Poincare conjecture

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

TheoremEvery 3-manifold without boundarythat is homotopy equivalent to the $\mathrm{3}$-sphereis in fact homeomorphic to it.Or, in a more elementary form:every simply-connected compact $\mathrm{3}$-manifold without boundaryis homeomorphic to $S^{\mathrm{3}}$.

The first statement is also truewhen $3$ 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 Prizefor resolution of the Poincaré Conjecture, but has also declined this prize.(The long delay in the awarding of the Millennium Prizewas largely due to the way that Perelman chose to publish his results.Full details of his proof did not appearin a peer reviewed mathematical publication until 2006,and the Millennium Prize rules require a waiting period of two yearsafter 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.