What is a bundle of fibers

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(Forwarded by Basis_ (fiber bundle))

A Fiber bundle is a space in mathematics, especially in topology, that looks locally like a product: fiber bundles can be understood as continuous urjective mappings \ ({\ displaystyle \ pi \ colon E \ to B} \) with a local triviality condition: For each point of \ ({\ displaystyle B} \) there is an environment \ ({\ displaystyle U} \) for which the restriction \ ({\ displaystyle \ pi | _ {\ pi ^ {- 1} (U)}} \) Projection of a product \ ({\ displaystyle U \ times F} \) with a fiber \ ({\ displaystyle F} \).

\ ({\ displaystyle E, B} \) and \ ({\ displaystyle F} \) are called Total space, Base and fiber of the bundle. There are various generalizing terms of the term of the fiber bundle of fibers, in particular Serre and Hurewicz fibers.

Special fiber bundles are z. B. vector bundles, which play a fundamental role in the mathematical formulation of physical gauge theory.

Table of Contents

definition


A fiber bundle is specified by the data \ ({\ displaystyle (E, B, \ pi, F)} \), where \ ({\ displaystyle E} \), \ ({\ displaystyle B} \) and \ ({ \ displaystyle F} \) are topological spaces and the projection \ ({\ displaystyle \ pi \ colon E \ to B} \) is a continuous surjective mapping, which is locally trivializable, i.e. H. for every point \ ({\ displaystyle x} \) in \ ({\ displaystyle B} \) there is an open environment \ ({\ displaystyle U} \), so that \ ({\ displaystyle \ pi ^ {- 1 } (U)} \) is homeomorphic to the space \ ({\ displaystyle U \ times F} \), provided with the product topology, and the following diagram commutes:

where \ ({\ displaystyle \ operatorname {proj} _ {1} \ colon U \ times F \ to U} \) is the natural projection onto the first factor and \ ({\ displaystyle \ phi \ colon \ pi ^ {- 1 } (U) \ to U \ times F} \) is a homeomorphism. The set of all such \ ({\ displaystyle \ {(U_ {i}, \ phi _ {i}) \}} \) is called local trivialization of the bundle, which cover \ ({\ displaystyle U_ {i}} \) by definition the base.

For each \ ({\ displaystyle x} \) from \ ({\ displaystyle B} \) the archetype \ ({\ displaystyle \ pi ^ {- 1} (x)} \) is homeomorphic to \ ({\ displaystyle F } \) and is called fiber over \ ({\ displaystyle x} \). A fiber bundle \ ({\ displaystyle (E, B, \ pi, F)} \) is often represented by the short exact sequence \ ({\ displaystyle F \ longrightarrow E \ {\ xrightarrow {\, \ \ pi \}} \ B} \). Note that every fiber bundle \ ({\ displaystyle \ pi \ colon E \ to B} \) is an open map, since projections of products are open mappings. Therefore \ ({\ displaystyle B} \) carries the quotient topology induced by the mapping \ ({\ displaystyle \ pi} \).

A fiber bundle is a special case of Serre fiber, that is, it has the so-called Homotopy elevation property for images of CW complexes.

Examples


Let \ ({\ displaystyle E = B \ times F} \) and \ ({\ displaystyle \ pi \ colon E \ to B} \) be the projection onto the first factor. Then \ ({\ displaystyle E} \) is a fiber bundle over \ ({\ displaystyle B} \) with fiber \ ({\ displaystyle F} \). In this case \ ({\ displaystyle E} \) is not only locally a product space, but even global. Such a fiber bundle is called trivial bundle.

A simple example of a nontrivial bundle is the Möbius strip. The base \ ({\ displaystyle B} \) is here \ ({\ displaystyle S ^ {1}} \) (the circular line), the fiber \ ({\ displaystyle F} \) a closed interval. The corresponding trivial bundle would be a cylinder from which the Möbius strip extends through a Twisting the fiber is different. This rotation is only visible globally, the cylinder and Möbius strip are locally identical.

A similar nontrivial bundle is the Klein bottle, which is a \ ({\ displaystyle S ^ {1}} \) fiber over \ ({\ displaystyle S ^ {1}} \). The corresponding trivial bundle would be a torus. Every fiber bundle over \ ({\ displaystyle S ^ {1}} \) is a mapping torus.

Each overlay of a contiguous space is a fiber bundle with a discrete fiber.

A special class of fiber bundles, the vector bundles, are characterized by the fact that their fibers are vector spaces and the trivializations are linear fiber by fiber. Important examples here are the tangential and cotangential bundles of a manifold.

Another special class of fiber bundles are the principal or main fiber bundles.

Cuts


Main article: Cut (bundle of fibers)

Under a global cut one understands a continuous mapping \ ({\ displaystyle f \ colon B \ to E} \), so that \ ({\ displaystyle \ pi (f (x)) = x} \) for all \ ({\ displaystyle x} \) from \ ({\ displaystyle B} \). The theory of characteristic classes in algebraic topology deals with the existence of global intersections.

Often you can only define sections locally. A local cut is a continuous mapping \ ({\ displaystyle f \ colon U \ to E} \), where \ ({\ displaystyle U} \) is an open set in \ ({\ displaystyle B} \) and \ ( {\ displaystyle \ pi (f (x)) = x} \) for all \ ({\ displaystyle x} \) from \ ({\ displaystyle U} \). For a local trivialization \ ({\ displaystyle (U, \ phi)} \) this is always possible. These cuts are equivalent to continuous mappings \ ({\ displaystyle U \ to F} \), which form a sheaf.

Structural groups


With the exception of topological equivalence, fiber bundles are characterized by "atlases", which indicate how their local trivializations are "glued together": Let \ ({\ displaystyle G} \) be a topological group, which by means of an effective effect on the fiber \ ({\ displaystyle F} \) acts from the left. An \ ({\ displaystyle G} \) atlas of the bundle \ ({\ displaystyle (E, B, \ pi, F)} \) consists of local trivializations, so that for every two overlapping maps \ ({\ displaystyle ( U_ {i}, \ phi _ {i})} \) and \ ({\ displaystyle (U_ {j}, \ phi _ {j})} \) the endomorphism

\ ({\ displaystyle \ phi _ {i} \ phi _ {j} ^ {- 1} \ colon (U_ {i} \ cap U_ {j}) \ times F \ to (U_ {i} \ cap U_ { j}) \ times F} \)

by

\ ({\ displaystyle \ phi _ {i} \ phi _ {j} ^ {- 1} (x, \ xi) = (x, t_ {ij} (x) \ xi)} \)

is given, where \ ({\ displaystyle t_ {ij} \ colon U_ {i} \ cap U_ {j} \ to G} \) is a continuous mapping. Two \ ({\ displaystyle G} \) - atlases are equivalent if their union is also a \ ({\ displaystyle G} \) - atlas. A \ ({\ displaystyle G} \) - bundle is a fiber bundle together with an equivalence class of \ ({\ displaystyle G} \) - atlases. The group \ ({\ displaystyle G} \) is called the Structure group of the bundle. Each fiber bundle can be described by an \ ({\ displaystyle G} \) - atlas if we choose the automorphism group of the fiber as the structural group; if we choose \ ({\ displaystyle G} \) smaller, the fiber bundle gains additional structure.

Because the Card change \ ({\ displaystyle t_ {ij}} \) describe the transition between local trivializations, they suffice Cozykel condition \ ({\ displaystyle t_ {ik} (x) = t_ {ij} (x) t_ {jk} (x)} \) (see also Čech cohomology); in particular, \ ({\ displaystyle t_ {ii} (x) = \ operatorname {id} _ {U_ {i}}} \) and \ ({\ displaystyle t_ {ij} (x) = t_ {ji} (x ) ^ {- 1}} \).

A principal bundle is a \ ({\ displaystyle G} \) - bundle in which the fiber is identified with \ ({\ displaystyle G} \) and on which a fiber-preserving right - \ ({\ displaystyle G} \) - Effect on total space is explained.

literature


  • Norman Steenrod: The Topology of Fiber Bundles. Princeton University Press, Princeton NJ 1951 (Princeton Mathematical Series 14, ISSN 0079-5194), (7. printing, and 1. paperback printing. Ibid 1999, ISBN 0-691-00548-6 (Princeton Landmarks in Mathematics and Physics. = Princeton Paperbacks)).
  • David Bleecker: Gauge Theory and Variational Principles. Addison-Wesley publishing, Reading MA 1981, ISBN 0-201-10096-7, chapter 1.
  • Martin Schottenloher: Geometry and Symmetry in Physics. The leitmotif of mathematical physics. Vieweg, Braunschweig et al. 1995, ISBN 3-528-06565-6 (Vieweg textbook Mathematical Physics).

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Categories:Algebraic topology | Differential topology




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