A characteristics for a surface sum of two handlebodies along an annulus or a once-punctured torus to be a handlebody

Abstract

The main results of the paper is that we give a characteristics for an annulus sum and a once-punctured torus sum of two handlebodies to be a handlebody as follows: 1. The annulus sum $H=H_1\cup_A H_2$ of two handlebodies $H_1$ and $H_2$ is a handlebody if and only if the core curve of $A$ is a longitude for either $H_1$ or $H_2$. 2. Let $H=H_1\cup_T H_2$ be a surface sum of two handlebodies $H_1$ and $H_2$ along a once-punctured torus $T$. Suppose that $T$ is incompressible in both $H_1$ and $H_2$. Then $H$ is a handlebody if and only if the there exists a collection $\{\delta, \sigma\}$ of simple closed curves on $T$ such that either $\{\delta, \sigma\}$ is primitive in $H_1$ or $H_2$, or $\{\delta\}$ is primitive in $H_1$ and $\{\sigma\}$ is primitive in $H_2$.