An assertion attributed to H. Poincaré and stating: Any closed simply-connected three-dimensional manifold is homeomorphic to the three-dimensional sphere. A natural generalization is the following assertion (the generalized Poincaré conjecture): Any closed -dimensional manifold which is homotopy equivalent to the -dimensional sphere is homeomorphic to it; at present (1991) it has been proved for all (and for smooth manifolds also when ).
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Poincaré conjecture. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Poincar%C3%A9_conjecture&oldid=23466