Poincaré conjecture

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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 $n$-dimensional manifold which is homotopy equivalent to the $n$-dimensional sphere $S^n$ is homeomorphic to it; at present (1991) it has been proved for all $n\geq5$ (and for smooth manifolds also when $n=4$).



[a1] S. Smale, "Generalized Poincaré's conjecture in dimensions greater than four" Ann. of Math. (2) , 74 (1961) pp. 199–206
[a2] S. Smale, "On the structure of manifolds" Amer. J. Math. , 84 (1962) pp. 387–399
[a3] J.R. Stallings, "Polyhedral homotopy-spheres" Bull. Amer. Math. Soc. , 66 (1960) pp. 485–488
[a4] M.H. Freedman, "The topology of four-dimensional manifolds" J. Diff. Geometry , 17 (1982) pp. 357–453
[a5] J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)
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This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article