Poincaré Conjecture

A Simply Connected 3-Manifold is Homeomorphic to the 3-Sphere. The generalized Poincaré conjecture is that a Compact $n$-Manifold is Homotopy equivalent to the $n$-sphere Iff it is Homeomorphic to the $n$-Sphere. This reduces to the original conjecture for $n=3$.

The $n=1$ case of the generalized conjecture is trivial, the $n=2$ case is classical, $n=3$ remains open, $n=4$ was proved by Freedman (1982) (for which he was awarded the 1986 Fields Medal), $n=5$ by Zeeman (1961), $n=6$ by Stallings (1962), and $n\geq 7$ by Smale in 1961 (Smale subsequently extended this proof to include $n\geq 5$.)

See also Compact Manifold, Homeomorphic, Homotopy, Manifold, Simply Connected, Sphere, Thurston's Geometrization Conjecture

References

Freedman, M. H. ``The Topology of Four-Differentiable Manifolds.'' J. Diff. Geom. 17, 357-453, 1982.

Stallings, J. ``The Piecewise-Linear Structure of Euclidean Space.'' Proc. Cambridge Philos. Soc. 58, 481-488, 1962.

Smale, S. ``Generalized Poincaré's Conjecture in Dimensions Greater than Four.'' Ann. Math. 74, 391-406, 1961.

Zeeman, E. C. ``The Generalised Poincaré Conjecture.'' Bull. Amer. Math. Soc. 67, 270, 1961.

Zeeman, E. C. ``The Poincaré Conjecture for $n\geq 5$.'' In Topology of 3-Manifolds and Related Topics, Proceedings of the University of Georgia Institute, 1961. Englewood Cliffs, NJ: Prentice-Hall, pp. 198-204, 1961.