# Poincaré conjecture

From Encyclopedia of Mathematics

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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 $n=4$).

#### 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:**

Poincare conjecture.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Poincare_conjecture&oldid=23467