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 $EG\longrightarrow BG$ is a $G$-principal fibre bundle.
Proof There exists an injection of $G$ into a unitary group $U(n)$ for $n$ big enough. If we find $EU(n)$ then we can take $EG$ to be $EU(n)$.

Let $F_{n}(\mathbb {C} ^{k})$ be the space of orthonormal families of $n$ vectors in $\mathbb {C} ^{k}$. The group $U(n)$ acts freely on $F_{n}(\mathbb {C} ^{k})$ and the quotient is the Grassmannian $G_{n}(\mathbb {C} ^{k})$ of $n$-dimensional subvector spaces of $\mathbb {C} ^{k}$. The map
${\begin{aligned}F_{n}(\mathbb {C} ^{k})&\longrightarrow &S^{2k-1}\\(e_{1},\ldots ,e_{n})&\longmapsto &e_{n}\end{aligned}}$
is a fibre bundle of fibre $F_{n-1}(\mathbb {C} ^{k-1})$. Thus because $\pi _{p}(S^{2k-1})$ is trivial and because of the long exact sequence of the fibration, we have
$\pi _{p}(F_{n}(\mathbb {C} ^{k}))=\pi _{p}(F_{n-1}(\mathbb {C} ^{k-1}))$
whenever $p\leq 2k-2$. By taking $k$ big enough, precisely for $k>{\frac {1}{2}}p+n-1$, we can repeat the process and get
$\pi _{p}(F_{n}(\mathbb {C} ^{k}))=\pi _{p}(F_{n-1}(\mathbb {C} ^{k-1}))=\ldots =\pi _{p}(F_{1}(\mathbb {C} ^{k+1-n}))=\pi _{p}(S^{k-n})$.
This last group is trivial for $k>n+p$. Let
$EU(n)={\lim _{\rightarrow }}\;_{k\rightarrow \infty }F_{n}(\mathbb {C} ^{k})$
be the direct limit of all the $F_{n}(\mathbb {C} ^{k})$ (with the induced topology), which we will also denote by $F_{n}(\mathbb {C} ^{\infty })$.
Lemma
The group $\pi _{p}(F_{n}(\mathbb {C} ^{\infty }))$ is trivial for all $p$.
Proof Let $\gamma$ be a map from the sphere $S^{p}$ to $F_{n}(\mathbb {C} ^{\infty })$. As $S^{p}$ is compact, there exists $k$ such that $\gamma (S^{p})$ is included in $F_{n}(\mathbb {C} ^{k})$. By taking $k$ big enough, we see that $\gamma$ is homotopic, with respect to the base point, to the constant map. $\Box$

In addition, $U(n)$ acts freely on $F_{n}(\mathbb {C} ^{\infty })$. The spaces $F_{n}(\mathbb {C} ^{k})$ and $G_{n}(\mathbb {C} ^{k})$ are CW-complexes. One can find a decomposition of these spaces into CW-complexes such that the decomposition of $F_{n}(\mathbb {C} ^{k})$, resp. $G_{n}(\mathbb {C} ^{k})$, is induced by restriction of the one for $F_{n}(\mathbb {C} ^{k+1})$, resp. $G_{n}(\mathbb {C} ^{k+1})$. Thus $F_{n}(\mathbb {C} ^{\infty })$ (and also $G_{n}(\mathbb {C} ^{\infty })$) is a CW-complexe. By Whitehead Theorem and the above Lemma, $F_{n}(\mathbb {C} ^{\infty })$ is contractible.$\Box$

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 $f:M\longrightarrow BG$, well defined up to homotopy, such that $P$ is isomorphic to $f^{*}(EG)$, 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 $P\times _{G}EG\longrightarrow BG$ is the bundle $P\times G$. On the other hand, the pull-back of the principal $G$-bundle $P\longrightarrow M$ by the projection $p:P\times _{G}EG\longrightarrow M$ is also $P\times EG$

${\begin{aligned}P&\longleftarrow &P\times EG&\longrightarrow &EG\\\downarrow &&\downarrow &&\downarrow \pi \\M&\longleftarrow ^{\!\!\!\!\!\!\!p}&P\times _{G}EG&\longrightarrow &BG.\end{aligned}}$
Since $p$ is a fibration with contractible fibre $EG$, sections of $p$ exist. To such a section $s$ we associate the composition with the projection $P\times _{G}EG\longrightarrow BG$. 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 $f:M\longrightarrow BG$ such that $f^{*}EG\longrightarrow M$ 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 between $f^{*}EG$ and $P$
$\Phi :\{(x,u)\in M\times EG\mid \,f(x)=\pi (u)\}\longrightarrow P$.
Now, simply define a section by
${\begin{aligned}M&\longrightarrow &P\times _{G}EG\\x&\longrightarrow &\lbrack \Phi (x,u),u\rbrack .\end{aligned}}$
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)

See also

• Chern class

External link

• PlanetMath page of universal bundle examples