for questions about fiber bundles, including structure groups, principal bundles, and spaces of sections.

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.

See also Wikipedia