Topology and manifolds, part five

Back to topology.

Last time, I was talking about how according to Thurston’s Geometrization Conjecture, any three-manifold can be broken into pieces in such a way that each piece has a geometric structure. Furthermore, out of the eight possible geometric structures, hyperbolic geometry is the only one for which the corresponding geometric three-manifolds haven’t been completely classified, and that’s one reason why hyperbolic three-manifolds are interesting.

Here’s a question: is the hyperbolic structure on such a three-manifold unique? Or, alternatively, can a manifold have two different hyperbolic structures?

This is a question with consequences, particularly if the answer is “hyperbolic structures are unique”. A manifold, in general, is a topological thing, not a geometric one. A geometric structure, at first glance, is something that we impose on the manifold: a choice that we make, of how to define “distance” and “angle” on that manifold. But if hyperbolic structures are unique, then that means we aren’t making a choice at all, but instead that “distance” and “angle” are determined by the manifold itself. Even though there’s nothing geometric in a manifold’s topological definition, somehow geometry springs forth from it anyway, unbidden.

Mathematicians love that kind of thing.

Unfortunately, hyperbolic structures aren’t quite unique. It is possible for a manifold to have two different hyperbolic structures. Fortunately, those structures are going to be really, really similar. Mostow’s Rigidity Theorem implies that any two such structures are going to be isometric: that there’s going to be a map from the manifold to itself which converts distances and angles from the first hyperbolic structure to distances and angles from the second. That doesn’t imply that the two hyperbolic structures are the same (note that the map from the manifold to itself doesn’t have be the identity map), but it does imply that the two structures are going to have many of the same properties.

For example, we can use a hyperbolic geometric structure to define the hyperbolic volume of the entire manifold. The volume isn’t necessarily finite, but when it is Mostow’s Rigidity Theorem guarantees that the volume is well-defined: it doesn’t depend on which hyperbolic structure we choose to define. The hyperbolic volume is a property of the manifold itself, not the hyperbolic structure that we choose to impose upon it.

In other words, hyperbolic volume is an invariant of hyperbolic three-manifolds. Invariants are the mathematician’s version of measurements: they’re how mathematicians distinguish and classify mathematical objects, as if they were species of butterflies or types of stars.

Hyperbolic volume is and was my particular favourite mathematical playground. We’re getting closer, ever closer, to my answer to “So what do you do?” but we’re not quite there yet…

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