principal bundle

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.

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 .