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.
Existence of a universal bundle
In the CW complex category
For compact Lie groups
We will first prove:
Let be a compact Lie group. There exists a contractible space on which acts freely. The projection is a -principal fibre bundle.
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 EU(n) is given in classifying space for U(n).
The following Theorem is a corollary of the above Proposition.
If is a paracompact manifold and is a principal -bundle, then there exists a map , well defined up to homotopy, such that is isomorphic to , the pull-back of the -bundle by .
Proof On one hand, the pull-back of the bundle by the natural projection 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 between and
Now, simply define a section by
Because all sections of are homotopic, the homotopy class of is unique.
Use in the study of group actions
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 or homotopy orbit space 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 consider instead the action on
- Y = X×EG,
and corresponding quotient. See equivariant cohomology for more detailed discussion.
- J. J. Duistermaat and J. A. Kolk, -- Lie Groups, Universitext, Springer. Corollary 4.6.5
- A.~Dold -- Partitions of Unity in the Theory of Fibrations,Annals of Math., vol. 78, No 2 (1963)