Connected Primitive Disk Complexes and Genus Two Goeritz Groups of Lens Spaces

Abstract

Given a stabilized Heegaard splitting of a three-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-2 Heegaard splitting of each lens space. In particular, we show that the complex for the genus-2 splitting for the lens space L(p, q) with 1 ≤ q ≤ p/2 is connected if and only if p ≡ ±1 (mod q), and describe the combinatorial structure of each of those complexes. As an application, we obtain a finite presentation of the genus-2 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-2 Heegaard splitting of it.