Archive for June, 2010|Monthly archive page

Frightening statistic of my life

Not topology related today…

We’re moving from Australia back to Canada shortly. Our cat, who we brought with us to Australia four years ago, is already back in Canada; we shipped her ahead. According to the Great Circle Mapper, our cat has flown 32,833 kilometres in her lifetime. The circumference of the earth is only 40,075 kilometres. Our cat…our cat…has effectively flown 80% of the way around the planet.

I shudder to think how many times I’ve effectively circumnavigated the globe.

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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.

Topology and manifolds, part three

Continuing!

So, what do spherical, Euclidean, and hyperbolic geometry have to do with manifolds? Well, it turns out that all two-dimensional manifolds have a natural geometric structure.

That statement will require some unpacking.

Let’s start with the torus, i.e. the surface of a donut, which is commonly described as follows: take a rectangle (imagine it to be made of some thin, flexible material) and then glue two opposite edges together to form a tube. Then bend the tube, and glue the ends of the tube together to get a torus. This is an example of a Euclidean geometric structure on a torus, which is a fancy way of saying that we’ve constructed a torus out of pieces of the Euclidean plane…namely, the rectangle…in such a way that all of the edges and corners glue together neatly.

This “gluing together neatly” part is important. Specifically, it’s very important that the corners come together neatly. In the case of the torus above, the rectangle had four corners, each of which was a right angle, that is, π/2 radians or 90 degrees. Those four corners all come together to form a single point after all of that gluing, and when they do those four angles add up to 2π radians or 360 degrees, i.e. a full circle. That’s exactly what we want to happen: the single “corner” of the resulting torus lies flat.

Let’s take another example. Instead of a single rectangle, let’s take six squares and glue them together to form the surface of a cube. Topologically, a cube is the same as a sphere, but geometrically we have a problem: the corners don’t lie flat. At each corner of the cube there are three right angles coming together, and that only adds up to 3π/2 radians, less than a full circle. Gluing six squares together in this way does not result in a Euclidean geometric structure on a sphere.

But not surprisingly, spherical geometry steps in here. In spherical geometry, the angles of every triangle, and indeed every polygon, add up to more than they would in Euclidean geometry. It’s possible to construct a square in spherical geometry whose angles are each 2π/3 radians, or 120 degrees…all four of them. If you take six of those squares and glue them together, you’ll get a spherical “cube” where the corners do lie flat: each corner is made up of three angles of 2π/3 radians, which adds up to a full circle. A sphere has, not surprisingly, a spherical geometric structure: we can construct it neatly out of pieces of the sphere.

One last example. This is the fun one, but you might want to draw some pictures as you go along.

Imagine two pairs of pants. Stitch all the holes together: waist to waist, leg to leg, and other leg to other leg. What you’ll end up with is a two-holed torus. Now, cut the two-holed torus back into two pairs of pants. Then, cut each pair of pants into two pieces with three cuts, as follows: from the left edge of the waist down the outside of the left leg to the bottom, from the right edge of the waist down the outside of the right leg to the bottom, and finally along the inside of both legs, from the left hem up to the crotch and down to the right hem. (Basically, cut the pants into the front half and the back half.) You’ve now cut your two-holed torus into four pieces in total. Each piece is a hexagon, made of three cuts and three half-hems (if you consider the waist to be a hem). When you stitch the hexagons back together to make the two-holed torus, the corners of the hexagons will come together in sets of four. If we want those corners to lie flat, the simplest way would be to ensure that each corner of each hexagon is a right angle; four of those will add up to a full circle, just like in the torus example at the beginning.

Hexagons with six right angles? Try drawing that. It’s just crazy talk…in Euclidean geometry. But we can construct right-angled hexagons in hyperbolic geometry. Thus, the two-holed torus has a hyperbolic geometric structure.

Whew! Let’s cut to the chase: every two-dimensional manifold has some kind of geometric structure, either spherical, Euclidean, or hyperbolic. Talking about the compact manifolds I described in part two, the sphere and the projective plane have spherical geometric structures, the torus and the Klein bottle have Euclidean geometric structures, and everything else has a hyperbolic geometric structure.

But I said in the interlude that all of this geometry had something to do with three-dimensional manifolds, didn’t I? Yes, yes I did…