Fiber bundle

In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is "locally" a product space, but "globally" may have a different topological structure. Specifically, the similarity between a space formula_1 and a product space formula_2 is defined using a continuous surjective map, formula_3, that in small regions of "E" behaves just like a projection from corresponding regions of formula_2 to formula_5. The map formula_6, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space formula_1 is known as the total space of the fiber bundle, formula_5 as the base space, and formula_9 the fiber.
In the "trivial" case, formula_1 is just formula_2, and the map π 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 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 as projection) to formula_1 is called a section of formula_1. 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, acting on the fiber formula_9.
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 Seifert, Heinz Hopf, Jacques Feldbau, Whitney, Norman 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.
A fiber bundle is a structure formula_15, where formula_1, formula_5, and formula_9 are topological spaces and formula_19 is a continuous surjection satisfying a "local triviality" condition outlined below. The space formula_5 is called the base space of the bundle, formula_1 the total space, and formula_9 the fiber. The map "π" is called the projection map (or bundle projection). We shall assume in what follows that the base space formula_5 is connected.
We require that for every formula_24, there is an open neighborhood formula_25 of formula_26 (which will be called a trivializing neighborhood) such that there is a homeomorphism formula_27 (where formula_28 is the product space) in such a way that "π" agrees with the projection onto the first factor. That is, the following diagram should commute:
where formula_29 is the natural projection and formula_27 is a homeomorphism. The set of all formula_31 is called a local trivialization of the bundle.
Thus for any formula_32, the preimage formula_33 is homeomorphic to formula_9 (since proj1−1({"p"}) clearly is) and is called the fiber over "p". Every fiber bundle formula_19 is an open map, since projections of products are open maps. Therefore formula_5 carries the quotient topology determined by the map "π".
A fiber bundle formula_15 is often denoted
 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 smooth fiber bundle is a fiber bundle in the category of smooth manifolds. That is, formula_1, formula_5, and formula_9 are required to be smooth manifolds and all the functions above are required to be smooth maps.
Examples.
Trivial bundle.
Let formula_41 and let formula_19 be the projection onto the first factor. Then formula_1 is a fiber bundle (of formula_9) over formula_5. Here formula_1 is not just locally a product but "globally" one. Any such fiber bundle is called a trivial bundle. Any fiber bundle over a contractible CW-complex is trivial.
Nontrivial bundles.
Möbius strip.
Perhaps the simplest example of a nontrivial bundle formula_1 is the Möbius strip. It has the circle that runs lengthwise along the center of the strip as a base formula_5 and a line segment for the fiber formula_9, so the Möbius strip is a bundle of the line segment over the circle. A neighborhood formula_50 of formula_51 (where formula_52) is an arc; in the picture, this is the length of one of the squares. The preimage formula_53 in the picture is a (somewhat twisted) slice of the strip four squares wide and one long.
A homeomorphism (formula_54 in the Formal Definition section) exists that maps the preimage of formula_50 (the trivializing neighborhood) to a slice of a cylinder: curved, but not twisted. This pair locally trivializes the strip. The corresponding trivial bundle formula_56 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, formula_57.
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 by a group formula_58 is given, so that each fiber is a principal homogeneous space. The bundle is often specified along with the group by referring to it as a principal formula_58-bundle. The group formula_58 is also the structure group of the bundle. Given a representation formula_61 of formula_58 on a vector space formula_63, a vector bundle with formula_64 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. Given a vector bundle formula_1 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 formula_26 is the set of all unit vectors in formula_67. When the vector bundle in question is the tangent bundle formula_68, the unit sphere bundle is known as the unit tangent bundle. 
A sphere bundle is partially characterized by its Euler class, which is a degree formula_69 cohomology class in the total space of the bundle. In the case formula_70 the sphere bundle is called a circle bundle and the Euler class is equal to the first Chern class, which characterizes the topology of the bundle completely. For any formula_71, given the Euler class of a bundle, one can calculate its cohomology using a long exact sequence called the Gysin sequence.
Mapping tori.
If "X" is a topological space and formula_72 is a homeomorphism then the mapping torus formula_73 has a natural structure of a fiber bundle over the circle with fiber formula_74. Mapping tori of homeomorphisms of surfaces are of particular importance in 3-manifold topology.
Quotient spaces.
If formula_58 is a topological group and formula_76 is a closed subgroup, then under some circumstances, the quotient space formula_77 together with the quotient map formula_78 is a fiber bundle, whose fiber is the topological space formula_76. A necessary and sufficient condition for (formula_80) to form a fiber bundle is that the mapping formula_6 admit local cross-sections .
The most general conditions under which the quotient map will admit local cross-sections are not known, although if formula_58 is a Lie group and formula_76 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, formula_84, which is a fiber bundle over the sphere formula_85 whose total space is formula_86. From the perspective of Lie groups, formula_86 can be identified with the special unitary group formula_88. The abelian subgroup of diagonal matrices is isomorphic to the circle group formula_89, and the quotient formula_90 is diffeomorphic to the sphere.
More generally, if formula_58 is any topological group and formula_76 a closed subgroup that also happens to be a Lie group, then formula_93 is a fiber bundle.
Sections.
A section (or cross section) of a fiber bundle formula_6 is a continuous map formula_95 such that formula_96 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 of a fiber bundle is a continuous map formula_97 where "U" is an open set in "B" and formula_96 for all "x" in "U". If formula_99 is a local trivialization chart then local sections always exist over "U". Such sections are in 1-1 correspondence with continuous maps formula_100. 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 for the bundle ("E", "B", "π", "F") is a set of local trivialization charts formula_101 such that for any formula_102 for the overlapping charts formula_103 and formula_104 the function
is given by
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 transition functions "t""ij" satisfy the following conditions
The third condition applies on triple overlaps "Ui" ∩ "Uj" ∩ "Uk" 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.
Bundle maps.
It is useful to have notions of a mapping between two fiber bundles. Suppose that "M" and "N" are base spaces, and formula_110 and formula_111 are fiber bundles over "M" and "N", respectively. A bundle map (or bundle morphism) consists of a pair of continuous functions
such that formula_113. That is, the following diagram commutes:
For fiber bundles with structure group "G" and whose total spaces are (right) "G"-spaces (such as a principal bundle), bundle morphisms are also required to be "G"-equivariant on the fibers. This means that formula_114 is also "G"-morphism from one "G"-space to another, i.e., formula_115 for all formula_116 and formula_117.
In case the base spaces "M" and "N" coincide, then a bundle morphism over "M" from the fiber bundle formula_110 to formula_119 is a map formula_120 such that formula_121. This means that the bundle map formula_120 covers the identity of "M". That is, formula_123 and the diagram commutes
Assume that both formula_110 and formula_119 are defined over the same base space "M". A bundle isomorphism is a bundle map formula_126 between "π""E" : "E" → "M" and "π""F" : "F" → "M" such that formula_123 and such that φ is also a homeomorphism.
Differentiable fiber bundles.
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.
If "M" and "N" are compact and connected, then any submersion "f" : "M" → "N" gives rise to a fiber bundle in the sense that there is a fiber space "F" diffeomorphic to each of the fibers such that ("E", "B", "π", "F") = ("M", "N", ƒ, "F") is a fiber bundle. (Surjectivity of ƒ follows by the assumptions already given in this case.) More generally, the assumption of compactness can be relaxed if the submersion ƒ : "M" → "N" is assumed to be a surjective proper map, meaning that ƒ−1("K") is compact for every compact subset "K" of "N". Another sufficient condition, due to , is that if ƒ : "M" → "N" is a surjective submersion with "M" and "N" differentiable manifolds such that the preimage ƒ−1{"x"} is compact and connected for all "x" ∈ "N", then ƒ admits a compatible fiber bundle structure .