Let $X$ be a smooth complex projective variety, and $Y\hookrightarrow X$ be a smooth projective subvariety. Let $\pi:\tilde{X}\rightarrow X$ be the blow-up along $Y$, and let $j:E\hookrightarrow \tilde{X}$ be the exceptional divisor.

Let $Z$ be a $k$-dimensional subvariety of $E$ such that $dim\,\pi(Z)=dim\,Z$. Now, $[Z]\in CH_k(E)$ is a $k$-cycle, and let $Z':=j_*([Z])$.

**Question:** What is the relation between $\pi^*\pi_*(Z')$ and $Z'$ ? Are they equal?