What is a fibre bundle

What is a Fibre Bundle? A 5 Minute Introduction

Estimated Read Time: 2 minute(s)
Common Topics: space, fibre, bundle, product, called

Definition/Summary

Intuitively speaking, a fibre bundle is space E which ‘locally looks like’ a product space B×F, but globally may have a different topological structure.

Extended explanation

Definition:

A fibre bundle is the data group , where , and are topological spaces called the total space, the base space, and the fibre space, respectively and is a continuous surjection, called the projection, or submersion of the bundle, satisfying the local triviality condition.

(We assume the base space B to be connected.)

The local triviality condition states the following:

we require that for any that there exist an open neighborhood, of such that is homeomorphic to the product space in such a manner as to have carry over to the first factor space of the product.

is called the trivialization neighborhood, and the set of all is called to local trivialization of the bundle, where is a homeomorphism.

Visualization

The easiest way of visualizing a fibre bundle is one of the most ordinary household objects: the hairbrush.

In this case the base space is the cylinder, the fibre space are line fragments, and the projection : takes any point on a given fibre to the point where the fibre attaches to the cylinder.

In the trivial case is simply the product , and the map is just the projection from the product space to the first factor (B). This structure is called the trivial bundle.

Examples

Examples of non-trivial bundles are the Möbius strip and the Klein bottle.

In the case of the Möbius strip, the fibre bundle ‘locally looks like’ the flat Euclidean space R^2, however, the overall topology is markedly different.

A smooth fibre bundle is easily constructed with the above definition using smooth manifolds as , , and and the given functions are required to be smooth maps.

Generalization of fibre bundles may be given in a variety of ways. The most common is to require that the transition between the local trivial neighbourhoods conform to a certain topological group known as the structure group (or gauge group) acting on the fibre space .

See also: https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/ (esp. part 3)

Comments Thread

3 replies
  1. fresh_42 says:
    If you look at a small neighborhood of the Möbius strip, you will find a flat neighborhood with a one dimensional fiber at each point. This is the same as in the Euclidean plane with perpendicular one dimensional vector spaces attached at each point. However, if you consider the entire total space, then walking along a closed curve on the Möbius strip changes the direction (sign) of a vector in the fiber, whereas it does not on the Euclidean plane.

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