Topology and the Million-Dollar Poincare Conjecture

It's not often that the abstract mathematical discipline of topology is in the news, but a respected British mathematician announced last month that he has proved a nearly century-old puzzle, the Poincaré conjecture, about spheres and their generalizations to higher-dimensional spaces.

The theorem has been shown to hold for every dimension except three, and should Southampton University Professor Martin Dunwoody's proposed proof (available through his Web site) withstand the scrutiny of his peers, it will finally settle this famous unsolved question. The conjecture is so well-known and its proof has been so elusive that the Clay Institute of Boston last year offered $1 million for its solution. (It is one of only seven problems on the institute's list.)

Topological Tidbits

Before getting to the Poincaré conjecture, let me describe a few ideas from topology that provide a flavor of its results.

Topology is that branch of geometry concerned only with those basic properties of geometric figures that remain unchanged when the figures are twisted and distorted, stretched and shrunk, subjected to any "schmooshing" at all as long as they're not ripped or torn.

Size and shape are not topological properties since clay balls, dice, and oranges, for example, can be contracted, expanded or transformed into one another without ripping.

Whether a closed curve in space — say an unbroken piece of thread — has a knot in it or not is, however, a topological property of the curve in space. That a closed curve lying on a flat plane, no matter how convoluted it is, divides the plane into two parts — the inside and the outside — is a topological property of the curve in the plane. The dimensionality of a geometric figure (how many dimensions it posseses), whether or not it has a boundary and if so of what sort, these too are topological properties.

Also a matter of some significance is the genus of a figure — the number of holes it contains. A sphere has genus 0 since it contains no holes, a torus (a doughnut or tire-shaped figure) has genus 1, and eyeglass frames without the lenses or pretzels have genus 2, and so on.

Genus 0 objects such as cubes and oranges are topologically equivalent. Less obviously, so are a doughnut and a coffee cup with a handle, both figures of genus 1. To see this, imagine a coffee cup made of clay. Flatten the body of the cup and expand the size of its handle by squeezing clay from the body into the handle. The finger hole of the cup's handle is in this way transformed into the hole of the doughnut.

Mountain Climbing

There are applications of these ideas. Often, for example, what is important is knowing that a solution exists, not necessarily having a method for finding it.

To get a feel for such so-called existence proofs, imagine that a mountain climber begins his ascent at 6 a.m. Monday and arrives at the summit at noon. Tuesday morning he starts down at 6 a.m. and reaches the base at noon. We make no other assumptions about how fast or how smoothly the climber travels on the two days in question. He might, for all we know, have climbed at a slow pace and rested often on his way up on Monday, and after a leisurely stroll around the summit Tuesday morning descended at a literally breakneck pace, managing to fall the last 1000 feet.

The question now is: Can we be certain that no matter how he climbs there will necessarily exist some instant between 6 a.m. and noon on the two days when the climber is at exactly the same elevation? (Answer below)

The Poincaré Conjecture

So, with these appetizers under our belts, we can proceed to Dunwoody's possible accomplishment.

Poincaré's conjecture says that a certain property of a sphere in space holds for higher-dimensional analogues of a sphere. To understand the property, imagine stretching a rubber band around an orange. We can contract this rubber band slowly, making sure it neither breaks nor loses contact with the surface of the orange. In this way we can shrink the rubber band to a point. We can't do this with a rubber band stretched around a doughnut (either around the hole or around the body).

The orange, but not the doughnut, is said to be "simply connected." Henri Poincaré, who, incidentally, almost discovered relativity theory before Einstein, was aware of the fact that a ball like an orange could be characterized by this property of simple connectivity. He wondered if this held true for balls in higher-dimensional space.

If Dunwoody's proof survives the vetting of other topologists, Poincaré can stop wondering and rest in peace. And with his million dollars, Dunwoody can buy doughnuts and oranges for all the world's topologists.

Answer to question: The answer is Yes, and the proof is vivid and convincing. Imagine the ascent and descent, exact in every detail, being made simultaneously by two climbers. One climber starts at the base and the other at the summit and they both begin their journeys at 6 a.m. of the same day mimicking what the original climber did on Monday and Tuesday, respectively. It's clear that these two climbers will pass each other going in opposite directions and at that instant their elevations will be the same. Since they're only re-enacting the original climber's ascent and descent, we can be certain that the original climber was at the same elevation at the same time on the two successive days.

Professor of mathematics at Temple University and adjunct professor of journalism at Columbia University, John Allen Paulos is the author of several best-selling books, including Innumeracy and A Mathematician Reads the Newspaper. His Who’s Counting? column on appears the first weekend of every month.