Poincaré conjecture

Millennium Prize Problems
P versus NP problem
Hodge conjecture
Poincaré conjecture (solution)
Riemann hypothesis
Yang–Mills existence and mass gap
Navier–Stokes existence and smoothness
Birch and Swinnerton-Dyer conjecture
 
For compact 2-dimensional surfaces without boundary, if every loop can be continuously tightened to a point, then the surface is topologically homeomorphic to a 2-sphere (usually just called a sphere). The Poincaré conjecture asserts that the same is true for 3-dimensional surfaces.

In mathematics, the Poincaré conjecture (French, pronounced: [pwɛ̃kaʁe])[1] is a theorem about the characterization of the three-dimensional sphere among three-dimensional manifolds. Originally conjectured by Henri Poincaré, the claim concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaré 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 necessarily a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv.org. The proof followed the program of Richard Hamilton. Several high-profile teams of mathematicians have since verified the correctness of Perelman's proof.

The Poincaré conjecture, before being proven, was one of the most important open questions in topology. It is one of the seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for the first correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered a Fields Medal, which he declined. Perelman was awarded the Millennium Prize on 18 March, 2010.[2] The Poincaré conjecture is the first and, as of 2010, only solved Millennium problem.

On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year", the first time this had been bestowed in the area of mathematics.[3]

Contents

History

Poincaré's question

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. However, in a 1904 paper he described a counterexample to this claim, a space now called the Poincaré homology sphere. The Poincaré sphere was the first example of a homology sphere, a manifold that had the same homology as a sphere, of which many others have since been constructed. To establish that the Poincaré sphere was different from the 3-sphere, Poincaré introduced a new topological invariant, the fundamental group, and showed that the Poincaré sphere had a fundamental group of order 120, while the 3-sphere had a trivial fundamental group. In this way he was able to conclude that these two spaces were, indeed, different.

In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere. Poincaré's new condition—i.e., "trivial fundamental group"—can be re-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 3-manifold is homeomorphic to the 3-sphere.

Attempted solutions

This problem seems to have lain dormant for a time, 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 simply connected non-compact 3-manifolds not homeomorphic to R3, the prototype of which is now called the Whitehead manifold.

In the 1950s and 1960s, other mathematicians were to claim proofs only to discover a flaw. Influential mathematicians such as Bing, Haken, Moise, and Papakyriakopoulos attacked the conjecture. In 1958 Bing proved a weak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere.[4] Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.[5]

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."[6] Work on the conjecture improved understanding of 3-manifolds. Experts in the field were often reluctant to announce proofs, and tended to view any such announcement with skepticism. The 1980s and 1990s witnessed some well-publicized fallacious proofs (which were not actually published in peer-reviewed form).[7][8]

An exposition of attempts to prove this conjecture can be found in the non-technical book Poincaré's Prize by George Szpiro.[9]

Dimensions

The classification of closed surfaces gives an affirmative answer to the analogous question in two dimensions. For dimensions greater than three, one can pose the Generalized Poincaré conjecture: is a homotopy n-sphere homeomorphic to the n-sphere? A stronger assumption is necessary; in dimensions four and higher there are simply-connected manifolds which are not homeomorphic to an n-sphere.

Historically, while the conjecture in dimension three seemed plausible, the generalized conjecture was thought to be false. In 1961 Stephen Smale shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem. In 1982 Michael Freedman proved the Poincaré conjecture in dimension four. Freedman's work left open the possibility that there is a smooth four-manifold homeomorphic to the four-sphere which is not diffeomorphic to the four-sphere. This so-called smooth Poincaré conjecture, in dimension four, remains open and is thought to be very difficult. Milnor's exotic spheres show that the smooth Poincaré conjecture is false in dimension seven, for example.

These earlier successes in higher dimensions left the case of three dimensions in limbo. The Poincaré conjecture was essentially true in both dimension four and all higher dimensions for substantially different reasons. In dimension three, the conjecture had an uncertain reputation until the geometrization conjecture put it into a framework governing all 3-manifolds. John Morgan wrote:[10]

It is my view that before Thurston's work on hyperbolic 3-manifolds and . . . the Geometrization conjecture there was no consensus among the experts as to whether the Poincaré conjecture was true or false. After Thurston's work, notwithstanding the fact that it had no direct bearing on the Poincaré conjecture, a consensus developed that the Poincaré conjecture (and the Geometrization conjecture) were true.

Hamilton's program and Perelman's solution

Several stages of the Ricci flow on a two-dimensional manifold.

Hamilton's program was started in his 1982 paper in which he introduced the Ricci flow on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture.[11] In the following years he extended this work, but was unable to prove the conjecture. The actual solution was not found until Grigori Perelman published his papers using ideas from Hamilton's work.

In late 2002 and 2003 Perelman posted three papers on the arXiv.[12][13][14] In these papers he sketched a proof of the Poincaré conjecture and a more general conjecture, Thurston's geometrization conjecture, completing the Ricci flow program outlined earlier by Richard Hamilton.

From May to July 2006, several groups presented papers that filled in the details of Perelman's proof of the Poincaré conjecture, as follows:

All three groups found that the gaps in Perelman's papers were minor and could be filled in using his own techniques.

On August 22, 2006, the ICM awarded Perelman the Fields Medal for his work on the conjecture, but Perelman refused the medal.[19][20][21] John Morgan spoke at the ICM on the Poincaré conjecture on August 24, 2006, declaring that "in 2003, Perelman solved the Poincaré Conjecture."[22]

In December 2006 Science magazine honored the proof of Poincaré conjecture as the Breakthrough of the Year and featured it on its cover.[3]

Ricci flow with surgery

Hamilton's program for proving the Poincaré conjecture involves first putting a Riemannian metric on the unknown simply connected closed 3-manifold. The idea is to try to improve this metric; for example, if the metric can be improved enough so that it has constant curvature, then it must be the 3-sphere. The metric is improved using the Ricci flow equations;

\partial_t g_{ij}=-2 R_{ij}

where g is the metric and R its Ricci curvature, and one hopes that as the time t increases the manifold becomes easier to understand. Ricci flow expands the negative curvature part of the manifold and contracts the positive curvature part.

In some cases Hamilton was able to show that this works; for example, if the manifold has positive Ricci curvature everywhere he showed that the manifold becomes extinct in finite time under Ricci flow without any other singularities. (In other words, the manifold collapses to a point in finite time; it is easy to describe the structure just before the manifold collapses.) This easily implies the Poincaré conjecture in the case of positive Ricci curvature. However in general the Ricci flow equations lead to singularities of the metric after a finite time. Perelman showed how to continue past these singularities: very roughly, he cuts the manifold along the singularities, splitting the manifold into several pieces, and then continues with the Ricci flow on each of these pieces. This procedure is known as Ricci flow with surgery.

A special case of Perelman's theorems about Ricci flow with surgery is given as follows.

The Ricci flow with surgery on a closed oriented 3-manifold is well defined for all time. If the fundamental group is a free product of finite groups and cyclic groups then the Ricci flow with surgery becomes extinct in finite time, and at all times all components of the manifold are connected sums of S2 bundles over S1 and quotients of S3.

This result implies the Poincaré conjecture because it is easy to check it for the possible manifolds listed in the conclusion.

The condition on the fundamental group turns out to be necessary (and sufficient) for finite time extinction, and in particular includes the case of trivial fundamental group. It is equivalent to saying that the prime decomposition of the manifold has no acyclic components, and turns out to be equivalent to the condition that all geometric pieces of the manifold have geometries based on the two Thurston geometries S2×R and S3. By studying the limit of the manifold for large time, Perelman proved Thurston's geometrization conjecture for any fundamental group: at large times the manifold has a thick-thin decomposition, whose thick piece has a hyperbolic structure, and whose thin piece is a graph manifold, but this extra complication is not necessary for proving just the Poincaré conjecture.[23]

Notes

  1. ↑ "Poincaré, Jules Henri". The American Heritage Dictionary of the English Language (fourth edition ed.). Boston: Houghton Mifflin Company. 2000. ISBN 0-395-82517-2. http://www.bartleby.com/61/3/P0400300.html. Retrieved 2007-05-05. .
  2. ↑ Clay Mathematics Institute (March 18, 2010). "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (PDF). Press release. http://www.claymath.org/poincare/millenniumPrizeFull.pdf. Retrieved March 18, 2010. "The Clay Mathematics Institute (CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of the Poincaré conjecture." 
  3. ↑ 3.0 3.1 Mackenzie, Dana (2006-12-22). "The Poincaré Conjecture--Proved". Science (American Association for the Advancement of Science) 314 (5807): 1848–1849. doi:10.1126/science.314.5807.1848. ISSN: 0036-8075. http://www.sciencemag.org/cgi/content/full/314/5807/1848. 
  4. ↑ Bing, RH (1958). "Necessary and sufficient conditions that a 3-manifold be S3". The Annals of Mathematics, 2nd Ser. 68 (1): 17–37. doi:10.2307/1970041. http://links.jstor.org/sici?sici=0003-486X%28195807%292%3A68%3A1%3C17%3ANASCTA%3E2.0.CO%3B2-1. 
  5. ↑ Bing, RH (1964). "Some aspects of the topology of 3-manifolds related to the Poincaré conjecture". Lectures on Modern Mathematics, Vol. II. New York: Wiley. pp. 93–128. 
  6. ↑ Milnor, John (2004). "The Poincaré Conjecture 99 Years Later: A Progress Report" (PDF). http://www.math.sunysb.edu/~jack/PREPRINTS/poiproof.pdf. Retrieved 2007-05-05. 
  7. ↑ Taubes, Gary (July 1987). "What happens when hubris meets nemesis". Discover 8: 66–77. http://www.findarticles.com/p/articles/mi_m1511/is_v8/ai_4995863. 
  8. ↑ Matthews, Robert (9 April 2002). "$1 million mathematical mystery "solved"". NewScientist.com. http://www.newscientist.com/article.ns?id=dn2143. Retrieved 2007-05-05. 
  9. ↑ Szpiro, George (July 29, 2008). Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles. Plume. ISBN 978-0-452-28964-2. 
  10. ↑ Morgan, John W., Recent progress on the Poincaré conjecture and the classification of 3-manifolds. Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 1, 57–78
  11. ↑ Hamilton, Richard (1982). "Three-manifolds with positive Ricci curvature". Journal of Differential Geometry 17: 255–306.  Reprinted in: Cao, H.D.; et al. (Editors) (2003). Collected Papers on Ricci Flow. International Press. ISBN 978-1571461100. 
  12. ↑ Perelman, Grigori (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. 
  13. ↑ Perelman, Grigori (2003). Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109. 
  14. ↑ Perelman, Grigori (2003). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245. 
  15. ↑ Kleiner, Bruce; John W. Lott (2006). Notes on Perelman's Papers. arXiv:math.DG/0605667. 
  16. ↑ Cao, Huai-Dong; Xi-Ping Zhu (June 2006). "A Complete Proof of the Poincaré and Geometrization Conjectures – application of the Hamilton-Perelman theory of the Ricci flow" (PDF). Asian Journal of Mathematics 10 (2). http://www.intlpress.com/AJM/p/2006/10_2/AJM-10-2-165-492.pdf.  Erratum. Revised version (December 2006): Cao, Huai-Dong; Xi-Ping Zhu (2006). Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture. arXiv:math.DG/0612069. 
  17. ↑ Morgan, John; Gang Tian (2006). Ricci Flow and the Poincaré Conjecture. arXiv:math.DG/0607607. 
  18. ↑ Morgan, John; Gang Tian (2007). Ricci Flow and the Poincaré Conjecture. Clay Mathematics Institute. ISBN 0821843281. 
  19. ↑ Nasar, Sylvia; David Gruber (August 28, 2006). "Manifold destiny". The New Yorker: pp. 44–57.  On-line version at the New Yorker website.
  20. ↑ Chang, Kenneth (August 22, 2006). "Highest Honor in Mathematics Is Refused". New York Times. http://www.nytimes.com/2006/08/22/science/22cnd-math.html?hp&ex=1156305600&en=aa3a9d418768062c&ei=5094&partner=homepage. 
  21. ↑ "Reclusive Russian solves 100-year-old maths problem". China Daily: p. 7. 23 August 2006. http://www.chinadaily.com.cn/cndy/2006-08/23/content_671442.htm. 
  22. ↑ A Report on the Poincaré Conjecture. Special lecture by John Morgan.
  23. ↑ Terence Tao wrote an exposition of Ricci flow with surgery in: Tao, Terence (2006). Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective. arXiv:math.DG/0610903. 

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