In the fiber over a point \({\pi^{-1}(x)}\) in the intersection of two trivializing neighborhoods on a bundle \({(E,M,F)}\), we have a homeomorphism \({f_{i}f_{j}^{-1}\colon F\rightarrow F}\). If each of these homeomorphisms is the (left) action of an element \({g_{ij}(x)\in G}\), where \({G}\) is a subgroup of the group of homeomorphisms from \({F}\) to itself, then \({G}\) is called the structure group of \({E}\). This action is usually required to be faithful, so that each \({g\in G}\) corresponds to a distinct homeomorphism of \({F}\). The map \({g_{ij}\colon U_{i}\cap U_{j}\rightarrow G}\) is called a transition function; the existence of transition functions for all overlapping charts makes \({\{U_{i}\}}\) a G-atlas and turns the bundle into a G-bundle.
Applying the action of \({g_{ij}}\) to an arbitrary \({f_{j}(p)}\) yields
\(\displaystyle f_{i}(p)=g_{ij}\left(f_{j}(p)\right). \)
For example, the Möbius strip in the previous figure has a structure group \({G=\mathbb{Z}_{2}}\), where the action of \({0\in G}\) is multiplication by \({+1}\), and the action of \({1\in G}\) is multiplication by \({-1}\). In the top intersection \({U_{i}\cap U_{j}}\), \({g_{ij}=0}\), so that \({f_{i}}\) and \({f_{j}}\) are identical, while in the lower intersection \({g_{ij}=1}\), so that \({f_{i}(p)=g_{ij}\left(f_{j}(p)\right)=1\left(f_{j}(p)\right)=-f_{j}(p)}\).
At a point in a triple intersection \({U_{i}\cap U_{j}\cap U_{k}}\), the cocycle condition \({g_{ij}g_{jk}=g_{ik}}\) can be shown to hold, which implies \({g_{ii}=e}\) and \({g_{ji}=g_{ij}^{-1}}\). Going the other direction, if we start with transition functions from \({M}\) to \({G}\) acting on \({F}\) that obey the cocycle condition, then they determine a unique \({G}\)-bundle \({E}\).
| Δ It is important to remember that the left action of \({G}\) is on the abstract fiber \({F}\), which is not part of the entire space \({E}\), and whose mappings to \({E}\) are dependent upon local trivializations. A left action on \({E}\) itself based on these mappings cannot in general be consistently defined, since for non-abelian \({G}\) it will not commute with the transition functions. |
A given \({G}\)-atlas may not need all the possible homeomorphisms of \({F}\) between trivializing neighborhoods, and therefore will not “use up” all the possible values in \({G}\). If there exists trivializing neighborhoods on a \({G}\)-bundle whose transition functions take values only in a subgroup \({H}\) of \({G}\), then we say the structure group \({G}\) is reducible to \({H}\). For example, a trivial bundle’s structure group is always reducible to the trivial group consisting only of the identity element.
If each transition function is constant within its neighborhood intersection, the fiber bundle is called locally constant; in this case the foliations in each neighborhood with leaves \({U_{i}}\), defined by the right hand side of \({\phi_{i}\colon\pi^{-1}(U_{i})\to U_{i}\times F}\), may be stitched together to form a foliation of \({E}\). Again, an example is the foliation of the Möbius strip by circles.