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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073000/p0730001.png" />-dimensional manifold which is homotopy equivalent to the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073000/p0730002.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073000/p0730003.png" /> is homeomorphic to it; at present (1991) it has been proved for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073000/p0730004.png" /> (and for smooth manifolds also when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073000/p0730005.png" />).
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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 $n\geq 5$ (and for smooth manifolds also when $n=4$).
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====References====
 
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<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>
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<table>
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<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>
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<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>
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<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>
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<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>
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</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=22917
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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