TITLE: What morphisms induce injective/surjective maps on (Weil) cohomology?
QUESTION [10 upvotes]: Let $k$ be a field, let $f \colon X \to Y$ be a morphism of $k$-varieties, and assume $X$ and $Y$ are smooth and projective. Let $H(\_)$ be a classical Weil cohomology theory (i.e. one of $\ell$-adic étale, Betti, algebraic de Rham, crystalline). Of course it depends on $k$ which of the theories are applicable, but I do not want to spell all the cases out here.

What conditions can one impose on $f$ to assert that $f^{*} \colon H^{i}(Y) \to H^{i}(X)$ is injective/surjective?

Currently I can only think of the rather trivial: if $f$ admits a section (resp. retraction), then $f^{*}$ is injective (resp. surjective).
[Edit] For example, is it true that $f^{*}$ is injective if $f$ is dominant? [/Edit]

REPLY [6 votes]: For posterity, and as it was a bit tricky to find a reference, note also that the statement holds for possibly non-projective, quasi-projective smooth varieties, or, more generally for non-compact smooth Kähler manifolds:
Theorem (Wells): Let $f:X\to Y$ be a proper surjective morphism of smooth complex manifolds and assume that $X$ is Kähler. Then the following induced homomorphisms are injective

$f^*:H^q(Y,\Omega^p_Y)\to H^q(X,\Omega^p_X)$
$f^*:H^k(Y,\mathbb{C})\to H^k(X,\mathbb{C})$
$f^*:H^k(Y,\Omega^p_Y(E))\to H^k(X,\Omega^p_X(f^*E))$ for a holomorphic vector bundle $E$.

The reference is Wells Comparison of de Rham and Dolbeault cohomology for proper surjective mappings.