Poincaré conjecture

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In mathematics, the Poincaré conjecture (IPA: [pwɛ̃kaˈʀe])<ref>Poincaré pronunciation example at Bartleby.com</ref> is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. The conjecture concerns a space that locally looks like ordinary three dimensional space but is finite in size 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 just a three-dimensional sphere. An analogous result has been known in higher dimensions for some time.

After nearly a century of effort by mathematicians, a series of papers made available in 2002 and 2003 by Grigori Perelman, following the program of Richard Hamilton, sketched a solution. Three groups of mathematicians have produced works filling in the details of Perelman's proof.

The Poincaré conjecture is 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. Perelman's work is under review and the prize money could be awarded if the proof is considered valid two years after publication.<ref>"Interview with Jim Carlson" in ICM 2006 Daily News, Madrid August 29 2006, p. 1</ref><ref>"Before consideration, a proposed solution must be published in a refereed mathematics publication of worldwide repute (or such other form as the SAB shall determine qualifies), and it must also have general acceptance in the mathematics community two years after." from Rules for the Millennium Prizes, Revision of January 19, 2005, available at the website of the Clay Mathematics Institute</ref>

1 History
2 Ricci flow with surgery
3 In other dimensions
4 References
5 External links

[edit] History

[edit] 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. 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 compact 3-manifold (without boundary) is homeomorphic to a 3-sphere.

[edit] 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 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. <ref> R.H. Bing Necessary and Sufficient Conditions that a 3-Manifold Be S³ The Annals of Mathematics 2nd Ser., Vol. 68, No. 1 (Jul., 1958), pp. 17-37 </ref> Bing also described some of the pitfalls in trying to prove the Poincaré conjecture. <ref>Bing, R. H. Some aspects of the topology of 3-manifolds related to the Poincaré conjecture. 1964 Lectures on modern mathematics, Vol. II pp. 93--128 Wiley, New York</ref>

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."<ref>"The Poincaré Conjecture 99 Years Later: A Progress Report" (PDF file) by John Milnor, February 2003</ref> 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).

[edit] Hamilton and Perelman's solution

Hamilton's program was started in the paper

R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom 17 (1982), 255-306 (reprinted in Collected Papers on Ricci Flow ISBN 1571461108).

He introduced the Ricci flow on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture. In the following years he extended this work, introducing many ideas later used by Perelman.

In late 2002 and 2003 Grigori Perelman of the Steklov Institute of Mathematics, Saint Petersburg posted 3 papers on the arXiv:<ref>By April 2003 the press was reporting on these developments Mathematical Digest</ref>

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:

Bruce Kleiner and John Lott, Notes on Perelman's Papers (May 2006) filled in the details of Perelman's proof of the geometrization conjecture.
Zhu Xiping and Cao Huaidong, A Complete Proof of the Poincaré and Geometrization Conjectures - Application of the Hamilton-Perelman theory of the Ricci flow (Asian Journal of Mathematics Volume 10, Number 2, p. 165-492 (June 2006)), Erratum.

John Morgan and Gang Tian Ricci Flow and the Poincaré Conjecture (July 2006) gave a detailed proof of just the Poincaré Conjecture (which is somewhat easier than the full geometrization conjecture).

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

On August 22, 2006, the ICM awarded Perelman the Fields Medal for his work on the conjecture, but Perelman refused the medal.<ref name="NY">Sylvia Nasar and David Gruber, "Manifold destiny", The New Yorker, August 28, 2006, pp. 44–57. On-line version at the New Yorker website</ref><ref name="NYT">Highest Honor in Mathematics Is Refused by Kenneth Chang in the New York Times, August 22, 2006</ref><ref name="CD">Reclusive Russian solves 100-year-old maths problem, China Daily, 23 August 2006, page 7</ref> John Morgan spoke at the ICM on the Poincaré conjecture on August 24 2006, declaring that "in 2003, Perelman solved the Poincaré Conjecture."<ref>A Report on the Poincaré Conjecture. Special lecture by John Morgan.</ref>

The August 2006 issue of The New Yorker contains an article, titled "Manifold Destiny", on some of the issues surrounding Perelman's work.

[edit] Ricci flow with surgery

Main article: Ricci flow

Hamilton's program for proving the Poincaré conjecture involves first putting a Riemannian metric on the unknown simply connected 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, the manifold is easy to identify as all constant curvature manifolds are well understood. The metric is improved using the Ricci flow equations;

<math>\partial_t g_{ij}=-2 R_{ij}</math>

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 strucutre, and whose thin piece is a graph manifold, but this extra complication is not necessary for proving just the Poincaré conjecture.

Terence Tao wrote an exposition of Ricci flow with surgery in

T. Tao, Perelman's proof of the Poincaré conjecture: a nonlinear PDE perspective

[edit] In other dimensions

The Poincaré conjecture in other dimensions states:

Every homotopy sphere (a 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 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. Grigori Perelman solved the last case n=3 in 2003.

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.<ref>Michel A. Kervaire; John W. Milnor. "Groups of Homotopy Spheres: I" in The Annals of Mathematics, 2nd Ser., Vol. 77, No. 3. (May, 1963), pp. 504-537. This paper calculates the structure of the group of smooth structures on an n-sphere for n > 4.</ref> 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. <ref> Herman Gluck, The embedding of two-spheres in the four-sphere,, Trans. Amer. Math. Soc. 104 (1962) 308-333.</ref>

For piecewise linear manifolds, the Poincaré conjecture is true except possibly in 4 dimensions, where the answer is unknown, and equivalent to the smooth case. 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. See Fragments of Geometric Topology from the Sixties by Sandro Buoncristiano, in Geometry & Topology Monographs, Vol. 6 (2003)