In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group , is a specific bundle over a classifying space , such that every bundle with the given structure group over is a pullback by means of a continuous map .
When the definition of the classifying space takes place within the homotopy category of CW complexes, existence theorems for universal bundles arise from Brown's representability theorem.
We will first prove:
Proof. There exists an injection of into a unitary group for big enough. If we find then we can take to be . The construction of is given in classifying space for.
The following Theorem is a corollary of the above Proposition.
Proof. Define to be the quotient of the product space by the equivalence relation . On one hand, the pull-back of the bundle by projection onto the second factor is the bundle . On the other hand, the pull-back of the principal -bundle by the projection is also
Since is a fibration with contractible fibre , sections of exist. To such a section we associate the composition with the projection . The map we get is the we were looking for.
For the uniqueness up to homotopy, notice that there exists a one-to-one correspondence between maps such that is isomorphic to and sections of . We have just seen how to associate a to a section. Inversely, assume that is given. Let be an isomorphism:
Now, simply define a section by
Because all sections of are homotopic, the homotopy class of is unique.
The total space of a universal bundle is usually written . These spaces are of interest in their own right, despite typically being contractible. For example, in defining the homotopy quotient or homotopy orbit space of a group action of , in cases where the orbit space is pathological (in the sense of being a non-Hausdorff space, for example). The idea, if acts on the space , is to consider instead the action on , and corresponding quotient. See equivariant cohomology for more detailed discussion.
If is contractible then and are homotopy equivalent spaces. But the diagonal action on , i.e. where acts on both and coordinates, may be well-behaved when the action on is not.