Unstabilized and uncritical dual Heegaard splitting of product I-bundle

Let M be a 3-manifold, F be an incompressible surface in M   with χ(F)≤0χ(F)≤0, which cuts M   into two 3-manifolds M1M1 and M2M2. Suppose Mi=Vi∪SiWiMi=Vi∪SiWi(i=1,2)(i=1,2) is a Heegaard splitting of MiMi, M=V∪SWM=V∪SW is the dual Heegaard splitting of M1=V1∪S1W1M1=V1∪S1W1 and M2=V2∪S2W2M2=V2∪S2W2 along F  . If at least one of M1=V1∪S1W1M1=V1∪S1W1 and M2=V2∪S2W2M2=V2∪S2W2 is some ∂-stabilizations along a component of ∂M1∪∂M2∂M1∪∂M2, which contains F  , then M=V∪SWM=V∪SW is stabilized. As a corollary, if M2=F1×I=V2∪S2W2M2=F1×I=V2∪S2W2 is a nontrivial Heegaard splitting, then M=V∪SWM=V∪SW is stabilized. We also prove that if M2=F1×I=V2∪S2W2M2=F1×I=V2∪S2W2 is a trivial Heegaard splitting and d(S1)≥5d(S1)≥5, then M=V∪SWM=V∪SW is unstabilized and S is uncritical. As a corollary, we give a condition of the criticality of S.