# A twisted first homology group of the Goeritz group of 3-sphere

## TL;DR

This paper computes the twisted first homology group of the genus-2 Goeritz group of the 3-sphere, providing new algebraic insights into the symmetries of Heegaard splittings.

## Contribution

It is the first to explicitly determine the twisted first (co)homology group of the genus-2 Goeritz group of the 3-sphere.

## Key findings

- Computed the twisted first homology group of the genus-2 Goeritz group.
- Provided algebraic structure insights into the symmetries of genus-2 Heegaard splittings.
- Enhanced understanding of the mapping class group actions on 3-sphere splittings.

## Abstract

Given a genus-g Heegaard splitting of a 3-sphere, the genus-g Goeritz group is defined to be the group of the isotopy classes of orientation preserving homeomorphism of the 3-sphere that preserve the splitting. In this paper, we determine the twisted first (co)homology group of the genus-2 Goeritz group of 3-sphere.