Fiber bundle

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 : EB

such that E looks locally like the product space

B × F

for some space F (called the fiber space). (Locally here means locally on B.) A bundle that can be so expressed globally (i.e. by a homeomorphism respecting π) is called a trivial bundle. The theory of bundles was set up to express what it means for a bundle to be non-trivial, in ways that are easier to handle than the direct definition.

Fiber bundles generalize vector bundles of which the main example is the tangent bundle of a manifold. 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.

 Contents

Formal definition

A fiber bundle consists of the data (E, B, π, F), where E, B, and F are topological spaces and π : EB 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:

Missing image
FiberBundle-01.png
Local triviality condition

where proj1 : U × FU 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

Missing image
FiberBundle-02.png
Image:FiberBundle-02.png

to indicate a short exact sequence of spaces. Note that every fiber bundle π : EB 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.

Examples

Let E = B × F and let π : EB 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.

Missing image
MobiusStrip-01.png
The Möbius strip is a nontrivial bundle over the circle.

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 [itex]x \in B[itex] is an arc; in the picture, this is the length of one of the squares. The preimage [itex]\pi^{-1}(U)[itex] 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.

Another special class of fiber bundles are called principal bundles. See that article for more examples.

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.

Sections

A section (or cross section) of a fiber bundle is a continuous map f : BE 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 : UE 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 UF. Sections form a sheaf.

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

[itex]\phi_i\phi_j^{-1} : (U_i \cap U_j) \times F \to (U_i \cap U_j) \times F[itex]

is given by

[itex]\phi_i\phi_j^{-1}(x, \xi) = (x, t_{ij}(x)\xi)[itex]

where [itex]t_{ij} : U_i \cap U_j \to G[itex] 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. [itex]t_{ii}(x) = 1[itex]
2. [itex]t_{ij}(x) = t_{ji}(x)^{-1}[itex]
3. [itex]t_{ik}(x) = t_{ij}(x)t_{jk}(x)[itex]

The third condition applies on triple overlaps [itex]U_i \cap U_j \cap U_k[itex] 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.

References

• Norman Steenrod, The Topology of Fiber Bundles, Princeton University Press (1951). ISBN 0-691-00548-6.
• David Bleecker, Gauge Theory and Variational Principles, Addison-Wesley publishing, Reading, Mass (1981). ISBN 0-201-10096-7. See chapter one.es:Fibrado

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