Faserbündelabbildung

A Faserbündelabbildung, or fiber bundle map, is a morphism between two fiber bundles that respects their structure over a common base space. Formally, let \(E \xrightarrow{\pi_E} B\) and \(F \xrightarrow{\pi_F} B\) be fiber bundles over the same base \(B\), each with typical fiber \(V\). A bundle map \(f:E \to F\) is a continuous (or smooth) map such that \(\pi_F \circ f = \pi_E\); in other words, \(f\) sends a point in \(E\) to a point in \(F\) lying over the same base point. When the fibers carry additional structure—such as a vector space or group structure—one typically requires \(f\) to be linear or a group homomorphism on each fiber.

In categorical terms, fiber bundles over a fixed base \(B\) form a category where objects are the bundles and morphisms are fiber bundle maps. The defining commutative diagram

\[

\begin{array}{ccc}

E & \xrightarrow{\,f\,} & F \\

\pi_E \downarrow & & \downarrow \pi_F \\

B & = & B

\end{array}

\]

encapsulates this structure. Fiber bundle maps are central in differential topology and gauge theory. For instance, the tangent bundle map induced by a smooth map between manifolds, or the inclusion of a subbundle such as a vertical distribution in a principal bundle, are all fiber bundle maps. In physics, gauge transformations are represented as bundle automorphisms—fiber bundle maps from a principal bundle to itself that preserve the group structure on each fiber.

An important special case is a *bundle morphism covering the identity* on the base, where \( \pi_F = \pi_E\). These maps are often used to compare different bundle structures over the same space. When the total and base spaces are smooth manifolds and the fibers are vector spaces, such morphisms are called smooth vector bundle maps and are studied extensively in the theory of differential operators and sheaf cohomology. Key references include Forster’s “Lectures on Riemann Surfaces” and Steenrod’s “The Topology of Fibre Bundles“.