Louis Hainaut: A classifying space for the handlebody mapping class group

Abstract

A classical problem in algebraic geometry and algebraic topology is to understand the moduli space \(\mathcal{M}_g\). This space has a multitude of interpretations, but for this talk we will focus specifically on two descriptions: on the one hand it is the classifying space of the mapping class group \(\operatorname{Mod}_g\), defined as the group of isotopy classes of (orientation-preserving) diffeomorphisms of a fixed surface \(\Sigma_g\) of genus \(g\); on the other hand it parametrizes hyperbolic surfaces of genus \(g\). Let \(V_g\) denote a 3-dimensional handlebody with \(g\) handles, or in other words a 3-dimensional manifold whose boundary is a surface of genus \(g\). We can define the handlebody mapping class group \(\operatorname{HMod}_g\) as the group of (orientation-preserving) diffeomorphisms of \(V_g\) up to isotopy and it is known that this handlebody mapping class group is a subgroup of the (surface) mapping class group. In recent joint work with Dan Petersen we constructed a classifying space for the handlebody mapping class group as an explicit open subspace of \(\mathcal{M}_g\). I will explain this construction and present some consequences for the homology of the handlebody mapping class group.