On symplectic stabilisations and mapping classes

TL;DR

This paper explores the relationship between symplectic mapping class groups in higher dimensions and classical surface mapping class groups, focusing on configurations of Lagrangian spheres and their associated Dehn twists.

Contribution

It establishes conditions under which relations among Dehn twists in higher-dimensional Lagrangian spheres imply similar relations among surface Lagrangians, linking symplectic topology across dimensions.

Findings

Relations between Dehn twists in higher dimensions mirror those in surfaces.

Configurations of Lagrangian spheres with matching plumbing graphs are analogous.

Corollaries provide insights into subgroups of symplectic mapping class groups.

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.