Good Morning. Atiyah Patodi and Singer [Spectral asymmetry and Riemannian geometry III] write that if $E$ is a complex flat bundle (non holomorphic, just smooth and complex) on a compact manifold $X$ (more generally a CW-complex) there must be some $m$ such that a direct sum $E^{\oplus m}$ is the trivial bundle.

Of course since the Chern character is isomorphic modulo torsion we know the bundle is torsion in topological K-theory but this is not sufficient by stabilization.

I checked an impressive amount of literature on characteristic classes without finding a clue. Several authors cite directly Atiyah who does not prove the claim.

Someone says that it cannot be true.

Does someone know the answer?