Topology and manifolds, part two

Continuing my detailed answer to the question, “So what do you do?”…

When a mathematician sees the definition of a mathematical object, they want to know what all the possible examples of that object are. It’s a pretty hard-wired reflex. So what are all the possible examples of manifolds?

Well, that’s a rather huge question. One-dimensional manifolds are easy: the circle and the real line. That’s it. Two-dimensional manifolds, i.e. surfaces, are harder, but the question has largely been answered. If we restrict ourselves to compact two-dimensional manifolds, then the question was answered at the beginning of the twentieth century, although the i’s weren’t dotted and the t’s weren’t crossed until the 1920’s.

I haven’t explained what compact means yet. Compact is, loosely speaking, a geometrical notion of finiteness. Of the two one-dimensional manifolds, the circle is compact, but the real line is not, because of the way it extends to infinity. Compact manifolds are an important subclass of manifolds, if only because they’re easier to study.

The list of compact two-dimensional manifolds starts with the sphere, e.g. the surface of the earth. Next is the torus, e.g. the surface of a doughnut. After that, picture the surface of a doughnut with two holes. Mathematicians usually just call this a “two-holed torus” or a “genus-two surface” if they’re getting technical. (I once misread an old topology book which used the term “quoit” and thought the word referred to something with the shape of a genus-two surface, but it turns out a “quoit” is just a ring. Pity.) Then there’s a three-holed torus, a four-holed torus, and so on.

That’s not the complete list. Have you ever seen a Klein bottle? Wonderful stuff. A Klein bottle is an example of a non-orientable surface, a class of two-dimensional manifolds that can’t actually be constructed in three-dimensional space. (The examples at the link cheat a bit by letting the surface pass through itself.) There’s a list of non-orientable two-dimensional manifolds that parallels the list of many-holed tori, of which the Klein bottle is the second on the list. The first is the projective plane, which is a Moebius band glued to a disk in a wonderfully mind-bending way which I won’t go into here.

So: sphere, torus, two-holed torus, and so on, plus projective plane, Klein bottle, and so on. Two simple, if infinite, lists containing every possible compact two-dimensional manifold. What’s more, if you present a mathematician with some arbitrary compact two-dimensional manifold, it takes only a few very simple calculations before the mathematician can tell you where in the two lists the manifold has to appear. It’s the best possible outcome to this kind of mathematical problem.

What about three-dimensional manifolds?

Oy. You have no idea.

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2 comments so far

  1. Daniel Trebbien on

    Peter, I just came from Stack Overflow to read your blog. It was highly enjoyable reading your stuff and I look forward to reading more.

    Cheers,
    Daniel


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