# Universal bundle

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

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.

### 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.^{[1]} 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.^{[2]} 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.

## Examples

## See also

## External links

## Notes

- ↑ 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)