Say that I have a morphism of projective algebraic varieties $f: X \to Y$, which is birational. There is a pushforward of cycles morphism $f_*: N_*(X) \to N_*(Y)$.
Now, if I could pull back cycles and if I had a projection formula then I could say that $f_*$ is surjective. In fact, given a cycle $\alpha \in N_*(Y)$ I could consider $f_*f^*\alpha = f_*f^*(\alpha \cdot [Y]) = f_*(f^*\alpha \cdot [X]) = \alpha \cdot f_*[X] = \alpha ,$ giving me surjectivity of $f_*$.
In my situation $X$ is regular and $Y$ is Gorenstein (and I am working over $\mathbb{C}$): can I still say that $f_*$ is surjective?
EDIT: if it helps, I'm happy to assume f to be an isomorphism in codimension one.