Comparing Heegaard splittings of non-haken 3-manifolds

Abstract

Cerf theory can be used to compare two strongly irreducible Heegaard splittings of the same closed orientable 3-manifold. Any two splitting surfaces can be isotoped so that they intersect in a non-empty collection of curves, each of which is essential in both splitting surfaces. More generally, there are interesting isotopies of the splitting surfaces during which this intersection property is preserved. As sample applications we give new proofs of Waldhausen's theorem that Heegaard splittings of S³ are standard, and of Bonahon and Otal's theorem that Heegaard splittings of lens spaces are standard. We also present a solution to the stabilization problem for irreducible non-Haken 3-manifolds: If p ⩽ q are the genera of two splittings of such a manifold, then there is a common stabilization of genus 5p + 8q − 9.