COMPARING HEEGAARD-SPLITTINGS OF NON-HAKEN 3-MANIFOLDS

CERF THEORY can be used to compare two strongly irreducible Heegaard s plittings of the same closed orientable 3-manifold. Any two splitting surfaces can be isotoped so that they intersect in a non-empty collect ion of curves, each of which is essential in both splitting surfaces. More generally, there are interesting isotopies of the splitting surfa ces during which this intersection property is preserved. As sample ap plications we give new proofs of Waldhausen's theorem that Heegaard sp littings of S-3 are standard, and of Bonahon and Otal's theorem that H eegaard splittings of lens spaces are standard. We also present a solu tion to the stabilization problem for irreducible non-Haken 3-manifold s: If p less than or equal to q are the genera of two splittings of su ch a manifold, then there is a common stabilization of genus 5p + 8q - 9. Copyright (C) 1996 Elsevier Science Ltd