Q:

Nontrivial flat vector bundle over manifold with amenable fundamental group

Given a smooth, closed, oriented $n$-manifold $M$ with amenable fundamental group $\pi_1(M)$. Is every $n$-dimensional flat orientable vector bundle $\xi = (p: E \rightarrow M)$ over $M$ trivial? 
By an $n$-dim orientable flat vector bundle I mean a vector bundle arising as $ E= \tilde{M} \times \mathbb{R}^n / \pi_1(M)$, where the fundamental group acts on the universal covering $\tilde{M}$ by deck transformations and on $\mathbb{R}^n$ via a homomorphism $\rho: \pi_1(M) \rightarrow GL_n^+(\mathbb{R}) $ and with $p$ induced by the canonical projection. 
I don't see why it should be true and I am searching for a counterexample with some manifold and bundle satisfying the assumptions, but since the Euler class always vanishes and so do all characteristic classes in cohomology with real coefficients (e.g. by M. Karlsson in her thesis "characteristic classes and bounded cohomology"), since the real bounded cohomology of $M$ is trivial, I haven't been able to come up with a counterexample myself yet. But my approach would remain to have a bundle some nontrivial characteristic classes that are torsion. I appreciate any suggestions. 

A:

Like you say, we cannot get real characteristic class obstructions because of the lack of curvature. So we look to the second Stiefel-Whitney class. Googling I find this paper. (I have not verified any claims in there, only extracted the relevant results.) Proposition 4.4 claims to construct a flat oriented non-spin manifold $M$. See section 2 for their notion of "acting diagonally" and the beginning of section 4 for the (complicated-looking) construction of the action they want. 
$M$ is given as a quotient of $T^n$ by a free action of $\Bbb Z_2^d$; in particular, its fundamental group fits into the exact sequence $$1 \to \Bbb Z^n \to \pi_1(M) \to \Bbb Z_2^d \to 1.$$ Abelian groups are amenable and extensions of amenable groups are amenable so $\pi_1(M)$ is amenable.

