Decomposing Heegaard splittings along separating incompressible surfaces in 3-manifolds

TL;DR

This paper studies how separating incompressible surfaces can be used to decompose 3-manifolds with Heegaard splittings, revealing simplified hierarchical structures especially when the splitting has high Hempel distance.

Contribution

It introduces a method to decompose 3-manifolds along separating incompressible surfaces in the context of Heegaard splittings, especially for high-distance cases.

Findings

Existence of incompressible subsurfaces after decomposition

Presence of paired subsurfaces on both sides for high-distance splittings

Simplification of the 3-manifold hierarchy

Abstract

In this paper, by putting a separating incompressible surface in a 3-manifold into Morse position relative to the height function associated to a strongly irreducible Heegaard splitting, we show that an incompressible subsurface of the Heegaard splitting can be found, by decomposing the 3-manifold along the separating surface. Further if the Heegaard surface is of Hempel distance at least 4, then there is a pair of such subsurfaces on both sides of the given separating surface. This gives a particularly simple hierarchy for the 3-manifold.