# Principal bundles

A principal bundle (AKA principal $${G}$$-bundle) $${(P,M,\pi,G)}$$ has a topological group $${G}$$ as both abstract fiber and structure group, where $${G}$$ acts on itself via left translation as a transition function across trivializing neighborhoods, i.e.

$$\displaystyle f_{i}(p)=g_{ij}f_{j}(p),$$

where the operation of $${g_{ij}}$$ is the group operation. Note that the fiber over a point $${\pi^{-1}(x)}$$ is only homeomorphic as a space to $${G}$$ in a given trivializing neighborhood, and so is missing a unique identity element and is a $${G}$$-torsor, not a group (see the section on group actions).

A principal bundle lets us introduce a consistent right action of $${G}$$ on $${\pi^{-1}(x)}$$ (as opposed to the left action on the abstract fiber). This right action is defined by

\displaystyle \begin{aligned}g(p) & \equiv f_{i}^{-1}\left(f_{i}(p)g\right)\\ \Rightarrow f_{i}\left(g(p)\right) & =f_{i}(p)g \end{aligned}

for $${p\in\pi^{-1}(U_{i})}$$, where in an intersection of trivializing neighborhoods $${U_{i}\cap U_{j}}$$ we see that

\displaystyle \begin{aligned}g(p) & =f_{j}^{-1}\left(f_{j}(p)g\right)\\ & =f_{i}^{-1}f_{i}f_{j}^{-1}\left(f_{j}(p)g\right)=f_{i}^{-1}\left(g_{ij}f_{j}(p)g\right)\\ & =f_{i}^{-1}\left(f_{i}(p)g\right)=g(p), \end{aligned}

i.e. $${g(p)}$$ is consistently defined across trivializing neighborhoods. Via this fiber-wise action, $${G}$$ then has a right action on the bundle $${P}$$.

The above depicts how a principal bundle has the same group $${G}$$ as both abstract fiber and structure group, where $${G}$$ acts on itself via left translation. $${G}$$ also has a right action on the bundle itself, which is consistent across trivializing neighborhoods. The identity sections (defined below) are also depicted.

 Δ It is important to remember that $${M}$$ is not part of $${E}$$, and that the depiction of each fiber in the bundle $${\pi^{-1}(x)\in E}$$ as “hovering over” the point $${x\in M}$$ is only valid locally.
 Δ Note that from its definition and basic group properties, the right action of $${G}$$ on $${\pi^{-1}(x)}$$ is automatically free and transitive (making $${\pi^{-1}(x)}$$ a “right $${G}$$-torsor”). An equivalent definition of a principal bundle excludes $${G}$$ as a structure group but includes this free and transitive right action of $${G}$$. Also note that the definition of the right action is equivalent to saying that $${f_{i}\colon\pi^{-1}(x)\rightarrow G}$$ is equivariant with respect to the right action of $${G}$$ on $${\pi^{-1}(x)}$$ and the right action of $${G}$$ on itself.
 Δ A principal bundle is sometimes defined so that the structure group acts on itself by right translation instead of left. In this case the action of $${G}$$ on the bundle must be a left action.
 Δ A principal bundle can also be denoted $${P(M,G)}$$ or $${G\hookrightarrow P\overset{\pi}{\longrightarrow}M}$$.

Since the right action is an intrinsic operation, a principal bundle map between principal $${G}$$-bundles (e.g. a principal bundle automorphism) is required to be equivariant with regard to it, i.e. we require $${\Phi_{E}(g(p))=g(\Phi_{E}(p))}$$, or in juxtaposition notation, $${\Phi_{E}(pg)=\Phi_{E}(p)g}$$. In fact, any such equivariant map is automatically a principal bundle map, and if the base spaces are identical and unchanged by $${\Phi_{E}}$$, then $${\Phi_{E}}$$ is an isomorphism. For a principal bundle map $${\Phi_{E}\colon(P^{\prime},M^{\prime},G^{\prime})\rightarrow(P,M,G)}$$ between bundles with different structure groups, we must include a homomorphism $${\Phi_{G}\colon G^{\prime}\rightarrow G}$$ between structure groups so that the equivariance condition becomes $${\Phi_{E}(g(p))=\Phi_{G}(g)(\Phi_{E}(p))}$$, or in juxtaposition notation, $${\Phi_{E}(pg)=\Phi_{E}(p)\Phi_{G}(g)}$$.

 Δ Note that the right action of a fixed $${g\in G}$$ is thus not a principal bundle automorphism, since for non-abelian $${G}$$ it will not commute with another right action.

A principal bundle has a global section iff it is trivial. However, within each trivializing neighborhood on a principal bundle we can define a local identity section

$$\displaystyle \sigma_{i}(x)\equiv f_{i}^{-1}(e),$$

where $${e}$$ is the identity element in $${G}$$. In $${U_{i}\cap U_{j}}$$, we can then use $${f_{i}(\sigma_{i})=e}$$ to see that the identity sections are related by the right action of the transition function:

\displaystyle \begin{aligned}g_{ij}(\sigma_{i}) & =f_{i}^{-1}\left(f_{i}(\sigma_{i})g_{ij}\right)\\ & =f_{i}^{-1}\left(g_{ij}\right)\\ & =f_{i}^{-1}\left(g_{ij}f_{j}(\sigma_{j})\right)\\ & =f_{i}^{-1}\left(f_{i}(\sigma_{j})\right)\\ & =\sigma_{j}, \end{aligned}

or in juxtaposition notation,

$$\displaystyle \sigma_{j}=\sigma_{i}g_{ij}.$$

 Δ The different actions of $${G}$$ are a potential source of confusion. $${g_{ij}}$$ has a left action on the abstract fiber of a $${G}$$-bundle, which on a principal bundle becomes left group multiplication, and also has a right action on the bundle itself that relates the elements in the identity section.

If $${G}$$ is a closed subgroup of a Lie group $${P}$$ (and thus also a Lie group by Cartan’s theorem), then $${(P,P/G,G)}$$ is a principal $${G}$$-bundle with base space the (left) coset space $${P/G}$$. The right action of $${G}$$ on the entire space $${P}$$ is just right translation.