A fiber bundle with base $$B$$ and fiber $$F$$ is map of topological spaces $$\pi: E \to B$$ such that $$B$$ admits an open cover $$\{U_i\}$$ on which $$\pi$$ can be locally identified with the projection from $$U_i \times F$$ to $$F$$. Fiber bundles are often equipped with structure groups, so that a group $$G$$ acts on $$F$$ and $$E$$, and the local identification with the product is $$G$$-equivariant.