Reducible and $\partial$-reducible handle additions

Abstract:

Let $M$ be a simple 3-manifold with $F$ a component of $\partial M$ of genus at least two. For a slope $\alpha$ on $F$, we denote by $M(\alpha )$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $\alpha$ on $F$. Suppose that $\alpha$ and $\beta$ are two separating slopes on $F$ such that $M(\alpha )$ and $M(\beta )$ are reducible. Then the distance between $\alpha$ and $\beta$ is at most 2. As a corollary, if $g(F)=2$, then there is at most one separating slope $\gamma$ on $F$ such that $M(\gamma )$ is either reducible or $\partial$-reducible.