On symplectic stabilisations and mapping classes

Abstract

We are interested in comparing properties of symplectic mapping class groups of symplectic manifolds of dimension four or higher with properties of classical mapping class groups of surfaces. For $n \geq 2$, consider a configuration of Lagrangian $S^n$s in a Weinstein domain $M^{2n}$. If it is analogous, in some sense that we make precise, to a configuration of exact Lagrangian $S^1$s on a surface $\Sigma$, we show that any relation between Dehn twists in the $S^n$s must also hold between the $S^1$s. Such analogous pairs of configurations include plumbings of $T^\ast S^1$s and $T^\ast S^n$s with the same plumbing graph, and vanishing cycles for a two-variable singularity and for its stabilisation. We give a number of corollaries for subgroups of symplectic mapping class groups.