Bounding the stable genera of Heegaard splittings from below

TL;DR

This paper constructs specific 3-manifolds demonstrating lower bounds on the genus of common stabilizations of Heegaard splittings, advancing understanding of their minimal genus configurations.

Contribution

It introduces explicit examples of 3-manifolds with Heegaard splittings that establish new lower bounds on stabilization genus, highlighting limitations of stabilization processes.

Findings

Existence of 3-manifolds with Heegaard surfaces of genus 2k and 2k-1 with high stabilization genus

Construction of 3-manifolds with multiple non-isotopic Heegaard splittings of the same genus

Lower bounds on stabilization genus for certain pairs of Heegaard surfaces

Abstract

We describe for each postive integer $k$ a 3-manifold with Heegaard surfaces of genus $2k$ and $2k-1$ such that any common stabilization of these two surfaces has genus at least $3k-1$. We also show that for every positive $n$, there is a 3-manifold that has $n$ pairwise non-isotopic Heegaard splittings of the same genus all of which are stabilized.