In this talk, I will present a generalization of the Dax invariant for closed embedded surfaces in 4-manifolds. As an application, we classify the isotopy classes of all embeddings of $\Sigma$ in $S^2 \times \Sigma$ that are geometrically dual to $S^2\times pt$; when $\Sigma$ has a positive genus, there are infinitely many such isotopy classes.  Since $\pi_1(\Sigma)$ has no 2-torsion, this answered a question of Gabai. Using the Dax invariant, we also show that the mapping class group of $S^2\times \Sigma$ has a surjection to $Z^\infty$ whose restriction to the subgroup of homotopically trivial elements is also of infinite rank.  The talk is based on joint work with Jianfeng Lin and Yi Xie.