M.Sc Student M.Sc Thesis Shacham Hagay Combinatorial Conditions which Affect the Heegaard Distance Department of Mathematics PROFESSOR EMERITUS Yoav Moriyah

Abstract

Every closed orientable 3-dimensional manifold M admits a Heegaard splitting, i.e. a decomposition into two handlebodies  which meet along their boundary. This common boundary is called a Heegaard surface in M, and is usually considered only up to isotopy in M. The genus g of the Heegaard surface is said to be the genus of the handlebodies. A Heegaard splitting gives us the Heegaard distance, which is defined using the curve complex. The fact that a Heegaard splitting is high distance has important consequences for the geometry of the 3-manifold determined by it. In this research thesis we discuss two previously introduced combinatorial conditions on the Heegaard distance - the rectangle condition, first introduced be Casson and Gordon, and the double rectangle condition, first introduced by Lustig and Moriah - and their effect on the Heegaard distance, and hence on the geometry of the 3-manifold. Both of these conditions mean high complexity of the Heegaard splitting, and so it seems reasonable to ask whether they also imply a large Heegaard distance, namely Heegaard distance strictly larger than 2. We will show, using induction on the genus g  of the handlebodies which define the Heegaard splitting, that for each of these combinatorial conditions, and for every genus g, there exists a 3-manifold Mg with Heegaard splitting of genus g with two complete decomposing systems which satisfy the relevant condition and implies Heegaard distance of exactly 2.