In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F₁ to F₂, which are both topological spaces with a group action of G. For a fibre bundle F with structure group G, the transition functions of the fibre (i.e., the cocycle) in an overlap of two coordinate systems U_α and U_β are given as a G-valued function g_αβ on U_α∩U_β. One may then construct a fibre bundle F′ as a new fibre bundle having the same transition functions, but possibly a different fibre. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, in particular in topology, a fiber bundle is a space which locally looks like a product of two spaces but may possess a different global structure. ... 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. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, a symmetry group describes all symmetries of objects. ...

An example

A simple case comes with the Möbius band, for which G is a cyclic group of order 2. We can take as F any of: the real number line , the interval , the real number line less the point 0, or the two-point set . The action of G on these (the non-identity element acting as in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles and together: what we really need is the data to identify to itself directly at one end, and with the twist over at the other end. This data can be written down as a patching function, with values in G. The associated bundle construction is just the observation that this data does just as well for as for . A Möbius strip made with a piece of paper and tape. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a. ...

Construction

In general it is enough to explain the transition from a bundle with fiber F, on which G acts, to the principal bundle (namely the bundle where the fiber is G, considered to act by translation on itself). For then we can go from from F₁ to F₂, via the principal bundle. Details in terms of data for an open covering are given as a case of descent. 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 idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in topology. ...

Fiber bundle associated to a principal bundle

Let π : P → X be a principal G-bundle and let ρ : G → Homeo(F) be an continuous left action of G on a space F (in the smooth category, we should have a smooth action on a smooth manifold). Without loss of generality, we can take this action to be effective (ker(ρ) = 1). 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. ...

Define a right action of G on P × F via

We then identify by this action to obtain the space E = P ×_ρ F = (P × F)/G. Denote the equivalence class of (p,f) by [p,f]. Note that In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or gluing together certain points of a given space. ...

Define a projection map π_ρ : E → X by π_ρ([p,f]) = π(p). Note that this is well-defined. In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...

Then π_ρ : E → X is a fiber bundle with fiber F and structure group G. The transition functions are given by ρ(t_ij) where t_ij are the transition functions of the principal bundle P.

Reduction of structure group

The companion concept to associated bundles is the reduction of the structure group of a G-bundle B. We ask whether there is an H-bundle C, such that the associated G-bundle is B, up to isomorphism. More concretely, this asks whether the transition data for B can consistently be written with values in H. In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor). In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

Examples of reduction

Examples for vector bundles include: the introduction of a metric (equivalently, reduction to an orthogonal group from GL_n); and the existence of complex structure on a real bundle (from to .) 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 orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

Another important case is the reduction from to , the latter sitting inside as block matrices. A reduction here is a consistent way of taking complementary k- and n − k-dimensional subspaces; in other words, finding a decomposition of a vector bundle V as a Whitney sum (direct sum) of sub-bundles of the specified fiber dimensions. In the mathematical discipline of matrix theory, a block matrix or a partitioned matrix is a partition of a matrix into rectangular smaller matrices called blocks. ... 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). ...

One can also express the condition for a foliation to be defined as a reduction of the tangent bundle to a block matrix subgroup - but here the reduction is only a necessary condition, there being an integrability condition so that the Frobenius theorem applies. In mathematics, informally speaking, a foliation is a kind of clothing worn on a manifold, cut from a stripy fabric. ... 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, the Frobenius theorem states, in its smooth version of degree 1, the following: Let U be an open set in Rn and F a submodule of Ω1(U) of constant rank r in U. Then F is integrable if and only if for every p ∈ U the stalk...

See also

• spinor bundle