4 precise reference to the Atiyah Patodi and Singer statement added

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 positive integer $m$ such that the direct sum $E\oplus E\oplus\cdots\oplus E$ ($m$ times) is a trivial bundle. More precisely, they write on top of page 19, "The vector bundle $V_a$ defined by $a$ is flat so its real Chern classes vanish, hence some multiple $kV_a$ is (unitarily) trivial".

Of course since the Chern character is an isomorphism from $K^\bullet(X)\otimes \mathbb{Q}$ to $H^{2\bullet}(X;\mathbb{Q})$, we know the bundle $E$ (or, more precely its K-theory class) is torsion in topological K-theory but this only tells us that $E\oplus E\oplus\cdots\oplus E$ is stably trivial.

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

Other authors say that the above statement cannot be true.

Does someone know the answer ?

3 improved formatting

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 positive integer $m$ such that a the direct sum $E^{\oplus m}$ E\oplus E\oplus\cdots\oplus E$($m$times) is the a trivial bundle. Of course since the Chern character is isomorphic modulo torsion an isomorphism from$K^\bullet(X)\otimes \mathbb{Q}$to$H^{2\bullet}(X;\mathbb{Q})$, we know the bundle$E$(or, more precely its K-theory class) is torsion in topological K-theory but this only tells us that$E\oplus E\oplus\cdots\oplus E$is not sufficient by stabilizationstably trivial. I checked an impressive amount of literature on characteristic classes without finding a clue. Several authors cite directly Atiyah who, however, does not prove the claim. Someone says Other authors say that it the above statement cannot be true. Does someone know the answer ? 2 LaTeX Good Morning. Atiyah Patodi and Singer [Spectral asymmetry and Riemannian geometry III] write that if E$E$is a complex flat bundle (non holomorphic, just smooth and complex) on a compact manifold X$X$(more generally a CW-complex) there must be some m$m$such that a direct some multiple E++++++++++E (m times) sum$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.