AI 中文总结
该研究在单连通4维流形中构造无穷多类满足特定映射类群与Alexander模条件的嵌入曲面，借助补集群结构实现区分与刚性检测。
AI 中文摘要
对每个整数g≥3、每个闭连通定向单连通光滑4维流形X及每个纽结K⊂S³，我们构造无穷多个两两拓扑不等价的光滑嵌入定向亏格g曲面F⊂X；这些曲面的保定向可延拓映射类子群在拓扑与光滑范畴中均为平凡，且其一阶Alexander模与K的Alexander模同构，特别地存在无穷多个一阶Alexander模消失的此类曲面。该构造在一个4维球内完成，尽管其Alexander数据独立给定，但曲面可通过补集群的非阿贝尔中心化子结构区分，其映射类刚性也由此得以检测。
英文摘要
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.
Comments84 pages, 3 figures. Comments welcome
