Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization

Abstract

We construct families of pairs of Heegaard splittings that must be stabilized several times to become equivalent. The first such pair differs only by their orientation. These are genus n splittings of a closed 3-manifold that must be stabilized at least n-2 times to become equivalent. The second is a pair of genus n splittings of a manifold with toroidal boundary that must be stabilized at least n-4 times to become equivalent. The last example is a pair of genus n splittings of a closed 3-manifold that must be stabilized at least ${1/2}n -3$ times to become equivalent, regardless of their orientations. All of these examples are splittings of manifolds that are obtained from simpler manifolds by gluing along incompressible surfaces via "sufficiently complicated" maps.