principal bundle

In some sense, it is best to always think of a principal bundle as a structure that “shows” how a topological space is quotiented using a group action. The base of the principal bundle will be an orbit space, and the total space the topological space, which has a canonical projection to the orbit space.

Definition

For a group , a principal -bundle is a G-space with a continuous (smooth in the smooth manifold category) map such that for each point , there exits a neighborhood such that

the following diagram commutes. Furthermore, is -equivariant given that for and ,

These maps are called local trivializations.

Quick and dirty definition

Looking at the above, we could just define a principal -bundle as a locally trivial, right -space

since the equivariance condition above defines a right action of .

Notes

Since the local trivializations are equivariant with respect to the action of on , the action preserves the fibers of and acts freely and properly such that the map

is a homeomorphism.

That is, topologically, the bundle gives that is a bunch of copies of the group indexed by the base .

Examples

• Consider the double cover of by

This is a principal bundle.

• For a Lie group , and a normal subgroup , then the canonical projection map

is a principal bundle.