# The Meyer function on the handlebody group

## TL;DR

This paper provides an explicit formula for the signature of handlebody bundles over the circle, linking homological monodromy to Meyer's signature cocycle, and offers topological insights into the hyperelliptic handlebody group.

## Contribution

It introduces a new explicit formula for the signature of handlebody bundles and interprets the generator of the first cohomology of the hyperelliptic handlebody group.

## Key findings

- Explicit signature formula in terms of homological monodromy
- Cobounding function for Meyer's signature cocycle on handlebody group
- Topological interpretation of the hyperelliptic handlebody group's cohomology

## Abstract

We give an explicit formula for the signature of handlebody bundles over the circle in terms of the homological monodromy. This gives a cobounding function of Meyer's signature cocycle on the mapping class group of a $3$-dimensional handlebody, i.e., the handlebody group. As an application, we give a topological interpretation for the generator of the first cohomology group of the hyperelliptic handlebody group.