Stabilization, amalgamation, and curves of intersection of Heegaard splittings

TL;DR

This paper investigates the stabilization process of Heegaard splittings in Haken 3-manifolds, providing bounds on stabilization steps and demonstrating the existence of manifolds with arbitrarily many intersection curves between essential tori and Heegaard surfaces.

Contribution

It establishes an upper bound on stabilizations needed for Heegaard splittings to become isotopic to an amalgamation and constructs examples with arbitrarily many intersection curves.

Findings

Bound on the number of stabilizations for isotopy to amalgamation.

Existence of 3-manifolds with arbitrarily many intersection curves.

First examples of lower bounds exceeding one for intersection curves.

Abstract

We address a special case of the Stabilization Problem for Heegaard splittings, establishing an upper bound on the number of stabilizations required to make a Heegaard splitting of a Haken 3-manifold isotopic to an amalgamation along an essential surface. As a consequence we show that for any positive integer $n$ there are 3-manifolds containing an essential torus and a Heegaard splitting such that the torus and splitting surface must intersect in at least $n$ simple closed curves. These give the first examples of lower bounds on the minimum number of curves of intersection between an essential surface and a Heegaard surface that are greater than one.