Principal Bundle

In mathematics, a principal bundle is a mathematical object which formalizes some of the essential features of the Cartesian product X × 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

  1. An action of G on P, analogous to (x,g)h = (x, gh) for a product space.
  2. A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) → x.

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 × GG 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 FE 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 in the usual way on ordered bases. 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. Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group G determine a unique principal G-bundle from which the original bundle can be reconstructed.

Read more about Principal Bundle:  Formal Definition, Examples, Classification of Principal Bundles

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Spin Structure
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Clutching Construction - Definition - Classifying Map Construction
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Classification of Principal Bundles
... 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 ... 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 ...
Orthonormal Frame Bundle
... If a vector bundle E is equipped with a Riemannian bundle metric then each fiber Ex is not only a vector space but an inner product space ... The orthonormal frame bundle of E, denoted FO(E), is the set of all orthonormal frames at each point x in the base space X ... by a method entirely analogous to that of the ordinary frame bundle ...
Spin Structures On Vector Bundles
... topological manifold and E an oriented vector bundle on M of dimension n equipped with a fibre metric ... A spinor bundle of E is a prescription for consistently associating a spin representation to every point of M ... are topological obstructions to being able to do it, and consequently, a given bundle E may not admit any spinor bundle ...

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