Reducing weak H-systems

It is a result of Rego and Rourke [Topology 27 (1988) 137] that a Heegaard diagram defines a homotopy 3-sphere if and only if it is associated to an H-system. In this paper, we introduce the concept of the weak H-system and show that a Heegaard diagram defines a homotopy 3-sphere if and only if it is associated to a weak H-system. We further show that in some situations a weak H-system (therefore the corresponding Heegaard splitting for the homotopy 3-sphere) can be reduced.