2026/08/02 by Weizhe Niu
Mathematics · #math.GT
84 pages, 3 figures. Comments welcome
arxiv created 2026/08/02 · arxiv updated 2026/08/04
For every g≥ 3, every closed, connected, oriented, simply connected smooth 4-manifold X, and every knot K⊂ S3, we construct infinitely many pairwise topologically inequivalent smoothly embedded oriented genus-g surfaces F⊂ 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.