$\partial$-reducible handle additions

Abstract

Let $M$ be a simple 3-manifold, and $F$ be a component of $\partial M$ of genus at least 2. Let $\alpha$ and $\beta$ be separating slopes on $F$. Let $M(\alpha)$ (resp. $M(\beta)$) be the manifold obtained by adding a 2-handle along $\alpha$ (resp. $\beta$). If $M(\alpha)$ and $M(\beta)$ are $\partial$-reducible, then the minimal geometric intersection number of $\alpha$ and $\beta$ is at most 8.