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

Abstract

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

arXiv comment27 pages. Comments are welcome!