Poincaré conjectureThe Poincaré conjecture is widely considered the most important unsolved problem in topology. It was first formulated by Henri Poincaré in 1904. In 2000 the Clay Mathematics Institute selected the Poincaré conjecture as one of seven Millennium Prize Problems and offered a $1,000,000 prize for its solution. The conjecture states:
Every simply connected compact 3-manifold without boundary is homeomorphic to a 3-sphere.
Loosely speaking, this means that if a given "three-dimensional object" has a set of sphere-like properties (most notably that all loops in it can be shrunken to points), then it is really just a "deformed version" of a 3-sphere. (Note that "three-dimensional", in this context, refers to the intrinsic topological dimension of the object, the one the object has regardless of where it may live.)
The conjecture has induced a long list of false proofs, and some of them have led to a better understanding of low-dimensional topology.
Analogues of the Poincaré conjecture in dimensions other than 3 can also be formulated:
Every compact n-manifold which is homotopy equivalent to the n-sphere is homeomorphic to the n-sphere.
The Poincaré conjecture as given above is equivalent to the case n=3. The difficulty of low-dimensional 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 3-dimensional version of Poincaré's conjecture remains unsolved.
Its solution is related to the problem of classifying 3-manifolds. A classification of 3-manifolds is generally accepted to mean that one can generate a list of all 3-manifolds up to homeomorphism with no repetitions. Such a classification is equivalent to a recognition algorithm, which would be able to check if two 3-manifolds were homeomorphic or not.
One can regard the Poincaré Conjecture as a special case of Thurston's 25-year-old Geometrization Conjecture. The latter conjecture, if proven, would finish off the quest for a classification of 3-manifolds. 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 3-manifold with finite fundamental group has a spherical geometry, i.e. it is covered by the 3-sphere. 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.