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Conjuring with conjectures

The Clay Mathematics Institute is offering a million dollars for a solution to the Poincaré conjecture, and Grisha Perelman may have found one. What is the conjecture, and why does it matter?

In 1904, Henri Poincaré was making fundamental advances in topology — the multi-dimensional study of mathematical properties such as 'knotted' or 'connected', which are unchanged by continuous deformations. Buried in his work was an unjustified assumption about three-dimensional spaces. When he noticed it, and failed to find a proof, the assumption became first a question, and then a conjecture. Unsolved almost a century later, the Poincaré conjecture joined other problems (such as the Riemann hypothesis and Yang–Mills theory) on a list of 'Millennium Prize Problems' for whose solutions a substantial cash prize is offered1. Grisha Perelman of the Steklov Institute of Mathematics in St Petersburg, Russia, may now have found the answer2,3.

The Poincaré conjecture can be understood by analogy with the case in two dimensions. A two-dimensional space, or surface, is like a bubble made from an infinitely thin film of soap. If the bubble is round, the surface is a sphere, but many other shapes are possible. A torus, for instance, is shaped like an American doughnut; a two-holed torus is like two doughnuts joined together, and so on.

The early topologists proved that any surface is topologically equivalent to a torus with a finite number of holes. A lumpy potato is the same as a sphere, a coffee-cup is the same as a torus, and a teapot is the same as a two-holed torus4. Poincaré sought a comparable understanding of three-dimensional spaces. The simplest is the three-sphere, analogous to a sphere but having three dimensions instead of two. This is almost the same as a solid ball, but only if you pretend that the entire outer surface of the ball is a single point.

The key problem is to decide whether a given surface — a lumpy potato, say — is equivalent to a sphere, or to something more exotic. Poincaré's predecessors had solved that problem in two dimensions, by working out how closed loops in the surface behave when they are moved. Any loop on a sphere can be continuously deformed, or shrunk, to a point. But on a torus, or a more complicated surface, many loops cannot be shrunk (Fig. 1a). Better still, if every loop can be shrunk then the surface must be a sphere. So the 'shrinkability' of loops provides a definitive test for spherical topology.

Figure 1: Testing topology.
figure1

a, On the surface of a sphere, every loop can be deformed to a single point. But on any other surface, such as the two-holed torus shown here, some loops cannot be deformed to a point. Topologically, these are two-dimensional objects, but in three dimensions the 'shrinkability' of loops becomes the basis of Poincaré's conjecture. b, Perelman's solution2,3 to the Poincaré conjecture starts with 'Ricci flow': to achieve constant curvature of its surface, an object such as this dumbbell-shaped bubble splits into two round ones.

What about three dimensions? It is easy to prove that every loop in a three-sphere can be shrunk, but Poincaré tacitly assumed the converse: if every loop can be shrunk, then the space is topologically a three-sphere. Only later did it dawn on him that this statement — now called the Poincaré conjecture — was not obvious, and perhaps might not even be true.

Throughout the twentieth century, the conjecture remained as mysterious as ever. Analogous statements in all numbers of dimensions from four upwards had been proved true, but Poincaré's original three-dimensional problem still held out. The Poincaré conjecture had become a serious obstacle to any understanding of three-dimensional topology. Because topology is fundamental to many branches of mathematics, and to several areas of mathematical physics, this missing piece of the mathematical tool kit had become a serious embarrassment. How can we hope to understand the shape of our own Universe, for instance, when we don't even know the range of possibilities? And what about the even harder topological problems arising in quantum field theory (for example, in relation to string theory, where space-time has extra dimensions that curl up in some topological manner)?

Around 1983, William Thurston had devised an entirely new approach to the Poincaré conjecture, by relating it to classical geometry. Starting, again, in two dimensions, Thurston wondered why there are lots of shapes for a potato, but only one round sphere. What makes that shape special? It has constant (positive) curvature: every bit of a round sphere is bent the same amount as any other. Similarly, there is a 'canonical' geometry for a torus, and in this case the curvature is zero. The usual torus does not look flat, but a rectangle with opposite edges identified is flat, and topologically that is a torus. Finally, tori with two or more holes can be realized as surfaces of constant negative curvature.

Thurston suggested that something similar should happen in three dimensions. He proved that exactly eight different 'geometries' can occur. It turned out to be too much to expect every three-dimensional space to have a single canonical geometry, but Thurston's 'geometrization conjecture' asserted the next best thing: every three-dimensional space can be cut up, in a systematic way, so that each piece has precisely one of those eight geometries. This new conjecture was much more ambitious than Poincaré's: it aimed at understanding all three-dimensional spaces, not just the three-sphere. In particular, the Poincaré conjecture was an easy consequence of the geometrization conjecture: the condition on loops meant that only one piece would be needed, and the associated geometry must be that of the three-sphere.

Perelman's work2,3 is a new approach to the geometrization conjecture. If it pans out, it will constitute a huge leap forward in three-dimensional topology and mathematical physics. It is based on the Ricci flow, an idea of Richard Hamilton's5: if a soap bubble is deformed away from its normal round shape, surface tension will pull it back to a perfect sphere. The Ricci flow is an analogous way to deform any surface so that its curvature 'tries' to become constant. Along the way, though, it may have to split into separate pieces; for example, a dumbbell-shaped bubble must break up into two round ones (Fig. 1b).

Hamilton defined the Ricci flow in three dimensions, and developed a programme to prove that when a three-dimensional 'bubble' follows the flow, the pieces it breaks into are essentially those predicted by the geometrization conjecture. Perelman's papers do not carry out the entire Hamilton programme, but it looks as though they might establish enough of it to prove the geometrization conjecture. That in turn will prove the Poincaré conjecture.

If everything hangs together, a 99-year search is at an end. If not, Perelman's papers will still shed a huge amount of light on the Ricci flow, which is important in quantum field theory and pure mathematics, and may well pave the way to an eventual proof of the two conjectures. Either way, the spin-off is likely to make profound changes to how we think about topology, space-time and quantum field theory.

References

  1. 1

    http://www.claymath.org/Millennium_Prize_Problems

  2. 2

    Perelman, G. Preprint math.DG/0211159 at <http://arXiv.org> (2002).

  3. 3

    Perelman, G. Preprint math.DG/0303109 at <http://arXiv.org> (2003).

  4. 4

    Stewart, I. Flatterland (Macmillan, London, 2001).

  5. 5

    Hamilton, R. S. J. Diff. Geom. 17, 255–306 (1982).

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Correspondence to Ian Stewart.

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