Universal bundle

In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group G, is a specific bundle over a classifying space BG, such that every bundle with the given structure group G over M is a pullback by means of a continuous map

M → BG.

This definition usually takes place within a category such as the homotopy category of CW complexes. There, existence theorems for universal bundles arise from Brown's representability theorem.

The total space of a universal bundle is usually written EG. These spaces are of interest in their own right, despite typically being contractible. For example in defining the homotopy quotient of a group action of G, in cases where the orbit space is pathological (in the sense of being a non-Hausdorff space, for example). The idea, if G acts on the space X, is to considet the action on

Y = X×EG,

and corresponding quotient.

If EG is contractible then X and Y are homotopy equivalent spaces. But the diagonal action on Y, i.e. where G acts on both X and EG coordinates, may be well-behaved when the action on X is not.

See also

• Chern class

Exterrnal link

• PlanetMath page of universal bundle examples