Pullback bundle

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In mathematics, a pullback bundle or induced bundle is a common construction in the theory of fiber bundles. Let π : E → B be a fiber bundle with fiber F and let f : B′ → B be an arbitrary continuous map. Then f induces a fiber bundle f*E over B′ with fiber F in a natural way. Roughly speaking, all we have to say is that 'the fiber at x should be the fiber at f(x)'; and then make all those fibers disjoint by imposing a disjoint union.

More formally, define

f^{*}E = \{(x,e) \in B' \times E \mid f(x) = \pi(e)\}

The projection map π′ : f*E → B′ is given by

\pi'(x,e) = x.\,

The projection onto the second factor gives a map \tilde f \colon f^{*}E \to E such that the following diagram commutes:

If {Ui, φi) is a local trivialization of E then (f−1Ui, ψi) is a local trivialization of f*E where

\psi_i(x,e) = (x, \mbox{proj}_2(\phi_i(e))).\,

It then follows that f*E is a fiber bundle over B′ with fiber F. The bundle f*E is called the pullback bundle or the bundle induced by f. The map \tilde f is then a morphism of bundles covering f.

If the bundle E → B has structure group G with transition functions tij then the pullback bundle f*E also has structure group G. The transition functions in f*E are given by

f^{*}t_{ij} = t_{ij} \circ f.

If E → B is a vector bundle or principal 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

(x,e)\cdot g = (x,e\cdot g)

It then follows that the map \tilde f is right equivariant and so defines a morphism of principal bundles.

In the language of category theory, the pullback bundle construction is an example of the more general categorical pullback. As such it satisfies the corresponding universal property.

[edit] Bundle and sheaf

The pullback of bundles is straightforward, so bundles are inherently contravariant. In contrast, a sheaf is inherently covariant: the straightforward construction is the direct image of a sheaf. The variance is opposite, even though any bundle has a sheaf of sections. This tension can be an advantage, in many areas. It should however be observed that the direct image of a sheaf does not have a closure property with respect to bundles. Taking it will most often produce a sheaf that is not of the type 'sections of a bundle'.

Therefore, while 'pushforward of a bundle' is not an empty notion, and is indeed important, the objects it creates cannot in general be bundles.

[edit] References

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