Structure group

Structure groups and transition functions

Fiber bundles often come with a group of symmetries that describe the matching conditions between overlapping local trivialization charts. Specifically, let G be a topological group that acts continuously on the fiber space F on the left. We lose nothing if we require G to act faithfully on F so that it may be thought of as a group of homeomorphisms of F. A G-atlas for the bundle ${\displaystyle (E,B,\pi ,F)}$ is a set of local trivialization charts ${\displaystyle \{(U_{k},\,\varphi _{k})\}}$ such that for any ${\displaystyle \varphi _{i},\varphi _{j}}$ for the overlapping charts ${\displaystyle (U_{i},\,\varphi _{i})}$ and ${\displaystyle (U_{j},\,\varphi _{j})}$ the function $${\displaystyle \varphi _{i}\varphi _{j}^{-1}:\left(U_{i}\cap U_{j}\right)\times F\to \left(U_{i}\cap U_{j}\right)\times F}$$ is given by $${\displaystyle \varphi _{i}\varphi _{j}^{-1}(x,\,\xi )=\left(x,\,t_{ij}(x)\xi \right)}$$ where ${\displaystyle t_{ij}:U_{i}\cap U_{j}\to G}$ is a continuous map called a transition function. Two G-atlases are equivalent if their union is also a G-atlas. A G-bundle is a fiber bundle with an equivalence class of G-atlases. The group G is called the structure group of the bundle. This is related to the gauge group in physics, which is the group of automorphisms of the principal G-bundle that leave the base space unchanged.