Twisted homological stability for handlebody mapping class groups

TL;DR

This work proves twisted homological stability for handlebody mapping class groups with varying genus and boundary markings, via a categorical framework based on braided monoidal groupoids and the Quillen bracket, and introduces coefficient bisystems to handle boundary markings. It extends classic stability results to coefficient systems of finite degree and establishes stability ranges in terms of genus and marking data, with split refinements. The approach reduces stability questions to high-connectivity of destabilisation complexes, leveraging Hatcher–Wahl connectivity results, and culminates in applications to moduli spaces of handlebodies with tangential structures. The results yield stable homology for moduli spaces of 3D handlebodies with tangential structures over rational coefficients and provide tools for computing twisted homology in these settings, with explicit stability ranges and independence results for the number of marked discs.

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.

Figures (4)

• Figure 1.1: The handlebody $V_{g+1}$ is obtained from $V_{g}$ by gluing on a genus $1$ handlebody along the marked disc and the stabilisation map $\sigma_{g,1}$ is defined by extending diffeomorphisms to the new part by the identity.
• Figure 1.2: By gluing a "solid pair of pants" to the first marked disc of our handlebody, we obtain a handlebody with an additional marked disc.
• Figure 2.1: The monoidal product $V_{1}\natural_{r} V_{2}$ of two handlebodies in $\mathcal{G}_{\mathcal{G}}^{\ge r}$.
• Figure 2.2: Illustration of the braiding in $(\boldsymbol{\mathscr{H}}^{\ge r},\natural_{r},I_{r})$ in a neighborhood of one of the glued disc of $V_{1}\natural_{r} V_{2}$.

Theorems & Definitions (102)

• Theorem A: Theorem \ref{['thm:HS_stabilistation-1']}
• Remark 1.1
• Theorem B: Corollary \ref{['cor:independence-of-marked-discs']}
• Remark 1.2
• Theorem C: Theorem \ref{['thmC-detailed']}
• Remark 1.3
• Lemma 2.1
• proof
• Remark 2.2
• Definition 2.3