In mathematics, the Poincaré conjecture (pronounced pwăN-kä-rā') is a conjecture about the characterisation of the three-dimensional sphere amongst 3-manifolds. It is widely considered to be the most important unsolved problem in topology.
The Poincaré conjecture 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. As of 2005 a consensus amongst experts is developing that recent work by Grigori Perelman in 2003 may have disposed of this question, after nearly a century.
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 and based on prior work of 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 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. This period was important to the growth of what would later be called low-dimensional topology. Work on the Poincaré conjecture has resulted in many interesting mathematical contributions and broadened understanding of manifolds, in particular 3-manifolds.
Over time, the conjecture gained the reputation of being very subtle and difficult to tackle. 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.
The Poincaré conjecture may now attract the first Millennium Prize to be awarded. In late 2002, Grigori Perelman of the Steklov Institute of Mathematics, Saint Petersburg was rumoured to have found a proof. He claims to have proven a more general conjecture, Thurston's Geometrization Conjecture, carrying out a program outlined earlier by Richard Hamilton. In 2003, he posted a second preprint and gave a series of lectures in the United States. As of October 2005, several experts have announced that they have verified Perelman's proof of the Poincaré conjecture, and a consensus seems to be rapidly forming. His techniques have already generated deep interest; Perelman's attempt is considered the most significant, serious attack on the Poincaré Conjecture thus far.
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 have now all been proven (with dimension n = 4 being the hardest one by far), while the original 3-dimensional version of Poincaré's conjecture remains unsolved. 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 Poincare 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.
- Description of the Poincaré conjecture by the Clay Mathematics Institute:
- John Milnor: The Poincaré Conjecture 99 Years Later: A Progress Report
- John Milnor: Towards the Poincare conjecture and the classification of 3-manifolds
- Grisha Perelman: The entropy formula for the Ricci flow and its geometric applications, Preprint 2002
- Grisha Perelman: Ricci flow with surgery on three-manifolds, Preprint 2003
- Notes and commentary on Perelman's Ricci flow papers
- Boston Globe article about Perelman's work
- "Not Even Wrong"af:Poincaré-vermoede