Cubical Scaffolding
Wise became “crazy excited” about cubulating shapes, he said, but at first his mathematical friends just laughed at his monomania.

Then Kahn and Markovic proved the incompressible surface theorem, and Wise and Bergeron immediately published a paper showing that the existence of incompressible surfaces in a compact hyperbolic three-manifold gave a way to cubulate it — and in such a way that the surfaces in the three-manifold correspond precisely to the hyperplanes in the resulting cube complex.

The key to Wise and Bergeron’s construction was the fact that Kahn and Markovic had shown how to construct not one, but a multitude of surfaces. Following an approach to cubulation pioneered in 2003 by Michah Sageev, now at the Technion in Haifa, Israel, Wise and Bergeron started by taking a huge collection of Kahn-Markovic surfaces — enough to divide the three-manifold into compact polyhedra.

Now consider one of the intersection points of these surfaces — suppose, say, that n surfaces meet at this point. Sageev’s insight was to view such an intersection as the shadow, so to speak, of the intersection of n hyperplanes in an n-dimensional cube. The cube complex corresponding to the three-manifold is built, roughly, by putting in one n-dimensional cube for each intersection of n surfaces (the actual construction is a bit subtler, in order to deal with various topological contingencies). Two cubes in the complex are adjacent if their corresponding intersection points in the three-manifold are connected by a face of one of the polyhedra.

“The cube complex is there precisely to record how the surfaces intersect with themselves and each other,” Dunfield said.

Wise and Bergeron showed that this cube complex is “homotopy equivalent” to the original manifold, meaning that the cube complex can be squished and stretched around (perhaps with some dimensional flattening and unflattening) until the cube complex has turned into the manifold, and vice versa. What’s more, this homotopy equivalence transforms each surface in the three-manifold into a corresponding, homotopy-equivalent hyperplane in the cube complex.

The cube complex constructed in this way satisfies the geometric requirements of Wise’s conjecture, meaning that if Wise’s conjecture is true then this cube complex has a finite cover in which all the hyperplanes are embedded.

If such a finite cover indeed exists (let’s say, a cover with m sheets), then recall that the cover could be built from the cube complex by cutting the complex open in some way, making m copies of the complex, and gluing the copies together along the cuts. It’s not hard to show that this recipe for making the cover would carry directly over to a corresponding recipe for making a finite cover of the three-manifold, and that in this finite cover, the Kahn-Markovic surfaces that were used to build the cube complex would lift to embedded surfaces. In other words, if Wise’s conjecture is true, then so is the virtual Haken conjecture.

“The tradeoff is strange: your cube complex might be 10,000-dimensional, for example, so on some level it seems as if you’re making things worse,” Wise said. “But even though the cube complex is so big, many features about it are very easy to understand, so it’s very valuable. We prefer to have something that is big but well-organized rather than to have a three-manifold.”

Even after Wise and Bergeron made the connection between cube complexes and the virtual Haken conjecture, most three-manifold topologists kept their distance from cube complexes. Perhaps this was because Wise’s 200-page paper seemed so daunting, or because cube complexes were so different from the kinds of spaces they were used to studying.

“These ideas were quite esoteric for people coming from hyperbolic geometry,” Bergeron said.

But one mathematician was already fluent in both three-manifold topology and the more abstract, combinatorial considerations that were the currency of Wise’s approach.

“I think Ian Agol was the only three-manifold guy who understood very early on that Wise’s ideas were useful for three-manifold topology,” Bergeron said.

Agol dived into the study of Wise’s masterwork and became convinced that all the parts of it pertaining to Wise’s conjecture were indeed correct. Agol had been working for some time on the virtual Haken conjecture; he realized that Wise’s approach, which resolved the flabby surfaces into crystalline hyperplanes, was exactly what he needed.

“The cube complex gives a scaffolding on which to construct the finite cover,” he said.

To build a special finite cover of a Wise-Bergeron cube complex, Agol started by (abstractly) chopping up the cube complex into “Lego blocks,” cutting along the hyperplanes. He then assigned colors to the faces of the blocks so that any two faces that meet at a corner have different colors. Next, Agol showed, roughly speaking, that there is a way to glue together a finite number of copies of the Lego blocks along faces with matching colors, in such a way that the colors on the sides of those faces also match; that way, each extended hyperplane will be all one color. The resulting cube complex is a finite cover of the original one, and all of its hyperplanes are embedded, since any two hyperplanes that intersect have different colors and therefore are not the same hyperplane intersecting itself.

On March 12, Agol announced that he had proven Wise’s conjecture, and thus the virtual Haken conjecture.

“It was the most exciting news since Perelman proved the geometrization conjecture,” Dunfield said.

Word raced through the three-manifold community, and cube complexes suddenly became a common topic of conversation among three-manifold topologists.

“Until now, I don’t think the mathematics community had realized just how powerful Wise’s work is,” Agol said. “I think my result will make people more aware of what spectacular progress he has made.”

Now, Wise said, mathematicians are starting to realize that “any time you cubulate something, you’re going to reveal all kinds of structural secrets.”

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