The End of an Era
Agol’s proof of Wise’s conjecture was a four-for-one deal: it proved not just the virtual Haken conjecture, but the other three of Thurston’s 23 questions that were still unresolved. In the years leading up to his proof, Agol and other mathematicians had shown that all three of these questions — the virtual fibering conjecture and two more technical questions about hyperbolic three-manifolds — were also consequences of Wise’s conjecture.
In the case of the virtual fibering conjecture, recall that the goal was to show that every compact hyperbolic three-manifold has a finite cover that fibers over the circle, meaning that it is built by gluing the opposite ends of a thickened surface. We know from the virtual Haken theorem that the manifold has a finite cover that is Haken — that is, the manifold’s cover has an embedded, incompressible surface. If you cut the Haken manifold open along that surface, you get something that looks like a thickened surface at its ends but has who-knows-what topological features in its “guts.”
In 2008, in what Calegari calls “an astonishing breakthrough,”Agol showed that hyperbolic three-manifolds that satisfy a certain technical condition are guaranteed to be virtually fibered. The following year, Wise built on this finding to show that all Haken manifolds are virtually fibered; that is, there is a way to unroll a Haken manifold to produce a finite cover that opens up the complicated topology of the guts, resulting in a simple fibered manifold. Thus, if a manifold is virtually Haken, then it also must be virtually fibered.
“I think everyone had believed that the virtual Haken conjecture would turn out to be true, but the virtual fibering conjecture had seemed orders of magnitude farther out of reach,” Calegari said. “To me, the fact that the virtual fibering conjecture follows from the virtual Haken conjecture is one of the most shocking aspects of the story.”
With the proof of the virtual fibering conjecture, “you’re tempted to think that this means three-manifolds are really simple, because manifolds that fiber over the circle are simple,” Minsky said. “But I think it teaches us that manifolds that fiber over the circle are not simple after all — they’re more subtle than we expected.”
At the same time, the virtual fibering theorem does mean that there is a simple and informative recipe for generating all compact hyperbolic three-manifolds: start with a thickened surface, glue its inner and outer boundary surfaces to each other with your choice of twist, and then fold that manifold over itself a finite number of times.
“If you were to ask me for a hyperbolic three-manifold, I’d ask what kind you want — what kind of fibration and which finite covering?” Calegari said. “We know now that we’re not missing out on any three-manifolds by doing this.”
While it will take some time for mathematicians to check Agol’s work thoroughly, many are optimistic that it will stand up to scrutiny.
“Ian Agol’s not a sloppy guy,” Minsky said.
Now that the final questions on Thurston’s list have presumably been laid to rest, researchers are already starting to ask what the field of three-manifold topology will look like in this brave new post-Thurston world.
Mathematicians agree that they will have plenty to do in figuring out what insights Wise’s cube complexes have to offer for other shapes that can be cubulated. When it comes to three-manifolds themselves, mathematicians have reached the end of an era, Agol said, but also the beginning of a new one.
“Most fields of mathematics don’t have an all-encompassing vision to guide the field over twenty or thirty years, the way we’ve had,” he said. Now, he suggests, three-manifold topology and geometry may become more like those other fields, in which mathematicians “grope around” and manage to make progress even without the benefit of a big conjectural picture of what is going on.
“New generations of mathematicians will figure out what the next important questions are,”Agol said.
Reprinted with permission from Simons Science News, an editorially-independent division of SimonsFoundation.org whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the computational, physical and life sciences.