Every simply connected compact 3manifold without boundary is homeomorphic to a 3sphere.
Loosely speaking, this means that if a given "threedimensional object" has a set of spherelike properties (most notably that all loops in it can be shrunken to points), then it is really just a "deformed version" of a 3sphere.
The conjecture has induced a long list of false proofs, and some of them have led to a better understanding of lowdimensional topology.
Analogues of the Poincaré conjecture in dimensions other than 3 can also be formulated:
Every compact nmanifold which is homotopy equivalent to the nsphere is homeomorphic to the nsphere.
The Poincaré conjecture as given above is equivalent to the case n=3. The difficulty of lowdimensional topology is highlighted by the fact that these analogues have now all been proven (with dimension n=4 being the hardest one by far), while the original 3dimensional version of Poincaré's conjecture remains unsolved.
Its solution is related to the problem of classifying 3manifolds. A classification of 3manifolds is generally accepted to mean that one can generate a list of all 3manifolds up to homeomorphism with no repetitions. Such a classification is equivalent to a recognition algorithm, which would be able to check if two 3manifolds were homeomorphic or not.
One can regard the Poincaré Conjecture as a special case of Thurston's 25yearold Geometrization Conjecture. The latter conjecture, if proven, would finish off the quest for a classification of 3manifolds. The only parts of the Geometrization Conjecture left to be proven are called the Hyperbolization Conjecture and the Elliptization Conjecture.
The Elliptization Conjecture states that every closed 3manifold with finite fundamental group has a spherical geometry, i.e. it is covered by the 3sphere. The Poincaré Conjecture is exactly the subcase when the fundamental group is trivial.
In late 2002, reports surfaced that Grigori Perelman of Steklov Mathematical Institute, Saint Petersburg might have found a proof of the geometrization conjecture, carrying out a program outlined earlier by Richard Hamilton. In 2003, he posted a second preprint and gave a series of lectures in the United States. His proof is still being checked.
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