G-bundles

At a point \({x\in M}\) 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 \({g_{ij}(x)\in G}\), 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.

An Illustrated Handbook