Figuring out how to prove the geometrization conjecture for non-Haken manifolds stumped mathematicians for more than two decades. Finally, in 2002, Perelman set forth his proof, which drew on areas of mathematics far removed from those studied by most of Thurston’s followers. (Along the way, Perelman’s proof settled the century-old Poincaré conjecture, leading the Clay Mathematics Institute in 2010 to offer him a million-dollar prize — which he promptly rejected, for rather complicated reasons.)

Perelman’s landmark proof achieved Thurston’s dream of unifying topology and geometry. Now every topological question about three-manifolds had its geometric counterpart, and vice versa. But Perelman’s theorem left unresolved many important questions about what kinds of three-manifolds can exist.

In classifying compact two-manifolds (surfaces), mathematicians were able not only to show that each surface could be endowed with a geometric structure, but also to make a complete list of every possible two-manifold. In dimension three, such a list was lacking.

Seven of the eight three-dimensional geometries — all but hyperbolic geometry — are fairly easily understood, and even before Perelman’s work, three-manifold topologists had arrived at a complete description of the types of manifolds that can admit one of these seven geometries. Such shapes are relatively simple and few.

But just as with surfaces, in dimension three it turns out that most manifolds are in fact hyperbolic. Mathematicians had a much more tenuous grasp of the vast range of possibilities for hyperbolic three-manifolds than they had for the other seven geometries.

“Of the eight kinds of geometry, the hyperbolic manifolds are the most mysterious and rich,” said Nicolas Bergeron, of the Université Pierre et Marie Curie in Paris.

Perelman’s result told mathematicians that hyperbolic manifolds were indeed the final frontier — the only kind of three-manifold left to understand. But it didn’t tell them what these hyperbolic shapes actually look like.

Cover Story
Once again, mathematicians were able to turn to Thurston’s seminal paper for guidance. On his famous list of questions were many conjectures about the features of hyperbolic three-manifolds, including two conjectures that speak directly to what such manifolds can look like: the “virtual Haken” conjecture and the “virtual fibering” conjecture.

The virtual Haken conjecture proposes that every compact hyperbolic three-manifold is almost Haken, in a precise sense: it’s possible to convert the manifold into a Haken manifold simply by unrolling it a finite number of times, in a particular way. This new, unrolled manifold is called a “finite cover” of the original manifold.

Mathematicians say that one manifold N covers another manifold M if, roughly speaking, it’s possible to wrap N around M a certain number of times (perhaps infinitely many times) so that each part of M gets covered the same number of times as every other part. To be a covering, this wrapping should also have an assortment of other nice properties — for example, N should never fold over on itself or tear during this wrapping process. Each little piece of M is covered by a bunch of identical copies of it in the cover, N.

Figure 5. The six-petal flower covers the three-petal flower by wrapping around it two times.
Image: Courtesy of the Simons Foundation

For example, the six-petal flower in Figure 5 covers the three-petal flower: simply wrap the six-petal flower two times around the three-petal flower. Each point on the three-petal flower is covered by two points on the six-petal flower; mathematicians call this a two-sheeted covering.

Likewise, an infinitely long cylinder covers a torus: just keep wrapping the cylinder evenly around and around the torus, infinitely many times (see Figure 6). Every point on the torus is covered: Loop A on the torus is covered by an infinite collection of evenly spaced loops on the cylinder, while loop B unrolls on the cylinder to become a line that runs the length of the cylinder.

The topology of a manifold and its cover are intimately related. To reconstruct a manifold from an n-sheeted cover, you simply fold the cover over on itself n times. Likewise, to reconstruct the cover from the manifold, you slice open the manifold, make n copies of it, and glue the copies together along their boundaries (the particular cover you get may depend on your gluing choices).

A cover preserves some of the manifold’s topological features while unrolling others. The infinite cylinder, for example, remembers that loop A is a closed loop in the torus, but it forgets that loop B is also a closed loop.

Figure 6. An infinitely long cylinder covers a torus by wrapping around it again and again. Loop A on the torus “lifts” to the infinite collection of red loops on the cylinder. Loop B unrolls in the cylinder to become the green line. Point P on the torus lifts to the infinite collection of blue dots on the cylinder.
Image: Courtesy of the Simons Foundation

This unrolling process is precisely what led Thurston to hope that, given a three-manifold, it might be possible to produce a finite-sheeted cover that is Haken. As we’ve discussed, given an arbitrary compact, hyperbolic three-manifold, there is no reason to expect it to be Haken (that is, to have an embedded, incompressible surface). However, in 1968, German mathematician Friedhelm Waldhausen conjectured that such a manifold should at least contain an incompressible surface, although the surface might pass through itself in places, rather than being embedded.

If that is indeed the case, Thurston argued, there might well be a finite cover in which the surface unrolls in a way that eliminates all of its intersections with itself. Finite covers can often achieve such simplifications. For example, since the curve in the three-petal flower in Figure 7 goes around the central hole twice, no amount of stretching and shifting can prevent it from intersecting itself somewhere. But if we start unrolling this curve in the six-petal flower starting at a chosen point P, the resulting red curve (which mathematicians call a “lift” of the original curve) goes around the central hole only once and never intersects itself. (There is a second lift, the blue curve, which intersects the red curve at the two points that cover the intersection point in the three-petal flower.)

Figure 7. The green curve in the three-petal flower intersects itself, but its two lifts in the six-petal flower, the red and blue curves, never intersect themselves (though they intersect each other).
Image: Courtesy of the Simons Foundation

In his 1982 paper, Thurston proposed that given a compact, hyperbolic three-manifold, it should be possible to do a similar type of unrolling to produce embedded surfaces in some finite cover — in other words, the three-manifold should be “virtually” Haken.

A Haken manifold, as we’ve discussed, can be built by gluing the boundary walls of a polyhedron in a particular way. The virtual Haken conjecture implies, then, that any compact hyperbolic three-manifold can be built first by gluing up a polyhedron nicely, then by wrapping the resulting shape around itself a finite number of times.

Thurston went on to suggest something even stronger: that any compact hyperbolic three-manifold might be virtually fibered, meaning that it has a finite cover that is “fibered.” A manifold that “fibers over the circle” (as mathematicians say) is built by thickening a surface slightly to make it three-dimensional, then gluing the inner and outer boundary surfaces together according to any arrangement that matches the two surfaces up smoothly, point for point. (Such a gluing couldn’t be realized in ordinary space without parts of the resulting manifold passing through itself, but it can still be studied abstractly.) The manifold is said to be fibered because if you imagine stretching out the thickened surface so the boundary surfaces are far apart, then drawing the boundaries around to face each other before gluing them together, you can imagine that the resulting manifold is like a bracelet that has an infinitely thin surface-shaped bead at every point on the bracelet’s strand; these beads are the “fibers.”

Every fibered manifold is Haken, but the reverse is not true. Thus, the virtual fibering conjecture is a stronger statement than the virtual Haken conjecture, and Thurston was on the fence about whether it was indeed true. “This dubious-sounding question seems to have a definite chance for a positive answer,” was as far as he was willing to go in his 1982 paper.

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