Question

I want to know something about torsion in topological k-theory. So, consider complex bundle with chern classes lying in torsion part of integer homologies and my question is : does it admit a flat connection? In the case of line bundles (for wich flat structure does't implies triviality) this obviously true, all you need is just add to your connection with curvature $\Omega$ 1-form A with $dA=\Omega$, so it gives curvature-free connection.

Answer 1

No. A complex vector bundle on $S^5$ must have Chern classes zero, and in this case a flat bundle would have to be trivial, but there is a nontrivial bundle because $\pi_4U(2)$ is nontrivial.