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.