# 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 :
*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. 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 : *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 proj_{1} : *U* × *F* → *U* is the natural projection and φ : π^{−1}(*U*) → *U* × *F* is a homeomorphism. The set of all {(*U*_{i}, φ_{i})} 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

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.

## 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**.

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 **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in B}**
is an arc; in the picture, this is the length of one of the squares. The preimage **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi^{-1}(U)}**
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, *S*^{1} × *S*^{1}.

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 *E*_{x}.

## 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.

## 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 (

*U*

_{i}, φ

_{i}) and (

*U*

_{j}, φ

_{j}) the function

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_i\phi_j^{-1} : (U_i \cap U_j) \times F \to (U_i \cap U_j) \times F}**

is given by

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \phi_i\phi_j^{-1}(x, \xi) = (x, t_{ij}(x)\xi)}**

where **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{ij} : U_i \cap U_j \to G}**
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 *t*_{ij} satisfy the following conditions

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{ii}(x) = 1}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{ij}(x) = t_{ji}(x)^{-1}}****Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{ik}(x) = t_{ij}(x)t_{jk}(x)}**

The third condition applies on triple overlaps **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U_i \cap U_j \cap U_k}**
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.

## See also

## External links

## 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.