As Thurston’s article approached its 30th anniversary this year, all but four of the 23 main questions had been settled, including the geometrization conjecture, which the Russian mathematician Grigori Perelman proved in 2002 in one of the signal achievements of modern mathematics. The four unsolved problems, however, stubbornly resisted proof.
“The fact that we couldn’t solve them for so long meant that something deep was going on,” said Yair Minsky, of Yale University.
Finally, in March, Ian Agol, of the University of California at Berkeley, electrified the mathematics community by announcing a proof to “Wise’s conjecture,” which settled the last four of Thurston’s questions in one stroke.
Mathematicians are calling the result the end of an era.
“The vision of three-manifolds that Thurston articulated in his paper, which must have looked quite fantastic at the time, has now been completely realized,” said Danny Calegari, of the California Institute of Technology. “His vision has been remarkably vindicated in every way: every detail has turned out to be correct.”
“I used to feel that there was certain knowledge and certain ways of thinking that were unique to me,” Thurston wrote when he won a Steele mathematics prize this year, just months before he died in August at 65. “It is very satisfying to have arrived at a stage where this is no longer true — lots of people have picked up on my ways of thought, and many people have proven theorems that I once tried and failed to prove.”
Agol’s result means that there is a simple recipe for constructing all compact, hyperbolic three-manifolds — the one type of three-dimensional shape that had not yet been fully explicated.
“In a precise sense, we now understand what all three-manifolds look like,” said Henry Wilton, of University College London. “This is the culmination of a massive success story in mathematics.”
Thurston’s program tried to do for three-dimensional manifolds what mathematicians had successfully done more than a century earlier for two-dimensional manifolds. As a warm-up for understanding three-dimensional manifolds, let’s look under the hood at the classification of “compact, orientable” surfaces (finite surfaces with no punctures or gashes and a consistent sense of orientation).
To tackle this classification problem, mathematicians showed that, given an arbitrary surface, it is possible to progressively simplify it by cutting it open along curves until the surface completely opens out into a flat polygon.
Figure 1. Cutting a torus open along loop A yields a cylinder. Cutting further, along loop B, unfurls the cylinder into a square.
Image: Courtesy of the Simons Foundation
Figure 2. Cutting a double torus along loops A, B, C and D yields an octagon.
Image: Courtesy of the Simons Foundation
It’s easy to see how to do this with, say, a torus: first cut it open along loop A in Figure 1, producing a cylinder. Next, cut along loop B, flattening the cylinder out into a square. It’s a little harder to see, but cutting along the four curves in Figure 2 converts a double torus (a torus with two holes) into an octagon. Similarly, for any n-holed torus, we can cut along 2n loops to flatten out the surface into a 4n-gon.
Given an arbitrary, unidentified surface, we can try to simplify it (and ultimately identify it) by dissecting it in a similar way. Provided that the surface is not a sphere, topologists have shown that it must contain some embedded loops (loops that don’t intersect themselves) that cannot be pulled down to a single point, similar to loops A and B on the torus. Dissecting the surface along one of these loops removes some of the surface’s interesting topological features. Mathematicians have shown that there are only a finite number of times we can cut in this way before we have reduced the surface to a flat polygon.
Once we have simplified our surface down to a polygon, it’s fairly simple to see that when we re-glue the sides to recover our original surface, we must produce a torus, or a double torus, or a triple torus, and so on. After all, the first gluing will turn the polygon into a tunnel-shaped surface, and then each subsequent gluing will either introduce a new tunnel-shaped handle on the surface or simply sew up some open seams. When we’re finished, the result is a torus surface with some number of holes.
This approach does more than just show that the surface is topologically equivalent to a sphere or a torus of some type: it also gives a way to endow the surface with a simple, uniform geometric structure.