Fiber bundle

A fiber bundle is in mathematics, specifically in topology, a space that looks locally like a product: fiber bundle can be conceived as a continuous surjective pictures with a local Trivialitätsbedingung: For each point of B there is a neighborhood U for which the restriction projecting a product with a fiber F is.

Special fiber bundles such as vector bundles, which play a fundamental role in the mathematical formulation of the physical gauge theory.

Definition

A fiber bundle is specified by the data ( E, B, π, F), where E, ​​B and F are topological space and the projection π: E → B is a constant onto mapping which is locally trivialisierbar, i.e. exists for each point x in B has an open neighborhood U such that π -1 ( U) is homeomorphic to the space U × F, equipped with the product topology, is so the following diagram commutes:

Where proj1: U × F → U is the natural projection onto the first factor and Φ: π -1 ( U) → U × F is a homeomorphism. The set of all { (Ui, ? I ) } is called a local trivialization of the bundle.

For each x in B, the inverse image π -1 (x) is homeomorphic to F and is called the fiber over x. A fiber bundle ( E, B, π, F ) is often represented by the short exact sequence. It should be noted that each fiber bundle π: E → B is an open figure, since projections of products are open figures. Therefore, B transmits the π induced by the mapping quotient topology.

A fiber bundle is a special case of a Serre fibration, that is, it has the so-called homotopy high elevation property for pictures of CW - complexes.

Examples

Let E = B × F and π: E → B is the projection onto the first factor. Then E is a fiber bundle over B with fiber F. In this case E is not locally a product space, but even globally. Such a fiber bundle is called trivial bundle.

A simple example of a nontrivial bundle is the Möbius strip. The base B (the circle ), the fiber F is a closed interval here. The corresponding trivial bundle would be a cylinder from which the Möbius band differs by a rotation of the fiber. This rotation is only visible globally, locally cylinder and Möbius strip are identical.

A similar non-trivial bundle is the Klein bottle, which is a fibration over. The corresponding trivial bundle would be a torus. Each fiber bundle over a Abbildungstorus.

Each superposition of a coherent space is a fiber bundle with a discrete fiber.

A special class of fiber bundles, vector bundles are distinguished by the fact that its fibers are vector spaces and the trivialization fiber linear. Important examples are the tangential and cotangent bundle of a manifold.

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

Slice

→ Main article: Section ( fiber bundle)

Under a global section is defined as a continuous map such that for all x in B. The theory of characteristic classes in algebraic topology is concerned with the existence of global sections.

Often you can define sections only locally. A local interface is a continuous map, where U is an open set in B and for all x in U. For a local trivialization (U, φ ), this is always possible. These cuts are equivalent to continuous maps which form a sheaf.

Structure groups

Fiber bundles are up to topological equivalence characterized by " atlases " that specify how their local trivialization "glued " are: Let G be a topological group which acts by means of an effective action on the fiber F from the left. A G- atlas of the bundle (E, B, π, F) consists of local trivialization, so that for any two overlapping maps (Ui, ? I ) and ( Uj, φj ) of the endomorphism

By

Is given, which is a continuous map. Two G- atlases are equivalent if their union is also a G- atlas. A G- bundle is a fiber bundle together with an equivalence class of G- atlases. The group G is called the structure group of the bundle. Each fiber bundle can be described by a G- atlas, if we choose the automorphism group of the fiber as a structural group; we choose G smaller, so the fiber bundle gains additional structure.

Because the maps change tij the transition between local trivialization describe, they satisfy the Cozykel condition (see also Čech cohomology ); particular and follow.

A principal bundle is a G - bundle, where the fiber is identified with G and is explained on a fiber -preserving right -G - action on the total space.