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 π_E:E→ M and π_F:F→ M be fiber bundles over a space M. Then a bundle map from E to F over M is a continuous map φ:E→ F such that . That is, the diagram

should commute. Equivalently, for any point x in M, φ maps the fiber E_x = π_E⁻¹({x}) of E over x to the fiber F_x = π_F⁻¹({x}) of F over x.

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 φ:E→ F is called a bundle map from E to F if there is a continuous map f:M→ N such that the diagram