- Pullback bundle
In
mathematics , a pullback bundle or induced bundle is a useful construction in the theory offiber bundle s. Given a fiber bundle "π" : "E" → "B" and a continuous map "f" : "B"′ → "B" one can define a "pullback" of "E" by "f" as a bundle "f"*"E" over "B"′. The fiber of "f"*"E" over a point "x" in "B"′ is just the fiber of "E" over "f"("x"). Thus "f"*"E" is thedisjoint union of all these fibers equipped with a suitable topology.Formal definition
Let "π" : "E" → "B" be a fiber bundle with abstract fiber "F" and let "f" : "B"′ → "B" be a continuous map. Define the pullback bundle by:and equip it with the
subspace topology and the projection map π′ : "f"*"E" → "B"′ given by the projection onto the first factor, i.e.,:The projection onto the second factor gives a map such that the following diagram commutes:If ("U", φ) is a
local trivialization of "E" then ("f"−1"U", ψ) is a local trivialization of "f"*"E" where:It then follows that "f"*"E" is a fiber bundle over "B"′ with fiber "F". The bundle "f"*"E" is called the pullback of "E" by "f" or the bundle induced by "f". The map is then abundle morphism covering "f".Properties
Any section "s" of "E" over "B" induces a section of "f"*"E", called the pullback section "f"*"s", simply by defining .
If the bundle "E" → "B" has
structure group "G" with transition functions "t""ij" (with respect to a family of local trivializations {("U""i", φ"i")} ) then the pullback bundle "f"*"E" also has structure group "G". The transition functions in "f"*"E" are given by:If "E" → "B" is a
vector bundle orprincipal bundle then so is the pullback "f"*"E". In the case of a principal bundle the right action of "G" on "f"*"E" is given by:It then follows that the map isequivariant and so defines a morphism of principal bundles.In the language of
category theory , the pullback bundle construction is an example of the more generalcategorical pullback . As such it satisfies the correspondinguniversal property .The construction of the pullback bundle can be carried out in subcategories of the category of
topological spaces , such as the category ofsmooth manifold s. The latter construction is useful indifferential geometry and topology Examples: It is illuminating to consider the pullback of the degree 2 map from the circle to itself over the degree 3 or 4 map from the circle to itself. In such examples one sometimes gets a connected and sometimes disconnected space, but always several copies of the circle.
Bundles and sheaves
Bundles may also be described by their sheaves of sections. The pullback of bundles then corresponds to the inverse image of sheaves, which is a
contravariant functor. A sheaf, however, is more naturally acovariant object, since it has apushforward , called thedirect image of a sheaf . The tension and interplay between bundles and sheaves, or inverse and direct image, can be advantageous in many areas of geometry. However, the direct image of a sheaf of sections of a bundle is "not" in general the sheaf of sections of some direct image bundle, so that although the notion of a 'pushforward of a bundle' is defined in some contexts (for example, the pushforward by a diffeomorphism), in general it is better understood in the category of sheaves, because the objects it creates cannot in general be bundles.References
*cite book | last = Steenrod | first = Norman | title = The Topology of Fibre Bundles | publisher = Princeton University Press | location = Princeton | year = 1951 | id = ISBN 0-691-00548-6
*cite book | last = Sharpe | first = R. W. | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer | location = New York | year=1997 | id=ISBN 0-387-94732-9External links
* [http://planetmath.org/encyclopedia/PullbackBundle.html Pullback Bundle] , PlanetMath
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