$\partial$-reducible handle additions

Han Lou, Mingxing Zhang

Published 2018-10-23Version 1

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 distance (intersection number) between $\alpha$ and $\beta$ is at most 8.