Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies

TL;DR

This paper demonstrates that a specific subset of the moduli space of hyperbolic surfaces, characterized by short geodesics, serves as a classifying space for handlebody mapping class groups, linking top weight cohomology to these groups.

Contribution

It establishes a new geometric realization of the handlebody mapping class group within the moduli space and connects top weight cohomology to this group.

Findings

The locus with many short geodesics is a classifying space for the handlebody mapping class group.

Top weight cohomology injects into the cohomology of the handlebody mapping class group.

Provides new insights into the structure of moduli spaces and their cohomological properties.

Abstract

We show that a certain locus inside the moduli space $M_g$ 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 $M_g$, studied by Chan-Galatius-Payne, maps injectively into the cohomology of the handlebody mapping class group.