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?



Question regarding Chow group of a blow-up
Say $Y$ is $\mathbb P^1$ and has codimension $2$, so $E$ is a $\mathbb P^1$-bundle on $\mathbb P^1$. Then this $\mathbb P^1$-bundle has many sections, which are not equivalent in the Chow group, but only equivalence class contains pullback of $Y$, which is the pullback of the pushforward of any section. 


