The Heegaard distances cover all non-negative integers

TL;DR

This paper demonstrates that all non-negative integers can be realized as Heegaard distances of closed 3-manifolds with specified genus, except for two small cases, and constructs infinitely many such manifolds for larger distances.

Contribution

It proves the existence of 3-manifolds with any given Heegaard distance for genus at least 2, extending the understanding of the spectrum of Heegaard distances.

Findings

Constructs manifolds for all distances n ≥ 0 (except two small cases)

Shows existence of infinitely many manifolds for large distances n ≥ 4

Provides hyperbolic examples for most cases

Abstract

In this paper, we prove that (1) For any integers $n\geq 1$ and $g\geq 2$, there is a closed 3-manifold $M_{g}^{n}$ which admits a distance $n$ Heegaard splitting of genus $g$ except that the pair of $(g, n)$ is $(2, 1)$. Furthermore, $M_{g}^{n}$ can be chosen to be hyperbolic except that the pair of $(g, n)$ is $(3, 1)$. (2) For any integers $g\geq 2$ and $n\geq 4$, there are infinitely many non-homeomorphic closed 3-manifolds admitting distance $n$ Heegaard splittings of genus $g$.