A four-dimensional mapping class group relation

TL;DR

This paper establishes a new relation in the symplectic mapping class group of 4-dimensional Weinstein domains using holomorphic curve techniques, with implications for understanding Dehn twists and symplectic isotopy problems.

Contribution

It introduces a novel relation between products of Dehn twists in 4D symplectic topology, utilizing advanced holomorphic curve methods and solving a key isotopy problem.

Findings

Derived a relation between Dehn twist products in 4D Weinstein domains.

Provided an alternative proof connecting fibered Dehn twists and Dehn twist products.

Advanced understanding of symplectic isotopy in del Pezzo surfaces.

Abstract

In the symplectic mapping class group of a $4$-dimensional Weinstein domain, we give a relation between two products of (right-handed) Dehn twists via holomorphic curve techniques. A key ingredient of the construction is a solution to the symplectic isotopy problem on symplectic submanifolds in del Pezzo surfaces. In the appendix, we provide an alternative proof of a relation between a fibered Dehn twist and a product of Dehn twists.