TITLE: Is there an alternative characterisation of vector bundles with vanishing characteristic classes?
QUESTION [16 upvotes]: This question came up yesterday during our index theory seminar.
Let $M$ be a 1-connected smooth manifold and let $E \to M$ be a finite-rank complex vector bundle over $M$.  If all the Chern classes of $E$ vanish, what else can one say about $E$?  In other words, is there an alternative characterisation of such $E$?
(For a similar question in the case of $M$ having nontrivial fundamental group, see this previous question.)

REPLY [20 votes]: Since no one has mentioned it yet, let me point out one possibly interesting observation.  If the base manifold $M$ is compact and has no torsion in its integral cohomology, then a vector bundle $E$ with vanishing Chern classes is stably trivial.  This was pointed out to me by Robert Lipshitz.  The reason is as follows: from looking at the Atiyah-Hirzebruch spectral sequence, one can see that there can't be any torsion in the complex K-theory of $M$.  Looking at the Chern character, one concludes that $[E]$ must be trivial in $\widetilde{K}^0(M)$, i.e. $E$ is stably trivial.