G-bundles

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 $${g_{ij}(x)\in G}$$, 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.