# 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 MBG.

## 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 EGBG 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 PM is a principal Template:Mvar-bundle, then there exists a map  f  : MBG, unique up to homotopy, such that Template:Mvar is isomorphic to  f (EG), the pull-back of the Template:Mvar-bundle EGBG by  f.

Proof. On one hand, the pull-back of the bundle π : EGBG by the natural projection P ×G EGBG is the bundle P × EG. On the other hand, the pull-back of the principal Template:Mvar-bundle PM by the projection p : P ×G EGM is also P × EG

${\displaystyle {\begin{array}{rcccl}P&\to &P\times EG&\to &EG\\\downarrow &&\downarrow &&\downarrow \pi \\M&\to _{\!\!\!\!\!\!\!s}&P\times _{G}EG&\to &BG\end{array}}}$

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 EGBG. 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  : MBG such that  f (EG) → M is isomorphic to PM 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:

${\displaystyle \Phi :\left\{(x,u)\in M\times EG\ :\ f(x)=\pi (u)\right\}\to P}$

Now, simply define a section by

${\displaystyle {\begin{cases}M\to P\times _{G}EG\\x\mapsto \lbrack \Phi (x,u),u\rbrack \end{cases}}}$

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.