I am looking for an example of an oriented rank 5 (or lower) real vector bundle $V$ over an oriented manifold such that the cup product $w\_2(V) w\_3(V)$ of Stiefel-Whitney classes does not vanish. It would be best if the manifold had dimension 7 or lower.



Vector bundle over an oriented manifold with non-vanishing w_2w_3
As far as I know the Wu manifold $X=SU(3)/SO(3)$ is orientable and has mod 2 cohomology ring $H^\*(X;\mathbb{Z}\_2)=\Lambda(\omega\_2(X),\omega\_3(X))$. Thus $\omega\_2(X)\cdot\omega\_3(X)\neq 0$, and in fact generates $H^5(X;\mathbb{Z}\_2)$.


