Principal Bundle - Formal Definition

Formal Definition

A principal G-bundle, where G denotes any topological group, is a fiber bundle π : P → X together with a continuous right action P × G → P such that G preserves the fibers of P and acts freely and transitively on them. This implies that the fiber of the bundle is homeomorphic to the group G itself. Frequently, one requires the base space X to be Hausdorff and possibly paracompact.

Since the group action preserves the fibers of π : P → X and acts transitively, it follows that the orbits of the G-action are precisely these fibers and the orbit space P/G is homeomorphic to the base space X. Because the action is free, the fibers have the structure of G-torsors. A G-torsor is a space which is homeomorphic to G but lacks a group structure since there is no preferred choice of an identity element.

An equivalent definition of a principal G-bundle is as a G-bundle π : P → X with fiber G where the structure group acts on the fiber by left multiplication. Since right multiplication by G on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by G on P. The fibers of π then become right G-torsors for this action.

The definitions above are for arbitrary topological spaces. One can also define principal G-bundles in the category of smooth manifolds. Here π : P → X is required to be a smooth map between smooth manifolds, G is required to be a Lie group, and the corresponding action on P should be smooth.