Universal bundle

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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 MBG.

Existence of a universal bundle[edit]

In the CW complex category[edit]

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[edit]

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 EGBG is a G-principal fibre bundle.

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).

The following Theorem is a corollary of the above Proposition.

Theorem. If M is a paracompact manifold and PM is a principal G-bundle, then there exists a map  f  : MBG, unique up to homotopy, such that P is isomorphic to  f (EG), the pull-back of the G-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 G-bundle PM by the projection p : P ×G EGM 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.[2] To such a section s 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 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[edit]

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.

See also: equivariant cohomology#Homotopy quotient.

Examples[edit]

See also[edit]

External links[edit]

Notes[edit]

  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 Math., vol. 78, No 2 (1963)