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M



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 Complex es, 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\longrightarrow BG is a
G-principal fibre bundle.

Proof
There exists an injection of G into a Unitary Group U(n) for n big enoughJ.~J.~Duistermaat and J.~A.~Kolk,
-- ''Lie Groups'', Universitext, Springer. Corollary 4.6.5.
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) .
\Box

The following Theorem is a corollary of the above Proposition.

Theorem

If M is a paracompact manifold and P\longrightarrow M is a principal G-bundle, then there exists a map
  • (EG), the pull-back

    • of the G-bundle EG\longrightarrow BG by f.

Proof
On one hand, the pull-back of the bundle \pi:EG\longrightarrow BG by the natural projection P imes_G EG\longrightarrow BG is the bundle P imes EG. On the other hand, the pull-back of the principal G-bundle P\longrightarrow M by the projection
p:P imes_G EG\longrightarrow M is also P imes EG


\begin{align}
P & \longleftarrow & P imes EG& \longrightarrow & EG \
\downarrow & & \downarrow & & \downarrow\pi\
M & \longleftarrow^{\!\!\!\!\!\!\!p} & P imes_G EG & \longrightarrow & BG.
\end{align}

Since p is a fibration with contractible fibre EG,
sections of p existA.~Dold
-- ''Partitions of Unity in the Theory of Fibrations'',Annals of Math., vol. 78, No 2 (1963). To such a section s
we associate the composition with the projection P imes_G EG\longrightarrow 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
  • EG\longrightarrow M is isomorphic to P\longrightarrow M and sections of p. We have just seen

    • how to associate a f to a section. Inversely, assume that f is given. Let \Phi be an isomorphism

  • EG and P


    • \Phi: \{(x,u)\in M imes EG\mid\,f(x)=\pi(u)\} \longrightarrow P.

Now, simply define a section by

\begin{align}
M & \longrightarrow & P imes_G EG \
x & \longrightarrow & \lbrack \Phi(x,u),u brack.
\end{align}

Because all sections of p are homotopic, the homotopy class of f is unique.
\Box



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

Y


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




SEE ALSO




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NOTES