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 $g' < \frac{d}{2}$ 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 $g \geq 2, d \geq 2$, 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 $d \geq 4$, a three-manifold with a genus $g$, distance $d$ Heegaard surface $\Sigma$ and a second Heegaard surface with genus $g' = \frac{1}{2} d + g - 1$ that is not a stabilization of $\Sigma$.