Boundary reducible handle additions on simple 3-manifolds

Abstract

Let $M$ be a simple manifold, and $F$ be a component of $\partial M$ of genus two. For a slope $γ$ on $F$, we denote by $M(γ)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $γ$ on $F$. In this paper, we shall prove that there is at most one separating slope $γ$ on $F$ so that $M(γ)$ is $\partial$-reducible.