Isotopies of surfaces in 4-manifolds via banded unlink diagrams

TL;DR

This paper develops a comprehensive set of moves for relating banded unlink diagrams of isotopic surfaces in any 4-manifold, extending previous work and applying it to prove conjectures and classify surfaces in complex projective space.

Contribution

It introduces a complete set of moves for banded unlink diagrams in arbitrary 4-manifolds, extending prior results and applying to bridge trisections and isotopies in complex projective space.

Findings

Proved that bridge trisections of isotopic surfaces are related by perturbations.

Showed many unit surfaces in \\mathbb{C}P^2 are isotopic to the standard \\mathbb{C}P^1.

Strengthened results related to the Gluck twist in $S^4$.

Abstract

In this paper, 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 $S^4$. 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 $\mathbb{C}P^2$ (i.e. spheres in the generating homology class), proving that many explicit unit surfaces are isotopic to the standard $\mathbb{C}P^1$. This strengthens some previously known results about the Gluck twist in $S^4$, related to Kirby problem 4.23.