# Poincaré Conjecture

**Type:** Theorem  
**Domain:** Topology  
**Proposed:** 1904, Henri Poincaré  
**Proved:** 2002-2003, Grigori Perelman  
**Codex URL:** /mathematics/topology/poincare-conjecture/  
**Entry status:** Live — v1.0 (2026-05-27)

## Statement
Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere S³.

## Summary
Millennium Prize Problem — only one of seven solved so far. Perelman declined the Fields Medal (2006) and $1M Clay Prize (2010).

## Dimensional history (higher dimensions solved first!)
- Smale (1961): dimension ≥5, Fields Medal
- Freedman (1982): dimension 4, Fields Medal
- Perelman (2002-03): dimension 3 — the hardest, solved last

## Perelman's method
Built on Hamilton's Ricci flow (1980s). Perelman's key contribution: "Ricci flow with surgery" — systematically removing singularities. Actually proved Thurston's broader Geometrization Conjecture (1982).

## Verification
Posted on arXiv, not peer-reviewed journal. Verified by independent teams: Kleiner-Lott (2008), Cao-Zhu (2006), Morgan-Tian (2007).

## Sources
### Tier 1
- Perelman, G. (2002-2003). The entropy formula for the Ricci flow. arXiv.
- Morgan, J. and Tian, G. (2007). *Ricci Flow and the Poincaré Conjecture*. AMS/Clay.
### Tier 2
- Kleiner, B. and Lott, J. (2008). Notes on Perelman's papers. *Geometry & Topology*.
### Tier 3
- Szpiro, G. (2007). *Poincaré's Prize*. Dutton.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*