On Heegaard Splittings with Finitely Many Pairs of Disjoint Compression Disks

Abstract:

      Suppose $V\cup_S W$ is a genus-$g$ weakly reducible Heegaard splitting of a closed 3-manifold with finitely many pairs of disjoint compression disks on distinct sides up to isotopy 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 a unique separating compressing disk and $d(S_i)\geq 2$, for $i=1,2$. If there exist more than one but finitely many pairs of disjoint compression disks, at least one of $d(S_i)$ is 2 and $S$ is a critical Heegaard surface.