### Topology and manifolds, part four

So in part two I described all of the compact two-dimensional manifolds, and in part three I discussed how each two-dimensional manifold has a geometric structure. Let’s finally start talking about three-dimensional manifolds.

Simply put, three-dimensional manifolds are much, much harder than their two-dimensional cousins. Most of the clever tricks that can be used to classify two-dimensional manifolds are next to worthless in higher dimensions. But fortunately there exceptions to that rule. Geometric structures, it turns out, are as important to three-dimensional manifolds as to two-dimensional ones.

It’s helpful to have at least one example in the back of your head, so let’s construct a three-dimensional analogue of a torus, called a *three-torus*. We constructed a torus before by taking a rectangle and gluing opposite edges of the rectangle together, left to right and top to bottom. To construct a three-torus, start with a three-dimensional cube, and glue opposite *faces* together: left to right, top to bottom, and front to back.

It’s impossible to do this physically, of course. In six dimensions it’d be easy, but as we’re stuck in three dimensions we just have to pretend that the gluing has taken place without actually doing it, so that if we were standing in the cube and walked out through the left-hand face we would “wrap around” and reappear in the right-hand face, and similarly for front and back, and for top and bottom.

(Did you ever see that “Avengers” movie with Uma Thurman? Remember the scene where her character gets trapped in a room which seems to have two exits but every time she leaves through one she seems to reappear through the other? It’s like that. But I digress.)

If we can make that conceptual leap, then the result of all that gluing is a three-torus. There are annoying technical details to worry about concerning the vertices of the cube and what happens when we glue them together: while in the two-dimensional case the worst thing that could happen was that the vertices didn’t lay flat, in the three-dimensional case it’s possible that the vertices come together in such a way that we can’t even construct a chart around the resulting point, meaning that the resulting object isn’t a manifold at all!

Also, it’s the edges that might not lie flat in the three-dimensional case, not the vertices.

Did I mention that three-dimensional manifolds can be annoying?

Don’t worry about it. Instead, take my word for now that in the case of the three-torus the vertices behave, so we indeed have a three-manifold, and moreover the edges lay flat when we glue everything together, so the three-torus is a Euclidean manifold, i.e. it has a natural *Euclidean geometric structure* just like the two-dimensional torus does.

Recall that every two-dimensional manifold had some kind of geometric structure, either spherical, Euclidean, or hyperbolic. Is the same true in three dimensions? Sadly, no.

However, in 1982 William Thurston proposed Thurston’s Geometrization Conjecture, which says the next best thing. Oversimplifying hugely for a moment, Thurston’s Geometrization Conjecture says any three-manifold which doesn’t have a geometric structure can be cut into pieces in a canonical way, such that that the resulting pieces *do* have a geometric structure.

Thurston’s Geometrization Conjecture was proved quite recently by Grigori Perelman, who completed a program first proposed by Richard Hamilton to prove the conjecture. And since proving the Geometrization Conjecture also proves a long-standing open problem called the Poincaré Conjecture, which was one of the Clay Millennium Prize problems, Grigori Perelman became one of the first person to win one of those. William Thurston is also my academic grandfather: my Ph.D. advisor’s Ph.D. advisor. So you can see why I’m interested in this sort of thing.

It turns out that there are eight different kinds of geometrical structure you have to worry about in three-dimensional geometry, but three of them are very familiar: the three-dimensional analogues of spherical, Euclidean, and hyperbolic geometries. It also turns out that geometric three-manifolds (i.e. manifolds which have geometric structures even before you cut them into pieces) have all been classified, *except* for hyperbolic three-manifolds. That, in my opinion, is the last great frontier when it comes to classifying three-dimensional manifolds. And it’s a huge frontier. Next I’ll try to describe my tiny piece of it.

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[…] 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. […]