Poincar.C3.A9 conjecture Conjecture

in mathematics, poincaré conjecture theorem characterization of 3-sphere, hypersphere bounds unit ball in four-dimensional space. conjecture states:

every connected, closed 3-manifold homeomorphic 3-sphere.

an equivalent form of conjecture involves coarser form of equivalence homeomorphism called homotopy equivalence: if 3-manifold homotopy equivalent 3-sphere, homeomorphic it.

originally conjectured henri poincaré, theorem concerns space locally looks ordinary three-dimensional space connected, finite in size, , lacks boundary (a closed 3-manifold). poincaré conjecture claims if such space has additional property each loop in space can continuously tightened point, three-dimensional sphere. analogous result has been known in higher dimensions time.

after century of effort mathematicians, grigori perelman presented proof of conjecture in 3 papers made available in 2002 , 2003 on arxiv. proof followed on program of richard s. hamilton use ricci flow attempt solve problem. hamilton later introduced modification of standard ricci flow, called ricci flow surgery systematically excise singular regions develop, in controlled way, unable prove method converged in 3 dimensions. perelman completed portion of proof. several teams of mathematicians have verified perelman s proof correct.

the poincaré conjecture, before being proven, 1 of important open questions in topology.