An upper bound on common stabilizations of Heegaard splittings

TL;DR

This paper establishes an upper bound on the genus of common stabilizations of two Heegaard splittings of a 3-manifold, providing a new theoretical limit that improves understanding of their relationships.

Contribution

The paper introduces a new upper bound on the genus of common stabilizations for any two Heegaard splittings, advancing the theoretical framework in 3-manifold topology.

Findings

Upper bound of 3/2 p + 2q - 1 on common stabilization genus

Comparison with examples showing minimal common stabilization can be larger

Provides theoretical limits for the relationship between different Heegaard splittings

Abstract

We show that for any two Heegaard splittings of genus $p$ and $q$ for the same closed 3-manifold, there is a common stabilization of genus at most 3/2 p + 2q - 1. One may compare this to recent examples of Heegaard splittings whose smallest common stabilizations have genus at least $p+q$ or $p + 1/2 q$ depending on the notion of equivalence.