Poincaré Conjecture

A Simply Connected 3-Manifold is Homeomorphic to the 3-Sphere. The generalized Poincaréconjecture is that a Compact -Manifold is Homotopy equivalent to the -sphereIff 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 byFreedman (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 .)

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.

• Orthologic
• Orthonormal Basis
• Orthonormal Functions
• Orthonormal Vectors
• Orthopole
• Orthoptic Curve
• Orthotomic
• Orthotope
• Osborne's Rule
• Oscillation
• Oscillation Land
• Osculating Circle
• Osculating Curves
• Osculating Interpolation
• Osculating Plane
• Osculating Sphere
• Osedelec Theorem
• Ostrowski's Inequality
• Ostrowski's Theorem
• Otter's Tree Enumeration Constants
• Outdegree
• Outer Automorphism Group
• Outer Product
• Oval
• Oval of Descartes