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I have two questions about what I write below (which honestly seems pretty elementary).

  1. Is it true (more or less)?
  2. Is there a clean reference that I can cite.

Let $G$ be a compact Lie group, $M$ a smooth manifold, and $P\to M$ a principal $G$-bundle. One may ask: "does the principal $G$-bundle $P$ admit a flat ($G$-)connection?" In general, this is supposed to be not easy, though there is a necessary condition given by Chern-Weil theory: if $P$ admits a flat connection, then "all" characteristic classes in $\mathbb{R}$-coefficient cohomology vanish.

Now suppose the compact Lie group $G$ is such that its identity component $G_0$ is abelian. I want a necessary-and-sufficient characteristic class criterion for admitting a flat $G$-connection, which I think goes like this:

Given $P\to M$, pick a connection $A$ on $P$, which will have a curvature form $F_A\in \Omega^2(M, \mathfrak{g}_P)$, where $\mathfrak{g}_P=P\times_G \mathfrak{g}_{\mathrm{ad}}$, the associated vector bundle of $P$ via the adjoint representation.

Because $G_0$ is abelian, the adjoint representation factors through the discrete group $G/G_0$, and thus the vector bundle $\mathfrak{g}_P$ is actually a local system of vector spaces on $M$. Thus, we can consider the local cohomology groups $H^*(M,\mathfrak{g}_P)$. The claim is

  • $[F_A]\in H^2(M,\mathfrak{g}_P)$ only depends on $P$ (not $A$), so write $F(P):=[F_A]$, and

  • $P$ admits a flat connection iff $F(P)=0$.

Given this, you expect that

  • there exists a universal element $F\in H^2(BG, \mathfrak{g}_{\mathrm{ad}})$ giving this class.

The Chern-Weil theorem says that $H^*(BG,\mathbb{R})\approx \mathrm{Sym}(\mathfrak{g}_{\mathrm{ad}}^*)^G$. Presumably this generalizes: for any local system $V$ of $\mathbb{R}$-vector spaces on $BG$ (i.e., representation of $\pi_1BG=G/G_0$), we should have a natural isomorphism $H^*(BG, V)\approx \mathrm{Hom}(\mathrm{Sym}(\mathfrak{g}_{\mathrm{ad}}), V)^G$. So we expect

  • to identify $F\in H^2(BG,\mathfrak{g}_{\mathrm{ad}}) \approx \mathrm{Hom}(\mathfrak{g}_{\mathrm{ad}},\mathfrak{g}_{\mathrm{ad}})^G$ with the idenity map of $\mathfrak{g}_{\mathrm{ad}}$ (up to maybe a scalar).

When $G$ is abelian, this seems to be pretty well known: $F(P)=0$ is the same as $\kappa(P)=0$ for all $\kappa\in H^2(BG,\mathbb{R})$. Unfortunately, I can't find any references that talk about non-abelian $G$ of this type.

Example: For $G=O(2)$, we have $H^2(BO(2), \mathfrak{g}_{\mathrm{ad}})\approx \mathbb{R}$, whose generator $e$ is a non-orientable version of the Euler class. Consider the tangent bundle to $\mathbb{RP}^2$ with a metric, reducing the structure group to $O(2)$. Then, $e(T\mathbb{RP}^2)\neq0$ in $H^2(\mathbb{RP}^2,\mathbb{R}_{\mathrm{sgn}})$ so $T\mathbb{RP}^2$ does not admit a flat $O(2)$-connection, though $H^*(BO(2),\mathbb{R})\approx \mathbb{R}[p_1]$ and of course $p_1(T\mathbb{RP}^2)=0$.

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