From Poincaré's question to Perelman's proof — topology through geometry
If every loop on a closed 3D space can shrink to a point, must the space be a 3-sphere? The question stood for a century until Ricci flow answered it.
Add a few loops, then press Contract All. On the sphere every loop shrinks to a point; switch to the torus and see which loops get stuck around the hole.
Add loops to the surface and contract them. On the sphere, every loop shrinks to a point. On the torus, loops that go around the hole cannot be contracted.
In 1904, Henri Poincaré posed a deceptively simple question: Is every simply connected, closed 3-manifold homeomorphic to the 3-sphere S³? In other words, if a closed 3-dimensional shape has no “holes” — every loop on it can be continuously shrunk to a point — must it be a 3-sphere?
This question remained open for nearly a century and became one of the seven Millennium Prize Problems. It was finally resolved by Grigori Perelman in 2002–2003 using Richard Hamilton's Ricci flow, combined with revolutionary new ideas about entropy, non-collapsing, and geometric surgery.
The Poincaré Conjecture was proved in higher dimensions first, with the hardest case — dimension 3 — solved last:
Poincaré poses the conjecture for 3-manifolds.
Stephen Smale proves the generalized conjecture for dimensions ≥ 5 using h-cobordism theory (Fields Medal, 1966).
Michael Freedman proves the 4-dimensional case using intricate topological techniques (Fields Medal, 1986).
Richard Hamilton introduces Ricci flow and proves the conjecture for manifolds with positive Ricci curvature.
Grigori Perelman completes the proof for dimension 3 using Ricci flow with surgery. Awarded the Fields Medal (2006) and Millennium Prize (2010) — he declined both.
Step through, or auto-play, a sketch of Perelman's proof: the flow hits a singularity, surgery splits the space, and each piece either becomes a round S³ or goes extinct.
Richard Hamilton introduced Ricci flow to attack the conjecture. The idea: start with any Riemannian metric on the 3-manifold and evolve it by
This flow smooths out the geometry, analogous to how the heat equation smooths temperature. Hamilton proved that on manifolds with positive Ricci curvature, the flow converges (after rescaling) to a metric of constant positive curvature — proving the conjecture in that special case. The challenge: in general, singularities can form before the flow reaches constant curvature. Neckpinches and other degeneracies can develop at finite time, blocking the flow from completing its work.
Grigori Perelman posted three papers to arXiv that resolved the conjecture by introducing several revolutionary ideas:
The proof proceeds: run Ricci flow until a singularity forms, perform surgery, continue the flow on each piece. After finitely many surgeries, every remaining piece either becomes round (homeomorphic to S³) or goes extinct. If the original manifold was simply connected, only S³ pieces remain — proving the conjecture.
Press play, or drag the time slider, and watch Perelman's entropy W(g, f, τ) climb as the flow moves through the landscape. It never goes down.
Left: a particle traverses an energy landscape under Ricci flow. Right: Perelman's W-entropy increases monotonically, providing the crucial analytic control that makes the proof work. The entropy prevents local collapsing near singularities.
Perelman's key innovation was a monotonic entropy functionalW(g, f, τ)that never decreases under Ricci flow. This prevents “local collapsing” and gives fine control over singularity formation.
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