The Heegaard distances cover all nonnegative integers

Ruifeng Qiu, Yanqing Zou and Qilong Guo

Vol. 275 (2015), No. 1, 231–255

DOI: 10.2140/pjm.2015.275.231

Abstract

We prove two main results: (1) For any integers $n\ge 1$ and $g\ge 2$, there is a closed 3-manifold $M_g^n$ admitting a distance-$n$, genus-$g$ Heegaard splitting, unless $\left(g,n\right)=\left(2,1\right)$. Furthermore, $M_g^n$ can be chosen to be hyperbolic unless $\left(g,n\right)=\left(3,1\right)$. (2) For any integers $g\ge 2$ and $n\ge 4$, there are infinitely many nonhomeomorphic closed 3-manifolds admitting distance-$n$, genus-$g$ Heegaard splittings.

Keywords

attaching handlebody, Heegaard distance, subsurface projection

Mathematical Subject Classification 2010

Primary: 57M27

Secondary: 57M50

Milestones

Received: 18 June 2014

Revised: 10 September 2014

Accepted: 21 September 2014

Published: 12 April 2015