A refinement of Johnson's bounding for the stable genera of Heegaard splittings

TL;DR

This paper refines Johnson's bounds on the stable genera of Heegaard splittings, providing a new construction that improves the lower bounds for common stabilizations in 3-manifolds.

Contribution

It introduces a modified construction that establishes a higher lower bound for the genus of common stabilizations of Heegaard splittings.

Findings

Constructs 3-manifolds with Heegaard splittings of genus 2k

Shows any common stabilization has genus at least 3k

Improves previous bounds on stable genera

Abstract

For each integer k > 1, Johnson gave a 3-manifold with Heegaard splittings of genera 2k and 2k-1 such that any common stabilization of these two surfaces has genus at least 3k-1. We modify his argument to produce a 3-manifold with two Heegaard splitings of genus 2k such that any common stabilization of them has genus at least 3k.