Let $X$ and $Y$ be compact oriented smooth 4-manifolds. There is a well-defined intersection form $H_2(X)\times H_2(X)\to \Bbb Z$, $(\alpha,\beta)\mapsto \alpha \cdot \beta$, and similarly for $Y$ (https://en.wikipedia.org/wiki/Intersection_form_of_a_4-manifold).
Suppose $f:X\to Y$ is an $n$-fold covering map. For a homology class $\alpha\in H_2(X)$, is there a relation between the self intersection $\alpha^2=\alpha \cdot \alpha$ and $(f_*\alpha)^2$? The reason I am interested in this problem is that I think it may help the question: Intersection of two lines in weighted projective plane.
Any related comments will be appreciated.