fiber bundle
nounEtymology (mathematics): Coined as fibre bundle by American mathematician Norman Steenrod in 1951, The Topology of Fibre Bundles. The related usages fiber and fiber space probably derive (as calques respectively of German Faser and gefaserter Räume) from 1933, Herbert Seifert, “Topologie dreidimensionaler gefaserter Räume,” Acta Mathematica, 60, (1933), 147-238.
Category theory, Computing, Engineering, Mathematics, Natural sciences, Physical sciences, Sciences, Topology US 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).
- 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.
3 more examples
- 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.1995, Sunny Y. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, page 214:
- 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.2001, John M. May, Parallel I/O for High Performance Computing, Morgan Kaufmann Publishers, page 236:
- 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].2013, Patrick Iglesias-Zemmour, Diffeology, American Mathematical Society, page 229:
- Related terms
- associated bundle, base space, fiber space, fibration, principal bundle, structure group, total space, trivial bundle
- Hypernyms
- bundle
- Hyponyms
- vector bundle
- Meronyms
- base space, fiber, cross section