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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.
In 1904, Henri Poincaré posed the question of whether an analogous statement holds true for three-dimensional shapes. This came to be known as the Poincaré conjecture, the precise formulation of which states: Any three-dimensional topological manifold which is closed and simply-connected must be homeomorphic to the 3-sphere.
The generalized Poincaré conjecture is true topologically, but false smoothly in some dimensions. This results from the construction of the exotic spheres, manifolds that are homeomorphic, but not diffeomorphic, to the standard sphere, which can be interpreted as non-standard smooth structures on the standard (topological) sphere.
Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and mathematicians in general, each prize problem includes a million-dollar bounty. As with the Hilbert problems, one of the prize problems (the Poincaré conjecture) was solved relatively soon after the problems were announced.
"Manifold Destiny" is an article in The New Yorker written by Sylvia Nasar and David Gruber and published in the 28 August 2006 issue of the magazine. [1] It claims to give a detailed account (including interviews with many mathematicians) of some of the circumstances surrounding the proof of the Poincaré conjecture, one of the most important accomplishments of 20th and 21st century ...
The solution to the smooth Poincaré conjecture is known in all dimensions other than 4 (it is usually false in dimensions at least 7; see exotic sphere). The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4. In 4 dimensions, the PL Poincaré conjecture is equivalent to the smooth Poincaré conjecture, and ...
BBC in Our Time, "Discussion of the Poincaré conjecture," 2 November 2006, hosted by Melvyn Bragg. Poincare Contemplates Copernicus at MathPages High Anxieties – The Mathematics of Chaos (2008) BBC documentary directed by David Malone looking at the influence of Poincaré's discoveries on 20th Century mathematics.
Poincaré conjecture, 1904: 3-manifolds: 2003: Grigori Perelman: geometrization conjecture of Thurston: 3-manifolds: ⇒spherical space form conjecture: 2003: Ben Green; and independently by Alexander Sapozhenko: Cameron–Erdős conjecture: sum-free sets: 2003: Nils Dencker: Nirenberg–Treves conjecture: pseudo-differential operators: 2004 ...