The Poincaré Conjecture is first and only of the Clay Millennium problems to be solved. It was proved by Grigori Perelman who subsequently turned down the $1 million prize money, left mathematics, and moved in with his mother in Russia. Most explanations of the problem are overly-simplistic or overly-technical, but on this blog I try to hit a nice middle-ground. Here is the statement of the conjecture from wikipedia:
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.
This is a statement about topological spaces. Let’s define each of the terms in the conjecture:
simply connected space – This means the space has no “holes.” A football is simply connected, but a donut is not. Technically we can say but I’ll explain this notation further on.
closed space – The space is finite and has no boundaries. A sphere (more technically a 2-sphere or ) is closed, but the plane () is not because it is infinite. A disk is also not because even though it is finite, it has a boundary.
manifold – At every small neighborhood on the space, it approximates Euclidean space. A standard sphere is called a 2-sphere because it is actually a 2-manifold. Its surface resembles the 2d plane if you zoom into it so that the curvature approaches 0. Continuing this logic the 1-sphere is a circle. A 3-sphere is very difficult to visualize because it has a 3d surface and exists in 4d space.
homeomorphic – If one space is homeomorphic to another, it means you can continuously deform the one space into the other. The 2-sphere and a football are homeomorphic. The 2-sphere and a donut are not; no matter how much you deform a sphere, you can’t get that pesky hole in the donut, and vice-versa.
What the conjecture is basically saying then is this:
Any finite 3-dimensional space, which doesn’t have any “holes” in it, can be continuously deformed into a 3-sphere.
This was proven for all dimensions > 3, but 3-manifolds proved to be the trickiest.
Now, how do you show that 2 spaces are homeomorphic to each other? One way is to show that the spaces share the same fundamental group. Hopefully you remember what a group is, but basically it’s a collection of elements which includes an identity element, and a product. Also each element has an inverse. The fundamental group basically describes how many ways you can draw a path in a space. It is organized as follows, where each element is a path which starts and ends at the same point. The identity fundamental group is the trivial path of length 0, i.e. a single point. The product in the fundamental group is just combining 2 paths end-to-end. Since all paths start and end and the same point you can always take their product and get a new element in the group.
The fundamental group of the n-sphere is equivalent to the group with one element (the identity element), because any path can always “shrink” down into a single point due to the fact the n-sphere has no hole. If a space’s fundamental group is trivial, then that means it is simply connected, as described above. To say the fundamental group for any n-sphere is trivial you use the following notation:
Where means this group contains elements of 1-dimensional closed paths. A little bit off topic but even more fascinating, is that you can also examine higher dimensional paths and these non-fundamental groups develop an interesting and complex structure.
Anyway, the Poincaré conjecture says that if for any 3-manifold, , if , then can be continuously deformed into .
Perelman finally solved this for 3-manifolds by using Ricci flow techniques to show that these manifolds are homeomorphic.
Update: 6 years after his proof of the Poincare Conjecture and 5 days after this post, Clay Mathematics officially awarded the Millennium prize to Grigori Perelman. If he declines, this could be the first time in modern history where someone living at home refuses $1 million.