The Poincare Conjecture

Well, after a year of looking at the background mathematics, we're finally ready to state the Poincaré Conjecture:

Every simply-connected, closed 3-manifold is homeomorphic to the 3-sphere.

This isn't exactly how Poincaré put it (for a start, he said it in French) but this is the best way to express it given the terminology we've used up until this point.

Poincaré's conjecture has to do with three-dimensional manifolds but it might be easiest to start off thinking about the two-dimensional version:

Every simply-connected, closed 2-manifold is homeomorphic to the 2-sphere.

Consider the surface of a ball and a torus. Both are closed 2-manifolds. But only one is simply-connected. A torus isn't simply-connected so it can't be homeomorphic to the 2-sphere. The surface of a ball is simply-connected and it is homeomorphic to the 2-sphere.

The big question is: is it possible to find a closed 2-manifold that is simply-connected but is not homeomorphic to the 2-sphere? In a nutshell: no, it's not. If it's a closed 2-manifold and it's simply-connected then there isn't any topological property that will distinguish it from the 2-sphere.

The Poincaré Conjecture is that this is true for 3-manifolds as well.

Of course, this is really just the beginning of our journey. Mathematicians have spent the last century trying to prove this so we still have a lot to cover.

Interestingly, it's already been proven that it's true for dimensions greater than 3 (as well as for 1 and 2 dimensions). Stephen Smale proved it in 1960 for dimensions greater than 6 and then extended his proof to cover dimensions greater than 4. In 1966, he was awarded the highest prize in mathematics, the Fields Medal, for this proof. Michael Freedman then proved in 1982 that it's true for 4 dimensions which won him the Fields Medal in 1986.

UPDATE: next post

The original post was in the category: poincare_project but I'm still in the process of migrating categories over.

The original post had 2 comments I'm in the process of migrating over.