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Wikipedia / Fiber bundle

In mathematics, and particularly topology, a fiber bundle (Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure is defined using a continuous surjective map called the projection or submersion is known as the total space as the base space the fiber.

In the trivial is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a manifold and other more general vector bundles, play an important role in differential geometry and differential topology, as do principal bundles.

Mappings between total spaces of fiber bundles that "commute" with the projection maps are known as bundle maps, and the class of fiber bundles forms a category with respect to such mappings. A bundle map from the base space itself (with the identity mapping is called a section Fiber bundles can be specialized in a number of ways, the most common of which is requiring that the transition maps between the local trivial patches lie in a certain topological group, known as the structure group.

History

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, 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 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle.

The theory of fibered spaces, of which vector bundles, principal bundles, topological fibrations and fibered manifolds are a special case, is attributed to Herbert Seifert, Heinz Hopf, Jacques Feldbau, Whitney, Norman Earl Steenrod, Charles Ehresmann, Jean-Pierre Serre, 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.

Whitney came to the general definition of a fiber bundle from his study of a more particular notion of a sphere bundle, that is a fiber bundle whose fiber is a sphere of arbitrary dimension.

Formal definition

are topological spaces is a continuous surjection satisfying a local triviality is called the the the is called the (or is connected.

, there is an open neighborhood (which will be called a trivializing neighborhood) such that there is a homeomorphism is given the subspace topology agrees with the projection onto the first factor. That is, the following diagram should commute:

is the natural projection. The set is called a of the bundle.

, the preimage) and is called the is an open map carries the quotient topology

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 is a fiber bundle in the category of smooth manifolds are required to be smooth manifolds and all the functions above are required to be smooth maps.

Examples

Trivial bundle

is not just locally a product but globally one. Any such fiber bundle is called a . Any fiber bundle over a contractible CW-complex is trivial.

Nontrivial bundles

Möbius strip

is the Möbius strip. It has the circle and a line segment, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood) is an arc; in the picture, this is the length of one of the squares. The preimage).

in would be a cylinder, but the Möbius strip has an overall "twist". 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).

Klein bottle

A similar nontrivial bundle is the Klein bottle, which can be viewed as a "twisted" circle bundle over another circle. The corresponding non-twisted (trivial) bundle is the 2-torus.

Covering map

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.

Vector and principal bundles

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).

Another special class of fiber bundles, called principal bundles, are bundles on whose fibers a free and transitive action is given, so that each fiber is a principal homogeneous space is also the structure group of the bundle. Given a representation as a structure group may be constructed, known as the associated bundle.

Sphere bundles

A sphere bundle is a fiber bundle whose fiber is an n-sphere with a metric (such as the tangent bundle to a Riemannian manifold) one can construct the associated unit sphere bundle is the set of all unit vectors, the unit sphere bundle is known as the unit tangent bundle.

A sphere bundle is partially characterized by its Euler class cohomology the sphere bundle is called a circle bundle and the Euler class is equal to the first Chern class, given the Euler class of a bundle, one can calculate its cohomology using a long exact sequence called the Gysin sequence.

Mapping tori

is a topological space is a homeomorphism then the mapping torus has a natural structure of a fiber bundle over the circle Mapping tori of homeomorphisms of surfaces are of particular importance in 3-manifold topology.

Quotient spaces

is a topological group is a closed subgroup, then under some circumstances, the quotient space. A necessary and sufficient condition admits local cross-sections .

The most general conditions under which the quotient map is a Lie group a closed subgroup (and thus a Lie subgroup by Cartan's theorem), then the quotient map is a fiber bundle. One example of this is the Hopf fibration can be identified with the special unitary group. The abelian subgroup of diagonal matrices is isomorphic to the circle group is diffeomorphic to the sphere.

is a fiber bundle.

Sections

A (or cross section 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. The obstruction to the existence of a section can often be measured by a cohomology class, which leads to the theory of characteristic classes in algebraic topology.

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.

Often one would like to define sections only locally (especially when global sections do not exist). A local section where U is an open set in B for all x in U is a local trivialization chart then local sections always exist over U. Such sections are in 1-1 correspondence. Sections form a sheaf.

Structure groups and transition functions

Fiber bundles often come with a group of symmetries that describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group that acts continuously on the fiber space F on the left. We lose nothing if we require G to act faithfully on F so that it may be thought of as a group of homeomorphisms of F. A G-atlas is a continuous map called a . 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 of the bundle. This is related to the gauge group in physics, which is the group of automorphisms of the principal G-bundle that leave the base space unchanged.

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.

satisfy the following conditions

The third condition applies on triple overlaps Ui ∩ Uj ∩ Uk and is called the (Čech) cocycle condition (see also Čech cohomology). The importance of this is that the transition functions determine the fiber bundle (if one assumes the 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.

Bundle maps

, respectively. A or consists of a pair of continuous. That is, the following diagram is commutative:

-spaces (such as a principal bundle), bundle morphisms-equivariant.

covers and the following diagram commutes:

. A bundle isomorphism is also a homeomorphism.

Differentiable fiber bundles

In the category of differentiable manifolds, fiber bundles arise naturally as submersions from a differentiable manifold M to another differentiable manifold N is called a fibered manifold. However, this necessary condition is not quite sufficient, and there are a variety of sufficient conditions in common use.

If M and N are compact and connected gives rise to a fiber bundle in the sense that there is a fiber space F is assumed to be a surjective proper map is compact for every compact subset K of N. Another sufficient condition, due to is a surjective submersion with M and N differentiable manifolds admits a compatible fiber bundle structure .

Generalizations

  • The notion of a bundle applies to many more categories in mathematics, at the expense of appropriately modifying the local triviality condition; cf. principal homogeneous space and torsor (algebraic geometry).

  • In topology, a fibration that has certain homotopy-theoretic properties in common with fiber bundles. Specifically, under mild technical assumptions a fiber bundle always has the homotopy lifting property or homotopy covering property (see for details). This is the defining property of a fibration.

  • A section of a fiber bundle is a "function whose output range is continuously dependent on the input." This property is formally captured in the notion of dependent type.

See also

Notes

References