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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
- 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 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.[1]
If we find then we can take to be .
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 is a paracompact manifold and is a principal -bundle, then there exists a map
, well defined up to homotopy, such that is isomorphic to , the pull-back
of the -bundle by .
Proof
On one hand, the pull-back of the bundle by the natural projection is the bundle . On the other hand, the pull-back of the principal -bundle by the projection
is also
Since is a fibration with contractible fibre ,
sections of exist.[2] 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
between and
.
Now, simply define a section by
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 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.
Examples
See also
External links
Notes
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- ↑ J. J. Duistermaat and J. A. Kolk, -- Lie Groups, Universitext, Springer. Corollary 4.6.5
- ↑ A.~Dold -- Partitions of Unity in the Theory of Fibrations,Annals of Math., vol. 78, No 2 (1963)