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Poincaré conjecture

Historically, while the conjecture in dimension three seemed plausible, the generalized conjecture was thought to be false. In 1961 [Stephen Smale] shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental [h-cobordism theorem] . In 1982 [Michael Freedman] proved the Poincaré conjecture in dimension four. Freedman's work left open the possibility that there is a smooth four-manifold homeomorphic to the four-sphere which is not diffeomorphic to the four-sphere. This so-called smooth Poincare conjecture , in dimension four, remains open and is thought to be very difficult. [Milnor] 's [exotic sphere] s show that the smooth Poincare conjecture is false in dimension seven, for example.

These earlier successes in higher dimensions left the case of three dimensions in limbo. The Poincaré conjecture was essentially true in both dimension four and all higher dimensions for substantially different reasons. In dimension three, the conjecture had an uncertain reputation until the [geometrization conjecture] put it into a framework governing all 3-manifolds. [John Morgan] wrote:Morgan, John W., Recent progress on the Poincaré conjecture and the classification of 3-manifolds.
Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 1, 57–78

Hamilton's program and Perelman's solution
upright 250px Several stages of the Ricci flow on a two-dimensional manifold.
Hamilton's program was started in his 1982 paper in which he introduced the [Ricci flow] on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture. Reprinted in: In the following years he extended this work, but was unable to prove the conjecture. The actual solution was not found until [Grigori Perelman] published his papers using ideas from Hamilton's work.

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