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Henri Poincaré (1854–1912) was one of the most eminent French mathematicians of the past two centuries. Together with his German contemporary, David Hilbert, Poincaré was one of the last mathematicians who not only had a deep understanding of all areas of mathematics but also was active in all of them. After Hilbert and Poincaré, mathematics became so vast that nobody could hope to grasp more than a minute part of it.

One of Poincaré’s best-known problems is what is today called the Poincaré conjecture. It has baffled and challenged several generations of mathematicians. In the spring of 2002 Michael Dunwoody from the University of Southampton believed—albeit only for a few weeks—that he had been successful in finding a proof for the conjecture.

The Poincaré conjecture is considered so important that the Clay Mathematics Institute named it one of the seven Millennium Prize problems. Each person who is the first to solve one of the problems will be awarded $1 million. The prize committee believed it would take decades for the first of the prizes to be allocated; but here, a mere two years after the announcement, the Clay Foundation seemed to be faced with the possibility of having to award its first prize. But doubts arose about the validity of the proof provided by Dunwoody—and as it turned out there was ample reason.

The Poincaré conjecture falls within the realm of topology. This branch of mathematics focuses, roughly speaking, on the issue of whether one body can be deformed into a different body through pulling, squashing, or rotating, without tearing or gluing pieces together. A ball, an egg, and a flowerpot are, topologically speaking, equivalent bodies,

since any one of them can be deformed into any of the others without performing any of the “illegal” actions. A ball and a coffee cup, on the other hand, are not equivalent, since the cup has a handle, which could not have been formed out of the ball without poking a hole through it. The ball, egg, and flowerpot are said to be “simply connected” as opposed to the cup, a bagel, or a pretzel. Poincaré sought to investigate such issues not by geometric means but through algebra, thus becoming the originator of “algebraic topology.”

In 1904 he asked whether all bodies that do not have a handle are equivalent to spheres. In two dimensions this question refers to the surfaces of eggs, coffee cups, and flowerpots and can be answered yes. (Surfaces like the leather skin of a football or the crust of a bagel are two-dimensional objects floating in three-dimensional space.) For three-dimensional surfaces in four-dimensional space, the answer is not quite clear. While Poincaré was inclined to believe that the answer was yes, he was not able to provide a proof.

Interestingly enough, within several decades mathematicians were able to prove the equivalent of Poincaré’s conjecture for all bodies of dimension greater than four. This is because higher-dimensional spaces provide more elbowroom, so mathematicians find it simpler to prove the Poincaré conjecture. Christopher Zeeman in Cambridge started the race in 1961 by proving Poincaré’s conjecture for five-dimensional bodies. In the same year Stephen Smale from Berkeley announced a proof for bodies of seven and all higher dimensions. John Stallings, also from Berkeley, demonstrated a year later that the conjecture was correct for six-dimensional bodies. Finally, in 1982, Michael Freedman from San Diego provided proof for four-dimensional bodies. All that was left now were three-dimensional bodies floating in four-dimensional space. This was all the more frustrating since four-dimensional space represents the space-time continuum in which we live.

Michael Dunwoody thought he had found a proof. On April 7, 2002, he posted a preprint entitled “Proof for the Poincaré Conjecture” on the Web. Reputable mathemati-

cians called it the first serious attempt at solving the Poincaré conjecture for a long time. In higher dimensions it is really not at all easy to recognize a sphere when you meet one, despite all the additional elbowroom. To understand the difficulty, just think of olden times, when pirates and adventurers did not realize that the world was round, despite all their expeditions and discovery trips. Dunwoody based his work on previous work by Hyam Rubinstein, an Australian mathematician who had studied the surfaces of four-dimensional spheres. (Remember: The surface of a four-dimensional object is a three-dimensional object.)

Dunwoody required no more than five pages to develop his argument, which concluded with the statement that all simply connected, closed, three-dimensional surfaces can be converted into the surface of a sphere by means of pulling, squishing, and squashing—but without tearing. This statement is equivalent to a proof of Poincaré’s conjecture.

Alas, only a few weeks after posting his findings on the Web, Dunwoody was forced to append a question mark to the title of his paper. One of his colleagues had discovered that the proof had a hole. The title now read “Proof for the Poincaré Conjecture?” and, though Dunwoody immediately attempted to fill the hole, he was unsuccessful. So were friends and colleagues who tried the same. A few weeks later the paper disappeared from the Web. Poincaré’s conjecture remained as elusive as ever. (See, however, Chapter 13.)