In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group Template:Mvar, is a specific bundle over a classifying space Template:Mvar, such that every bundle with the given structure group Template:Mvar over Template:Mvar 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:
- Proposition. Let Template:Mvar be a compact Lie group. There exists a contractible space Template:Mvar on which Template:Mvar acts freely. The projection EG → BG is a Template:Mvar-principal fibre bundle.
Proof. There exists an injection of Template:Mvar into a unitary group U(n) for Template:Mvar big enough. If we find EU(n) then we can take Template:Mvar to be EU(n). The construction of EU(n) is given in classifying space for U(n).
The following Theorem is a corollary of the above Proposition.
- Theorem. If Template:Mvar is a paracompact manifold and P → M is a principal Template:Mvar-bundle, then there exists a map f : M → BG, unique up to homotopy, such that Template:Mvar is isomorphic to f ∗(EG), the pull-back of the Template:Mvar-bundle EG → BG by f.
Proof. On one hand, the pull-back of the bundle π : EG → BG by the natural projection P ×G EG → BG is the bundle P × EG. On the other hand, the pull-back of the principal Template:Mvar-bundle P → M by the projection p : P ×G EG → M is also P × EG
Since Template:Mvar is a fibration with contractible fibre Template:Mvar, sections of Template:Mvar exist. To such a section Template:Mvar we associate the composition with the projection P ×G EG → BG. The map we get is the f we were looking for.
For the uniqueness up to homotopy, notice that there exists a one to one correspondence between maps f : M → BG such that f ∗(EG) → M is isomorphic to P → M and sections of Template:Mvar. We have just seen how to associate a f to a section. Inversely, assume that f is given. Let Φ : f ∗(EG) → P be an isomorphism:
Now, simply define a section by
Because all sections of Template:Mvar are homotopic, the homotopy class of f is unique.
Use in the study of group actions
The total space of a universal bundle is usually written Template:Mvar. 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 Template:Mvar, in cases where the orbit space is pathological (in the sense of being a non-Hausdorff space, for example). The idea, if Template:Mvar acts on the space Template:Mvar, is to consider instead the action on Y = X × EG, and corresponding quotient. See equivariant cohomology for more detailed discussion.
If Template:Mvar is contractible then Template:Mvar and Template:Mvar are homotopy equivalent spaces. But the diagonal action on Template:Mvar, i.e. where Template:Mvar acts on both Template:Mvar and Template:Mvar coordinates, may be well-behaved when the action on Template:Mvar is not.
See also: equivariant cohomology#Homotopy quotient.
- 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)