# Fiber bundle

In topology, the terms * fiber* (German:

*Faser*) and

*(*

**fiber space***gefaserter Raum*) appeared for the first time in a paper by Herbert Seifert in 1933,

^{[1]}

^{[2]}but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the

**base space**(topological space) of a fiber (topological) space

*E*was not part of the structure, but derived from it as a quotient space of

*E*. The first definition of

**fiber space**was given by Hassler Whitney in 1935

^{[3]}under the name

**sphere space**, but in 1940 Whitney changed the name to

**sphere bundle**.

^{[4]}

The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Seifert, Heinz Hopf, Jacques Feldbau,^{[5]} Whitney, Norman Steenrod, Charles Ehresmann,^{[6]}^{[7]}^{[8]} Jean-Pierre Serre,^{[9]} and others.

Fiber bundles became their own object of study in the period 1935–1940. The first general definition appeared in the works of Whitney.^{[10]}

Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle,^{[11]} that is a fiber bundle whose fiber is a sphere of arbitrary dimension.^{[12]}

that, in analogy with a short exact sequence, indicates which space is the fiber, total space and base space, as well as the map from total to base space.

A **covering space** is a fiber bundle such that the bundle projection is a local homeomorphism. It follows that the 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 principal bundle (see below).

The most well-known example is the hairy ball theorem, where the Euler class is the obstruction to the tangent bundle of the 2-sphere having a nowhere vanishing section.

where *t*_{ij} : *U*_{i} ∩ *U*_{j} → *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; the analogous term in physics is gauge group.

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 third condition applies on triple overlaps *U _{i}* ∩

*U*∩

_{j}*U*and is called the

_{k}**cocycle condition**(see Čech cohomology). The importance of this is that the transition functions determine the fiber bundle (if one assumes the Čech cocycle condition).

A principal *G*-bundle is a *G*-bundle where the fiber *F* is a principal homogeneous space for the left action of *G* itself (equivalently, one can specify that the action of *G* on the fiber *F* is free and transitive, i.e. regular). In this case, it is often a matter of convenience to identify *F* with *G* and so obtain a (right) action of *G* on the principal bundle.

In the category of differentiable manifolds, fiber bundles arise naturally as submersions of one manifold to another. Not every (differentiable) submersion ƒ : *M* → *N* from a differentiable manifold *M* to another differentiable manifold *N* gives rise to a differentiable fiber bundle. For one thing, the map must be surjective, and (*M*, *N*, ƒ) is called a fibered manifold. However, this necessary condition is not quite sufficient, and there are a variety of sufficient conditions in common use.