Mapping class group actions from Hopf monoids and ribbon graphs

Abstract

We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $\mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g\ge 1$ with $n\ge 1$ boundary components. These mapping class group actions are given by group homomorphisms into the group of automorphisms of certain Yetter–Drinfeld modules over $H$. They are associated with edge slides in embedded ribbon graphs that generalise chord slides in chord diagrams. We give a concrete description of these mapping class group actions in terms of generating Dehn twists and defining relations. For the case where $\mathcal{C}$ is finitely complete and cocomplete, we also obtain actions of mapping class groups of closed surfaces by imposing invariance and coinvariance under the Yetter–Drinfeld module structure.