Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies

Abstract

We show that a certain locus inside the moduli space ${\mathcal{M}}_g$ of hyperbolic surfaces, given by surfaces with “sufficiently many” short geodesics, is a classifying space of the handlebody mapping class group. A consequence of the construction is that the top weight cohomology of ${\mathcal{M}}_g$, studied by Chan–Galatius–Payne, maps injectively into the cohomology of the handlebody mapping class group.