Symplectic Mapping Class Group Relations Generalizing the Chain Relation

In this paper, we examine mapping class group relations of some symplectic manifolds. For each $n\ge 1$ and $k\ge 1$, 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.