In mathematics, the universal bundle in the theory of fiber bundles with structure group a given topological group , is a specific bundle over a classifying space , such that every bundle with the given structure group over is a pullback by means of a continuous map .

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

Proof. There exists an injection of into a unitary group for big enough. If we find then we can take to be . The construction of is given in classifying space for .

The following Theorem is a corollary of the above Proposition.

:Theorem. If is a paracompact manifold and is a principal -bundle, then there exists a map , unique up to homotopy, such that is isomorphic to , the pull-back of the -bundle by .

Proof. Define <math display="inline">P \times_G EG</math> to be the quotient of the product space <math display="inline">P \times EG</math> by the equivalence relation <math>(g\cdot p, e) \sim (p, g \cdot e)</math>. On one hand, the pull-back of the bundle by projection onto the second factor is the bundle . On the other hand, the pull-back of the principal -bundle by the projection is also

:<math>\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}</math>

Since is a fibration with contractible fibre , sections of exist. To such a section we associate the composition with the projection . The map we get is the we were looking for.

For the uniqueness up to homotopy, notice that there exists a one-to-one correspondence between maps such that is isomorphic to and sections of . We have just seen how to associate a to a section. Inversely, assume that is given. Let be an isomorphism:

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

Now, simply define a section by

:<math>\begin{cases}

M \to P\times_G EG \\

x \mapsto \lbrack \Phi(x,u),u \rbrack

\end{cases}</math>

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

Use in the study of group actions

The total space of a universal bundle is usually written . 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 , in cases where the orbit space is pathological (in the sense of being a non-Hausdorff space, for example). The idea, if acts on the space , is to consider instead the action on , and corresponding quotient. See equivariant cohomology for more detailed discussion.

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

Examples

  • Classifying space for U(n)

See also

  • Chern class
  • tautological bundle, a universal bundle for the general linear group.
  • PlanetMath page of universal bundle examples

Notes