Note on primitive disk complexes

TL;DR

This paper investigates the connectivity of the primitive disk complex in Heegaard splittings of the 3-sphere, proving that a related quotient complex, the homotopy primitive disk complex, is connected.

Contribution

It introduces the homotopy primitive disk complex and proves its connectivity, advancing understanding of the structure of primitive disk complexes in high-genus splittings.

Findings

The homotopy primitive disk complex is connected.

Connectivity of the primitive disk complex remains open for genus greater than three.

The quotient construction provides new insights into the topology of disk complexes.

Abstract

Given a Heegaard splitting of the $3$-sphere, the primitive disk complex is defined to be the full subcomplex of the disk complex for one of the handlebodies of the splitting. It is an open question whether the primitive disk complex is connected or not when the genus of the splitting is greater than three. In this note, we prove that a quotient of the primitive disk complex, called the homotopy primitive disk complex, is connected.