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

Abstract

Let $Σ_g^0\subset S^4$, $g\ge 3$, be the standard unknotted closed oriented surface, and let $a\subsetΣ_g^0$ be an oriented nonseparating curve. For a nontrivial knot $J\subset S^3$, let $Σ_{g,a,J}\subset S^4$ be the surface obtained by ordinary untwisted rim surgery along $a$. Assuming a meridian-longitude rigidity condition on the knot group of $J$, we compute the extendable mapping-class subgroup exactly: \[ E(Σ_{g,a,J})= \operatorname{Stab}_{\operatorname{Mod}(Σ_g)}(q_0)\cap \operatorname{Stab}_{\operatorname{Mod}(Σ_g)}([a]), \] where $q_0$ is the Rokhlin quadratic form of the standard embedding and $[a]\in H_1(Σ_g;\mathbb Z)$ is the oriented homology class.