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ContentsArticlesPoincar conjecture P versus NP problem Hodge conjecture Riemann hypothesis NavierStokes existence and smoothness Birch and Swinnerton-Dyer conjecture YangMills existence and mass gap 1 7 17 21 44 48 51

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Poincar conjecture

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Poincar conjectureMillennium Prize Problems P versus NP problem Hodge conjecture Poincar conjecture (solution) Riemann hypothesis YangMills existence and mass gap NavierStokes existence and smoothness Birch and Swinnerton-Dyer conjecture

In mathematics, the Poincar conjecture ([pwkae],[1] English:/pwn.kre/ pwen-kar-ay) is a theorem about the characterization of the three-dimensional sphere (3-sphere), which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states:

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 Poincar conjecture asserts that the same is true for 3-dimensional surfaces.

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it. Originally conjectured by Henri Poincar, the theorem 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 Poincar 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. The proof followed the program of Richard Hamilton. Several high-profile teams of mathematicians have verified that Perelman's proof is correct. The Poincar 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 March 18, 2010.[2] On July 1, 2010, he turned down the prize saying that he believes his contribution in proving the Poincar conjecture was no greater than that of U.S. mathematician Richard Hamilton (who first suggested a program for the solution).[3] [4] The Poincar conjecture is the first and, as of April 2011, the only solved Millennium problem. On December 22, 2006, the journal Science honored Perelman's proof of the Poincar conjecture as the scientific "Breakthrough of the Year", the first time this had been bestowed in the area of mathematics.[5]

Poincar conjecture

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HistoryPoincar's questionAt the beginning of the 20th century, Henri Poincar was working on the foundations of topologywhat would later be called combinatorial topology and then algebraic topology. He was particularly interested in what topological properties characterized a sphere. Poincar 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 Poincar homology sphere. The Poincar 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 Poincar sphere was different from the 3-sphere, Poincar introduced a new topological invariant, the fundamental group, and showed that the Poincar 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, Poincar wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere. Poincar's new conditioni.e., "trivial fundamental group"can be restated 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? Poincar never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the Poincar conjecture. Here is the standard form of the conjecture: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Attempted solutionsThis 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 Poincar 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.[6] Bing also described some of the pitfalls in trying to prove the Poincar conjecture.[7] 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."[8] 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).[9] [10] An exposition of attempts to prove this conjecture can be found in the non-technical book Poincar's Prize by George Szpiro.[11]

Poincar conjecture

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DimensionsThe 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 Poincar 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 Poincar conjecture for dimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem. In 1982 Michael Freedman proved the Poincar 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 Poincar conjecture, in dimension four, remains open and is thought to be very difficult. Milnor's exotic spheres show that the smooth Poincar conjecture is false in dimension seven, for example. These earlier successes in higher dimensions left the case of three dimensions in limbo. The Poincar 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:[12] 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 Poincar conjecture was true or false. After Thurston's work, notwithstanding the fact that it had no direct bearing on the Poincar conjecture, a consensus developed that the Poincar conjecture (and the Geometrization conjecture) were true.

Hamilton's program and Perelman's solutionHamilton'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 Poincar conjecture.[13] 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.[14] [15] [16] In these papers he sketched a proof of the Poincar 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 Poincar conjecture, as follows: Bruce Kleiner and John W. Lott posted a paper on the arXiv in May 2006 which filled in the details of Perelman's proof of the geometrization conjecture.[17] Huai-Dong Cao and Xi-Ping Zhu published a paper in the June 2006 issue of the Asian Journal of Mathematics giving a complete proof of the Several stages of the Ricci flow on a two-dimensional manifold. Poincar and geometrization conjectures.[18] They initially claimed t