Credit: Andrey Krasnov
In a 1904 paper, the French mathematician Jules Henri Poincaré stated that a sphere is a sphere is a sphere. You can punch, kick and throw it; you can inflate or deflate it; you can mold the sphere into another shape. But in the world of topology, no matter what you do to it, the resulting deformed, twisted and complicated form is still a sphere.
Also, you cannot poke a hole in it. You cannot, for example, turn your sphere into a donut. You cannot turn it into a coffee cup with a handle, frames for your eyeglasses or a key ring. You can stick your finger into it, but you can’t actually puncture the surface or reach inside. If you break the surface in any way you’ve ventured into a different genus of topological objects.
Say you’re walking down a street, and you encounter a strange and complicated shape whose surface sports peaks and valleys, mountains and molehills, but no holes. If you were a mathematician, you may want to study the way that functions behave on it. Poincaré‘s conjecture says that no matter what it looks like, it’s a sphere. The conjecture gives mathematicians a short and easy way to identify a deformed blob as a sphere in disguise.
There is one more complication: When most people think of a sphere, they generally consider the space that a sphere occupies—a ping-pong ball, for example. When topologists talk about a sphere, they are talking exclusively about its surface.
A 1-sphere, for example, is the outside of a circle. A 2-sphere is the curved surface of what we consider a sphere. It is two-dimensional because, if you stand on the surface and look around, it appears that you are in a two-dimensional space. The surface of the Earth serves as a rough analogy—the world essentially appears flat when we stand on the ground and survey the horizon.
In its original form, however, the Poincaré conjecture concerned three-dimensional spheres (i.e., 3-spheres). These shapes are difficult, perhaps impossible, to visualize—the universe, for instance, is thought of as a 3-sphere. Even without being able to picture it, draw it or know that it exists, we can do math on a 3-sphere. We can calculate distances between points. Any system that can be characterized by three numbers automatically determines a three-dimensional shape. In baseball, for example, if you tally the numbers of runs, pitches and fouls for each inning of a game that doesn’t go into extra innings, you have established nine data points in a three-dimensional space. With those nine points, you can make statements about the “shape” you have created.
Now imagine that this 3-sphere universe is distorted, wadded, dilated and deformed (but not punctured). If we lived on this deformed 3-sphere, you could feasibly walk across the Golden Gate Bridge and end up to Mars.
Or, you could think about it this way: You tie a lasso around your blob and tighten it until the string lies on the surface. If, for every different way you can tie the lasso, you can slip it off, then the blob is a sphere. The term for this is “simply connected.” If it is possible to tie the lasso in such a way that it proves impossible to remove the lasso without breaking either the rope or the blob, it is not a sphere.
A donut, for example, is not “simply connected.” If your lasso passes through the center of a donut, you cannot remove it without either altering the shape of the donut or cutting the rope. Though this is not its mathematically precise term, “breaking the donut” is absolutely, positively not allowed in the world of topology.
To illustrate Perelman’s work, Jim Carlson, president of the Clay Mathematical Institute, draws a very complicated, closed squiggle. “This is really a circle, but it’s a very wild circle,” he says. (If his squiggle were an island, it would be an island ringed with fjords.)
“The idea is, in some sense, to apply heat to the shape and to allow the heat to simplify it. Take this very complicated wild circle, and imagine putting a little air hose in here and inflating it,” he said, drawing a little box next to the squiggle. “It will dilate, and eventually it will achieve a round shape. Imagine a crinkled up balloon - you want to know what its real shape is, well blow it up with air, and then look at it. It achieves the simplest possible shape after you blow it up enough.”
This notion of adding air, or heat, to a complicated shape was first developed by Richard Hamilton in the 1980s and is called Ricci Flow. Hamilton came close to solving the Poincaré conjecture, but he failed to successfully account for all singularities that may arise on the object. Singularities might be thought of as places where the fabric of the object is ‘pinched.’ Imagine that the balloon, for example, turned out to be shaped like a barbell. In that case, the two sides of the barbell would continue to inflate, while the connecting rod became thinner and thinner. As time goes on, it will not resemble a sphere. This connecting rod would be considered a singularity.
Perelman’s insight was to essentially ‘snip’ the rod. By utilizing abstract scissors, Perelman’s method allowed each side of the barbell to become its own sphere. The two resulting spheres would be topologically indistinguishable. Mathematicians refer to this process as ‘surgery’ on a 3-dimensional object.
Perelman’s use of surgery on these complicated surfaces was unprecedented and unexpected.
Originally published August 24, 2006