Heegaard splittings and virtually special square complexes

Abstract

We give a new characterization of irreducibility of Heegaard splittings for most 3-manifolds: a Heegaard splitting is irreducible if and only if it has an augmented Heegaard diagram that is a surface. A Heegaard splitting is a decomposition of a closed orientable 3-manifold into two isomorphic handle bodies that have a shared boundary surface. Usually, a number of curves on the shared boundary surface, called a Heegaard diagram, are used to describe a Heegaard splitting. We define a more complete object, the augmented Heegaard diagram, by building on a method used by Stallings to encode the information of a Heegaard splitting.Augmented Heegaard diagrams have several desirable properties: each 2-cell is a square, they have non-positive combinatorial curvature and they are virtually special. Restricting to manifolds that do not have S 1 ×S 2 as a connect summand, augmented Heegaard diagrams lend themselves well to understanding the decomposition of a 3-manifold via connect sum, leading to the characterization of irreducibility above. Along the way to proving this result, we find combinatorial methods to simplify Stallings's description of Heegaard splittings and find a connection to Guirardel's notion of a core.