Poincar� conjecture (French, IPA-frpw??ka?epron) is a theorem about the characterization (Characterization (mathematics)) of the three-dimensional sphere (3-sphere) among three-dimensional manifolds (3-manifold). Originally conjectured by Henri Poincar�, the claim concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed (Closed manifold) 3-manifold). The Poincar� conjecture claims that if such a space has the additional property that each loop (path (topology)) in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An analogous result (generalized Poincar� conjecture) has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv.org. The proof followed the program of Richard Hamilton (Richard Hamilton (professor)). Several high-profile teams of mathematicians have since verified the correctness of Perelmans proof.

The Poincar� conjecture, before being proven, was one of the most important open questions in topology. It is one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. Perelmans work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined. Perelman was awarded the Millennium Prize on 18 March, 2010. The Poincar� conjecture is the first and, as of 2010, only solved Millennium problem.

On December 22, 2006, the journal Science (Science (journal)) honored Perelmans proof of the Poincar� conjecture as the scientific "Breakthrough of the Year", the first time this had been bestowed in the area of mathematics.