Poincaré conjecture
Field | Geometric topology |
---|---|
Conjectured by | Henri Poincaré |
Conjectured in | 1904 |
First proof by | Grigori Perelman |
First proof in | 2002 |
Implied by | |
Generalizations | Generalized Poincaré conjecture |
Millennium Prize Problems |
---|
In the
Originally conjectured by Henri Poincaré in 1904, the theorem concerns spaces that locally look like ordinary three-dimensional space but which are finite in extent. Poincaré hypothesized 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. Attempts to resolve the conjecture drove much progress in the field of geometric topology during the 20th century.
The eventual proof built upon Richard S. Hamilton's program of using the Ricci flow to solve the problem. By developing a number of new techniques and results in the theory of Ricci flow, Grigori Perelman was able to modify and complete Hamilton's program. In papers posted to the arXiv repository in 2002 and 2003, Perelman presented his work proving the Poincaré conjecture (and the more powerful geometrization conjecture of William Thurston). Over the next several years, several mathematicians studied his papers and produced detailed formulations of his work.
Hamilton and Perelman's work on the conjecture is widely recognized as a milestone of mathematical research. Hamilton was recognized with the
Overview
The Poincaré conjecture was a mathematical problem in the field of geometric topology. In terms of the vocabulary of that field, it says the following:
Poincaré conjecture.
Every three-dimensionalhomeomorphic to the three-dimensional sphere.
Familiar shapes, such as the surface of a ball (which is known in mathematics as the two-dimensional sphere) or of a torus, are two-dimensional. The surface of a ball has trivial fundamental group, meaning that any loop drawn on the surface can be continuously deformed to a single point. By contrast, the surface of a torus has nontrivial fundamental group, as there are loops on the surface which cannot be so deformed. Both are topological manifolds which are closed (meaning that they have no boundary and take up a finite region of space) and connected (meaning that they consist of a single piece). Two closed manifolds are said to be homeomorphic when it is possible for the points of one to be reallocated to the other in a continuous way. Because the (non)triviality of the fundamental group is known to be invariant under homeomorphism, it follows that the two-dimensional sphere and torus are not homeomorphic.
The two-dimensional analogue of the Poincaré conjecture says that any two-dimensional topological manifold which is closed and connected but non-homeomorphic to the two-dimensional sphere must possess a loop which cannot be continuously contracted to a point. (This is illustrated by the example of the torus, as above.) This analogue is known to be true via the classification of closed and connected two-dimensional topological manifolds, which was understood in various forms since the 1860s. In higher dimensions, the closed and connected topological manifolds do not have a straightforward classification, precluding an easy resolution of the Poincaré conjecture.
History
Poincaré's question
In the 1800s,
The primary purpose of Poincaré's paper was the interpretation of the Betti numbers in terms of his newly-introduced
In order to avoid making this work too prolonged, I confine myself to stating the following theorem, the proof of which will require further developments:
Each polyhedron which has all its Betti numbers equal to 1 and all its tables Tq orientable is simply connected, i.e., homeomorphic to a hypersphere.
(In a modern language, taking note of the fact that Poincaré is using the terminology of
However, after publication he found his announced theorem to be incorrect. In his fifth and final supplement, published in 1904, he proved this with the counterexample of the
One question remains to be dealt with: is it possible for the fundamental group of V to reduce to the identity without V being simply connected? [...] However, this question would carry us too far away.
In this remark, as in the closing remark of the second supplement, Poincaré used the term "simply connected" in a way which is at odds with modern usage, as well as his own 1895 definition of the term.[12][16] (According to modern usage, Poincaré's question is a tautology, asking if it is possible for a manifold to be simply connected without being simply connected.) However, as can be inferred from context,[18] Poincaré was asking whether the triviality of the fundamental group uniquely characterizes the sphere.[14]
Throughout the work of Riemann, Betti, and Poincaré, the topological notions in question are not defined or used in a way that would be recognized as precise from a modern perspective. Even the key notion of a "manifold" was not used in a consistent way in Poincaré's own work, and there was frequent confusion between the notion of a
For this reason, it is not possible to read Poincaré's questions unambiguously. It is only through the formalization and vocabulary of topology as developed by later mathematicians that Poincaré's closing question has been understood as the "Poincaré conjecture" as stated in the preceding section.However, despite its usual phrasing in the form of a conjecture, proposing that all manifolds of a certain type are homeomorphic to the sphere, Poincaré only posed an open-ended question, without venturing to conjecture one way or the other. Moreover, there is no evidence as to which way he believed his question would be answered.[14]
Solutions
In the 1930s, J. H. C. Whitehead claimed a proof but then retracted it. In the process, he discovered some examples of simply-connected (indeed contractible, i.e. homotopically equivalent to a point) non-compact 3-manifolds not homeomorphic to , the prototype of which is now called the Whitehead manifold.
In the 1950s and 1960s, other mathematicians attempted proofs of the conjecture only to discover that they contained flaws. Influential mathematicians such as Georges de Rham, R. H. Bing, Wolfgang Haken, Edwin E. Moise, and Christos Papakyriakopoulos attempted to prove the conjecture. In 1958, R. H. 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.[20] Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.[21]
Włodzimierz Jakobsche showed in 1978 that, if the Bing–Borsuk conjecture is true in dimension 3, then the Poincaré conjecture must also be true.[22]
Over time, the conjecture gained the reputation of being particularly tricky to tackle.
An exposition of attempts to prove this conjecture can be found in the non-technical book Poincaré's Prize by George Szpiro.[26]
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 than simply-connectedness is necessary; in dimensions four and higher there are simply-connected, closed manifolds which are not homotopy equivalent to an n-sphere.
Historically, while the conjecture in dimension three seemed plausible, the generalized conjecture was thought to be false. In 1961,
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:[27]
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 solution
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.[28] 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.
In late 2002 and 2003, Perelman posted three papers on
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 W. Lott posted a paper on arXiv in May 2006 which filled in the details of Perelman's proof of the geometrization conjecture, following partial versions which had been publicly available since 2003.[32] Their manuscript was published in the journal "Geometry and Topology" in 2008. A small number of corrections were made in 2011 and 2013; for instance, the first version of their published paper made use of an incorrect version of Hamilton's compactness theorem for Ricci flow.
- Xi-Ping Zhu published a paper in the June 2006 issue of the Asian Journal of Mathematics with an exposition of the complete proof of the Poincaré and geometrization conjectures.[33]The opening paragraph of their paper stated
In this paper, we shall present the Hamilton-Perelman theory of Ricci flow. Based on it, we shall give the first written account of a complete proof of the Poincaré conjecture and the geometrization conjecture of Thurston. While the complete work is an accumulated efforts of many geometric analysts, the major contributors are unquestionably Hamilton and Perelman.
- Some observers interpreted Cao and Zhu as taking credit for Perelman's work. They later posted a revised version, with new wording, on arXiv.[34] In addition, a page of their exposition was essentially identical to a page in one of Kleiner and Lott's early publicly available drafts; this was also amended in the revised version, together with an apology by the journal's editorial board.
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 Ricci flow, but Perelman refused the medal.[38][39] John Morgan spoke at the ICM on the Poincaré conjecture on August 24, 2006, declaring that "in 2003, Perelman solved the Poincaré Conjecture."[40]
In December 2006, the journal Science honored the proof of Poincaré conjecture as the Breakthrough of the Year and featured it on its cover.[5]
Ricci flow with surgery
Hamilton's program for proving the Poincaré conjecture involves first putting a
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, his original breakthrough was to show that if the Riemannian manifold has positive Ricci curvature everywhere, then the above procedure can only be followed for a bounded interval of parameter values, with , and more significantly, that there are numbers such that as , the Riemannian metrics smoothly converge to one of constant positive curvature. According to classical Riemannian geometry, the only simply-connected compact manifold which can support a Riemannian metric of constant positive curvature is the sphere. So, in effect, Hamilton showed a special case of the Poincaré conjecture: if a compact simply-connected 3-manifold supports a Riemannian metric of positive Ricci curvature, then it must be diffeomorphic to the 3-sphere.
If, instead, one only has an arbitrary Riemannian metric, the Ricci flow equations must lead to more complicated singularities. Perelman's major achievement was to show that, if one takes a certain perspective, if they appear in finite time, these singularities can only look like shrinking spheres or cylinders. With a quantitative understanding of this phenomenon, 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.
Perelman provided a separate argument based on
This condition on the fundamental group turns out to be necessary and sufficient for finite time extinction. 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. In the context that one makes no assumption about the fundamental group whatsoever, Perelman made a further technical study of the limit of the manifold for infinitely large times, and in so doing, proved Thurston's geometrization conjecture: at large times, the manifold has a
Solution
This section needs additional citations for verification. (October 2013) |
On November 13, 2002, Russian mathematician
Perelman proved the conjecture by deforming the manifold using the Ricci flow (which behaves similarly to the
The first step is to deform the manifold using the
Hamilton created a list of possible singularities that could form, but he was concerned that some singularities might lead to difficulties. He wanted to cut the manifold at the singularities and paste in caps and then run the Ricci flow again, so he needed to understand the singularities and show that certain kinds of singularities do not occur. Perelman discovered the singularities were all very simple: consider that a cylinder is formed by 'stretching' a circle along a line in another dimension, repeating that process with spheres instead of circles essentially gives the form of the singularities. Perelman proved this using something called the "Reduced Volume", which is closely related to an
Sometimes, an otherwise complicated operation reduces to multiplication by a
Completing the proof, Perelman takes any compact, simply connected, three-dimensional manifold without boundary and starts to run the Ricci flow. This deforms the manifold into round pieces with strands running between them. He cuts the strands and continues deforming the manifold until, eventually, he is left with a collection of round three-dimensional spheres. Then, he rebuilds the original manifold by connecting the spheres together with three-dimensional cylinders, morphs them into a round shape, and sees that, despite all the initial confusion, the manifold was, in fact, homeomorphic to a sphere.
One immediate question posed was how one could be sure that infinitely many cuts are not necessary. This was raised due to the cutting potentially progressing forever. Perelman proved this cannot happen by using
References
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- ^ S2CID 121869167.
- ^ "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (Press release). Clay Mathematics Institute. March 18, 2010. Archived from the original (PDF) on March 22, 2010. Retrieved November 13, 2015.
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.
- ^ a b "Последнее 'нет' доктора Перельмана" [The last "no" Dr. Perelman]. Interfax (in Russian). July 1, 2010. Retrieved 5 April 2016. Google Translated archived link at [1] (archived 2014-04-20)
- ^ Ritter, Malcolm (1 July 2010). "Russian mathematician rejects million prize". The Boston Globe.
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- ^ a b c cf. Stillwell's commentary in Poincaré (2010)
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- ^ The opening paragraphs of Poincaré (1904) refer to "simply connected in the true sense of the word" as the condition of being homeomorphic to a sphere.
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- ^ Bing, R. H. (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.
- arXiv:0811.0886.)
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- ^ 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
- ISBN 1-57146-110-8.
- arXiv:math.DG/0211159.
- arXiv:math.DG/0303109.
- arXiv:math.DG/0307245.
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- 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). Archived from the original(PDF) on 2012-05-14.
- arXiv:math.DG/0612069.
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- ^ Chang, Kenneth (August 22, 2006). "Highest Honor in Mathematics Is Refused". The New York Times.
- ^ A Report on the Poincaré Conjecture. Special lecture by John Morgan.
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- ^ "Poincaré Conjecture". Clay Mathematics Institute. Retrieved 2018-10-04.
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Further reading
- S2CID 119133773.
- Huai-Dong Cao; Xi-Ping Zhu (December 3, 2006). "Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture". arXiv:math.DG/0612069.
- MR 2334563.
- ISBN 978-0-8027-1654-5.
- Perelman, Grisha (November 11, 2002). "The entropy formula for the Ricci flow and its geometric applications". arXiv:math.DG/0211159.
- Perelman, Grisha (March 10, 2003). "Ricci flow with surgery on three-manifolds". arXiv:math.DG/0303109.
- Perelman, Grisha (July 17, 2003). "Finite extinction time for the solutions to the Ricci flow on certain three-manifolds". arXiv:math.DG/0307245.
- Szpiro, George (2008). Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles. ISBN 978-0-452-28964-2.
- MR 2958930.
- MR 3930611.
External links
- "The Poincaré Conjecture" – In Our Time, 2 November 2006. Contributors June Barrow-Green, Lecturer in the History of Mathematics at the Open University, Ian Stewart, Professor of Mathematics at the University of Warwick, Marcus du Sautoy, Professor of Mathematics at the University of Oxford, and presenter Melvyn Bragg.