, Grisha Perelman announced results which would imply the Poincar\'e conjecture if they prove to be true, but since they are highly technical, they are still being reviewed. |
In 2003, Grisha Perelman announced results which would imply the Poincar\'e conjecture if they prove to be true, but since they are highly technical, they are still being reviewed. |
