The Generalized Poincaré Conjecture extends the classical Poincaré Conjecture, which is a statement about the topology of 3-dimensional manifolds. The original Poincaré Conjecture, proposed by Henri Poincaré in 1904, asserts that any simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
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There are two cases:
- (topological) manifolds
- differential manifolds
Questions: are all compact manifolds / differential manifolds homotopic / diffeomorphic to the sphere in that dimension?
- Original problem posed, for topological manifolds.AKA: classification of compact 3-manifolds. The result turned out to be even simpler than compact 2-manifolds: there is only one, and it is equal to the 3-sphere.
- for differential manifolds:Counter examples are called exotic spheres.Totally unpredictable count table:is an open problem, there could even be infinitely many. Again, why are things more complicated in lower dimensions??