Principal bundle

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product $X \times G$ of a space $X$ with a group $G$ . In the same way as with the Cartesian product, a principal bundle $P$ is equipped with

Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of $(x, e)$ . Likewise, there is not generally a projection onto $G$ generalizing the projection onto the second factor, $X \times G \to G$ which exists for the Cartesian product. They may also have a complicated topology, which prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space.

A common example of a principal bundle is the frame bundle $F E$ of a vector bundle $E$ , which consists of all ordered bases of the vector space attached to each point. The group $G$ in this case is the general linear group, which acts on the right in the usual way: by changes of basis. Since there is no preferred way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section.

Principal bundles have important applications in topology and differential geometry. They have also found application in physics where they form part of the foundational framework of gauge theories.

A principal $G$ -bundle, where $G$ denotes any topological group, is a fiber bundle $π : P \to X$ together with a continuous right action $P \times G \to P$ such that $G$ preserves the fibers of $P$ (i.e. if $y \in P x$ then $yg \in P x$ for all $g \in G$ ) and acts freely and transitively on them. This implies that each fiber of the bundle is homeomorphic to the group $G$ itself. Frequently, one requires the base space $X$ to be Hausdorff and possibly paracompact.

...
Wikipedia