Unstabilized and uncritical dual Heegaard splitting of product I-bundle

Abstract

Let M be a 3-manifold, F be an incompressible surface in M with χ(F)≤0, which cuts M into two 3-manifolds M1 and M2. Suppose Mi=Vi∪SiWi(i=1,2) is a Heegaard splitting of Mi, M=V∪SW is the dual Heegaard splitting of M1=V1∪S1W1 and M2=V2∪S2W2 along F. If at least one of M1=V1∪S1W1 and M2=V2∪S2W2 is some ∂-stabilizations along a component of ∂M1∪∂M2, which contains F, then M=V∪SW is stabilized. As a corollary, if M2=F1×I=V2∪S2W2 is a nontrivial Heegaard splitting, then M=V∪SW is stabilized. We also prove that if M2=F1×I=V2∪S2W2 is a trivial Heegaard splitting and d(S1)≥5, then M=V∪SW is unstabilized and S is uncritical. As a corollary, we give a condition of the criticality of S.