The Poincaré conjecture

1Spaces that look flat up close

A manifold is a space that looks flat up close, wherever you stand. Earth's surface is a 2-manifold: every small patch looks like a flat map, even though the whole thing curves round. Ancient people could not tell the planet was round just by looking at the ground around them.

A 3-manifold is the same idea one dimension up. Every small region looks like ordinary 3D space, with up-down, left-right and forwards-back. But the whole thing might curve back on itself in ways we cannot step outside to see. Our own universe could be one: if you flew a rocket far enough in one direction, you might come back from the other side.

Topologists study these spaces while ignoring size and exact shape. Stretching and bending are allowed; tearing and gluing are not. Two spaces that can be turned into each other this way are called homeomorphic, and count as the same.

2The rubber-band test

Lay a loop anywhere in a space. If every loop, wherever you lay it, can slide and shrink to a single point without leaving the space, the space is simply connected. On a ball's surface every loop shrinks. On a doughnut's surface, called a torus, a loop running round the hole, or through it, gets stuck.

For surfaces, the 2D case, this test settles everything. Every closed surface is a sphere, a torus, a surface with several holes, or one of a few one-sided cousins. Of all of these, only the sphere passes the rubber-band test. So in 2D, "closed and simply connected" means "sphere". That was known in the 1800s.

Mathematicians keep track of loops with an object called the fundamental group, written $\pi_1(M)$. When every loop shrinks, the group is trivial, which we write as $\pi_1(M) = 1$.

3The 3-sphere, a space you can stand inside

The ordinary sphere is every point at distance 1 from a centre in 3D space: $x^2 + y^2 + z^2 = 1$. It is a 2D surface. Add one more coordinate and you get the 3-sphere, $S^3$: every point at distance 1 from a centre in 4D space.

You cannot see $S^3$ whole, because it lives in four dimensions. But you could live inside it. Up close it looks like ordinary space. Walk in a straight line and you eventually come back to where you started, from the opposite direction, just as a ship sailing west round Earth comes back from the east.

Poincaré's question: is the rubber-band test enough to pin $S^3$ down? If you are inside a closed 3D space and every loop you could ever lay shrinks to a point, are you certainly in $S^3$, perhaps stretched and squashed?

4Hamilton's idea, a heat equation for shape

For decades people attacked the problem with pure topology: cutting, gluing and rearranging. It kept failing. In 1982 Richard Hamilton tried something different. Give the space a metric, a ruler that says how far apart nearby points are, then let the ruler change over time so that the space smooths itself out.

His rule is called Ricci flow. The Ricci curvature, $\mathrm{Ric}(g)$, measures how much the space bends at each point, averaged over directions. The flow shrinks regions that bulge outwards, like a ball, and swells regions that curve like a saddle. Curvature spreads out and evens out, the way heat spreads through a metal bar until it is the same temperature everywhere.

The hope was simple. Start with any simply connected closed 3-manifold and any metric. Run the flow. If the space ends up perfectly round, it was $S^3$ all along. In his 1982 paper Hamilton proved this works when the space starts with positive Ricci curvature everywhere.

5The loop you just played with

The widget shows the one-dimensional cousin. For a curve, the "metric" is just how the curve sits in the plane, and the flow says each point moves inward at a speed equal to its curvature $\kappa$, which is one divided by the radius of the circle that best fits the bend there.

In 1986 Michael Gage and Hamilton proved that a convex loop, one with no dents, shrinks to a round point under this flow. Matthew Grayson extended it in 1987 to any loop that does not cross itself: dents and all, it first becomes convex, then round, then vanishes.

The area law is a lovely bonus. As you walk once round any simple closed curve, your direction turns through exactly one full turn, $2\pi$ radians. Adding up the inward speed $\kappa$ all the way round therefore always gives $2\pi$, so the area falls at exactly $2\pi$ per unit time. A loop of area 20 vanishes after $20 / 2\pi \approx 3.2$ seconds, whatever its shape.

To compute it, the widget keeps about 160 points spaced $h$ apart and estimates the curvature with a second difference, $(p_{i+1} - 2p_i + p_{i-1}) / h^2$. It moves each point by a tiny time step no bigger than $0.4h^2$, which keeps the sums stable, and re-spaces the points evenly after every step.

6Necks, pinches and surgery

In 3D the flow can go wrong before it finishes. Picture a dumbbell: two round lumps joined by a thin neck. Under Ricci flow the neck can pinch off, and the curvature there shoots up to infinity in a finite time. This is called a singularity, and the equation stops making sense there.

Hamilton proposed surgery: just before a neck pinches, cut it out, cap both open ends with round balls, and carry on flowing. The trouble was control. Nobody could prove that every possible singularity looks like a neat neck, or that surgeries would not pile up endlessly.

Perelman supplied the control. He found a quantity, a kind of entropy, that only ever moves one way under the flow, and used it to rule out the worst kinds of collapse. He proved that near any point of very high curvature the space always looks like one of a few standard models, so there is always a clean place to cut. And he showed that only finitely many surgeries happen in any finite time.

7How it ends, and what else fell

In his third paper Perelman showed that for a simply connected space, the flow with surgery dies out in finite time. The whole manifold shrinks away in pieces, and each piece is known to be a round 3-sphere. Undo the surgeries, gluing the spheres back together, and you find that the space you started with was $S^3$.

His first two papers went further and sketched a proof of Thurston's whole geometrization conjecture. It says every closed 3-manifold can be cut along spheres and tori into pieces, each carrying one of eight standard geometries, such as spherical, flat or hyperbolic. It is a complete catalogue of possible 3D spaces, and Poincaré drops out as a one-line corollary: a simply connected piece must be spherical.

Perelman never sent his papers to a journal. Other mathematicians spent years checking them in public. Bruce Kleiner and John Lott posted detailed notes in 2006, and John Morgan and Gang Tian posted a 493-page account the same year. By then the community accepted that the proof was right.