Extendable mapping classes of knotted surfaces obtained by rim surgery in $S^4$

Abstract

Let $\Sigma_g^0\subset S^4$, $g\ge3$, be the standard unknotted closed oriented surface, and let $a\subset\Sigma_g^0$ be an oriented nonseparating curve. For every nontrivial knot $J\subset S^3$, let $\Sigma_{g,a,J}\subset S^4$ be the surface obtained from $\Sigma_g^0$ by ordinary untwisted rim surgery along $a$. We compute its extendable mapping-class subgroup exactly: $$ E(\Sigma_{g,a,J}) = \operatorname{Stab}_{\operatorname{Mod}(\Sigma_g)}(q_0) \cap \operatorname{Stab}_{\operatorname{Mod}(\Sigma_g)} (\Gamma_\mu(J)\cdot[a]). $$ Here $q_0$ is the Rokhlin quadratic form of the standard embedding, $[a]\in H_1(\Sigma_g;\mathbb{Z})$ is the oriented rim homology class, and $\Gamma_\mu(J)\subset\{\pm1\}$ records whether a meridian-preserving diffeomorphism of the knot exterior can preserve or reverse the preferred longitude. Thus ordinary rim surgery cuts Hirose's unknotted extendable subgroup by the stabilizer of the rim homology class, with the only additional ambiguity coming from this peripheral symmetry of $J$. We also prove a prescribed-mapping-class classification for such ambient pairs $(S^4,\Sigma_{g,a,J})$. More precisely, given two such pairs and $f\in\operatorname{Mod}(\Sigma_g)$, we characterize when $f$ is induced by an orientation-preserving pair diffeomorphism in terms of the Rokhlin quadratic form, the rim homology classes, and the meridian--longitude symmetries of the knot exteriors.