On reduction complexity of Heegaard splittings

Let ∪F W be a Heegaard splitting of a closed, connected, orientable 3-manifold M with genus n. We introduce a reduction complexity, δ, for the Heegaard splitting, and conclude that: 1. ⩾ max{2, n} if and only if V ∪F W is reducible; ⩾ 2 if and only if V ∪F W is weakly reducible, > n if and only if M has exactly δ − n connected sum factors which are homeomorphic to S1 × S2.