Bundle map

In mathematics, a bundle map (or bundle morphism) 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 which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces. Then in the fourth section, some other examples will be given.

Bundle maps over a common base

Let $\pi _{E}\colon E\to M$ and $\pi _{F}\colon F\to M$ be fiber bundles over a space M. Then a bundle map from E to F over M is a continuous map $\varphi \colon E\to F$ such that $\pi _{F}\circ \varphi =\pi _{E}$ . That is, the diagram

should commute. Equivalently, for any point x in M, $\varphi$ maps the fiber $E_{x}=\pi _{E}^{-1}(\{x\})$ of E over x to the fiber $F_{x}=\pi _{F}^{-1}(\{x\})$ of F over x.^[1]

General morphisms of fiber bundles

Let π_E:E→ M and π_F:F→ N be fiber bundles over spaces M and N respectively. Then a continuous map $\varphi :E\to F$ is called a bundle map from E to F if there is a continuous map f:M→ N such that the diagram

commutes, that is, $\pi _{F}\circ \varphi =f\circ \pi _{E}$ . In other words, $\varphi$ is fiber-preserving, and f is the induced map on the space of fibers of E: since π_E is surjective, f is uniquely determined by $\varphi$ . For a given f, such a bundle map $\varphi$ is said to be a bundle map covering f.^[2]

Relation between the two notions

It follows immediately from the definitions that a bundle map over M (in the first sense) is the same thing as a bundle map covering the identity map of M.

Conversely, general bundle maps can be reduced to bundle maps over a fixed base space using the notion of a pullback bundle. If π_F:F→ N is a fiber bundle over N and f:M→ N is a continuous map, then the pullback of F by f is a fiber bundle f^*F over M whose fiber over x is given by (f^*F)_x = F_f(x). It then follows that a bundle map from E to F covering f is the same thing as a bundle map from E to f^*F over M.

Variants and generalizations

There are two kinds of variation of the general notion of a bundle map.

First, one can consider fiber bundles in a different category of spaces. This leads, for example, to the notion of a smooth bundle map between smooth fiber bundles over a smooth manifold.

Second, one can consider fiber bundles with extra structure in their fibers, and restrict attention to bundle maps which preserve this structure. This leads, for example, to the notion of a (vector) bundle homomorphism between vector bundles, in which the fibers are vector spaces, and a bundle map φ is required to be a linear map on each fiber.^[3] In this case, such a bundle map φ (covering f) may also be viewed as a section of the vector bundle Hom(E,f^*F) over M, whose fiber over x is the vector space Hom(E_x,F_f(x)) (also denoted L(E_x,F_f(x))) of linear maps from E_x to F_f(x).

Notes

^ Husemoller, Fibre bundles, Definition 3.2
^ Husemoller, Fibre bundles, Definition 3.2
^ Lee, Introduction to smooth manifolds, Page 261

References

Husemoller, Dale (1994). Fibre bundles. Graduate Texts in Mathematics. Vol. 20 (Third ed.). Springer-Verlag, New York. doi:10.1007/978-1-4757-2261-1. ISBN 0-387-94087-1. MR 1249482.
Lee, John M. (2013). Introduction to smooth manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). Springer, New York. ISBN 978-1-4419-9981-8. MR 2954043.
Steenrod, Norman (1951). The topology of fibre bundles. Princeton Mathematical Series. Vol. 14. Princeton University Press, Princeton, NJ. ISBN 978-1-4008-8387-5. MR 0039258.

[1] Husemoller, Fibre bundles, Definition 3.2

[2] Husemoller, Fibre bundles, Definition 3.2

[3] Lee, Introduction to smooth manifolds, Page 261

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