Difference between revisions of "Poincaré conjecture"
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| − | An assertion attributed to H. Poincaré and stating: Any closed simply-connected [[ | + | 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 $n\geq 5$ (and for smooth manifolds also when $n=4$). |
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Smale, "Generalized Poincaré's conjecture in dimensions greater than four" ''Ann. of Math. (2)'' , '''74''' (1961) pp. 199–206</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Smale, "On the structure of manifolds" ''Amer. J. Math.'' , '''84''' (1962) pp. 387–399</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> J.R. Stallings, "Polyhedral homotopy-spheres" ''Bull. Amer. Math. Soc.'' , '''66''' (1960) pp. 485–488</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> M.H. Freedman, "The topology of four-dimensional manifolds" ''J. Diff. Geometry'' , '''17''' (1982) pp. 357–453</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Smale, "Generalized Poincaré's conjecture in dimensions greater than four" ''Ann. of Math. (2)'' , '''74''' (1961) pp. 199–206</TD></TR> | ||
| + | <TR><TD valign="top">[a2]</TD> <TD valign="top"> S. Smale, "On the structure of manifolds" ''Amer. J. Math.'' , '''84''' (1962) pp. 387–399</TD></TR> | ||
| + | <TR><TD valign="top">[a3]</TD> <TD valign="top"> J.R. Stallings, "Polyhedral homotopy-spheres" ''Bull. Amer. Math. Soc.'' , '''66''' (1960) pp. 485–488</TD></TR> | ||
| + | <TR><TD valign="top">[a4]</TD> <TD valign="top"> M.H. Freedman, "The topology of four-dimensional manifolds" ''J. Diff. Geometry'' , '''17''' (1982) pp. 357–453</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J. Dieudonné, "A history of algebraic and differential topology 1900–1960" , Birkhäuser (1989)</TD></TR> | ||
| + | </table> | ||
Latest revision as of 17:53, 10 April 2023
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 $n\geq 5$ (and for smooth manifolds also when $n=4$).
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=53763
Poincaré conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_conjecture&oldid=53763
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article