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 Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ... A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ... A surjective function. ... In mathematics and related technical fields, the term map or mapping is often a synonym for function. ...

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, or in other words twisted in the large. In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ...

Fiber bundles generalize vector bundles, where the main example is the tangent bundle of a manifold, as well as principal bundles. 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. Fiber bundles specialize the more general bundle. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, a bundle is a generalization of a fiber bundle dropping the condition of a local product structure. ...

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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...

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: This is a glossary of some terms used in the branch of mathematics known as topology. ... This word should not be confused with homomorphism. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

where proj₁ : 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 In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...

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. Image File history File links FiberBundle-02. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In topology, an open map is a function between two topological spaces which maps open sets to open sets. ... For quotient spaces in linear algebra, see quotient space (linear algebra). ...

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. In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a smooth function is one that is infinitely differentiable, i. ...

Examples

Trivial bundle

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.

Mobius strip

The Möbius strip is a nontrivial bundle over the circle.

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 x ∈ B is an arc; in the picture, this is the length of one of the squares. The preimage π ^{− 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. Mobius strip created with Mathematica. ... Mobius strip created with Mathematica. ... A Möbius strip made with a piece of paper and tape. ... Circle illustration In Euclidean geometry, a circle is the set of all points in a plane at a fixed distance, called the radius, from a fixed point, the centre. ...

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 right circular cylinder An elliptic cylinder In mathematics, a cylinder is a quadric, i. ...

Klein bottle

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, S¹ × S¹. The Klein bottle immersed in three-dimensional space. ... A torus. ...

Covering map

A covering space is a fiber bundle whose fiber is a discrete space. In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

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). In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... In mathematics, the idea of a frame in the theory of smooth manifolds is understood in terms meaning it can vary from point to point. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...

Another special class of fiber bundles, called principal bundles, are bundles on whose fibers a free and transitive group action by G 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 G-bundle. The group G is also the structure group of the bundle. Given a representation ρ of G on a vector space V, a vector bundle with ρ(G)⊆Aut(V) as a structure group may be constructed, known as the associated bundle. In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, a principal homogeneous space, or G-torsor, for a group G is a set X on which G acts freely and transitively. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ...

Sphere bundles

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. When the vector bundle in question is the tangent bundle T(M), the unit sphere bundle is known as the unit tangent bundle, and is denoted UT(M).
In mathematics, a hypersphere is a sphere which has dimension 3 or higher. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In mathematics, the unit tangent bundle of a Riemannian manifold (M, g), denoted by UT(M) or simply UTM, is a fiber bundle given by the disjoint union where Tx(M) denotes the tangent space to M at x. ...

A sphere bundle is partially characterized by its Euler class, which is a degree n+1 cohomology class in the total space of the bundle. In the case n=1 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 n, given the Euler class of a bundle, one can calculate its cohomology using a long exact sequence called the Gysin sequence. In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. ... In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a cochain complex. ... In mathematics, a circle bundle is a fiber bundle where the fiber is the circle , or more precisely, a principal U(1)-bundle with fiber U(1). ... This article needs to be cleaned up to conform to a higher standard of quality. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In the field of mathematics known as algebraic topology, the Gysin sequence is a long exact sequence which relates the cohomology classes of the base space, the fiber and the total space of a sphere bundle. ...

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. In mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is twisted — particularly, whether it possesses sections or not. ... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...

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. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and...

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 In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, a topological group G is a group that is also a topological space such that the group multiplication G × G → G and the inverse operation G → G are continuous maps. ... In mathematics, a symmetry group describes all symmetries of objects. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... In topology, an atlas describes how a complicated space is glued together from simpler pieces. ...

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.

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. In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ...

The transition functions t_ij satisfy the following conditions

1. t_ii(x) = 1
2. t_ij(x) = t_ji(x) ^{− 1}
3. t_ik(x) = t_ij(x)t_jk(x)

The third condition applies on triple overlaps U_i ∩ U_j ∩ U_k and is called the cocycle condition (see Čech cohomology). ÄŒech cohomology is a particular type of cohomology in mathematics. ...

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. In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ... In differential geometry, a G-structure on a n-manifold M, for a given structure group G (which is a Lie subgroup of the general linear group GL(n)) is a G-subbundle of the frame bundle on M. The notion of G-structures includes many other structures on manifolds...

Bundle maps

It is useful to have notions of mapping between bundles on B for the same type of fiber F. The most common are the mappings between vector bundles that are linear on each fiber; these occur also for different bases, for example from one tangent bundle to another. Another kind is the bundle map of principal bundles, which is G-equivariant fiber by fiber. In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics, an equivariant map is a function between two sets that commutes with the action of a group. ...

See also

• Covering map
• Fibration
• Gauge theory
• Hopf bundle
• Principal bundle
• Pullback bundle
• Universal bundle
• Vector bundle

In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open... This article may be too technical for most readers to understand. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In mathematics, the Hopf bundle (or Hopf fibration) is a particular fiber bundle with base space S2, total space S3, and fiber S1: S1 → S3 → S2 It was discovered by Heinz Hopf in 1931. ... In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G. Principal G-bundles are G-bundles in the sense that the group G also serves... In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. ... In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group G, is a specific bundle over a classifying space BG, such that every bundle with the given structure group G over M is a pullback by means of a continuous map M... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ...

References and external links

• Norman Steenrod, The Topology of Fibre 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.
• Fiber Bundle, PlanetMath
• Weisstein, Eric W., Fiber Bundle at MathWorld.