It's been a while since I've written a topology post. Rest assured - there's plenty more topology to come. For instance, today, I'm going to talk about something called a fiber bundle. I like to say that a fiber bundle is a cross between a product and a manifold. (There's a bit of a geeky pun in there, but it's too pathetic to explain.)
The idea of a fiber bundle is very similar to the idea of a manifold. Remember, a manifold is a topological space where every point is inside of a neighborhood that appears to be euclidean, but the space as a whole may be very non-euclidean. There are all sorts of interesting things that you can do in a manifold because of that property of being locally almost-euclidean - things like calculus.
A fiber bundle is based on a similar sort of idea: a local property that does not necessarily hold globally - but instead the local property being a property of individual points, it's based on a property of regions of the space.
So what is a fiber bundle, and why should we care? It's something that looks almost like a product of two topological spaces. The space can be divided into regions, each of which is a small piece of a product space - but the space as a whole may be twisted in all sorts of ways that would be impossible for a true product space.
For example, what's the difference between a cylinder and a mobius strip? They're both formed by taking a square, and joining opposite edges. But the cylinder is a product space - it's the product of a line and a circle; the mobius strip is not a product space, because it's got a twist in it. But in most respects, it behaves very much like a product space - in particular, in any small local region, it's indistinguishable from the cylinder. The mobius strip is a fiber bundle.
To get into the formal definition, when we define a fiber bundle, we talk about three different topological spaces, and a mapping function. The three spaces are usually called B (the base space), E (the total space, and F (the Fiber); and the mapping function is called π (the projection map). π : E → B is a continuous onto function mapping from the total space to the base space. Every small region of the total space E looks like a region of the product of the base space B and the fiber F. (The onto part means that for every point b ∈ B, there is at least one point e ∈ E where π(e)=b.)
As usual, there's a bit more to it. The projection map and the spaces need to satisfy some conditions that define what it means to have the property of locally looking like a product space. To understand this, it will help to remember what a product space looks like. A product space is a space formed from a pair (or a collection) of component spaces, with continuous projection maps from the product space to each of its components. We're doing the same sort of thing here with the fiber bundle - except that we're only going to look at one projection function, and we're going to let it be a bit funky.
Here's the tricky part. Suppose we have a point p in B. For the projection function to work property, there must be an open neighborhood N including p where π-1 is homeomorphic to N × F and the projection function from N × F to N correspond. That's actually easier to understand using a diagram in the category of topological spaces. (It's things like this that make me appreciate category theory; as a primarily visual thinker, I find it much easier to grasp the meaning of this constraint by thinking about the diagram.) In the following diagram, h is a homeomorphism from π-1(N) to N×F, and pN is the projection function from N×F to N. The requirement on the fiber bundle's projection function is that the following diagram commutes:
So now that we have both an intuitive and a formal sense of what a fiber bundle is, let's look at a couple of examples. I'm going to hit my favorite wierdos: the Mobius strip, and the Klein bottle.
We know that a cylinder is a product space - the product of a finite line segment and a circle. A Mobius strip is just a cylinder with a twist. So if we try to represent it as a fiber bundle, we'll find that it's a fiber bundle where the base space is a circle, and the fiber is a line segment. Looking at a Mobius strip, you can easily see the circular base; the base of a fiber bundle is generally pretty clear. The fiber is the part that can be twisted - and looking at a Mobius strip, the twisted fiber is clear. At any particular point, or any simple segment of the strip, it's obviously homoemorphic to a similar segment of a cylinder. So any neighborhood obviously looks like a cylindrical product space. But it's got that nasty twist in it, which means we can't build a proper projection function for the fiber, so it's not a product. But it does have most of the properties of a product, at least with respect to the circular base.
The Klein bottle is very similar to the Mobius strip, only it's a lot harder to visualize - which is pretty natural since a true Klein bottle can't exist in three dimensions A Klein bottle is a fiber bundle with a circular base space, and a circular fiber space - the twisted version of a Torus. Again, looking at a figure of a Klein bottle, you can (if you stretch a bit) see the circular base. And looking at it, you should also be able to see that any finite section of it is homeomorphic to a cylinder - just like a section of a torus. But when you look at the full closed loop of the bottle, it's got that bizarre twist in it - so again, because of the twist, it can't be a product space; but it's got a clean projection to the base, and a clear homeomorphism to the toroidal product space for neighborhoods. So it's a very nice fiber bundle.