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Poincaré conjecture

In the 1950s and 1960s, other mathematicians were to claim proofs only to discover a flaw. Influential mathematicians such as [Bing] , [Haken] , [Moise] , and [Papakyriakopoulos] attacked the conjecture. In 1958 Bing proved a weak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere. Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.

Over time, the conjecture gained the reputation of being particularly tricky to tackle. [John Milnor] commented that sometimes the errors in false proofs can be "rather subtle and difficult to detect." Work on the conjecture improved understanding of 3-manifolds. Experts in the field were often reluctant to announce proofs, and tended to view any such announcement with skepticism. The 1980s and 1990s witnessed some well-publicized fallacious proofs (which were not actually published in [peer-reviewed] form).

An exposition of attempts to prove this conjecture can be found in the non-technical book * Poincaré's Prize * by George Szpiro.

** Dimensions **

The [classification of closed surfaces] gives an affirmative answer to the analogous question in two dimensions. For dimensions greater than three, one can pose the ** Generalized Poincaré conjecture ** : is a homotopy * n * -sphere homeomorphic to the * n * -sphere? The stronger assumption is necessary; in dimensions four and higher there are simply-connected manifolds which are not homeomorphic to an * n * -sphere.

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