On classification of Heegaard splittings and triangulations

Daniel J. Heath

Vol. 178 (1997), No. 2, 241–264

DOI: 10.2140/pjm.1997.178.241

Abstract

In this paper we consider Heegaard splittings of 3-manifolds. By using Gabai’s concept of thin position on the 1-skeleton of some polyhedral decomposition, together with Casson- Gordon’s concept of strong irreducibility, we prove the Main Theorem (4.0). This theorem will allow us to classify the Heegaard splittings of manifolds whose polyhedral decompositions are particularily nice, which we demonstrate via examples. Specifically, we use it to classify Heegaard splittings of several hyperbolic spaces, including the figure-8 knot complement (Example 6.4) and the genus 2 case of the 5₂-knot complement (Example 6.7).

Milestones

Received: 18 July 1995

Revised: 13 March 1996

Published: 1 April 1997