Symplectic Mapping Class Group Relations Generalizing the Chain Relation

TL;DR

This paper explores symplectic mapping class group relations for certain Weinstein domains, generalizing classical chain relations and providing explicit descriptions of symplectomorphisms in specific cases.

Contribution

It introduces new symplectic relations for Weinstein domains derived from polynomial-defined manifolds, extending classical chain relations to higher dimensions.

Findings

Decomposition of boundary Dehn twists into Lagrangian sphere twists

Explicit symplectomorphism descriptions for n=2 case

Generalization of classical chain relation to higher dimensions

Abstract

In this paper, we examine mapping class group relations of some symplectic manifolds. For each $n\geq 1$ and $k \geq 1$, we show that the $2n$-dimensional Weinstein domain $W = \{f=\delta\} \cap B^{2n+2}$, determined by the degree $k$ homogeneous polynomial $f\in \mathbb{C}[z_0,\dots,z_n]$, has a Boothby-Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case $n=2$ recovering the classical chain relation for the torus with two boundary components.