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

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.