Conjecture (Poincaré) Every 3-manifold without boundary that is homotopy equivalent to the $3$ sphere is in fact homeomorphic to it. Or, in a more elementary form: <</SPAN>#72#>every simply-connected compact $3$ manifold without boundary is homeomorphic to $S^3$ .

The first statement is known to be true when $3$ is replaced by any other positive integer, but for a long time resisted proof in the 3-dimensional case. However, in 2003 Grigori Perelman announced a proof which is now generally accepted to be correct. For this work Perelman was offered a Fields Medal, though he declined it.

The Poincaré Conjecture is one of the Clay Mathematics Institute Millennium Prize Problems. For more information, see their page on the Poincaré Conjecture. Due to the way he chose to publish his results, full details of Perelman's proof did not appear in a peer reviewed mathematical publication until 2006. The Millennium Prize rules require a waiting period of two years after such publication before the prize can be awarded. Consequently, the prize for the Poincaré Conjecture has not been awarded at the time of writing. See also the Clay Mathematics Institute's page on Perelman's work.