Поанкаре болжамы: Геометриялық топологиядағы теорема

Theorem in geometric topology

In the mathematical field of geometric topology, the Poincaré conjecture (/UK/'/p//w//æ̃//k//ær//eɪ/, /US/,/p//w//æ̃//k//ɑː/'/r//eɪ/, pwɛ̃kaʁe) is a theorem about the characterization of the 3 sphere, which is the hypersphere that bounds the unit ball in four dimensional space. 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 Shaw Prize and the Leroy P. Steele Prize for Seminal Contribution to Research. The journal Science marked Perelman's proof of the Poincaré conjecture as the scientific Breakthrough of the Year in 2006. The Clay Mathematics Institute, having included the Poincaré conjecture in their well known Millennium Prize Problem list, offered Perelman their prize of US$1 million for the conjecture's resolution. He declined the award, saying that Hamilton's contribution had been equal to his own.

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 dimensional topological manifold which is closed, connected, and has trivial fundamental group is homeomorphic 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.

In the 1800s, Bernhard Riemann and Enrico Betti initiated the study of topological invariants of manifolds. They introduced the Betti numbers, which associate to any manifold a list of nonnegative integers. Riemann had showed that a closed connected two dimensional manifold is fully characterized by its Betti numbers. As part of his 1895 paper Analysis Situs (announced in 1892), Poincaré showed that Riemann's result does not extend to higher dimensions. To do this he introduced the fundamental group as a novel topological invariant, and was able to exhibit examples of three dimensional manifolds which have the same Betti numbers but distinct fundamental groups. He posed the question of whether the fundamental group is sufficient to topologically characterize a manifold (of given dimension), although he made no attempt to pursue the answer, saying only that it would "demand lengthy and difficult study." The primary purpose of Poincaré's paper was the interpretation of the Betti numbers in terms of his newly introduced homology groups, along with the Poincaré duality theorem on the symmetry of Betti numbers. Following criticism of the completeness of his arguments, he released a number of subsequent "supplements" to enhance and correct his work. The closing remark of his second supplement, published in 1900, said: this says that a closed connected oriented manifold with the homology of a sphere must be homeomorphic to a sphere. Poincaré was asking whether the triviality of the fundamental group uniquely characterizes the sphere. 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. Bing also described some of the pitfalls in trying to prove the Poincaré conjecture. Włodzimierz Jakobsche showed in 1978 that, if the Bing–Borsuk conjecture is true in dimension 3, then the Poincaré conjecture must also be true. 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." 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). An exposition of attempts to prove this conjecture can be found in the non technical book Poincaré's Prize by George Szpiro.

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, 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 four dimensions. 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: