Difference between revisions of "Poincaré conjecture"
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| − | An assertion attributed to H. Poincaré and stating: Any closed simply-connected [[Three-dimensional manifold|three-dimensional manifold]] is homeomorphic to the three-dimensional sphere. A natural generalization is the following assertion (the generalized Poincaré conjecture): Any closed | + | {{TEX|done}} |
| + | An assertion attributed to H. Poincaré and stating: Any closed simply-connected [[Three-dimensional manifold|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 $n=4$). | ||
Revision as of 15:42, 13 July 2014
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 $n=4$).
Comments
References
| [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) |
How to Cite This Entry:
Poincaré conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_conjecture&oldid=32433
Poincaré conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_conjecture&oldid=32433
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article