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

Abstract

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