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This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles.
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.
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
Dense connective tissue also known as fibrous tissue [8] is subdivided into dense regular and dense irregular connective tissue. [9] Dense regular connective tissue, found in structures such as tendons and ligaments, is characterized by collagen fibers arranged in an orderly parallel fashion, giving it tensile strength in one direction. Dense ...
C fiber axons are grouped together into what is known as Remak bundles. [3] These occur when a non-myelinating Schwann cell bundles the axons close together by surrounding them. [4] The Schwann cell keeps them from touching each other by squeezing its cytoplasm between the axons. [4] The condition of Remak bundles varies with age. [4] The ...
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 ...
The protein composition of neurofilaments varies widely across different animal phyla. Most is known about mammalian neurofilaments. Historically, mammalian neurofilaments were originally thought to be composed of just three proteins called neurofilament protein NF-L (low molecular weight; NF-L), NF-M (medium molecular weight; NF-M) and NF-H (high molecular weight; NF-H).
Bundles can be composed of polar filament arrays, in which all barbed ends point to the same end of the bundle, or non-polar arrays, where the barbed ends point towards both ends. A class of actin-binding proteins , called cross-linking proteins, dictate the formation of these structures.