Replies

This is from: http://www.answers.com/topic/poincar-conjecture :

At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology — what would later be called combinatorial topology and then algebraic topology. He was particularly interested in what topological properties characterized a sphere.

Poincaré claimed in 1900 that homology, a tool he had devised and based on prior work of Enrico Betti, was sufficient to tell if a 3-manifold was a 3-sphere. In a 1904 paper he described a counterexample, now called the Poincaré sphere, that had the same homology as a 3-sphere. Poincaré was able to show the Poincaré sphere had a fundamental group of order 120. Since the 3-sphere has trivial fundamental group, he concluded this was a different space. This was the first example of a homology sphere, and since then, many more have been constructed.

In this same paper, he wondered if a 3-manifold with the same homology as a 3-sphere but also trivial fundamental group had to be a 3-sphere. Poincaré's new condition, i.e. "trivial fundamental group", can be phrased as "every loop can be shrunk to a point".

The original phrasing was as follows:

Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?

Poincaré never declared whether he believed this additional condition could distinguish the 3-sphere, but nonetheless, the statement that it does has come down in history as the Poincaré conjecture. Here is the standard form of the conjecture:

Every simply connected closed (i.e. compact and without boundary) 3-manifold is homeomorphic to a 3-sphere. Loosely speaking, this means that if a given 3-manifold is "sufficiently like" a sphere (most importantly, that each loop in the manifold can be shrunk to a point), then it is really just a 3-sphere.

They never say too much about what Perelman actually did.

They never say too much about what Perelman actually did.

More accurately, from Wolfram Mathworld comes this statement of what he actually did, which is establish Thurston's Geometrization Conjecture's, a generalization of Poincare's Conjecture.

Thurston's Geometrization Conjecture

Thurston's conjecture proposed a complete characterization of geometric structures on three-dimensional manifolds.
Before stating Thurston's geometrization conjecture in detail, some background information is useful. Three-dimensional manifolds possess what is known as a standard two-level decomposition. First, there is the connected sum decomposition, which says that every compact three-manifold is the connected sum of a unique collection of prime three-manifolds.
The second decomposition is the Jaco-Shalen-Johannson torus decomposition, which states that irreducible orientable compact 3-manifolds have a canonical (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold removed by the tori is either "atoroidal" or "Seifert-fibered."
Thurston's conjecture is that, after you split a three-manifold into its connected sum and the Jaco-Shalen-Johannson torus decomposition, the remaining components each admit exactly one of the following geometries:
1. Euclidean geometry,
2. Hyperbolic geometry,
3. Spherical geometry,
4. The geometry of ,
5. The geometry of ,
6. The geometry of the universal cover of the Lie group ,
7. Nil geometry, or
8. Sol geometry.
Here, is the 2-sphere (in a topologist's sense) and is the hyperbolic plane. If Thurston's conjecture is true, the truth of the Poincaré conjecture immediately follows. Thurston shared the 1982 Fields Medal for work done in proving that the conjecture held in a subset of these cases.
Six of these geometries are now well understood, and there has been a great deal of progress with hyperbolic geometry (the geometry of constant negative scalar curvature). However, the geometry of constant positive curvature is still poorly understood, and in this geometry, the Thurston elliptization conjecture extends the Poincaré conjecture (Milnor).
Results due to Perelman (2002, 2003) appear to establish the geometrization conjecture, and thus also the Poincaré conjecture. Unlike a number of previous manuscripts attempting to prove the Poincaré conjecture, mathematicians familiar with Perelman's work describe it as well thought-out and expect that it will be difficult to locate any mistakes (Robinson 2003).