Disk complexes and genus two Heegaard splittings for non-prime 3-manifolds

TL;DR

This paper studies the structure of genus two Heegaard splittings in non-prime 3-manifolds, introducing a new contractible subcomplex of the disk complex, and classifying all such splittings with implications for the Goeritz group.

Contribution

It defines a new contractible subcomplex within the disk complex and classifies all genus two Heegaard splittings for non-prime 3-manifolds, extending previous results.

Findings

The special subcomplex of the disk complex is contractible.

The complex of Haken spheres is contractible, refining prior results.

All genus two Heegaard splittings for non-prime 3-manifolds are classified.

Abstract

Given a genus two Heegaard splitting for a non-prime 3-manifold, we define a special subcomplex of the disk complex for one of the handlebodies of the splitting, and then show that it is contractible. As applications, first we show that the complex of Haken spheres for the splitting is contractible, which refines the results of Lei and Lei-Zhang. Secondly, we classify all the genus two Heegaard splittings for non-prime 3-manifolds, which is a generalization of the result of Montesinos-Safont. Finally, we show that the mapping class group of the splitting, called the Goeritz group, is finitely presented by giving its explicit presentation.