# Universal bundle

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*.

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.

**Proof.** There exists an injection of G into a unitary group *U*(*n*) for n big enough.^{[1]} If we find *EU*(*n*) then we can take EG to be *EU*(*n*). The construction of *EU*(*n*) is given in classifying space for *U*(*n*).

**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 G-bundle *P* → *M* by the projection *p* : *P* ×_{G} *EG* → *M* is also *P* × *EG*

Since p is a fibration with contractible fibre EG, sections of p exist.^{[2]} To such a section s 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 p. 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:

Because all sections of p are homotopic, the homotopy class of *f* is unique.

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.

If EG is contractible then X and Y are homotopy equivalent spaces. But the diagonal action on Y, i.e. where G acts on both X and EG coordinates, may be well-behaved when the action on X is not.