Poincaré conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (UK: /ˈpwæ̃kæreɪ/,[2] US: /ˌpwæ̃kɑːˈreɪ/,[3][4] French: [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.[5] 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.[6] He declined the award, saying that Hamilton's contribution had been equal to his own.[7][8]
History[edit]
Poincaré's question[edit]
In the 1800s, Bernhard Riemann and Enrico Betti initiated the study of topological invariants of manifolds.[9][10] 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.[11][12][13] 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."[12][13][14]
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:[15][13]