the pullback of a bundle

Suppose someone has set up a vector bundle E over a space Y, and you have a map f: X -> Y from your own space into theirs. You would like to drag their bundle back to live over X. The pullback bundle f^*E does exactly that: over each point x in X you simply install the fiber that E had sitting over the image point f(x). No new geometry is invented — you reuse Y's fibers, re-indexed by where f sends your points.

Concretely f^*E = { (x, e) in X times E : f(x) = pi(e) }, the pairs of a point in X and a vector in the fiber above its image, with projection (x, e) -> x. This is the fibered product of X and E over Y, and the fiber of f^*E over x is canonically the fiber E_{f(x)}; there is a natural bundle map f^*E -> E covering f that is a linear isomorphism on each fiber. If you describe E by transition functions g_ij on Y, then f^*E is described over X by the composites g_ij composed with f — the same gluing data read through the map.

The single most important fact about pullbacks is homotopy invariance: if two maps f_0, f_1: X -> Y are homotopic and X is paracompact (e.g. a manifold or CW complex), then f_0^*E and f_1^*E are isomorphic bundles. So a bundle over X obtained by pulling back depends only on the homotopy class of the map used. This is the technical engine behind the entire classification scheme — it is exactly why bundles over X correspond to homotopy classes of maps into a classifying space, with the universal bundle being pulled back.