Principal bundle
They may also have a complicated topology that 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.
, which consists of all ordered bases of the vector space attached to each point.
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 natural 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 and mathematical gauge theory.
denotes any topological group, is a fiber bundle
) and acts freely and transitively (meaning each fiber is a G-torsor) on them in such a way that for each
Because the action is free and transitive, the fibers have the structure of G-torsors.
but lacks a group structure since there is no preferred choice of an identity element.
where the structure group acts on the fiber by left multiplication.
on the fiber commutes with the action of the structure group, there exists an invariant notion of right multiplication by
is required to be a Lie group, and the corresponding action on
unitary matrices, then a section can be constructed by considering four real-valued functions
This same procedure valids by taking a parameterization of a collection of matrices defining a Lie group
and by considering the set of functions from a patch of the base space
For instance, vector bundles always have a zero section whether they are trivial or not and sphere bundles may admit many global sections without being trivial.
The same fact applies to local trivializations of principal bundles.
Given a local trivialization one can define an associated local section where e is the identity in G. Conversely, given a section s one defines a trivialization Φ by The simple transitivity of the G action on the fibers of P guarantees that this map is a bijection, it is also a homeomorphism.
If we write in the form then the map satisfies Equivariant trivializations therefore preserve the G-torsor structure of the fibers.
On overlaps these must be related by the action of the structure group G. In fact, the relationship is provided by the transition functions By gluing the local trivializations together using these transition functions, one may reconstruct the original principal bundle.
It turns out that these properties completely characterize smooth principal bundles.
a smooth, free, and proper right action then Given a subgroup H of G one may consider the bundle
The reason for this name is that the (fiberwise) inverse image of the values of this section form a subbundle of
Reductions of the structure group do not in general exist.
-bundle may be rephrased as questions about the admissibility of the reduction of the structure group (from
vector fields that are linearly independent at each point if and only if its frame bundle admits a global section.
is a subgroup of the general linear group GL(
provides a reduction of structure group of the frame bundle of
The classifying space has the property that any G principal bundle over a paracompact manifold B is isomorphic to a pullback of the principal bundle EG → BG.
[5] In fact, more is true, as the set of isomorphism classes of principal G bundles over the base B identifies with the set of homotopy classes of maps B → BG.
