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In the mathematical field of geometric topology, the Poincaré conjecture ( UK: / ˈpwæ̃kæreɪ /, [2] US: / ˌpwæ̃kɑːˈreɪ /, [3] [4] French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
In its original form, the Poincaré conjecture states that every simply connected closed three-manifold is homeomorphic to the three-sphere (in a topologist's sense) S^3, where a three-sphere is simply a generalization of the usual sphere to one dimension higher. More colloquially, the conjecture says that the three-sphere is the only type of ...
Poincaré Conjecture. In 1904 the French mathematician Henri Poincaré asked if the three dimensional sphere is characterized as the unique simply connected three manifold. This question, the Poincaré conjecture, was a special case of Thurston’s geometrization conjecture.
Poincaré conjecture, in topology, conjecture—now proven to be a true theorem—that every simply connected, closed, three-dimensional manifold is topologically equivalent to S 3, which is a generalization of the ordinary sphere to a higher dimension (in particular, the set of points in four-dimensional space that are equidistant from the ...
The conjecture that the answer to Poincare's question is `No' has come to be known as the Poincare Conjecture: PC: If M is a 3-manifold with trivial fundamental group, and H_i(M)=0 for i=1,2 and = Z for i=0,3 (i.e., M has the homology groups of a 3-sphere), then M is homeomorphic to the 3-sphere.
Conjecture and the closely related 3-dimensional spherical space-form conjecture are then immediate. The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincaré Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture.
Dec 22, 2006 · The Poincaré Conjecture--Proved. This article has a correction. To mathematicians, Grigori Perelman's proof of the Poincaré conjecture qualifies at least as the Breakthrough of the Decade. But it has taken them a good part of that decade to convince themselves that it was for real.
