Connected primitive disk complexes and genus two Goeritz groups of lens spaces

TL;DR

This paper investigates the structure of primitive disk complexes in genus two Heegaard splittings of lens spaces, characterizing their connectivity and deriving finite presentations of associated Goeritz groups.

Contribution

It provides a detailed analysis of the primitive disk complex structure for lens spaces and offers finite presentations of their genus two Goeritz groups, a novel contribution.

Findings

Primitive disk complex is connected iff p ≡ ±1 mod q

Describes the combinatorial structure of the complexes

Provides finite presentations of the genus two Goeritz groups

Abstract

Given a stabilized Heegaard splitting of a $3$-manifold, the primitive disk complex for the splitting is the subcomplex of the disk complex for a handlebody in the splitting spanned by the vertices of the primitive disks. In this work, we study the structure of the primitive disk complex for the genus two Heegaard splitting of each lens space. In particular, we show that the complex for the genus two splitting for the lens space $L(p, q)$ with $1\leq q \leq p/2$ is connected if and only if $p \equiv \pm 1 \pmod q$, and describe the combinatorial structure of each of those complexes. As an application, we obtain a finite presentation of the genus two Goeritz group of each of those lens spaces, the group of isotopy classes of orientation preserving homeomorphisms of the lens space that preserve the genus two Heegaard splitting of it.