The distance between two separating, reducing slopes is at most 4

Abstract

Let $M$ be a simple 3-manifold such that one component of $\partial M$, say $F$, has 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 $F$. If $M(\alpha)$ is reducible, then $\alpha$ is called a reducing slope. In this paper, we shall prove that the distance between two separating, reducing slopes on $F$ is at most 4.