Stabilizations of Heegaard splittings of sufficiently complicated 3-manifolds (Preliminary Report)

TL;DR

This paper investigates the stabilization properties of Heegaard splittings in complex 3-manifolds, demonstrating conditions under which splittings are stabilized or remain unstabilized, and exploring their uniqueness and distance properties.

Contribution

It introduces new constructions of 3-manifolds with Heegaard splittings requiring multiple stabilizations and establishes conditions for the uniqueness of amalgamated splittings in complex manifolds.

Findings

Manifolds with pairs of genus g Heegaard splittings need about g stabilizations to become equivalent.

Amalgamation of boundary-unstabilized splittings via complex maps yields unstabilized splittings.

Existence of high-genus distance one Heegaard splittings in certain manifolds.

Abstract

We construct families of manifolds that have pairs of genus $g$ Heegaard splittings that must be stabilized roughly $g$ times to become equivalent. We also show that when two unstabilized, boundary-unstabilized Heegaard splittings are amalgamated by a "sufficiently complicated" map, the resulting splitting is unstabilized. As a corollary, we produce a manifold that has distance one Heegaard splittings of arbitrarily high genus. Finally, we show that in a 3-manifold formed by a sufficiently complicated gluing, a low genus, unstabilized Heegaard splitting can be expressed in a unique way as an amalgamation over the gluing surface.