Meaning of fiber bundle | Babel Free

Definitions

1. Synonym of vascular bundle.
2. An abstract object in topology where copies of one object are "attached" to every point of another, as hairs or fibers are attached to a hairbrush. Formally, a topological space E (called the total space), together with a topological space B (called the base space), a topological space F (called the fiber), and surjective map π from E to B (called the projection or submersion), such that every point of B has a neighborhood U with π⁻¹(U) homeomorphic to the product space U × F (that is, E looks locally the same as the product space B × F, although its global structure may be quite different).
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Equivalents

Examples

“A Möbius strip⟳ is a fiber bundle which looks locally (i.e., over a connected proper subset of its base⟳ space) like⟳ the corresponding part of a cylinder S¹#92;times#91;0,1#93; (a Möbius strip⟳ and a cylinder have⟳ isomorphic base⟳ spaces). A Klein bottle is a fiber bundle which looks locally like⟳ the corresponding part of a torus S¹#92;timesS¹ (again they could be thought of as sharing the same base⟳ space S¹; cutting out even a single point⟳ of that base⟳ space makes the cut⟳ Klein bottle isomorphic to the cut⟳ torus).”

“In general, a fiber bundle consists of a set⟳ of mutually disjoint fibers “over” a base⟳ space, which indexes the fibers; there is a copy⟳ of some fiber on top of, or projecting (“canonically”) onto each point⟳ of the base⟳ space.”

“In the 1960s, some physicists including E. Lubkin and A. Trautman recognized that interaction potentials can be represented by connections on principal fiber bundles. In 1975, T. T. Wu and C. N. Yang used the fiber bundle method to solve⟳ a problem on magnetic monopoles.”

“One proposed general model⟳ for high-level scientific data uses fiber bundles, which Butler and Pendley [22] proposed in 1989.[…] A fiber bundle is the Cartesian product of the fibers and the base⟳ space; in other words, it is the collection of valid data ranges for the base⟳ space.”

“Finding the right notion of fiber bundle for diffeology [Igl85] has been a question⟳ raised by the study⟳ of the irrational torus T#95;#92;alpha [Dolg85].”

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