The conjecture that every simply connected 3-manifold is homeomorphic
to the 3-sphere. This conjecture was first proposed in 1904 by H. Poincar¨¦
(Poincar¨¦ 1953, pp. 486 and 498), and subsequently generalized to the
conjecture that every compact -manifold is homotopy-equivalent to the
-sphere iff it is homeomorphic to the -sphere. The generalized statement
reduces to the original conjecture for .
The case of the generalized conjecture is trivial, the case is classical,
remains open, was proved by Freedman (1982) (for which he was awarded
the 1986 Fields medal), by Zeeman (1961), by Stallings (1962), and by
Smale in 1961. Smale subsequently extended his proof to include .
Compact Manifold, Homeomorphic, Homotopy, Manifold, Property P, Simply
Connected, Sphere, Thurston's Geometrization Conjecture
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