Fiber bundle

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In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. Every fiber bundle consists of a continuous surjective map

π : E → B

where small regions in the total space E look like small regions in the product space

B × F.

Here B is the base space while F is the fiber space. For example, the product space B × F, equipped with π equal to projection onto the first coordinate, is a fiber bundle. This is called the trivial bundle. One goal of the theory of bundles is to quantify, via algebraic invariants, what it means for a bundle to be non-trivial.

Fiber bundles generalize vector bundles, where the main example is the tangent bundle of a manifold, as well as principal bundles. They play an important role in the fields of differential topology and differential geometry. They are also a fundamental concept in the mathematical formulation of gauge theory. Fiber bundles specialize the more general bundle.

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Formal definition

A fiber bundle consists of the data (E, B, π, F), where E, B, and F are topological spaces and π : E → B is a continuous surjection satisfying a local triviality condition outlined below. B is called the base space of the bundle, E the total space, and F the fiber. The map π is called the projection map. We shall assume in what follows that the base space B is connected.

We require that for any x in B, there is an open neighborhood U of x such that π−1(U) is homeomorphic to the product space U × F, in such a way that π carries over to the projection onto the first factor. That is, the following diagram should commute:

where proj1 : U × F → U is the natural projection and φ : π−1(U) → U × F is a homeomorphism. The set of all {(Ui, φi)} is called a local trivialization of the bundle.

For any x in B, the preimage π−1(x) is homeomorphic to F and is called the fiber over x. A fiber bundle (E, B, π, F) is often denoted

Image:FiberBundle-02.png

to indicate a short exact sequence of spaces. Note that every fiber bundle π : E → B is an open map, since projections of products are open maps. Therefore B carries the quotient topology determined by the map π.

A smooth fiber bundle is a fiber bundle in the category of smooth manifolds. That is, E, B, and F are required to be smooth manifolds and all the functions above are required to be smooth maps. This is the most common context in which fiber bundles are studied and used.

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Examples

Let E = B × F and let π : E → B be the projection onto the first factor. Then E is a fiber bundle over B. Here E is not just locally a product but globally one. Any such fiber bundle is called a trivial bundle.

Image:MobiusStrip-01.png Perhaps the simplest example of a nontrivial bundle E is the Möbius strip. The Möbius strip has a circle for a base B and a line segment for the fiber F. A neighborhood U of a point <math>x \in B</math> is an arc; in the picture, this is the length of one of the squares. The preimage <math>\pi^{-1}(U)</math> in the picture is a (somewhat twisted) slice of the strip four squares wide and one long. The homeomorphism φ maps the preimage of U to a slice of a cylinder: curved, but not twisted.

The corresponding trivial bundle B × F would look like a cylinder, but the Möbius strip has an overall "twist". Note that this twist is visible only globally; locally the Möbius strip and the cylinder are identical (making a single vertical cut in either gives the same space).

A similar nontrivial bundle is the Klein bottle which can be viewed as a "twisted" circle bundle over another circle. The corresponding trivial bundle would be a torus, S1 × S1.

A covering space is a fiber bundle whose fiber is a discrete space.

A special class of fiber bundles, called vector bundles, are those whose fibers are vector spaces (to qualify as a vector bundle the structure group of the bundle — see below — must be a linear group). Important examples of vector bundles include the tangent bundle and cotangent bundle of a smooth manifold. From any vector bundle, one can construct the frame bundle of bases which is a principle bundle.

Another special class of fiber bundles, called principal bundles, are bundles on whose fibers a free and transitive group action by G is given, so that each fiber is a principal homogeneous space. The bundle is often specified along with the group by referring to it as a principle G-bundle. The group G is also the structure group of the bundle. Given a representation ρ of G on a vector space V, a vector bundle with ρ(G)⊆Aut(V) as a structure group may be constructed, known as the associated bundle.

A sphere bundle is a fiber bundle whose fiber is an n-sphere. Given a vector bundle E with a metric (such as the tangent bundle to a Riemannian manifold) one can construct the associated unit sphere bundle, for which the fiber over a point x is the set of all unit vectors in Ex.

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Sections

A section (or cross section) of a fiber bundle is a continuous map f : B → E such that π(f(x))=x for all x in B. Since bundles do not in general have globally-defined sections, one of the purposes of the theory is to account for their existence. This leads to the theory of characteristic classes in algebraic topology.

Often one would like to define sections only locally (especially when global sections do not exist). A local section of a fiber bundle is a continuous map f : U → E where U is an open set in B and π(f(x))=x for all x in U. If (U, φ) is a local trivialization chart then local sections always exist over U. Such sections are in 1-1 correspondence with continuous maps U → F. Sections form a sheaf.

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Structure groups and transition functions

Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group which acts continuously on the fiber space F on the left. We lose nothing if we require G to act effectively on F so that it may be thought of as a group of homeomorphisms of F. A G-atlas for the bundle (E, B, π, F) is a local trivialization such that for any two overlapping charts (Ui, φi) and (Uj, φj) the function

<math>\phi_i\phi_j^{-1} : (U_i \cap U_j) \times F \to (U_i \cap U_j) \times F</math>

is given by

<math>\phi_i\phi_j^{-1}(x, \xi) = (x, t_{ij}(x)\xi)</math>

where <math>t_{ij} : U_i \cap U_j \to G</math> is a continuous map called a transition function. Two G-atlases are equivalent if their union is also a G-atlas. A G-bundle is a fiber bundle with an equivalence class of G-atlases. The group G is called the structure group of the bundle.

In the smooth category, a G-bundle is a smooth fiber bundle where G is a Lie group and the corresponding action on F is smooth and the transition functions are all smooth maps.

The transition functions tij satisfy the following conditions

  1. <math>t_{ii}(x) = 1</math>
  2. <math>t_{ij}(x) = t_{ji}(x)^{-1}</math>
  3. <math>t_{ik}(x) = t_{ij}(x)t_{jk}(x)</math>

The third condition applies on triple overlaps <math>U_i \cap U_j \cap U_k</math> and is called the cocycle condition (see Čech cohomology).

A principal G-bundle is G-bundle where the fiber can be identified with G itself and where there is a right action of G on the total space which is fiber preserving. Questions about a bundle can often be turned into questions about the reduction of the structure group, or the G-structure of a manifold.

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See also

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References

es:Fibrado ja:ファイバー束 zh:纤维丛

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