Every principal $G$-bundle over a surface is trivial if $G$ is compact and simply connected: reference?

Question

I'm looking for a reference for the following result:

If $G$ is a compact and simply connected Lie group and $\Sigma$ is a compact orientable surface, then every principal $G$-bundle over $\Sigma$ is trivial.

The proof supposedly uses homotopy theory and classifying spaces. I'm not very familiar with either, so I don't know where to start looking.

Answer 1

The homotopy theoretic proof is as follows: Let $E \longrightarrow \Sigma$ be a principal $G$-bundle over a surface $\Sigma$. Such a bundle is determined by a homotopy class $[f_E] \in [\Sigma, BG]$ by classifying space theory. Since $G$ is simply connected (and presumably connected), the classifying space $BG$ is $2$-connected (i.e. connected, simply connected, and $\pi_2(BG) = 0$). A surface $\Sigma$ has the homotopy type of a $2$-dimensional CW-complex. Hence $[\Sigma, BG] = 0$ and it must be that $[f_E] = 0$, so that $E$ is the trivial bundle.

We have the following elementary fact in general:

Fact 1. If $X$ is an $n$-dimensional CW complex and $Y$ is $n$-connected, then $[X, Y] = 0$.

Furthermore, a basic property of classifying spaces is the following:

Fact 2. If $G$ is $(n-1)$-connected, then $BG$ is $n$-connected.

By the same argument as above, these two facts imply the following general fact:

Fact 3. If $G$ is an $(n-1)$-connected Lie group and $X$ is an $n$-dimensional manifold, then every principal $G$-bundle over $X$ is trivial.

For example, every $\mathrm{SU}(2)$-bundle over a $3$-manifold is trivial.

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I'll describe a few references here. First of all, if you want to know exactly how the classifying space $BG$ is constructed and that the principal $G$-bundle $EG \longrightarrow BG$ is universal, you can read sections 4.11-4.13 of Husemöller's Fibre Bundles. The main idea is the following:

For a topological group $G$, there is a topological space $BG$ and a principal $G$-bundle $EG \longrightarrow BG$ with $EG$ contractible such that for any paracompact space $X$, isomorphism classes of principal $G$-bundles over $X$ are in one-to-one correspondence with homotopy classes of maps from $X$ to $BG$ as follows: $$[X, BG] \leftrightarrow \text{Prin}_G(X),$$ $$[f] \leftrightarrow f^\ast EG.$$

Here $f^\ast EG$ is the pullback of $EG \longrightarrow BG$ by any map $f$ representing to homotopy class $[f] \in [X, BG]$. So to classify principal $G$-bundles over a paracompact space $X$ (e.g. a manifold), we can look at homotopy classes of maps from $X$ to $BG$. If we're lucky, some properties of $X$ and $BG$ can let us find $[X, BG]$ without doing much work.

I don't know of a place where Fact 1 appears as a theorem, but it is Exercise 117 in Davis and Kirk's Lecture Notes in Algebraic Topology. You can solve by just working cell-by-cell.

Fact 2 follows from the long exact sequence of a fibration in homotopy and the fact that $EG$ is contractible. The long exact sequence of a fibration goes as follows: if $F \hookrightarrow E \to B$ is a fibration, then we have a long exact sequence $$\cdots \to \pi_{k+1}(B) \to \pi_k(F) \to \pi_k(E) \to \pi_k(B) \to \pi_{k-1}(F) \to \cdots.$$ If we apply this to $G \hookrightarrow EG \to BG$ and use the fact that $EG$ is contractible, we find that $\pi_{k+1}(BG) \cong \pi_k(G)$, so if $G$ is $n$-connected, $BG$ is $(n+1)$-connected.

Finally, Fact 3 is just a corollary of Facts 1 and 2 as well as the basic properties of classifying spaces.