A formula on Heegaard genus of self-amalgamated 3-manifolds

Let M be an orientable compact irreducible and ∂-irreducible 3-manifold, and suppose ∂M consists of two boundary components F 1 and F 2 with g ( F 1 ) = g ( F 2 ) > 1 . Let M f be the closed orientable 3-manifold obtained by identifying F 1 and F 2 via a homeomorphism f : F 1 → F 2 . With the assumption that M is small or g ( M , F 1 ) = g ( M ) + g ( F 1 ) , we show that if f is sufficiently complicated, then g ( M f ) = g ( M , ∂ M ) + 1 .