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fiber bundle
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Definition/Summary
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| 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. |
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Breakdown
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Physics
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Extended explanation
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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, satisying 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 neighbourhood, [itex]U[/itex] of [itex]\pi (x)[/itex] such that [itex]\pi[/itex]-1[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 neighbourhood, 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 hair brush (see the first figure).
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 budles are the Möbius strip (second image), and the Klein bottle (third image).
In the case of the Möbius strip, the fibre bundle 'locally looks like' the flat euclidean space R2, 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]. |
Commentary
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