Poincaré conjecture
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
Article
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.