Heegaard splittings of distance exactly $n$

TL;DR

This paper proves that for any integers n ≥ 2 and g ≥ 2, there exist genus-g Heegaard splittings of compact 3-manifolds with distance exactly n, expanding understanding of 3-manifold topology.

Contribution

It establishes the existence of Heegaard splittings with prescribed exact distance for all relevant genus and distance values, filling a gap in 3-manifold topology.

Findings

Existence of genus-g Heegaard splittings with distance exactly n for all n ≥ 2 and g ≥ 2

Construction methods for such splittings

Implications for the classification of 3-manifolds

Abstract

In this paper, we show that, for any integers $n\geq 2$ and $g\geq 2$, there exist genus-$g$ Heegaard splittings of compact 3-manifolds with distance exactly $n$.