{"id":"poincare-conjecture","domain":"topology","type":"Theorem","title":"Poincaré Conjecture","slug":"poincare-conjecture","url":"/mathematics/topology/poincare-conjecture/","statement":"Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.","proposed":"1904, Henri Poincaré","proved":"2002-2003, Grigori Perelman","summary":"Millennium Prize Problem, only one of seven solved so far. Perelman declined Fields Medal (2006) and $1M prize (2010). Proved via Ricci flow with surgery.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Perelman, G. (2002-2003). The entropy formula for the Ricci flow and its geometric applications. arXiv preprints."},{"tier":1,"citation":"Morgan, J. and Tian, G. (2007). Ricci Flow and the Poincaré Conjecture. AMS/Clay Mathematics Institute."},{"tier":2,"citation":"Kleiner, B. and Lott, J. (2008). Notes on Perelman's papers. Geometry & Topology, 12(5), 2587-2855."},{"tier":3,"citation":"Szpiro, G. (2007). Poincaré's Prize. Dutton."}],"relationships":[{"type":"uses","target":"topological-space","label":"Topological Space"},{"type":"millennium_prize","target":null,"label":"Solved 2003 — only one of seven solved"},{"type":"contrasts_with","target":"riemann-hypothesis","label":"Riemann Hypothesis — still unsolved"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.6"}