Editing Associated bundle

In [[mathematics]], the theory of [[fiber bundle]]s with a [[structure group]] <math>G</math> (a [[topological group]]) allows an operation of creating an '''associated bundle''', in which the typical fiber of a bundle changes from <math>F_1</math> to <math>F_2</math>, which are both [[topological space]]s with a [[Group action (mathematics)|group action]] of <math>G</math>.  For a fiber bundle <math>F</math> with structure group <math>G</math>, the transition functions of the fiber (i.e., the [[cocycle (algebraic topology)|cocycle]]) in an overlap of two coordinate systems <math>U_\alpha</math> and <math>U_\beta</math> are given as a <math>G</math>-valued function <math>g_{\alpha\beta}</math> on <math>U_\alpha \cap U_\beta</math>.  One may then construct a fiber bundle <math>F'</math> as a new fiber bundle having the same transition functions, but possibly a different fiber.

==An example==

A simple case comes with the [[Möbius strip]], for which <math>G</math> is the [[cyclic group]] of order 2, <math>\mathbb{Z}_2</math>. We can take <math>F</math> to be any of the following: the real number line <math>\mathbb{R}</math>, the interval <math>[-1,\ 1]</math>, the real number line less the point 0, or the two-point set <math>\{-1,\ 1\}</math>. The action of <math>G</math> on these (the non-identity element acting as <math>x\ \rightarrow\ -x</math> in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles <math>[-1,\ 1] \times I</math> and <math>[-1,\ 1] \times J</math> together: what we really need is the data to identify <math>[-1,\ 1]</math> 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 <math>G</math>. The '''associated bundle''' construction is just the observation that this data does just as well for <math>\{-1,\ 1\}</math> as for <math>[-1,\ 1]</math>.

==Construction==
In general it is enough to explain the transition from a bundle with fiber <math>F</math>, on which <math>G</math> acts, to the associated [[principal bundle]] (namely the bundle where the fiber is <math>G</math>, considered to act by translation on itself). For then we can go from <math>F_1</math> to <math>F_2</math>, via the principal bundle.  Details in terms of data for an open covering are given as a case of [[Descent (category theory)|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.<ref>All of these constructions are due to [[Charles Ehresmann|Ehresmann]]  (1941-3). Attributed by Steenrod (1951) page 36</ref>

===Associated bundles in general===
Let <math display="inline">\pi:E\to X</math> be a fiber bundle over a [[topological space]] <math>X</math> with structure group <math>G</math> and typical fibre <math>F</math>.  By definition, there is a [[Group action (mathematics)|left action]] of <math>G</math> (as a [[transformation group]]) on the fibre <math>F</math>.  Suppose furthermore that this action is [[Group action (mathematics)#Notable properties of actions|effective]].<ref>Effectiveness is a common requirement for fibre bundles; see Steenrod (1951).  In particular, this condition is necessary to ensure the existence and uniqueness of the principal bundle associated with <math>E</math>.</ref>
There is a [[locally trivial|local trivialization]] of the bundle <math>E</math> consisting of an [[open cover]] <math>U_i</math> of <math>X</math>, and a collection of [[bundle map|fibre maps]]<math display="block">\varphi_i : \pi^{-1}(U_i) \to U_i \times F</math>such that the [[transition map]]s are given by elements of <math>G</math>.  More precisely, there are continuous functions <math>g_{ij} \colon U_i \cap U_j \to G</math> such that<math display="block">\psi_{ij}(u,f) := \varphi_i \circ \varphi_j ^{-1}(u,f) = \big(u, g_{ij}(u) f \big),\quad 
\text{for each } (u,f)\in (U_i \cap U_j)\times F\, .</math> 

Now let <math>F'</math> be a specified topological space, equipped with a continuous left action of <math>G</math>. Then the bundle '''associated''' with <math>E</math> with fibre <math>F'</math> is a bundle <math>E'</math> with a local trivialization subordinate to the cover <math>U_i</math> whose transition functions are given by<math display="block">\psi'_{ij}(u,f') = \big(u, g_{ij}(u) f' \big),\quad 
\text{for each } (u,f')\in (U_i \cap U_j)\times F'\,,</math>where the <math>G</math>-valued functions <math>g_{ij}(u)</math> are the same as those obtained from the local trivialization of the original bundle <math>E</math>. 
This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of <math>G</math>-valued functions. (Using another local trivialization, and passing to a common refinement if necessary, the <math>g_{ij}</math> transform via the same coboundary.) Hence, by the [[fiber bundle construction theorem]], this produces a fibre bundle <math>E'</math> with fibre <math>F'</math> as claimed.

===Principal bundle associated with a fibre bundle===
As before, suppose that <math>E</math> is a fibre bundle with structure group <math>G</math>.  In the special case when <math>G</math> has a [[Group action (mathematics)#Types of actions|free and transitive]] left action on <math>F'</math>, so that <math>F'</math> is a principal homogeneous space for the left action of <math>G</math> on itself, then the associated bundle <math>E'</math> is called the principal <math>G</math>-bundle associated with the fibre bundle <math>E</math>. If, moreover, the new fibre <math>F'</math> is identified with <math>G</math> (so that <math>F'</math> inherits a right action of <math>G</math> as well as a left action), then the right action of <math>G</math> on <math>F'</math> induces a right action of <math>G</math> on <math>E'</math>. With this choice of identification, <math>E'</math> 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 <math>G</math>, any two such actions will yield principal bundles which have the same underlying fibre bundle with structure group <math>G</math> (since this comes from the left action of <math>G</math>), and isomorphic as <math>G</math>-spaces in the sense that there is a <math>G</math>-equivariant isomorphism of bundles relating the two.

In this way, a principal <math>G</math>-bundle equipped with a right action is often thought of as part of the data specifying a fibre bundle with structure group <math>G</math>, 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 with a principal bundle===

Let <math>\pi \colon P \to X</math> be a [[principal bundle|principal ''G''-bundle]] and let <math>\rho \colon G \to \text{Homeo}(F)</math> be a continuous [[Group action (mathematics)|left action]] of <math>G</math> on a space <math>F</math> (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 <math>G</math> on <math>P \times F</math> via<ref>Husemoller, Dale (1994), p. 45.</ref><ref>Sharpe, R. W. (1997), p. 37.</ref>
:<math>(p,f)\cdot g = (p\cdot g, \rho(g^{-1})f)\, .</math>
We then [[Quotient space (topology)|identify]] by this action to obtain the space <math>E = P \times_\rho F = (P \times F) / G</math>. Denote the equivalence class of <math>(p, f)</math> by <math>[p, f]</math>. Note that
:<math>[p\cdot g,f] = [p,\rho(g)f] \mbox{ for all } g\in G.</math>
Define a projection map <math>\pi_\rho \colon E \to X</math> by <math>\pi_\rho([p, f]) = \pi(p)</math>. Note that this is [[well-defined]].

Then <math>\pi_\rho \colon E \to X</math> is a fiber bundle with fiber <math>F</math> and structure group <math>G</math>. The transition functions are given by <math>\rho(t_{ij})</math> where <math>t_{ij}</math> are the transition functions of the principal bundle <math>P</math>.

This construction can also be seen [[Category theory|categorically]].
More precisely, there are two continuous maps <math>P \times G \times F \to P \times F</math>, given by acting with <math>G</math> on the right on <math>P</math> and on the left on <math>F</math>.
The associated vector bundle <math>P \times_\rho F</math> is then the [[coequalizer]] of these maps.

==Reduction of the structure group==
{{details|reduction of the structure group}}

The companion concept to associated bundles is the '''reduction of the structure group''' of a <math>G</math>-bundle <math>B</math>. We ask whether there is an <math>H</math>-bundle <math>C</math>, such that the associated <math>G</math>-bundle is <math>B</math>, up to [[isomorphism]]. More concretely, this asks whether the transition data for <math>B</math> can consistently be written with values in <math>H</math>. In other words, we ask to identify the image of the associated bundle mapping (which is actually a [[functor]]).

===Examples of reduction===

Examples for [[vector bundle]]s include: the introduction of a ''metric'' resulting in reduction of the structure group from a [[general linear group]] <math>\mathrm{GL}(n)</math> to an [[orthogonal group]] <math>\mathrm{O}(n)</math>; and the existence of complex structure on a real bundle resulting in reduction of the structure group from real general linear group <math>\mathrm{GL}(2n, \mathbb{R})</math> to complex general linear group <math>\mathrm{GL}(n, \mathbb{C})</math>.

Another important case is finding a decomposition of a vector bundle <math>V</math> of rank <math>n</math> as a [[Whitney sum]] (direct sum) of sub-bundles of rank <math>k</math> and <math>n-k</math>, resulting in reduction of the structure group from <math>\mathrm{GL}(n, \mathbb{R})</math> to <math>\mathrm{GL}(k, \mathbb{R}) \times \mathrm{GL}(n-k, \mathbb{R})</math>.

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 (differential topology)|Frobenius theorem]] applies.

== See also ==
*[[Spinor bundle]]

== References ==
{{reflist}}

==Books==
*{{cite book | last = Steenrod | first = Norman | author-link=Norman Steenrod| title = The Topology of Fibre Bundles | url = https://archive.org/details/topologyoffibreb0000stee | url-access = registration | publisher = [[Princeton University Press]] | location = Princeton | year = 1951 | isbn = 0-691-00548-6}}
*{{cite book | last = Husemoller | first = Dale | author-link=Dale Husemoller |title = Fibre Bundles | publisher = [[Springer Science+Business Media|Springer]] | edition = Third |location = New York | year=1994 | isbn=978-0-387-94087-8}}
*{{cite book | last = Sharpe | first = R. W. | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = [[Springer Science+Business Media|Springer]] | location = New York | year=1997 | isbn=0-387-94732-9}}

{{Manifolds}}

[[Category:Algebraic topology]]
[[Category:Differential geometry]]
[[Category:Differential topology]]
[[Category:Fiber bundles]]