Universal bundle

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 G be a compact Lie group. There exists a contractible space EG on which G acts freely. The projection EG → BG is a G-principal fibre bundle.

Proof. There exists an injection of G into a unitary group U(n) for n big enough. 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).

The following Theorem is a corollary of the above Proposition.

:Theorem. If M is a paracompact manifold and P → M is a principal G-bundle, then there exists a map f : M → BG, unique up to homotopy, such that P is isomorphic to f ∗(EG), the pull-back of the G-bundle EG → BG by f.

Proof. Define P \times_G EG to be the quotient of the product space P \times EG by the equivalence relation (g\cdot p, e) \sim (p, g \cdot e). On one hand, the pull-back of the bundle π : EG → BG by projection onto the second factor 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

:\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 p is a fibration with contractible fibre EG, sections of p exist. 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:

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

Now, simply define a section by

:\begin{cases} M \to P\times_G EG \ x \mapsto \lbrack \Phi(x,u),u \rbrack \end{cases}

Because all sections of p 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 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 , 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.

Examples

• Classifying space for U(n)

Notes

References

1. [[J. J. Duistermaat]] and J. A. Kolk,-- ''Lie Groups'', Universitext, Springer. Corollary 4.6.5
2. A.~Dold -- ''Partitions of Unity in the Theory of Fibrations'', Annals of Mathematics, vol. 78, No 2 (1963)