Non-uniqueness of high distance Heegaard splittings

Abstract

Kevin Hartshorn showed that if a three-dimensional manifold $M$ admits a Heegaard surface $\Sigma$ with Hempel distance $d$ then every incompressible surface in $M$ has genus at least $\frac{d}{2}$. Scharlemann-Tomova generalized this, proving that in such a manifold, every other Heegaard surface for $M$ of genus $g^{\prime }<\frac{d}{2}$ is a stabilization of $\Sigma$. In the present paper, we show that Hartshorn's bound is sharp and Scharlemann-Tomova's bound is very close to sharp. In particular, for every pair of integers $g\ge 2,d\ge 2$, we construct a three-manifold $M$ with a genus $g$, distance $d$ Heegaard splitting and an incompressible surface of genus $\frac{d}{2}$. We also construct, for every $d\ge 4$, a three-manifold with a genus $g$, distance $d$ Heegaard surface $\Sigma$ and a second Heegaard surface with genus $g^{\prime }=\frac{1}{2}d+g-1$ that is not a stabilization of $\Sigma$.

34 pages, 18 figures

References in corpus (2)

• Multiple genus 2 Heegaard splittings: a missed case
• An upper bound on common stabilizations of Heegaard splittings

Cited by in corpus (2)

• On keen Heegaard splittings
• Heegaard distance of the link complements in $S^3$