Heegaard splittings with large subsurface distances

TL;DR

This paper demonstrates that high subsurface distances in Heegaard splittings impose strong constraints on the genus of other Heegaard surfaces, ensuring certain subsurfaces must appear.

Contribution

It establishes a new relationship between subsurface distances and the genus bounds of Heegaard surfaces, revealing structural limitations.

Findings

High subsurface distances force certain subsurfaces to appear in all Heegaard surfaces of bounded genus.

The genus of Heegaard surfaces is constrained by the relative Hempel distance.

Subsurface distances influence the topology and complexity of Heegaard splittings.

Abstract

We show that sub-surfaces of a Heegaard surface for which the relative Hempel distance of the splitting is sufficiently high have to appear in any Heegaard surface of genus bounded by half that distance.