LogFAQs > #934962171

Bundles at a first approximation

Been talking a lot about bundles but haven't really explained what they really do. Before I get into that, we should get a trivial example in our minds to help motivate our intuition. There are a lot of types of bundles but one of the easier ones to get is the vector bundle.

Wikipedia definition is:

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.

Translation: We can create supercharged coordinate systems that are parameterized by a space.

A bundle comes in three parts. You have a base space which is just a blob in your mind...can be any shape. And at each point of that space you "attach" what is called a fiber. This can be a different type of space. The two of those combined create what is called the total space. And there is a MAP from the base space to the fiber

This map is not a bijection as mentioned previously in the manifold section. It's onto but NOT one-to-one because each point on the base space maps to every point on that fiber.

An example helps clarify.

Let's take a circle as our base space, and our fiber will be R1 (or the real number line) from [0,1]. So at each point on the circle (S1) you have a line sticking up and down and this goes all the way around the circle. What do you have?

A cylinder. Simple enough and this is generalizing the S1 x R1 product. However, we could take that cylinder, cut it and flip it so that there is a point that maps to -1 and maintain the maps continuity around the circle.

That completely changes it to a mobius strip. So the bundle allows us to have it locally be a cross product of two manifolds even though the total space is not.

We'll extend this definition a bit later when we start attaching lie groups as our fibers to give us frame bundles.