$\partial$-reducible handle additions

TL;DR

This paper establishes an upper bound of 8 on the minimal intersection number of separating slopes on a boundary component of a simple 3-manifold, given that adding 2-handles along these slopes yields boundary-reducible manifolds.

Contribution

It proves a new bound on the intersection number of separating slopes leading to boundary-reducible manifolds after 2-handle addition.

Findings

The minimal intersection number is at most 8 for such slopes.

Provides conditions under which handle additions produce boundary-reducible manifolds.

Advances understanding of handle addition effects on 3-manifold boundary reducibility.

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.