Bundle map

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.

General morphisms of fiber bundles

Let πE:E→ M and πF: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 πE 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'''.

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 πF: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 fF over M whose fiber over x is given by (fF)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. In this case, such a bundle map φ (covering f) may also be viewed as a section of the vector bundle Hom(E,fF*) over M, whose fiber over x is the vector space Hom(Ex,F**f(x)) (also denoted L(Ex,F**f(x))) of linear maps from Ex to F**f(x).

Notes

References

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References

1. [[#refHusemoller1994. Husemoller, ''Fibre bundles'']], Definition 3.2
2. [[#refHusemoller1994. Husemoller, ''Fibre bundles'']], Definition 3.2
3. [[#refLee2013. Lee, ''Introduction to smooth manifolds'']], Page 261