Poincaré conjecture

The Poincaré conjecture is one of the most well-known for a long time unproven mathematical propositions and states that as long as a geometric object has no hole (ie, shrunk, shrunk, inflated, etc.) it can be deformed into a ball. And not only applies in the case of a two-dimensional surface in three-dimensional space, but also for three-dimensional surface in the four-dimensional space.

The Poincaré conjecture was long considered one of the most important unsolved problems in topology, a branch of mathematics. Henri Poincaré had erected in 1904. In 2000, the Clay Mathematics Institute counted the Poincare conjecture to the seven most important unsolved mathematical problems, the Millennium problems, and praised for their solution a million U.S. dollars. Grigori Perelman has proved the conjecture in 2002. In 2006, he was the Fields Medal for his proof obtained, but he refused. Him the Millennium Prize of the Clay Institute was awarded on 18 March 2010, which he also refused.

Wording and description

In addition, there is a generalization of the conjecture on n- dimensional manifolds in the following form:

In the case of this generalized conjecture with the original Poincaré conjecture agrees.

Simplified one can describe the Poincaré conjecture as follows: The surface of a sphere is two -dimensional, limited and borderless, and every closed curve can be contracted to a point which is also on the ball. She is ( topologically speaking) the only 2-dimensional structure with these properties. In the Poincaré conjecture is about the 3-dimensional analogue: This is about a 3-dimensional " surface " of a 4-dimensional body.

Notes

The conjecture in higher dimensions

We describe a manifold M as m- connected if every picture of a k- sphere to M for k <= m can move in together to a point. For m = 1, the results in exactly the term of the above described ' simply connected '. A formulation of the n-dimensional Poincare conjecture now says the following:

An argument with Poincaré duality shows that one can replace here also ( n-1) by (n -1 ) / 2. For n = 3, this results in exactly the above given formulation of the Poincaré conjecture.

There are a number of other equivalent formulations that are often found in the literature. One replaces the condition ( n-1) -connected by the fact that one demands that the manifold is already homotopy equivalent to the n- sphere. These two conditions are according to the theorem of Hurewicz equivalent. Homotopy equivalence is a coarser equivalence relation than homeomorphism, but which is often easier to check. The Poincaré conjecture states that these two relations then but fortunately coincide in the case of the sphere.

Another equivalent condition is that the manifold is simply connected and has the same homology as an n- sphere has. While this description is technical, it has the advantage that it is often relatively easy to compute the homology of a manifold.

As has long been known in dimension 3, that any manifold that is homeomorphic to the sphere, is also diffeomorphic to the sphere, which is in higher dimensions not so. From dimension 7, there are so-called exotic spheres which are homeomorphic but not diffeomorphic to the standard sphere. Thus, one can in the Poincaré conjecture of n> 6 is not ' homeomorphic ' replace ' diffeomorphic ' by.

History

Originally Poincaré had a slightly different assumption made: He believed that every 3 -dimensional closed manifold, which has the same homology as a 3- sphere, already topologically must be a sphere. While Poincaré initially believed to have a proof that can do with this weaker assumption, proved to be the requirement that the manifold is simply connected, as indispensable. Poincaré found himself with the Poincaré homology sphere a counterexample to its original presumption: But it has the same homology as a 3- sphere is not simply connected and therefore can not even be homotopy equivalent to a 3- sphere. Therefore he changed his guess on what is now known statement.

It is interesting that the n-dimensional Poincaré conjecture in various dimensions has very different evidence, while the formulation is general.

For the statement is considered classical; in this case are known 2- dimensional manifolds and classifying even all ( closed ).

In the case of the conjecture of Stephen Smale proved in 1960 ( for smooth and PL -manifolds ), for which he used techniques of Morse theory. Among other evidence for this he received the Fields Medal in 1966. Max Newman later expanded his argument on topological manifolds.

Michael Freedman solved the case in 1982. He also was awarded the Fields Medal in 1986.

The case has ( not surprisingly) proved to be the most difficult. Many mathematicians have provided evidence that but then turned out to be wrong. Nevertheless, some of these faulty evidence has expanded the understanding of low-dimensional topology.

Evidence

The end of 2002 appeared messages, Grigori Perelman of the Steklov Institute in St. Petersburg have the assumption proved. He uses developed by Richard Hamilton analytical method of Ricci flow to prove the more general presumption of geometrization of 3-manifolds by William Thurston, from which follows the Poincaré conjecture as a special case. Perelman published his over several publications and extending a total of about 70 pages long chain of evidence in the online archive arXiv. The work has since been verified by mathematicians worldwide and in recognition of the correctness of his proof Grigori Perelman was awarded in 2006 at the International Congress of Mathematicians in Madrid, the Fields Medal, which he did, as announced by him before, did not accept.

As Perelman himself has no interest in a more detailed presentation of its proof, different groups of mathematicians have taken this: So Bruce Kleiner and John Lott have already published their elaboration of many details soon after the announcement of Perelman's work and amended several times to become 192 pages. John Morgan and TianGang have published a complete elaboration of 474 pages in July 2006 on the arXiv. Also Huai -Dong Cao and Xi - Ping Zhu, published in 2006 a proof of the Poincaré conjecture and the geometrization, Holding forth elaborated on exactly 300 pages, the proof of Perelman.

Thus the Poincaré conjecture is proved.

Importance of the presumption

The proof of the Poincare conjecture is an important contribution to the classification of all 3-manifolds. This is because that Perelman actually the more general geometrization of closed 3-manifolds proves that contains the Poincaré conjecture as a special case.

By the Poincaré conjecture to general propositions about the nature of the universe can take.