Conjecture: every compact, simply-connected, smooth 3-manifold without boundary is homeomorphic to the 3-sphere S³. Equivalently: any loop on such a manifold can be contracted to a point if and only if the entire space can be continuously deformed into the standard sphere.
Posed by Poincaré in 1904 after discovering his own earlier characterization attempt failed. Higher-dimensional analogues fell first (Smale 1961, Freedman 1984 — both Fields work), making dimension three the stubborn holdout: Ricci curvature collapsed, surgeries misbehaved.
Proved by Grigori Perelman, 2002–2003, in three arXiv preprints proving Thurston's full Geometrization Conjecture via Ricci flow with surgery, overcoming the singularities that stalled Hamilton's programme. Verification occupied three independent teams through 2006. Perelman declined the Fields Medal (2006) and the Clay Millennium Prize (2010) — the only person ever offered either for this problem, declining both. Purely human, purely individual, and possibly the last of its kind.
How to check it: Hamilton's Ricci flow plus Perelman's entropy functional and reduced volume, which rule out the collapsing singularities that stalled Hamilton. Three independent verifications (Kleiner–Lott notes, Morgan–Tian monograph, Cao–Zhu) closed the case by 2006; the Clay prize followed in 2010. The dimension-four smooth case in this directory stays open.
Verified by curator — see the API for full claim provenance.
