Twisted homological stability for handlebody mapping class groups

TL;DR

This paper proves twisted homological stability for handlebody mapping class groups, extending previous results to include marked boundary discs and points, and applies this to moduli spaces with tangential structures.

Contribution

It extends twisted homological stability results for handlebody groups to include arbitrary boundary markings and introduces coefficient bisystems for this purpose.

Findings

Homology stabilizes with respect to genus and boundary markings.

Stability results apply to moduli spaces of 3D handlebodies with tangential structures.

Refines previous stability theorems for handlebodies.

Abstract

We prove twisted homological stability for handlebody mapping class groups. Using the categorical framework developed by Randal-Williams and Wahl, we establish that the homology of the handlebody groups stabilises with respect to both genus and the number of marked boundary discs, for all coefficient systems of finite degree. Our first main theorem refines and extends the twisted stability result for handlebodies outlined by Randal-Williams and Wahl, allowing any number of marked discs and boundary points. We then introduce the notion of coefficient bisystem to treat stability under variation of boundary markings. As an application, we deduce homological stability for moduli spaces of 3-dimensional handlebodies equipped with tangential structures.