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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.

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 ?

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 ?

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?

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.

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  ?

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 some multiple E++++++++++E (m times) 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  ?

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?