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. 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

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

Since Template:Mvar is a fibration with contractible fibre Template:Mvar, sections of Template:Mvar exist. 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:

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

Now, simply define a section by

$\{\begin{array}{l}M\to P{\times }_{G}EG\\ x\mapsto [\mathrm{\Phi }(x,u),u]\end{array}$

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.

Examples

• Classifying space for U(n)

See also

• Chern class
• tautological bundle, a universal bundle for the general linear group.

External links

• PlanetMath page of universal bundle examples