# Disk surgery and the primitive disk complexes of the $3$-sphere

## TL;DR

This paper investigates the structure of primitive disk complexes in genus-$g$ Heegaard splittings of the 3-sphere, revealing that these complexes are not closed under disk surgery for genus $g \,\ge\, 3$.

## Contribution

It demonstrates that the primitive disk complex for certain splittings is not weakly closed under disk surgery, providing new insights into the topology of 3-sphere splittings.

## Key findings

- Primitive disk complexes are not weakly closed under disk surgery for genus $g \ge 3$.
- Existence of primitive disks where surgery yields no primitive disks.
- Highlights limitations of disk surgery in primitive disk complexes.

## Abstract

Given a genus-$g$ Heegaard splitting of the $3$-sphere with $g \ge 3$, we show that the primitive disk complex for the splitting is not weakly closed under disk surgery operation. That is, there exist two primitive disks in one of the handlebodies of the splitting such that any disk surgery on one along the other one yields no primitive disks.