Suppose we know that all stable characteristic classes of the tangent bundle of a manifold $M$ vanish, i.e. the map $f:M\rightarrow BO$ stably classifying the tangent bundle is trivial on cohomology, i.e. $0=f^*: H^*(BO, A)\rightarrow H^*(M, A)$, for all coefficients $A$. Is it then possible to find a stable framing of $M$? In other words, can we lift $f$ to a map $\tilde{f}: M\rightarrow B\{1\}\simeq *$?

I know this is not the case for a general vector bundle (as discussed in Hatcher's "Vector bundles and K-theory", p.75-76), but I was wondering whether it might be true when restricting to the case where the bundle is the tangent bundle of a manifold. And how about if it is a normal bundle?

4more comments