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Tintin
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Question 1: Is there any known example for which the Gysin morphism $f_∗$ is surjective, at least when $k$ is even?

Yes, consider the immersion $\mathbb{P}^n \to \mathbb{P}^{n+1}$. It is surjective for all $k$ and $n$.

Question 2: Is there any known criteria to know if the Gysin morphism $f_∗$ is surjective?

This is precisely the Gysin long exact sequence: $$ \cdots \to H^{k}(Y)\to H^{k+2r}(X)\to H^{k+2r}(U)\to H^{k+1}(Y)\to\cdots $$ Apart from that I don't know of any other criteria.

Tintin
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