Scholar - SciOpen

Let $M$ be a compact connected orientable 3-manifold and $F$ be a compact connected orientable surface properly embedded in $M$ . If $F$ cuts $M$ into two handlebodies $X$ and $Y$ (i.e., $M = X \cup_{F} Y$ ), then we say that $F$ is an $H^{'}$ -splitting surface for $M$ and call $X \cup_{F} Y$ an $H^{'}$ -splitting for $M$ . When the $H^{'}$ -splitting surface $F$ is incompressible in a handlebody $H$ , a characteristic of an $H^{'}$ -splitting $H_{1} \cup_{F} H_{2}$ to denote $H$ is already known. In the present paper, we generalize the above result as follows: Let $H$ be a handlebody of genus $g \geq 1$ , $X \cup_{F} Y$ an $H^{'}$ -splitting for $H$ . Then, either $X \cup_{F} Y$ is stabilized, or there exists a reducing system $J_{1} \cup K_{1}$ of $F$ , such that $J_{1}$ is quasi-primitive in $Y$ and $K_{1}$ is quasi-primitive in $X$ . Combining the result with the known result, we obtain a characteristic of an $H^{'}$ -splitting $H_{1} \cup_{F} H_{2}$ to denote a handlebody.