Embedded surfaces with trivial extendable mapping class groups in simply connected $4$-manifolds

Abstract

For every $g\geq 3$, every closed, connected, oriented, simply connected smooth $4$-manifold $X$, and every knot $K\subset S^3$, we construct infinitely many pairwise topologically inequivalent smoothly embedded oriented genus-$g$ surfaces $F\subset X$ whose orientation-preserving extendable mapping class subgroups are trivial in both the topological and smooth categories and whose first Alexander modules are isomorphic to the Alexander module of $K$. In particular, there are infinitely many such surfaces with vanishing first Alexander module. The construction is supported in a $4$-ball. Although their Alexander data are prescribed independently, the surfaces are distinguished, and their mapping-class rigidity is detected, by the nonabelian centralizer structure of their exterior groups.