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

TL;DR

The paper studies genus two Goeritz equivalence for curves on the Heegaard surface of lens spaces $L(p,1)$ by encoding the Goeritz action as a $4\times4$ matrix representation $*: \mathcal{G}_p \to GL(4,\mathbb{Z})$ on a canonical homology basis. It establishes that the kernel of the star map is generated by $\beta^2$, and it provides a precise description of the image $*(\mathcal{G}_p)$, including explicit forms for small $p$ and a tractable subgroup description for large $p$ (via a projection to a top-left $2\times2$ block and a block-structure analysis). The paper then derives strong obstructions to Goeritz equivalence: homology obstructions yielding determinant and linear-relations constraints among homology vectors, and a homotopy obstruction showing that Goeritz equivalence among homologous curves is equivalent to being connected by reducing-sphere twists, with $\beta^2$ generating this twist subgroup. Together, these results provide concrete, computational criteria for deciding when two curves on the genus two Heegaard surface are Goeritz equivalent in $L(p,1)$, enriching the toolbox for studying lens-space knots and Dehn surgeries via Heegaard splittings.

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.

Figures (15)

• Figure 1: Standard homology basis for $F$, and the axis of rotation for the homeomorphisms $\alpha$ and $\beta$.
• Figure 2: Pants decomposition of $F$ used to describe $\delta$, together with the resulting curves and sub-arcs of the homology basis, and $\partial D$, $\partial E$, $\partial F$.
• Figure 3: The image $\delta(a)$, superimposed onto Figure \ref{['figure:deltabasepants']}.
• Figure 4: The image $\delta(x)$, superimposed onto Figure \ref{['figure:deltabasepants']}.
• Figure 5: The image $\delta(b)$, superimposed onto Figure \ref{['figure:deltabasepants']}.

Theorems & Definitions (29)

• Theorem 2.1: Cho ChoG2GGLS
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.8