Bundle Map - Relation Between The Two Notions

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.