Universal bundle

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 $G$ be a compact Lie group.There exists a contractible space $EG$ on which $G$ acts freely. The projection $EGlongrightarrow\; BG$ is a$G$-principal fibre bundle.
ProofThere exists an injection of $G$ into a unitary group $U(n)$ for $n$ big enough [J. 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 $Plongrightarrow\; M$ is a principal $G$-bundle, then there exists a map$f:Mlongrightarrow\; BG$, well defined up to homotopy, such that $P$ is isomorphic to $f^*(EG)$, the pull-backof the $G$-bundle $EGlongrightarrow\; BG$ by $f$.
ProofOn one hand, the pull-back of the bundle $pi:EGlongrightarrow\; BG$ by the natural projection $P\; imes\_G\; EGlongrightarrow\; BG$ is the bundle $P\; imes\; EG$. On the other hand, the pull-back of the principal $G$-bundle $Plongrightarrow\; M$ by the projection$p:P\; imes\_G\; EGlongrightarrow\; M$ is also $P\; imes\; EG$


Since $p$ is a fibration with contractible fibre $EG$,sections of $p$ exist [A.~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 $P\; imes\_G\; EGlongrightarrow\; BG$. The map we get is the $f$ we werelooking for.
For the uniqueness up to homotopy, notice that there exists a one to one correspondence between maps$f:Mlongrightarrow\; BG$ such that $f^*EGlongrightarrow\; M$ is isomorphic to $Plongrightarrow\; M$ and sections of $p$. We have just seenhow to associate a $f$ to a section. Inversely, assume that $f$ is given. Let $Phi$ be an isomorphismbetween $f^*EG$ and $P$
$Phi:\; \{(x,u)in\; M\; imes\; EGmid,f(x)=pi(u)\}\; longrightarrow\; P$.
Now, simply define a section by

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" = "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

* Classifying space for U(n)

ee also

* Chern class

External links

* [http://planetmath.org/?op=getobj&from=objects&id=3663 PlanetMath page of universal bundle examples]

Notes