Poincaré conjecture

• 1. Introduction

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  The Poincaré conjecture is a theorem about the deepest nature of three-dimensional space, and for most of the twentieth century no one could prove it. Henri Poincaré first posed the question in 1904, in the closing remarks of a mathematical supplement. He was not even sure which way the answer would go. A century later, a reclusive Russian mathematician posted his solution to an online preprint server and then walked away from every prize he was offered. What was the question Poincaré asked? Why did it take a hundred years to answer it? And why did the man who solved it refuse a million dollars?

• 2. The Question Poincaré Asked

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  Bernhard Riemann and Enrico Betti began the study of what are called topological invariants in the 1800s. They introduced a set of numbers, now called the Betti numbers, that could describe the shape of any manifold. Riemann showed that for two-dimensional surfaces these numbers told you everything. Poincaré, in his landmark 1895 paper Analysis Situs, showed that in higher dimensions this was no longer true. To illustrate the failure, he invented a new invariant called the fundamental group and displayed three-dimensional shapes that had matching Betti numbers but different fundamental groups. His question was whether the fundamental group alone could fully identify a shape.

  In the closing remarks of his second supplement, published in 1900, Poincaré announced a theorem claiming that any closed connected oriented manifold with the same homology as a sphere must itself be homeomorphic to a sphere. He published it before he had verified it. By the time he reached his fifth and final supplement, in 1904, he had found his own theorem to be wrong. His counterexample was the Poincaré homology sphere: a closed connected three-dimensional manifold whose homology matched the sphere exactly, but whose fundamental group contained 120 elements. Homology, he now understood, was too blunt an instrument.

  In the closing remarks of that fifth supplement, Poincaré rewrote his question using the fundamental group instead of homology. He asked whether a manifold could have a trivial fundamental group without being simply connected in the sense he meant, which in modern language asks whether triviality of the fundamental group uniquely identifies the three-dimensional sphere. He added only that pursuing the answer "would carry us too far away." There is no evidence, in any of his writing, as to which answer he believed was correct.

• 3. A Century of Failed Proofs

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  In the 1930s, J. H. C. Whitehead announced that he had proved the conjecture. He then retracted that claim. In the process of finding his error, he discovered an entirely new class of mathematical objects: simply connected, non-compact three-dimensional manifolds that are not homeomorphic to ordinary three-dimensional space. The prototype of this class is now called the Whitehead manifold.

  Through the 1950s and 1960s, several distinguished mathematicians took their turn, among them Georges de Rham, R. H. Bing, Wolfgang Haken, Edwin E. Moise, and Christos Papakyriakopoulos. In 1958, Bing succeeded in proving a restricted version of the conjecture: if every simple closed curve inside a compact three-dimensional manifold can be contained in a three-dimensional ball, then the manifold must be homeomorphic to the three-sphere. Bing also catalogued many of the traps that awaited anyone attempting a full proof.

  John Milnor later noted that errors in false proofs of the conjecture could be "rather subtle and difficult to detect." The 1980s and 1990s brought a new round of well-publicized false claims, none of which survived peer review. The conjecture developed a reputation in the mathematical community as something to approach with caution. Experts who had spent careers in the area tended to view any new announcement of a proof with open skepticism. An account of these attempts was eventually gathered in a non-technical book called Poincaré's Prize by George Szpiro.

• 4. Higher Dimensions First

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  In 1961, Stephen Smale produced a result that surprised much of the mathematical world. He proved the generalized version of the Poincaré conjecture for all dimensions greater than four. The very case that seemed most intuitive, three dimensions, turned out to be the last to yield. Michael Freedman then proved the conjecture for four dimensions in 1982.

  Freedman's work left an unresolved question: whether a smooth four-dimensional manifold homeomorphic to the four-sphere must also be diffeomorphic to it. That question, called the smooth Poincaré conjecture in dimension four, remains open. John Milnor's discovery of exotic spheres showed that the smooth version is false in dimension seven. So the situation across all dimensions was peculiar: the conjecture was settled in dimensions two, four, and higher than four, by substantially different methods in each case, but the original three-dimensional question remained unresolved.

  What finally gave mathematicians reason to expect the conjecture was true in three dimensions was not a proof but a framework. Mathematician John Morgan later wrote that before William Thurston's work on hyperbolic three-dimensional manifolds and his geometrization conjecture, there was no consensus among experts as to whether the Poincaré conjecture was true or false. After Thurston's work, a consensus formed that both it and the geometrization conjecture were likely true, even though Thurston's results had no direct bearing on the conjecture itself.

• 5. Hamilton's Program

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  Richard S. Hamilton introduced the Ricci flow in a 1982 paper and immediately put it to work on the Poincaré conjecture. The Ricci flow is a set of equations that deforms the metric of a manifold over time, following a rule analogous to the heat equation that describes how heat diffuses through a solid object. Just as heat flow smooths out temperature differences, Ricci flow tends to smooth out irregularities in the curvature of a shape. Hamilton showed that if a compact manifold has positive Ricci curvature everywhere, the flow brings it toward a round sphere.

  The problem was singularities: places where the manifold pinches or stretches in ways that make the flow break down. Hamilton constructed a list of possible singularities and worried that some of them might make the program unworkable. One troublesome case was what he called the cigar soliton, a shape resembling a strand protruding from the manifold with nothing on the other end. Hamilton spent years extending his results but could not resolve the singularity problem. The program stalled.

• 6. Perelman's Solution

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  On the 11th of November 2002, Grigori Perelman posted the first of three papers to the arXiv preprint server. He did not submit them to a journal. In these papers, he outlined a proof not only of the Poincaré conjecture but of Thurston's more general geometrization conjecture.

  Perelman's key contribution was to characterize every singularity that Ricci flow can produce. He proved that singularities can only look like shrinking spheres or cylinders. To establish this, he used a quantity he called the Reduced Volume, which is closely related to an eigenvalue of a certain elliptic equation. An eigenvalue is the number by which a particular operation simply scales its input, the way a musical note is a fundamental frequency of a vibrating object. Perelman showed this eigenvalue rises as the manifold is deformed by the Ricci flow, which ruled out the cigar soliton and other singularities that had concerned Hamilton.

  With singularities understood, Perelman introduced what is called Ricci flow with surgery. When a singularity appears, the manifold is cut at that location, caps are glued onto the open ends, and the flow resumes on the resulting pieces. The danger was that this surgery might need to be performed infinitely many times and never terminate. Perelman addressed this using minimal surfaces, which are surfaces on which any small deformation increases area, like a soap film spanning a bent wire. Hamilton had shown that Ricci flow shrinks the area of minimal surfaces. Perelman showed that once the area drops below a certain threshold, every subsequent cut can only remove three-dimensional spheres, not more complicated shapes. This guaranteed that the process terminates. The argument appeared in Perelman's third paper.

  From May to July 2006, three separate groups of mathematicians published papers filling in the details of Perelman's proof. Bruce Kleiner and John W. Lott posted their paper in May 2006 and it was published in the journal Geometry and Topology in 2008. Huai-Dong Cao and Xi-Ping Zhu published in the June 2006 issue of the Asian Journal of Mathematics; their paper initially drew controversy because its opening paragraph appeared to take credit for Perelman's work, and a page of their exposition was found to be essentially identical to an early publicly available draft by Kleiner and Lott. A revised version was subsequently posted. John Morgan and Gang Tian posted their paper in July 2006, focusing specifically on the Poincaré conjecture rather than the full geometrization conjecture, and expanded it into a book. All three groups concluded that the gaps in Perelman's papers were minor and could be filled using his own techniques.

• 7. The Prizes He Refused

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  On the 22nd of August 2006, the International Congress of Mathematicians awarded Perelman the Fields Medal for his work on the Ricci flow. He declined it. Two days later, on August 24, John Morgan addressed the congress and declared that Perelman had solved the Poincaré conjecture in 2003. In December 2006, the journal Science named the proof the scientific Breakthrough of the Year and featured it on its cover.

  The Clay Mathematics Institute had placed the Poincaré conjecture on its Millennium Prize Problem list, a set of seven unsolved problems each carrying a prize of one million US dollars. On the 18th of March 2010, the institute awarded Perelman that prize. He rejected it as well. His stated reason was that Hamilton's contribution to the solution had been equal to his own, and accepting the prize alone would not reflect that.

  Hamilton's own work was recognized separately. He received the Leroy P. Steele Prize for Seminal Contribution to Research in 2009 and the Shaw Prize in 2011. The Fields Medal, worth fifteen thousand Canadian dollars, and the million-dollar Millennium Prize both remain uncollected.