On the Heegaard Genera of 3-manifolds Containing Non-separating Surfaces

Abstract

Let M be a compact orientable 3-manifold which contains a non-separating closed incompressible surface F . Let M ′ = M −η(F ) where η(F ) is an open regular neighborhood of F in M . In the paper we show that if M ′ has a Heegaard splitting V ′ ∪S′ W ′ with d(S ′ ) > 2g(M ′ ), then g(M) ≥ g(M ′)−g(F ). Furthermore, if F is a torus, then g(M) ≥ g(M ′)+1.