On the union stabilization of two Heegaard splittings

TL;DR

This paper proves that any two Heegaard splittings of a 3-manifold have a union stabilization, introduces bounds on the minimal genus, and presents an example where union stabilization genus exceeds the stable genus.

Contribution

It establishes the existence of union stabilization for any pair of Heegaard splittings and explores the relationship between union stabilization genus and stable genus.

Findings

Existence of union stabilization for any two Heegaard splittings.

Numerical bounds on minimal genus of union stabilization.

An example where union stabilization genus exceeds stable genus.

Abstract

Let two Heegaard splittings $V_1 \cup W_1$ and $V_2 \cup W_2$ of a 3-manifold $M$ be given. We consider the union stabilization $M=V \cup W$ which is a common stabilization of $V_1 \cup W_1$ and $V_2 \cup W_2$ having the property that $V=V_1 \cup V_2$. We show that any two Heegaard splittings of a 3-manifold have a union stabilization. We also give some examples with numerical bounds on the minimal genus of union stabilization. On the other hand, we give an example of a candidate for which the minimal genus of union stabilization is strictly larger than the usual stable genus -- the minimal genus of common stabilization.