View source for Poincaré conjecture

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" />).



====Comments====


====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>