Genus two Goeritz equivalence in lens spaces $L(p,1)$

TL;DR

This paper investigates the action of the genus two Goeritz group on the homology of lens spaces $L(p,1)$, providing matrix descriptions and obstructions for curve equivalence.

Contribution

It offers a matrix-based description of the Goeritz group's action and identifies homology and homotopy obstructions for curve equivalence in lens spaces.

Findings

Matrix representation of the Goeritz group's action in $GL(4, \mathbb Z)$

Homology obstructions for Goeritz equivalence of curves

Homotopy obstructions for curve equivalence

Abstract

In this paper, we consider the action of the Goeritz group $\mathcal G_p$ for the genus two Heegaard splitting of the lens space $L(p,1)$ with $p\ge 2$ on the homology of the Heegaard surface. We describe the action in terms of matrices in $GL(4, \mathbb Z)$, and provide homology and homotopy obstructions for when two curves in the Heegaard surface are Goeritz equivalent.