What is a fibre bundle

What is a Fibre Bundle? A 5 Minute Introduction

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 [itex](E, B,\pi, F)[/itex], where [itex]E, B[/itex], and [itex]F[/itex] are topological spaces called the total space, the base space, and the fibre space, respectively and [itex]\pi : E \rightarrow B[/itex] 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 [itex]x \in E[/itex] that there exist an open neighborhood, [itex]U[/itex] of [itex]\pi (x)[/itex] such that [itex]\pi^-1[/itex][itex](x)[/itex] is homeomorphic to the product space [itex]U×F[/itex] in such a manner as to have [itex]\pi[/itex] carry over to the first factor space of the product.

[itex]U[/itex] is called the trivialization neighborhood, and the set of all [itex]\{U_i, \varphi_i\}[/itex] is called to local trivialization of the bundle, where [itex]\varphi_i:\pi^{-1}\rightarrow U×F[/itex] 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 [itex]B[/itex]is the cylinder, the fibre space [itex]F[/itex] are line fragments, and the projection [itex]\pi[/itex]: [itex]E[/itex] [itex]\rightarrow [/itex][itex]B[/itex] takes any point on a given fibre to the point where the fibre attaches to the cylinder.

In the trivial case [itex]E[/itex] is simply the product [itex]B×F[/itex], and the map [itex]\pi[/itex] 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 [itex]B[/itex], [itex]F[/itex], and [itex]E[/itex] 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 [itex]G[/itex]topological group known as the structure group (or gauge group) acting on the fibre space [itex]F[/itex].

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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