A twisted first homology group of the handlebody mapping class group

TL;DR

This paper computes the twisted first homology group of the handlebody mapping class group with coefficients in the boundary surface's first homology, providing new algebraic insights into the structure of these groups.

Contribution

It determines the twisted first homology group of the handlebody mapping class group with specific coefficients, a novel calculation in geometric topology.

Findings

Explicit description of the twisted first homology group for genus g ≥ 2.

New algebraic invariants for handlebody mapping class groups.

Enhanced understanding of the group's structure through homological computations.

Abstract

Let $H_{g}$ be a 3-dimensional handlebody of genus $g$. We determine the twisted first homology group of the mapping class group of $H_{g}$ with coefficients in the first integral homology group of the boundary surface $\partial H_{g}$ for $g\geq 2$.