Let $X$ be a smooth, projective variety and $i:Y \hookrightarrow X$ a smooth divisor. Let $Z \subset X$ be a proper, closed subvariety disjoint from $Y$. Let $\pi:\widetilde{X} \to X$ be the blow-up of $X$ along $Z$. Assume that $\widetilde{X}$ is regular. Denote by $j: Y \hookrightarrow \widetilde{X}$ (as $Z$ does not intersect $Y$, the strict transform of $Y$ is isomorphic to itself). Let $\tau \in H^m(Y,\mathbb{C})$. Is $(\pi^* \circ i_*)\tau=j_*\tau$? Here $i_*$ and $j_*$ denotes the associated Gysin morphisms.

Any reference/idea will be most welcome.