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 For compact 2-dimensional surfaces without boundary, if every loop can be continuously tightened to a point, then the surface is topologically homeomorphic to a 2-sphere (usually just called a sphere). The Poincaré conjecture, proved by Grigori Perelman, asserts that the same is true for 3-dimensional spaces. | |
| Type | Theorem |
|---|---|
| Field | Geometric topology |
| Statement | Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. |
| Conjectured by | Henri Poincaré |
| Conjectured in | 1904 |
| First proof by | Grigori Perelman |
| First proof in | 2006 |
| Implied by | |
| Open problem | No |
| Generalizations | Generalized Poincaré conjecture |