3-Manifolds with Heegaard Splittings of Distance Two

Abstract

It is well-known that the distance of a Heegaard splitting, as defined by Hempel [ 4 ], is related to global properties of the underlying 3-manifold. For example, Casson-Gordon [ 1 ] show that existence of a splitting of distance one implies that the manifold is Haken, i.e. has an embedded incompressible surface. Distance zero splittings are either stabilized, or the underlying manifold is reducible. In this paper, we study the case of distance two splittings. These are interesting as many known examples of splittings of distance greater than one are actually of distance two. Our aim is to describe how simple con..