Associated bundle
Encyclopedia
In mathematics
, the theory of fiber bundle
s 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 space
s with a group action
of
. 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 Uα and Uβ are given as a G-valued function gαβ on Uα∩Uβ. One may then construct a fibre bundle F′ as a new fibre bundle having the same transition functions, but possibly a different fibre.
, for which
is the cyclic group
of order 2,
. We can take as 
any of: the real number line 
, the interval 
, the real number line less the point 0, or the two-point set 
. The action of 
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 
.
, on which 
acts, to the associated principal bundle
(namely the bundle where the fiber is
, considered to act by translation on itself). For then we can go from 
to 
, via the principal bundle. Details in terms of data for an open covering are given as a case of descent
.
This section is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fibre, from a given fibre bundle. This then specializes to the case when the specified fibre is a principal homogeneous space
for the left action of the group on itself, yielding the associated principal bundle. If, in addition, a right action is given on the fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction.
X with structure group G and typical fibre F. By definition, there is a left action
of G (as a transformation group) on the fibre F. Suppose furthermore that this action is effective.
There is a local trivialization of the bundle E consisting of an open cover Ui of X, and a collection of fibre maps
such that the transition maps are given by elements of G. More precisely, there are continuous functions gij : (Ui ∩ Uj) → G such that
Now let F′ be a specified topological space, equipped with a continuous left action of G. Then the bundle associated to E with fibre F′ is a bundle E′ with a local trivialization subordinate to the cover Ui whose transition functions are given by
where the G-valued functions gij(u) are the same as those obtained from the local trivialization of the original bundle E.
This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of G-valued functions. (Using another local trivialization, and passing to a common refinement if necessary, the gij transform via the same coboundary.) Hence, by the fiber bundle construction theorem
, this produces a fibre bundle E′ with fibre F′ as claimed.
In this way, a principal G-bundle equipped with a right action is often thought of as part of the data specifying a fibre bundle with structure group G, since to a fibre bundle one may construct the principal bundle via the associated bundle construction. One may then, as in the next section, go the other way around and derive any fibre bundle by using a fibre product.
and let ρ : G → Homeo(F) be a 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.
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
Define a projection map πρ : E → X by πρ([p,f]) = π(p). Note that this is well-defined
.
Then πρ : E → X is a fiber bundle with fiber F and structure group G. The transition functions are given by ρ(tij) where tij are the transition functions of the principal bundle P.
-bundle 
. We ask whether there is an 
-bundle 
, such that the associated 
-bundle is 
, up to isomorphism
. More concretely, this asks whether the transition data for
can consistently be written with values in 
. In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor
).
s include: the introduction of a metric resulting in reduction of the structure group from a general linear group
GL(n) to an orthogonal group
O(n); and the existence of complex structure on a real bundle resulting in reduction of the structure group from real general linear group GL(2n,R) to complex general linear group GL(n,C).
Another important case is finding a decomposition of a vector bundle V of rank n as a Whitney sum (direct sum) of sub-bundles of rank k and n-k, resulting in reduction of the structure group from GL(n,R) to GL(k,R) × GL(n-k,R).
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the theory of fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
s with a structure group
(a topological groupTopological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from
to , which are both topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s with a group action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of
. 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 Uα and Uβ are given as a G-valued function gαβ on Uα∩Uβ. One may then construct a fibre bundle F′ as a new fibre bundle having the same transition functions, but possibly a different fibre.An example
A simple case comes with the Möbius stripMöbius strip
The Möbius strip or Möbius band is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface...
, for which
is the cyclic groupCyclic group
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 g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order 2,
. We can take as any of: the real number line , the interval , the real number line less the point 0, or the two-point set . The action of 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 .Construction
In general it is enough to explain the transition from a bundle with fiber, on which acts, to the associated principal bundlePrincipal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
(namely the bundle where the fiber is
, considered to act by translation on itself). For then we can go from to , via the principal bundle. Details in terms of data for an open covering are given as a case of descentDescent (category theory)
In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated...
.
This section is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fibre, from a given fibre bundle. This then specializes to the case when the specified fibre is a principal homogeneous space
Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G such that the stabilizer subgroup of any point is trivial...
for the left action of the group on itself, yielding the associated principal bundle. If, in addition, a right action is given on the fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction.
Associated bundles in general
Let π : E → X be a fibre bundle over a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X with structure group G and typical fibre F. By definition, there is a left action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
of G (as a transformation group) on the fibre F. Suppose furthermore that this action is effective.
There is a local trivialization of the bundle E consisting of an open cover Ui of X, and a collection of fibre maps
Bundle map
In mathematics, a bundle map is a morphism in the category of fiber bundles. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common base space. There are also several variations on the basic theme, depending on precisely...
- φi : π-1(Ui) → Ui × F
such that the transition maps are given by elements of G. More precisely, there are continuous functions gij : (Ui ∩ Uj) → G such that
- ψij(u,f) := φi o φj-1(u,f) = (u,gij(u)f) for each (u,f) ∈ (Ui ∩ Uj) × F.
Now let F′ be a specified topological space, equipped with a continuous left action of G. Then the bundle associated to E with fibre F′ is a bundle E′ with a local trivialization subordinate to the cover Ui whose transition functions are given by
- ψ′ij(u,f′) = (u, gij(u) f′) for (u,f′) ∈(Ui ∩ Uj) × F′
where the G-valued functions gij(u) are the same as those obtained from the local trivialization of the original bundle E.
This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of G-valued functions. (Using another local trivialization, and passing to a common refinement if necessary, the gij transform via the same coboundary.) Hence, by the fiber bundle construction theorem
Fiber bundle construction theorem
In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic...
, this produces a fibre bundle E′ with fibre F′ as claimed.
Principal bundle associated to a fibre bundle
As before, suppose that E is a fibre bundle with structure group G. In the special case when G has a free and transitive left action on F′, so that F′ is a principal homogeneous space for the left action of G on itself, then the associated bundle E′ is called the principal G-bundle associated to the fibre bundle E. If, moreover, the new fibre F′ is identified with G (so that F′ inherits a right action of G as well as a left action), then the right action of G on F′ induces a right action of G on E′. With this choice of identification, E′ becomes a principal bundle in the usual sense. Note that, although there is no canonical way to specify a right action on a principal homogeneous space for G, any two such actions will yield principal bundles which have the same underlying fibre bundle with structure group G (since this comes from the left action of G), and isomorphic as G-spaces in the sense that there is a globally defined G-valued function relating the two.In this way, a principal G-bundle equipped with a right action is often thought of as part of the data specifying a fibre bundle with structure group G, since to a fibre bundle one may construct the principal bundle via the associated bundle construction. One may then, as in the next section, go the other way around and derive any fibre bundle by using a fibre product.
Fiber bundle associated to a principal bundle
Let π : P → X be a principal G-bundlePrincipal bundle
In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × G of a space X with a group G...
and let ρ : G → Homeo(F) be a continuous left action
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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.
Define a right action of G on P × F via
We then identify
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
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
Define a projection map πρ : E → X by πρ([p,f]) = π(p). Note that this is well-defined
Well-defined
In mathematics, well-definition is a mathematical or logical definition of a certain concept or object which uses a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy...
.
Then πρ : E → X is a fiber bundle with fiber F and structure group G. The transition functions are given by ρ(tij) where tij are the transition functions of the principal bundle P.
Reduction of the structure group
The companion concept to associated bundles is the reduction of the structure group of a-bundle . We ask whether there is an -bundle , such that the associated -bundle is , up to isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
. More concretely, this asks whether the transition data for
can consistently be written with values in . In other words, we ask to identify the image of the associated bundle mapping (which is actually a functorFunctor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
).
Examples of reduction
Examples for vector bundleVector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s include: the introduction of a metric resulting in reduction of the structure group from a general linear group
General linear group
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. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible...
GL(n) to an orthogonal group
Orthogonal group
In mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
O(n); and the existence of complex structure on a real bundle resulting in reduction of the structure group from real general linear group GL(2n,R) to complex general linear group GL(n,C).
Another important case is finding a decomposition of a vector bundle V of rank n as a Whitney sum (direct sum) of sub-bundles of rank k and n-k, resulting in reduction of the structure group from GL(n,R) to GL(k,R) × GL(n-k,R).
One can also express the condition for a foliation
Foliation
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
to be defined as a reduction of the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
to a block matrix subgroup - but here the reduction is only a necessary condition, there being an integrability condition so that the Frobenius theorem
Frobenius theorem (differential topology)
In mathematics, Frobenius' theorem gives necessary and sufficient conditions for finding a maximal set of independent solutions of an overdetermined system of first-order homogeneous linear partial differential equations...
applies.