# Poincare conjecture

Until its proof in 2003, the Poincaré Conjecture was the central problem of low-dimensional topology. It is one of the http://www.claymath.orgClay Mathematics Institute http://www.claymath.org/millennium/Millennium Prize Problems, and so far the only one to be solved.

Theorem 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: every simply-connected compact $3$-manifold without boundary is homeomorphic to $S^{3}$.

The first statement is also true when $3$ is replaced by any other positive integer, but the 3-dimensional case turned out to be much harder than the other cases.

The Poincaré Conjecture was eventually proved by Grigoriy Perelman, who gave his proof in a series of preprints posted on arXiv.org in 2002 and 2003. For this work Perelman was offered a Fields Medal in 2006, though he declined it. On 18 March 2010, he was awarded the \$1,000,000 Millennium Prize for resolution of the Poincaré Conjecture, but has also declined this prize. (The long delay in the awarding of the Millennium Prize was largely due to the way that Perelman chose to publish his results. Full details of his proof did not appear in a peer reviewed mathematical publication until 2006, and the Millennium Prize rules require a waiting period of two years after such publication before the prize can be awarded.)

See also http://www.claymath.org/poincare/millenniumPrizeFull.pdfClay Mathematics Institute press release on the Millennium Prize for the Poincaré Conjecture (PDF file), and the http://www.claymath.org/millennium/Poincare_Conjecture/perelman+expositions.phpClay Mathematics Institute’s page on Perelman’s work.

Title Poincare conjecture PoincareConjecture 2013-03-22 13:56:16 2013-03-22 13:56:16 yark (2760) yark (2760) 20 yark (2760) Conjecture msc 57R60