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In mathematics, the Poincaré conjecture (IPA: [http://www.bartleby.com/61/wavs/3/P0400300.wav Poincaré pronunciation example at Bartlby.com is a conjecture about the characterisation of the three-dimensional sphere amongst three-dimensional manifolds. Loosely speaking, the conjecture surmizes that if a closed three-dimensional manifold is sufficiently like a sphere in that each loop in the manifold can be shrunk 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 to be 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. Many mathematicians believe that a recent series of papers by Grigori Perelman has proved the conjecture, after nearly a century. The prize has not yet been awarded.
The statement
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. This was the first example of a homology sphere, and since then, many more have been constructed.
In this same paper, he wondered if a 3-manifold with the same homology as 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 could distinguish the 3-sphere, but nonetheless, the statement that it does has come down in history 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.
History of attempted solutions
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. Milnor commented that sometimes the errors in false proofs can be "rather subtle and difficult to detect.""The Poincaré Conjecture 99 Years Later: A Progress Report" by John Milnor, February 2003 - PDF file Overall, work on the conjecture has improved understanding of 3-manifolds. Experts in the field have been most 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). Sometimes mathematicians obsessed with this problem are described as suffering from Poincaritis.
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.
In late 2002, Grigori Perelman of the Steklov Institute of Mathematics, Saint Petersburg was rumoured to have found a proof.By April 2003 the press was reporting on these developments Mathematical Digest 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 are on the agenda of the International Congress of Mathematicians (ICM), August 2006 in Madrid. Apparently in April 2006, at the time of the publication of the 20th Bulletin of the ICM, some of the content of Perelmans papers was still in the process of acquiring full endorsement by the scientific community:
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.[http://www.icm2006.org/?nav_id=863#poincare ICM, Bulletin 20, April 10, 2006: "Poincaré’s Conjecture Will Be the Highlight of the ICM2006"
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."
In its issue of June 2006,Asian Journal of Mathematics Volume 10, Number 2 (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 to
give a complete proof of the Poincaré and geometrization conjectures. [... the crowning achievement of the Hamilton-Perelman theory of Ricci flow.Abstract, table of contents and introduction (PDF file) to "A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman theory of the Ricci flow" by Huai-Dong Cao and Xi-Ping Zhu in Asian Journal of Mathematics Volume 10, Number 2 (June 2006), p.165-498According to the Fields medalist Shing-Tung Yau this was "putting the finishing touches to the complete proof of the Poincaré Conjecture"."Chinese mathematicians solve global puzzle", China View (Xinhua), 2006-06-03
The Poincaré conjecture in other dimensions
Analogues of the Poincaré conjecture in dimensions other than 3 can also be formulated:
Every closed n-manifold which is homotopy equivalent to the n-sphere is homeomorphic to the n-sphere.
The 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 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 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.
See also
References
External links
- Description of the Poincaré conjecture by the Clay Mathematics Institute:
- John Milnor: "The Poincaré Conjecture 99 Years Later: A Progress Report", February 2003 - PDF file
- John Milnor: "Towards the Poincaré Conjecture and Classification of 3-Manifolds" (version of 6-14-03) - PDF file
- Preprints of Grisha Perelman's papers at arXiv.org:
- Bruce Kleiner (Yale) and John Lott (University of Michigan): "Notes and commentary on Perelman's Ricci flow papers"
- Jascha Hoffman, Boston Globe Correspondent, 12/30/2003: "Century-old math problem may have been solved"
- Entry in the "Not Even Wrong" blog of the Columbia university, September 8th, 2004: "Perelman and the Poincare Conjecture"
- Mo Hong'e in China View, 2006-06-04: "Unraveling toughest puzzle outstanding"
- "Chinese Mathematicians Unravel Century-Old Mathematical Problem". The Epoch Times. Jun 06, 2006.
Geometric topology | 3-manifolds | Manifolds | Conjectures | Unsolved problems in mathematics | Millennium Prize Problems
Poincaré-vermoede | حدسية بوانكاريه | Poincaré-Vermutung | Conjetura de Poincaré | Conjecture de Poincaré | 푸앵카레 추측 | Congettura di Poincaré | Vermoeden van Poincaré | ポアンカレ予想 | Conjectura de Poincaré | Гипотеза Пуанкаре | Poincarén otaksuma | 庞加莱猜想
"Poincaré conjecture"