- Associated bundle
In
mathematics , the theory offiber bundle s with astructure group G (atopological group ) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are bothtopological space s with agroup 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.An example
A simple case comes with the
Möbius strip , for which G is thecyclic group of order 2, mathbb{Z}/2. We can take as F any of: the real number line mathbb{R}, the interval 1, 1] , the real number line less the point 0, or the two-point set 1, 1}. The action of G on these (the non-identity element acting as x ightarrow -x in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles 1, 1] imes I and 1, 1] imes J together: what we really need is the data to identify 1, 1] 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 1, 1} as for 1, 1] .Construction
In general it is enough to explain the transition from a bundle with fiber F, on which G acts, to the associated
principal bundle (namely the bundle where the fiber is G, considered to act by translation on itself). For then we can go from F_1 to F_2, 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 afibre product construction. [All of these constructions are due to Ehresmann (1941-3). Attributed by Steenrod (1951) p. 36.]Associated bundles in general
Let π : "E" → "X" be a fibre bundle over a
topological space "X" with structure group "G" and typical fibre "F". By definition, there is a left action of "G" (as atransformation group ) on the fibre "F". Suppose furthermore that this action is effective. [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 to "E".] There is a local trivialization of the bundle "E" consisting of anopen cover "U"i of "X", and a collection of fibre maps:φi : π-1("U"i) → "U"i × "F"such that thetransition map s are given by elements of "G". More precisely, there are continuous functions "g"ij : ("U"i ∩ "U"j) → "G" such that:ψij("u","f") := φi o φj-1("u","f") = ("u","g"ij("u")"f") for each ("u","f") ∈ ("U"i ∩ "U"j) × "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 "U"i whose transition functions are given by:ψ′ij("u","f"′) = ("u", "g"ij("u") "f"′) for ("u",f′) ∈("U"i ∩ "U"j) × "F"′where the "G"-valued functions "g"ij("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 "g"ij transform via the same coboundary.) Hence, by the
fiber bundle construction theorem , 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" left-acts freely and transitively 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"-bundle 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 (ker(ρ) = 1).
Define a right action of "G" on "P" × "F" via:p,f)cdot g = (pcdot g, ho(g^{-1})f).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:pcdot g,f] = [p, ho(g)f] mbox{ for all } gin G.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("t""ij") where "t""ij" 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 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 afunctor ).Examples of reduction
Examples for
vector bundle s include: the introduction of a "metric" resulting in reduction of the structure group from ageneral linear group GL("n") to anorthogonal 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(2"n",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 thetangent 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.See also
*
Spinor bundle References
*Cite book|title = Topology of Fibre Bundles|first = Norman|last = Steenrod| publisher = Princeton University Press|year=1951|id = ISBN 0-691-00548-6
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