## Poincaré Conjecture

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

The case of the generalized conjecture is trivial, the case is classical, remains open, was proved by Freedman (1982) (for which he was awarded the 1986 Fields Medal), by Zeeman (1961), by Stallings (1962), and by Smale in 1961 (Smale subsequently extended this proof to include .)

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 .'' 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.