EPILOGUE: THE MILLION-DOLLAR QUESTION
In the twentieth century topology rose to become one of the pillars of mathematics, sitting side by side with algebra and analysis. Many mathematicians who do not consider themselves topologists use topology on a daily basis. It is inescapable. Today, most first-year graduate students in mathematics are required take a full-year course in topology.
One way to gauge the importance of an academic field is to see the trail of awards for accomplishments in the discipline. There is no Nobel Prize in mathematics. The mathematical equivalent is the Fields Medal. Fields Medals have been awarded once every four years since 1936 (except during World War II). At each ceremony the medals are given to at most four mathematicians under the age of forty who made an outstanding contribution to mathematics. Of the forty-eight recipients, roughly a third were cited for their work in topology, and even more made contributions in closely related areas.
One specific topological problem is itself responsible for three Fields Medals. It was one of the most famous unsolved problems of the twentieth century—one that is still so important, and so difficult, that its resolution is worth $1 million to the mathematician who first proves it. The name of this thorny problem is the Poincaré conjecture.
The classification theorem for surfaces is one of the most elegant theorems in all of mathematics. It states that every surface is uniquely determined by its orientability, Euler number, and number of boundary components. Obviously, it would be nice to have a similar theorem for manifolds of every dimension, but this is a tall order. It is clear that if such a classification exists, the same checklist will not suffice since the Euler- Poincaré characteristic of every closed odd-dimensional manifold is zero (see chapter 23).
Poincaré dreamed of classifying higher-dimensional manifolds, but even classifying 3-manifolds was beyond his reach. The Poincaré conjecture was only the first step in this classification process.
The simplest closed n-manifold is the n-sphere, Sn. Poincaré was searching for a simple test to determine if a given a n-manifold is homeomorphic to Sn. In 1900 he thought he had such a test. He proved that any n-manifold that had the same homology as Sn must be homeomorphic to Sn.2 The homology of the n-sphere is especially simple. It has Betti numbers of 1 in dimensions 0 and n, 0 in all other dimensions, and it has no torsion.
Four years later he realized that his proof was flawed.3 Not only did he discover his own error, but he found a remarkable counterexample to his assertion. He constructed a pathological 3-manifold that had the same homology as S3, but was not homeomorphic to S3. He created the manifold by gluing together opposite faces of a solid dodecahedron, each with a 36° clockwise twist.
An interesting and surprising feature of Poincaré's dodecahedral space is that even though its first Betti number is zero, it is not simply connected. That is, every loop is zero in homology, but there exist loops in the manifold that cannot be shrunk to a point. In figure 23.3 we saw an example of a nontrivial loop on the double torus that is zero in homology, but in the dodecahedral space every loop that cannot be shrunk to a point is trivial in homology.
From this exotic example, Poincaré realized that homology alone was not sufficient to characterize Sn, not even S3. So he abandoned the n-dimensional question, focusing instead on 3-manifolds. He suspected that if all loops in a 3-manifold are topologically trivial, then the manifold must be homeomorphic to S3. This became the now-famous Poincaré conjecture.4
THE POINCARÉ CONJECTURE
Every simply connected closed 3-manifold is homeomorphic to the 3-sphere.
Actually, in Poincaré’s paper this statement was not a conjecture, but a question about whether it was true. He did not state his opinion on the direction it would go. Proving this theorem would be a far cry from classifying all 3-manifolds, but it would be an important first step.
Everyone likes a good challenge, and the Poincaré conjecture is as challenging as they come. It became one of a short list of problems—the four color theorem, Fermat's last theorem, the Riemann hypothesis—that attained mythical status. Just like these others, the Poincaré conjecture consumed those who worked on it. Countless young mathematicians entered the chase. As one journalist wrote, “Mathematicians speak of Poincaré's conjecture like Ahab expounding on the White Whale.”5 In the years since 1904 there have been many who claimed to have a proof. Until recently, every argument has possessed a flaw—sometimes a subtle mistake embedded in the middle of hundreds of pages of deep mathematics.
Eventually the conjecture was generalized to n-manifolds—every n-manifold that is sufficiently like the n-sphere must be homeomorphic to Sn. This generalization may seem ridiculously ambitious. How could we prove it for n = 100 if we are unable to prove it for n = 3? If I can’t bench press 175 pounds, what makes me think I can lift 500? Shockingly, the conjecture is easier for large n! It is often the case that low-dimensional topology is more challenging than high-dimensional topology. Roughly speaking, having more dimensions to play with gives more freedom to move things around without collisions.
The fiery young topologist Stephen Smale (b. 1930) from the University of California, Berkeley, gave the first proof of the generalized Poincaré conjecture. In 1960 he verified the conjecture for an important class of manifolds of dimensions n ≥ 5—the so-called smooth manifolds.6
Smale is quite a colorful character. He was a vocal critic of the Vietnam War and a vehement free speech activist. His protests, which included criticism of American foreign policy while visiting Moscow, earned him a subpoena by the House Un-American Activities Committee. Later he found himself in hot water over a six-month trip to Brazil paid for by the National Science Foundation. The science advisor to President Johnson wrote, “This blithe spirit leads mathematicians to seriously propose that the common man who pays the taxes ought to feel that mathematical creation should be supported with public funds on the beaches of Rio de Janeiro.”7
The impetus for this statement was Smale’s now-famous quote: “My best-known work was done on the beaches of Rio de Janeiro.”8 During his stay in Brazil, Smale not only proved the high-dimensional Poincaré conjecture, but he also discovered the Smale horseshoe, the template for chaotic dynamical systems.
Within two years Smale’s results for n > 5 were generalized to manifolds without the smoothness assumption.9 It seemed that the remaining cases would follow in no time. Then progress stalled. The n = 4 case did not yield until 1982, when it was proved by thirty-year-old Michael Freedman from the University of California, San Diego.10 Then progress stalled again. Each dimension was more difficult than the one before it. Dimension 3, that of the original conjecture, remained impenetrable. More false proofs came and went. The problem seemed untouchable.
In 1998, Smale published a list of the eighteen most important unsolved problems in mathematics11 (David Hilbert had done the same thing a century before). The classical Poincaré conjecture was on this list.
That same year the Clay Mathematics Institute offered a $1 million bounty for what they viewed as the seven most challenging unsolved problems in mathematics. The Poincaré conjecture was on this elite list. To win the prize a mathematician must give a proof of the theorem, and it must survive the intense scrutiny of the mathematical community for two years after it appears in print.
In 1982 Bill Thurston announced a plan to completely classify the geometry of all 3-manifolds. He theorized that every 3-manifold could be carved up into regions, each of which possess one of eight geometric structures.12 This became known as Thurston's geometrization conjecture. With these eight building blocks it would be possible to understand the geometry and topology of all 3-manifolds. In particular, it would imply that the only simply connected closed 3-manifold is the 3-sphere. It would prove the Poincaré conjecture.
That same year, Richard Hamilton, a mathematician at Cornell University, began a program that he believed would prove Thurston's geometrization conjecture.13 He introduced a means of taking any 3-manifold and, as if blowing up a balloon, continuously deform it into what he hoped would be a form that clearly fit the Thurston model. He made substantial progress toward this goal. Most experts expected that his techniques should work, but Hamilton and others were unable to rule out or adequately deal with singularities—parts of the manifold that did not get nicer over time, but instead pinched off to something worse.
In 2002 an unassuming Russian mathematician named Grigori “Grisha” Perelman (b. 1966) from the Steklov Institute in St. Petersburg surprised the mathematical community by posting to the Internet the first of three short, but extremely dense papers. The papers, which totaled only sixty-eight pages, claimed to finish Hamilton's two-decade-old research program.14 In them he showed that certain singularities would never occur and others could be carefuly eliminated. Taken together, they proved the geometrization conjecture, and in its wake, the classical Poincaré conjecture.
The mathematical community was skeptical—they had heard such proclamations before and the papers were extremely short on details—but they were guardedly optimistic. Perelman was a respected mathematician and he was carrying out Hamilton’s well-regarded plan.
Perelman’s arguments left much unsaid. Even the foremost experts in geometry and topology had difficulty assessing the legitimacy of the proofs. Independently of each other, three teams of mathematicians poured through his arguments, filling in the missing details.15 The average length of each analysis was over three hundred pages. They did not discover any major errors.
By the end of 2006 it was generally believed that Perelman’s proof was correct. That year, the journal Science named Perelman’s proof the “Breakthrough of the Year.”16 Like Smale and Freedman before him, the forty-year old Perelman was tapped to be a Fields Medals recipient for his contributions to the Poincaré conjecture (in fact, Thurston also received a Fields Medal for his work that indirectly led to the final proof). The countdown for the $1 million prize had begun (some wonder if Perelman and Hamilton will be offered the prize jointly).
It may be that one of the grand mathematical peaks has been summited, just like Fermat’s last theorem a decade before. The flag has been planted. One might assume that this accomplishment sounds the death knell for an area of mathematics. This most certainly is not the case. From atop this summit mathematicians behold a stunning vista of previously unseen peaks, all waiting to be climbed. Like Fermat’s last theorem, the result itself may not be as important as the huge body of mathematics that was created in an effort to prove it.
Great mathematics begets more great mathematics. Euler’s solution to the bridges of Königsberg problem and his proof of the polyhedron formula set in motion a voyage of discovery through many areas of beautiful mathematics that led to the creation of topology. The Poincaré conjecture is but one waypoint on this exciting journey. Topology is still a live and vibrant field of study.
A bizarre and unfortunate postscript to this otherwise wonderful story was the effect Perelman’s proof had on his life. It started well. In April of 2003 he embarked on a brief speaking tour. His lectures were attended by Andrew Wiles, John Forbes Nash Jr. (the subject of the Hollywood biopic A Beautiful Mind), John Conway, and other well-known mathematicians. But after he returned to Russia, the intense scrutiny of the mathematical community and the posturing of other mathematicians who wanted a share of his credit began to take its toll on him.17
The always-solitary Perelman became more reclusive. He wanted his work to speak for itself and did not want to be a part of the verification process. Eventually, his disenchantment with the mathematical community overwhelmed him and he left his academic position, stopped responding to correspondences, and by all accounts abandoned mathematics entirely. In an unprecedented move that shocked the scientific community, he declined to accept his Fields Medal.
At the end of the summer of 2006, Perelman was unemployed and living with his mother off her meager pension in a small apartment in St. Petersburg. When asked if he would accept the Clay Mathematics Institute’s prize money he responded, “I’m not going to decide whether to accept the prize until it is offered.”18
To many it is shocking that Perelman would decline the Fields Medal and possibly the prize money. But to him, solving the problem was the ultimate reward, fame and money were irrelevant. As he said, “If the proof is correct then no other recognition is needed.”19 Every researcher understands Perelman’s pure love for his subject and the profound satisfaction of a breakthrough discovery. It is not impossible to imagine that overwhelming personal attention would tarnish the accomplishment.
Surely it was this same unadulterated love of mathematics that drove Pythagoras, Kepler, Euler, Riemann, Gauss, Poincaré, and the rest to toil countless hours in pursuit of the perfect theorem and the perfect proof. We can only imagine Perelman’s elation when he realized that he proved the Poincaré conjecture, or Euler’s joy when he saw that V − E + F = 2.
As Poincaré so eloquently wrote, “The scientist does not study nature because it is useful; he studies it because he delights in it, and he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing, and if nature were not worth knowing, life would not be worth living.”20