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Poincaré conjecture
Sign in to saveAlso known as Гипотеза Пуанкаре, Теорема Пуанкаре, Теорема Перельмана
theorem in geometric topology that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere
Key facts
- Field
- Geometric topology
- Conjectured by
- Henri Poincaré
- First proof by
- Grigori Perelman
- Implied by
- Geometrization conjecture Zeeman conjecture
- Generalizations
- Generalized Poincaré conjecture
via Wikipedia infobox
Wikidata facts
- Instance of
- conjecture
- Part of
- Millennium Problems
- Named after
- Henri Poincaré
Show 6 more facts
- discoverer or inventor
- Henri Poincaré
- proved by
- Grigori Perelman
- on focus list of Wikimedia project
- Wikipedia:Vital articles/Level/4
- maintained by WikiProject
- WikiProject Mathematics
- uses
- Ricci flow
- studied by
- algebraic topology
Sources (5)
via Wikidata · CC0
~27 min read
Encyclopedic overview
In the mathematical field of geometric topology, the Poincaré conjecture ( UK: /ˈpwæ̃kæreɪ/, US: /ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere (the hypersphere that bounds the 4-ball in four-dimensional space).
Originally conjectured by Henri Poincaré in 1904, the theorem concerns spaces that locally look like ordinary three-dimensional Euclidean 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.
Excerpted from Wikipedia’s “Poincaré conjecture” article, available under the CC BY-SA 4.0 licence.