On Heegaard splittings with finite many pairs of disjoint compression disks

TL;DR

This paper investigates the structure of weakly reducible Heegaard splittings with finitely many disjoint compression disk pairs, showing they admit specific untelescoping decompositions and characterizing their criticality based on the number of disk pairs.

Contribution

It introduces a new decomposition (untelescoping) for such Heegaard splittings and characterizes their criticality when multiple disk pairs exist.

Findings

Existence of an untelescoping decomposition for the splitting.

Characterization of critical Heegaard surfaces when multiple disk pairs are present.

Conditions relating the number of disk pairs to the distance and criticality of the surface.

Abstract

Suppose $V\cup_S W$ is a weakly reducible Heegaard splitting of a closed 3-manifold which admits only $n$ pairs of disjoint compression disks on distinct sides and $g>2$. We show $V\cup_S W$ admits an untelescoping: $(V_1\cup_{S_1}W_1)\cup_F(W_2\cup_{S_2}V_2)$ such that $W_i$ has only one separating compressing disk and $d(S_i)\geq 2$, for $i=1,~2$. If $n>1$, at least one of $d(S_i)$ is 2 and $S$ is a critical Heegaard surface.