Universal bundle#Use in the study of group actions

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.

We will first prove:

:Proposition. Let {{mvar|G}} be a compact Lie group. There exists a contractible space {{mvar|EG}} on which {{mvar|G}} acts freely. The projection {{math|EG → BG}} is a {{mvar|G}}-principal fibre bundle.

Proof. There exists an injection of {{mvar|G}} into a unitary group {{math|U(n)}} for {{mvar|n}} big enough.J. J. Duistermaat and J. A. Kolk,-- Lie Groups, Universitext, Springer. Corollary 4.6.5 If we find {{math|EU(n)}} then we can take {{mvar|EG}} to be {{math|EU(n)}}. The construction of {{math|EU(n)}} is given in classifying space for U(n).

The following Theorem is a corollary of the above Proposition.

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

Proof. On one hand, the pull-back of the bundle {{math|π : EG → BG}} by the natural projection {{math|P ×_G EG → BG}} is the bundle {{math|P × EG}}. On the other hand, the pull-back of the principal {{mvar|G}}-bundle {{math|P → M}} by the projection {{math|p : P ×_G EG → M}} is also {{math|P × EG}}

:$\backslash begin\{array\}\{rcccl\}$

P & \to & P\times EG & \to & EG \\

\downarrow & & \downarrow & & \downarrow \pi \\

M & \to_{\!\!\!\!\!\!\!s} & P\times_G EG & \to & BG

\end{array}

Since {{mvar|p}} is a fibration with contractible fibre {{mvar|EG}}, sections of {{mvar|p}} exist.A.~Dold -- Partitions of Unity in the Theory of Fibrations, Annals of Mathematics, vol. 78, No 2 (1963) To such a section {{mvar|s}} we associate the composition with the projection {{math|P ×_G EG → BG}}. The map we get is the {{math| f }} we were looking for.

For the uniqueness up to homotopy, notice that there exists a one-to-one correspondence between maps {{math| f  : M → BG}} such that {{math| f ^∗(EG) → M}} is isomorphic to {{math|P → M}} and sections of {{mvar|p}}. We have just seen how to associate a {{math| f }} to a section. Inversely, assume that {{math| f }} is given. Let {{math|Φ :  f ^∗(EG) → P}} be an isomorphism:

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

Now, simply define a section by

:$\backslash begin\{cases\}$

M \to P\times_G EG \\

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

\end{cases}

Because all sections of {{mvar|p}} are homotopic, the homotopy class of {{math| f }} is unique.

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

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

{{See also|equivariant cohomology#Homotopy quotient}}

• Classifying space for U(n)

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

• [https://planetmath.org/someexamplesofuniversalbundles PlanetMath page of universal bundle examples]

{{reflist}}

{{Manifolds}}

Category:Fiber bundles

Category:Homotopy theory