A fiber bundle (also called simply a bundle) with fiber is a continuous surjection where is called the total space of the fiber bundle and the base space of the fiber bundle. The main condition for the map to be a fiber bundle is that every point in the base space has an open neighborhood such that is homeomorphic to in a special way. Namely, if

is the homeomorphism, then

where the map means the projection morphism onto the component. The homeomorphisms which "commute with the projection morphism" are called local trivializations for the fiber bundle . In other words, looks like the Cartesian product (at least locally), except that the fibers for may be a bit "twisted."

A fiber bundle is the most general kind of bundle. Special cases are often described by replacing the word "fiber" with a word that describes the fiber being used, e.g., vector bundles and principal bundles.

Examples of fiber bundles include any Cartesian product (which is a bundle over with fiber ), the Möbius strip (which is a fiber bundle over the circle with fiber given by the unit interval ; i.e., the base space is the circle), and (which is a bundle over with fiber ). A special class of fiber bundle is the vector bundle, in which the fiber is a vector space.

Some of the properties of function graphs of functions carry over to fiber bundles. A function graph of such a function sits in as . A function graph always projects onto the base space and is one-to-one.

A fiber bundle is a total space and, like , it has a projection . The preimage, , of any point is homeomorphic to . Unlike , there is no canonical projection from to . Instead, maps to only make sense locally on . Near any point in the base space , there is a trivialization of in which there are actual functions from a neighborhood to .

These local functions can sometimes be patched together to give a (global) bundle section such that the projection composed with is the identity map. This is analogous to the map from a domain of a function to its function graph in by .

A choice of structure group for a fiber bundle consists of a topological group acting continuously on the model fiber , together with local trivializations whose coordinate changes have the form for continuous maps . This group action on need not give a global group action on the total space . In topological examples, acts by homeomorphisms of ; for a vector bundle, it acts by invertible linear maps. The specified group and continuous transition functions are additional structural data (Mitchell 2011, pp. 5-6).

Fiber bundles are not always used to generalize functions. Sometimes they are convenient descriptions of interesting manifolds. A common example in geometric topology is a torus bundle on the circle.

See also

Bundle, Fiber Space, Fibration, Principal Bundle, Sheaf, Tangent Bundle, Trivial Bundle, Vector Bundle

Portions of this entry contributed by Todd Rowland

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References

Mitchell, S. A. Notes on Principal Bundles and Classifying Spaces. Jun. 2011. https://sites.math.washington.edu/~mitchell/Quillen/prin.pdf.

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Fiber Bundle

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "Fiber Bundle." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FiberBundle.html

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