TITLE: Cohomology classes annihilated by pullbacks
QUESTION [12 upvotes]: A friend of mine is interested in examples of the following situation:
an oriented smooth fiber bundle $\pi \colon M \to B$ with $M$ and $B$ compact 
and a non-zero class $a \in H^3(B; \mathbb{Q})$
such that $\pi^* a=0$  in $H^3(M; \mathbb{Q})$. 
It is easy to construct such an example if the class $a$ is a product of a degree $1$ class and degree $2$ class; are there examples not of this kind?

REPLY [3 votes]: The Becker--Gottlieb transfer implies that $\pi^*$ is rationally a (split) monomorphism unless the Euler characteristic of the fibre is zero. Thus any proposed example must have this property.