Poincaré conjecture

In mathematics, the Poincaré conjecture (IPA: [pw̃ɛkaˈʀe]) is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. Loosely speaking, the conjecture concerns a space that locally looks like ordinary three dimensional space but is finite and lacks any boundary (a closed 3-manifold). The conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is really just a three-dimensional sphere. The analogous result has been known to be true in higher dimensions for some time.

The Poincaré conjecture is widely considered one of the most important questions in topology. It is one of the seven Millennium Prize Problems for which the Clay Mathematics Institute is offering a $1,000,000 prize for a correct solution.

After nearly a century of effort by mathematicians all over the world, a series of papers made available in 2002 and 2003 by Grigori Perelman, following the program of Richard Hamilton, produced an outline for a solution. Following Perelman's work, several groups of mathematicians have produced works filling in the details for the full proof, though review by the mathematics community is ongoing.

Statement detail

At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology — what would later be called combinatorial topology and then algebraic topology. He was particularly interested in what topological properties characterized a sphere.

Poincaré claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient to tell if a 3-manifold was a 3-sphere. In a 1904 paper he described a counterexample, now called the Poincaré sphere, that had the same homology as a 3-sphere. Poincaré was able to show the Poincaré sphere had a fundamental group of order 120. Since the 3-sphere has trivial fundamental group, he concluded this was a different space. The Poincaré sphere was the first example of a homology sphere, of which many others have since been constructed.

In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere but also trivial fundamental group had to be a 3-sphere. Poincaré's new condition, i.e. "trivial fundamental group", can be phrased as "every loop can be shrunk to a point".

The original phrasing was as follows:

Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?

Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the Poincaré conjecture. Here is the standard form of the conjecture:

Every simply connected closed (i.e. compact and without boundary) 3-manifold is homeomorphic to a 3-sphere.

Attempted solutions

20th century

For a time, this problem seems to have lain dormant, until J. H. C. Whitehead revived interest in the conjecture, when in the 1930s he first claimed a proof, and then retracted it. In the process, he discovered some interesting examples of 3-manifolds, the prototype of which is now called the Whitehead manifold.

In the 1950s and 1960s, other famous mathematicians were to claim proofs only to discover a fatal flaw at the last minute. Influential mathematicians such as Bing, Haken, Moise, and Papakyriakopoulos expended great efforts at tackling the conjecture. This period was important to the growth of what would later be called low-dimensional topology.

Over time, the conjecture gained the reputation of being particularly tricky to tackle. John Milnor commented that sometimes the errors in false proofs can be "rather subtle and difficult to detect." Overall, work on the conjecture has improved understanding of 3-manifolds. Experts in the field have been reluctant to announce proofs, and have viewed any such announcement with skepticism. The 1980s and 1990s witnessed some well-publicized fallacious proofs (which were not actually published in peer-reviewed form).

21st century

In 2000, the Clay Mathematics Institute selected the Poincaré conjecture as one of seven Millennium Prize Problems and offered a $1,000,000 prize for its solution. Undoubtedly its difficulty and the expectation that a significant breakthrough would be needed were important factors in this selection.

Hamilton - Perelman - preparation of the 2006 International Congress of Mathematicians

In late 2002, Grigori Perelman of the Steklov Institute of Mathematics, Saint Petersburg was rumoured to have found a proof. He claimed to have proven a more general conjecture, Thurston's geometrization conjecture, carrying out a program outlined earlier by Richard Hamilton. Perelman had made a preprint paper available, which treated the geometrization conjecture by an application of the Ricci flow theory. In 2003, Perelman released two more "Ricci flow" preprint papers and gave a series of lectures in the United States.

The Poincaré conjecture, and Perelman's three publications on the subject were put on the agenda of the International Congress of Mathematicians (ICM), August 2006 in Madrid. In April 2006, at the time of the publication of the 20th Bulletin of the ICM, the content of Perelman's papers was still in the process of acquiring full endorsement by the scientific community:

Perelman already published three papers on the subject, general agreement having been reached on the fact that the first and much of the second are correct, leaving a “technically more difficult” part still to be checked.

However, according to Vicente Miquel, professor of Geometry and Topology at the University of Valencia, quoted in that April 2006 info bulletin regarding the ICM2006 Congress: "Everyone understands the third paper, which together with the verified parts of the first two would seem to provide a proof of the Poincaré Conjecture, and would be enough for Perelman to receive the Clay Institute million-dollar prize."

2006 papers detailing the Hamilton-Perelman approach

From May to July 2006, several groups presented papers that claimed to have filled in the details or complete the proof of the Poincaré conjecture, after which the home page of the website of the Clay Mathematics Institute listed the following papers that give a "detailed exposition" regarding Perelman's work:

• In May 2006, Bruce Kleiner and John Lott, both of the University of Michigan, posted a paper on arXiv titled "Notes on Perelman's Papers". They claim to fill in the details of Perelman's proof of the Geometrization conjecture.
• In June 2006, the Asian Journal of Mathematics published a paper by Zhu Xiping of Sun Yat-sen University in China and Cao Huaidong of Lehigh University in Pennsylvania, claiming:

  Abstract. In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.

  According to the Fields medalist Shing-Tung Yau this was "putting the finishing touches to the complete proof of the Poincaré Conjecture".
• In July 2006, John Morgan of Columbia University and Gang Tian of the Massachusetts Institute of Technology posted a paper on the arXiv titled, "Ricci Flow and the Poincaré Conjecture." In this paper, they claim to provide a "detailed proof of the Poincaré Conjecture".

The 2006 ICM - Perelman and awards

On August 22, 2006, the ICM awarded Perelman the Fields Medal for his work on the conjecture. Perelman refused the medal.

John Morgan spoke at the ICM in Madrid on the Poincaré conjecture on August 24 2006, declaring that "in 2003, Perelman solved the Poincaré Conjecture."

As for the $1 million Clay Institute prize, Perelman's work is under review: prize money could be awarded if the proof is considered valid two years after publication. According to the New York Times, Perelman said he wants no part of the Clay institute prize money. A Chinese newspaper cites however John Ball, president of the International Mathematical Union organising the ICM, who met Perelman in June:

Ball said he asked Perelman if he would accept the money. Perelman said that if he won, he would talk to the Clay institute.

Also the interview with Perelman published in the August 2006 issue of The New Yorker records that Perelman has made no decision yet what would happen if he were offered the Clay institute prize:

I’m not going to decide whether to accept the prize until it is offered.

An August 2006 article in The New Yorker

Further information: Manifold Destiny

The August 2006 article in The New Yorker, titled "Manifold Destiny", authored by Sylvia Nasar and David Gruber, criticised Yau's involvement. Yau is both an editor-in-chief of the Asian Journal of Mathematics as well as Cao's doctoral advisor. The article implies that Yau was intent on being associated, directly or indirectly, with the proof, and pressured the journal's editors to accept Zhu and Cao's paper on unusually short notice.

In an interview taken at the 2006 ICM congress, Cao stated that "Hamilton and Perelman have done the most important fundamental work...We just follow in the footsteps of Hamilton and Perelman and explain the details." He also notes that "Prof. Yau respected Perelman's contribution a great deal. In more than one occasion, including when talking to the reporters of the New Yorker...he has said that Perelman deserves the Fields medal."

Yau had his attorney send a letter to Nasar, Gruber, and the fact-checker who worked on the New Yorker article asking for immediate assistance “in undoing, to the extent possible, the literally world-wide damage” the story had done to Yau’s reputation.

In other dimensions

The Poincaré conjecture in other dimensions states:

Every closed n-manifold which is homotopy equivalent to the n-sphere is homeomorphic to the n-sphere.

It has now been proved in all dimensions. The original Poincaré conjecture as given above is equivalent to the case n = 3. The difficulty of low-dimensional topology is highlighted by the fact that these analogues had all been proven (with dimension n = 4 being the hardest one by far) by the 1980s, while an apparent solution to the original 3-dimensional version of Poincaré's conjecture was just submitted in 2003. The case n = 1 is easy and the case n = 2 has long been known. Stephen Smale solved the cases n ≥ 7 in 1960 and subsequently extended his proof to n ≥ 5; he received a Fields Medal for his work in 1966. Michael Freedman solved n = 4 in 1982 and received a Fields Medal in 1986.

An n-manifold homotopy equivalent to an n-sphere is sometimes called a homotopy sphere. Restated, the Poincaré conjecture states that the only homotopy spheres are actual spheres.

In the smooth category, the analogue of the Poincaré conjecture is usually false (see exotic sphere). For dimensions 1,2,3,5, and 6 there is only one smooth structure on the sphere, but Kervaire and Milnor showed that the oriented 7-sphere has 28 different smooth structures (or 15 ignoring orientations), and in higher dimensions there are usually many different smooth structures on a sphere. It is suspected that certain differentiable structures on the 4-sphere, called Gluck twists, are not isomorphic to the standard one, but at the moment there are no known invariants capable of distinguishing different smooth structures on a 4-sphere.

For piecewise linear manifolds, the Poincaré conjecture is true except possibly in 4 dimensions, where the answer is unknown. In other words, every compact PL manifold of dimension not equal to 4 that is homotopy equivalent to a sphere is PL isomorphic to a sphere.

External links

• Description of the Poincaré conjecture by the Clay Mathematics Institute
• Preprints of Grisha Perelman's papers at arXiv.org:
  • The entropy formula for the Ricci flow and its geometric applications, 2002
  • Ricci flow with surgery on three-manifolds, 2003
  • Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, 2003
• John Milnor: "Towards the Poincaré Conjecture and Classification of 3-Manifolds" (version of 6-14-03) - PDF file
• Bruce Kleiner (Yale) and John Lott (University of Michigan): "Notes and commentary on Perelman's Ricci flow papers"
• Eric W. Weisstein et al., "Poincaré conjecture" at MathWorld
• Jascha Hoffman, Boston Globe Correspondent, 30 December 2003: "Century-old math problem may have been solved"
• Mo Hong'e in China View, 2006-06-04: "Unraveling toughest puzzle outstanding"
• Sharon Begley, Major math problem is believed solved, Wall Street Journal, 21 July 2006
• Joshua Plotkin, a concise summary (PDF) of the proof of the generalized Poincaré conjecture in dimensions five and higher.
• New York Times story: Elusive Proof, Elusive Prover: A New Mathematical Mystery, August 15 2006
• Richard S Hamilton' Letter to Yau Shing-Tung' Attorney on September 25, 2006 Richard S Hamilton's letter in response to New Yorker's Article about Yau, etc.
• AOL News Story, 24 August 2006
• Stephen Ornes, What is The Poincaré Conjecture?, Seed Magazine, 25 August 2006