Topology and manifolds, part one

One additional caveat: this is not going to be as detailed as it could be. If you’re looking for a detailed technical online resource I figure Wikipedia, Wolfram MathWorld, and half a dozen other websites already have that covered. This is just me, writing my detailed answer to “So what do you do?” Onward.

What is a manifold? It’s something on which we can do a generalized form of geometry1.

“Geometry” is a good starting point. It’s something most people can visualize neatly: triangles, circles, and angles are familiar concepts. For many people it’s also their first and only exposure to proofs, the specialized, stilted form of language that is mathematicians’ bread and butter. Euclid of Alexandria is so famous for these kinds of formal arguments that the high school geometry of triangles, circles, and angles is properly known as Euclidean geometry.

The problem is, Euclidean geometry is a bit too abstract. It takes place on an infinite, flat plane, an ideal object bearing no resemblance to that thing we build skyscrapers, tennis courts, and particle accelerators on: the surface of the Earth, which is neither infinite nor flat. The surface of the Earth looks infinite…okay, really big…and it looks kinda-sorta flat depending on where you are, but that’s just because we’re really tiny and close to the ground. In reality, it’s a surface not at all like the ideal Euclidean plane. For that and other reasons, it’s necessary to generalize the mathematics of the Euclidean plane to other, more interesting shapes.

That’s where manifolds come in. A manifold is something which, like the surface of the Earth, looks like the Euclidean plane if you’re very small and standing right up next to it.

The technical definition of a manifold involves charts and atlases: a manifold is something which has an atlas made of charts, where each chart looks likes a portion of the Euclidean plane. The charts necessarily overlap, and together they map the entire manifold. On any one chart we can attempt to apply the mathematics of the Euclidean plane, but we can’t necessarily do that to the entire manifold, much like how we can lay flat any chart in an atlas of the Earth but can’t lay flat an entire globe.

The technical definition also has a very great deal to say about how those charts overlap. The short version is, how smoothly the charts overlap determines how much and what kind of mathematics you can do on the manifold. But let’s not go there.

I also should point out that what I’ve talking about so far are two-dimensional manifolds. We can define, say, three-dimensional manifolds by using “charts” which look like portions of Euclidean three-dimensional space. How such charts can come together to form an entire atlas is already much harder to visualize than in the two-dimensional case. It’s also a lot more interesting, in many ways. More on that later.

1And calculus. In fact calculus on manifolds is probably more important than geometry on manifolds. But the word “calculus” tends to frighten people.

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