Suppose we’ve got a topological space. So far, in our discussion of topology, we’ve tended to focus either very narrowly on local properties of **T** (as in manifolds, where locally, the space appears euclidean), or on global properties of **T**. We haven’t done much to *connect* those two views. How do we get from local properties to global properties?
One of the tools for doing that is a sheaf (plural “sheaves”). A sheaf is a very general kind of structure that provides ways of mapping or relating local information about a topological space to global information about that space. There are many different kinds of sheaves; rather than being exhaustive, I’ll pretty much stick to a simple sheaf of functions on the topological space. Sheaves show up *all over* the place, in everything from abstract algebra to algebraic geometry to number theory to analysis to differential calculus – pretty much every major abstract area of mathematics uses sheaves.
In my last topology post, I started talking about the fundamental group of a topological space. What makes the fundamental group interesting is that it tells you interesting things about the structure
of the space in terms of paths that circle around and end where they started. For example, if you’re looking at a basic torus, you can go in loops staying in a euclidean-looking region; you can loop around the donut hole, or you can loop around the donut-body.
Of course, in the comments, an astute reader (John Armstrong) leapt ahead of me, and mentioned the fundamental group*oid* of a topological space, and its connection with category theory. That’s
supposed to be the topic of this post.
In algebraic topology, one of the most basic ideas is *the fundamental group* of a point in the space. The fundamental group tells you a lot about the basic structure or shape of the group in a reasonably simple way. The easiest way to understand the fundamental group is to think of it as the answer to the question: “What kinds of different ways can I circle around part of the space?”
I’m going to start moving the topology posts in the direction of algebraic topology, which is the part of topology that I’m most interested in. There’s lots more that can be said about homology, homotopy, manifolds, etc., and I may come back to it as some point, but for now, I feel like moving on.
There’s some fun stuff in algebraic topology which comes from the intersection between group theory
and topology. To be able to talk about that, you need the concept of a *topological group*.
First, I’ll run through a very quick review of groups. I wrote a series of posts on group theory for GM/BM when it was at blogger; if you’re interested in details, you might want to [pop over there, and take a skim.](http://goodmath.blogspot.com/2006/06/group-theory-index.html). There are also some excellent articles on group theory [at Wolfram’s mathworld](http://mathworld.wolfram.com/GroupTheory.html),
and [wikipedia](http://en.wikipedia.org/wiki/Group_theory). Then I’ll show you the beginnings of how group theory, abstract algebra, and topology can intersect.
I thought it would be fun to do a couple of strange shapes to show you the interesting things that you can do with a a bit of glue in topology. There are a couple of standard *strange* manifolds, and I’m going to walk through some simple gluing constructions of them.
After my [initial post about manifolds](http://scienceblogs.com/goodmath/2006/10/manifolds_and_glue.php), I wanted to say a bit more about gluing.
You can form manifolds by gluing manifolds with an arbitrarily small overlap – as little as a single point along the point of contact between the manifolds. The example that I showed, forming a spherical shell out of two circles, used a minimal overlap. If all you want to do is show that the topology you form is a manifold, that kind of trivial gluing is sufficient, and it’s often the easiest way to splice things together.
But there are a lot of applications of manifolds where you need more than that. So today, I’m going to show you how to do proper gluing in a way that preserves things like metric properties of manifolds when they’re glued together.
Back in the early days of Good Math/Bad Math, when it was still at blogger, one of the most widely linked posts was one about the idea of dimension. At the time, I said that the easiest way to describe a dimension was as a direction. If you’ve got a point in a plane, and you want to say where it is, you can do it with two numbers – one for each of the fundamental directions in the plane. If you’ve set an origin, “(5,-2)” is enough to uniquely identify exactly one point. You con reach any point on the plane by moving in two directions: up/down and left/right.
If you’ve got a cube, you can’t uniquely specify a point using its distance in two directions. Up three and left two doesn’t give you one point – there are lots of points that are up three and left two. You need a third direction, forward/back, for depth. That’s the third dimension – a direction that could not be formed by any combination of the two directions you had in the plane.
Topology has its own sense of dimension – in fact, it has several. They’re interesting because, as happens so often in topology, they start with the intuition that we get from simple metric spaces like ℜn, and work it down to its bare essentials by figuring out what it means when you apply it to an arbitrary topological space – that is, an arbitrary structure formed from open sets.
Time to get back to some topology, with the new computer. Short post this morning, but at least it’s something. (I had a few posts queued up, just needing diagrams, but they got burned with the old computer. I had my work stuff backed up, but I don’t let my personal stuff get into the company backups; I like to keep them clearly separated. And I didn’t run my backups the way I should have for a few weeks.)
Last time, I started to explain a bit of patchwork: building manifolds from other manifolds using *gluing*. I’ll have more to say about patchwork on manifolds, but first, I want to look at another way of building interesting manifolds.
At heart, I’m really an algebraist, and some of the really interesting manifolds can be defined algebraically in terms of topological product. You see, if you’ve got two manifolds **S** and **T**, then their product topology **S×T** is also a manifold. Since we already talked about topological product – both in classic topological terms, and in categorical terms, I’m not going to go back and repeat the definition. But I will just walk through a couple of examples of interesting manifolds that you can build using the product.
The easiest example is to just take some lines. Just a simple, basic line. That’s a 1 dimensional manifold. What’s the product of two lines? Hopefully, you can easily guess that: it’s a plane. The standard cartesian metric spaces are all topological products of sets of lines: ℜn is the product of *n* lines.
To be a bit more interesting, take a circle – the basic, simple circle on a cartesian plane. Not the *contents* of the circle, but the closed line of the circle itself. In topological terms, that’s a 1-sphere, and it’s also a very simple manifold with no boundary. Now take a line, which is also a simple manifold.
What happens when you take the product of the line and the circle? You get a hollow cylinder.
What about if you take the product of the circle with *itself*? Thing about the definition of product: from any point *p* in the product **S×T**, you should be able to *project* an image of
**S** and an image of **T**. What’s the shape where you can make that work right? The torus.
In fact the torus is a member of a family of topological spaces called the toroids. For any dimensionality *n*, there is an *n*-toroid which the the product of *n* circles. The 1-toroid is a circle; the 2-toroid is our familiar torus; the 3-toroid is a mess. (Beyond the 2-toroid, our ability to visualize them falls apart; what kind of figure can be *sliced* to produce a torus and a circle? The *concept* isn’t too difficult, but the *image* is almost impossible.)
So, after the last topology post, we know what a manifold is – it’s a structure where the neighborhoods of points are *locally* homeomorphic to open spheres in some ℜn.
We also talked a bit about the idea of *gluing*, which I’ll talk about
more today. Any manifold can be formed by *gluing together* subsets of ℜn. But what does *gluing together* mean?
Let’s start with a very common example. The surface of a sphere is a simple manifold. We can build it by gluing together *two* circles from ℜ2 (a plane). We can think of that as taking each circle, and stretching it over a bowl until it’s shaped like a hemisphere. Then we glue the two hemispheres together so that the *boundary* points of the hemispheres overlap.
Now, how can we say that formally?
So far, we’ve been talking about topologies in the most general sense: point-set topology. As we’ve seen, there are a lot of really fascinating things that you can do using just the bare structure of topologies as families of open sets.
But most of the things that are commonly associated with topology aren’t just abstract point-sets: they’re *shapes* and *surfaces* – in topological terms, they’re things called *manifolds*.
Informally, a manifold is a set of points forming a surface that *appears to be* euclidean if you look at small sections. Manifolds include euclidean surfaces – like the standard topology on a plane; but they also include many non-euclidean surfaces, like the surface of a sphere or a torus.
