Horizontal Heegaard splittings of Seifert fibered spaces

TL;DR

This paper establishes a correspondence between strongly irreducible horizontal Heegaard splittings and elements of g in Seifert fibered spaces, revealing the structure of these splittings and their relation to the fundamental group.

Contribution

It introduces a classification of strongly irreducible horizontal Heegaard splittings in Seifert fibered spaces based on intersection slopes, and shows the existence of infinitely many non-isotopic splittings with Nielsen equivalent generators.

Findings

One-to-one correspondence between isotopy classes and g elements.

Existence of infinitely many non-isotopic splittings.

Splittings determine Nielsen equivalent generating systems.

Abstract

We show that if an orientable Seifert fibered space $M$ with an orientable genus $g$ base space admits a strongly irreducible horizontal Heegaard splitting then there is a one-to-one correspondence between isotopy classes of strongly irreducible horizontal Heegaard splittings and elements of $\mathbf{Z}^{2g}$. The correspondence is determined by the slopes of intersection of each Heegaard splitting with a collection of $2g$ incompressible tori in $M$. We also show that there are Seifert fibered spaces with infinitely many non-isotopic Heegaard splittings that determine Nielsen equivalent generating systems for the fundamental group of $M$.