Isotopies of surfaces in 4–manifolds via banded unlink diagrams

Abstract

© 2020, Mathematical Sciences Publishers. We study surfaces embedded in 4–manifolds. We give a complete set of moves relating banded unlink diagrams of isotopic surfaces in an arbitrary 4–manifold. This extends work of Swenton and Kearton–Kurlin in S4. As an application, we show that bridge trisections of isotopic surfaces in a trisected 4–manifold are related by a sequence of perturbations and deperturbations, affirmatively proving a conjecture of Meier and Zupan. We also exhibit several isotopies of unit surfaces in CP2 (ie spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard CP1. This strengthens some previously known results about the Gluck twist in S4, related to Kirby problem 4.23