On the Dax invariants of $S^2$-bundles over surfaces

This paper studies the nontrivial orientable $S^2$-bundle $\Sigma \ltimes S^2$ over a closed surface $\Sigma$ of genus $g \geq 1$. We have three main results as follows. We construct the relative Dax invariants for pointed embeddings of $\Sigma$ into $\Sigma \ltimes S^2$, which satisfies isotopy invariance, additivity, and naturality. For some embedded surfaces in $\Sigma \ltimes S^2$ with a fixed geometric dual, we establish a complete classification up to isotopy. We give an alternative construction of a surjective homomorphism $\hat{\Phi}: {\rm MCG}(\Sigma \ltimes S^2) \to \mathbb{Z}^\infty$.