Stabilization of Heegaard splittings

TL;DR

This paper demonstrates that for any genus greater than one, there exist 3-manifolds with pairs of Heegaard splittings that need as many as g stabilizations to become equivalent, expanding understanding of Heegaard splitting stabilization.

Contribution

It constructs examples of 3-manifolds with Heegaard splittings requiring g stabilizations, showing that previous bounds of at most one stabilization are not universal.

Findings

Existence of 3-manifolds with Heegaard splittings needing g stabilizations

Use of harmonic map deformation to control Heegaard surfaces

Extension of stabilization bounds in 3-manifold topology

Abstract

For each g greater than one there is a 3-manifold with two genus g Heegaard splittings that require g stabilizations to become equivalent. Previously known examples required at most one stabilization. Control of families of Heegaard surfaces is obtained through a deformation to harmonic maps.