# Poincaré conjecture

- kind: problem
- status: solved (evidence tier V3)
- year: 1904
- authors: Henri Poincaré

**Statement.** Every simply connected closed 3-manifold is homeomorphic to the 3-sphere.

Higher dimensions fell first: Smale in dimension 5 and up in 1961, Freedman in dimension 4 in 1982. Perelman finished dimension 3 by running Hamilton's Ricci flow, cutting out singularities as they formed, and proving Thurston's geometrization along the way. He declined both the Fields Medal and the Millennium Prize. The smooth four-dimensional version is still open.

*Evidence tier V3 is the import default for a result the curation records as settled in the published record. No formal proof or certificate is attached to this page yet.*

Posed 1904, Henri Poincaré. Closed 2003, Grigori Perelman.

## Prize

Clay Millennium Prize awarded 2010, declined

## Related problems

- The original curation links this problem to [Smooth 4-dimensional Poincaré conjecture](/wiki/Problem:smooth-poincare-4.md); the argument for the link has not been written yet.

## Sources

- [Wikipedia: Poincaré conjecture](https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture)

[Geometry and topology](/wiki/Space:geometry-topology.md) · [Mathematics](/wiki/Continent:mathematics.md)

## Relations

- related to [smooth-poincare-4](/wiki/Problem:smooth-poincare-4.md) — The original curation links this problem to Smooth 4-dimensional Poincaré conjecture; the argument for the link has not been written yet.

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