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

Let ${\mathrm{\Sigma }}_g^0\subset S^4$, $g\ge 3$, be the standard unknotted closed oriented surface, and let $a\subset {\mathrm{\Sigma }}_g^0$ be an oriented nonseparating curve. For every nontrivial knot $J\subset S^3$, let ${\mathrm{\Sigma }}_{g,a,J}\subset S^4$ be the surface obtained from ${\mathrm{\Sigma }}_g^0$ by ordinary untwisted rim surgery along $a$. We compute its extendable mapping-class subgroup exactly: $$E({\mathrm{\Sigma }}_{g,a,J})={\mathrm{Stab}}_{\mathrm{Mod}({\mathrm{\Sigma }}_g)}(q_0)\cap {\mathrm{Stab}}_{\mathrm{Mod}({\mathrm{\Sigma }}_g)}({\mathrm{\Gamma }}_{\mu }(J)\cdot [a])\mathrm{.}$$ Here $q_0$ is the Rokhlin quadratic form of the standard embedding, $[a]\in H_1({\mathrm{\Sigma }}_g;\mathbb{Z})$ is the oriented rim homology class, and ${\mathrm{\Gamma }}_{\mu }(J)\subset \{\pm 1\}$ 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 knot-dependent ambiguity coming from this peripheral longitude sign. We also classify the corresponding ambient pairs $(S^4,{\mathrm{\Sigma }}_{g,a,J})$. Two such pairs, obtained by one ordinary rim surgery, are orientation-preservingly diffeomorphic exactly when their Rokhlin forms, rim homology classes, and meridian-longitude data of the knot exteriors agree with the corresponding sign.