From the point of view of hyperbolic geometry, the shapes in Figure 4 are all regular octagons with straight edges. In one of these octagons, the angles are all 45 degrees — just what we need for a double torus. If we glue this octagon’s sides appropriately, the result will be a double torus with a perfect, uniform hyperbolic structure.
Figure 4. Regular octagons in hyperbolic space, such as the ones pictured above, can have any internal angle measure greater than zero and less than 135 degrees. The brown octagon, whose internal angles are all 45 degrees, can be glued together to form a double torus with smooth hyperbolic geometry.
Image: Courtesy of Silvio Levy
Similarly, we can equip a triple torus with a hyperbolic structure. A triple torus can be made by gluing the sides of a 12-sided polygon, so if we construct a hyperbolic dodecagon whose internal angles are all 30 degrees, its hyperbolic geometry can be carried over smoothly to the triple torus. Continuing in this way, we can equip a four-holed torus, a five-holed torus, and so on, with hyperbolic geometry. Our taxonomy of compact surfaces becomes: one surface with spherical geometry (the sphere), one surface with Euclidean geometry (the torus), and infinitely many surfaces with hyperbolic geometry (all the tori with more than one hole).
Over the past century, this taxonomy has given mathematicians an incredibly fruitful way to translate topological questions about surfaces into geometric ones, and vice versa. The classification of surfaces is the foundational concept in the study of two-dimensional shapes, a finding that all subsequent studies take as their starting point.
The Next Dimension
Three-manifolds are vastly more diverse than two-manifolds, and the questions are correspondingly harder. Even as simple-sounding a question as the famous Poincaré conjecture — which asks whether the three-dimensional version of the sphere is the only compact three-dimensional shape on which every loop can be pulled tight to a single point without getting snagged around a hole — remained unsolved for nearly a century after Henri Poincaré posed it in 1904.
Nevertheless, Thurston boldly conjectured that it should be possible to create a taxonomy for three-dimensional shapes similar to the one for two-dimensional shapes.
The two-dimensional Euclidean, spherical and hyperbolic geometries each have a counterpart in three dimensions. But in three dimensions, these are not the only “nice” geometries out there. For example, there are hybrid geometries that are hyperbolic or spherical in certain directions, and Euclidean in others. Altogether, there are eight different types of geometry in dimension three that are uniform, meaning that the geometry looks the same at every point in the space.
Thurston conjectured that, just as with surfaces, three-manifolds can be endowed with natural geometric structures. Specifically, he proposed that if you carve up any compact three-manifold into chunks in a particular way, each chunk can be endowed with one of the eight geometries.
“The goal was to completely unify topology and geometry in three dimensions,” Minsky said.
A natural approach to proving this “geometrization conjecture” would be to try to do something similar to what we did for surfaces, which we cut along curves until we had cut open all the interesting topological features and reduced the surface to a flat polygon. For a three-manifold, the corresponding approach would be to cut it open along surfaces until, hopefully, it reduced to a polyhedron, whose opposite sides could be glued together to recover the original shape. Then, if we could build the polyhedron with the right geometry, we could transfer that geometry to the original shape, just as we did with surfaces.
Remember that to make this work for surfaces, each curve we cut along must satisfy two properties: The curve should never cross itself (in mathematical lingo, it should be “embedded”), and it should be what we’ll call “topologically interesting,” meaning that it winds around some topological feature of the surface and can’t be tightened down to a single point (this requirement ensures that cutting along the curve simplifies the surface topologically).
In 1962, the mathematician Wolfgang Haken proved that it is indeed possible to simplify a three-manifold down to a polyhedron, provided the three-manifold contains a surface to cut along that satisfies two conditions: It must be embedded, and it must be “incompressible,” meaning that every topologically interesting curve on the surface is also topologically interesting in the larger context of the surrounding three-manifold.
So, for example, a torus is not incompressible in ordinary three-dimensional space, since the loop that dips through the hole of the torus is topologically interesting from the point of view of the torus, but in the full three-dimensional space it can be compressed down to a single point. By contrast, the torus is incompressible inside the three-manifold that you get just by thickening the torus surface slightly so that it is no longer infinitesimally thin. To be incompressible, every topological feature of the surface must truly reflect some of the three-manifold’s intrinsic topology. A three-manifold that has an embedded, incompressible surface is now known as a Haken manifold.
If our three-manifold does have an embedded, incompressible surface, then cutting along this surface will cut open some of the three-manifold’s interesting topology, leaving a simpler manifold in its place. What’s more, Haken showed that as long as the manifold contains one such surface, the new manifold produced by the cutting will itself be Haken: it will again have an embedded, incompressible surface to cut along. After a finite number of such steps, Haken showed, all the interesting topological features of the original manifold will have been cut away, leaving a polyhedron.
In the late 1970s, Thurston showed that it is possible to endow the resulting polyhedron with one of the eight three-dimensional geometries in such a way that the geometry transfers smoothly to the re-glued polyhedron, fitting together perfectly at the polyhedron’s corners and edges. In other words, Thurston proved his geometrization conjecture for manifolds whose standard decomposition yields chunks that are all Haken manifolds.
Unfortunately, given an arbitrary compact three-manifold, there’s no guarantee that it will indeed have such a surface. In fact, in the late 1970s and early 1980s, Thurston convinced the three-manifold community that three-manifolds that contain an embedded, incompressible surface (Haken manifolds) are the exception, rather than the rule.