I was wondering if we have some results on "naturality" of the genus of an homology class. More specifically:
Let $[C] \in H_2(X;\Bbb Z)$ be an homology class represented by an orientable submanfold $\Sigma \subset X$ of a fixed orientable $n$-dimensional manifold $X$. Assume $X$ is closed. If $[C]$ can be represented by an orientable surface embedded in $X$, we can ask what the minimum genus of such representative should be. The adjunction formula for complex surfaces let us compute it. We know something about it for symplectic manifold thanks to the solved Symplectic Thom conjecture (now a theorem). Call such minimum $g([C])$.
Let us assume that $\Sigma$ realise the minimum of the genus of representatives for $[C]$, i.e. $g(\Sigma)=g([C])$
Given a map $f \colon X \to X'$, we can consider $f_*[C] \in H_2(X';\Bbb Z)$. It's natural to ask what's the minimal genus of a representative for $f_*[C]$.
Clearly $g(f_*[C])$ could be lower (take $f=\text{const.}$), I'm wondering if we know more about it, like that cannot be higher than $g([C])$.