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 tij : UiUjG 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 UiUjUk and is called the 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.