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A bundle map from the base space itself (with the identity mapping as projection) to is called a section of . Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transition maps between the local trivial patches lie in a certain topological group, known as the structure group, acting on the fiber .
In differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle.
The Möbius strip can be constructed by a non-trivial gluing of two trivial bundles on open subsets U and V of the circle S 1.When glued trivially (with g UV =1) one obtains the trivial bundle, but with the non-trivial gluing of g UV =1 on one overlap and g UV =-1 on the second overlap, one obtains the non-trivial bundle E, the Möbius strip.
At each point in the fiber , the vertical fiber is unique. It is the tangent space to the fiber. The horizontal fiber is non-unique. It merely has to be transverse to the vertical fiber. In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle.
That surface has three I-bundles: the trivial bundle and two twisted bundles. Together with the Seifert fiber spaces, I-bundles are fundamental elementary building blocks for the description of three-dimensional spaces. These observations are simple well known facts on elementary 3-manifolds. Line bundles are both I-bundles and vector bundles ...
A mapping : between total spaces of two fibrations : and : with the same base space is a fibration homomorphism if the following diagram commutes: . The mapping is a fiber homotopy equivalence if in addition a fibration homomorphism : exists, such that the mappings and are homotopic, by fibration homomorphisms, to the identities and . [2]: 405-406
Bundles of fibers are known as nerve tracts, and consist of association tracts, commissural tracts, and projection tracts. The association fibers unite different parts of the same cerebral hemisphere , and are of two kinds: (1) short association fibers that connect adjacent gyri; (2) long association fibers that make connections between more ...
in vascular bundles, present in non-woody plants and non-woody parts of woody plants; in secondary xylem, laid down by a meristem called the vascular cambium in woody plants; as part of a stelar arrangement not divided into bundles, as in many ferns.