Let $X$, $Y$ be complex normal quasi-projective varieties and let $f\colon X\to Y$ be a dominant morphism. I would like to ask if the following statements are true:
$f_{*}\colon H_{1}(X; \mathbb{Q})\to H_{1}(Y; \mathbb{Q})$ is surjective.
Assume further that the fibers of $f$ are connected. Then $f_{*}\colon H_{1}(X; \mathbb{Z})\to H_{1}(Y; \mathbb{Z})$ is surjective.
For example, if $X$, $Y$ are projective curves then these assertions hold by the Hurwitz formula. I think I can prove assertion 1. using the answers to this question (after compactifying and resolving singularities), but I am not sure, and anyway I would like to ask if there is any simpler way to show this.
The additional assumption in (2) is clearly necessary, since we can kill torsion by a finite cover.
EDIT: as @KReiser points out below, it suffices to show it for an open inclusion $X\subset Y$. Indeed, since $f$ is generically smooth, there is a proper closed subset $D\subsetneq Y$ such that $f|_{X\setminus f^{-1}(D)}\colon X\setminus f^{-1}(D) \to Y\setminus D$ is a fiber bundle (in the smooth category), so it satisfies (1) and (2) by the Serre exact sequence. Now if the map $\iota_{*}\colon H_{1}(Y\setminus D;\mathbb{Z})\to H_{1}(Y;\mathbb{Z})$ induced by the inclusion is surjective, then (1) and (2) follow by the Mayer-Vietoris sequence.
I think that if $\mathrm{codim}\,_{Y} D\geq 2$ then $\iota_{*}$ is a isomorphism by the Nagata purity theorem. In any case, the surjectivity of $\iota_{*}$ should somehow follow from the fact that $D$ is of real codimension at least two. However, I find it difficult to fill in the details.