2 problems

Let $X$ and $Y$ each have a single incompressible boundary component of genus at least two, let $\varphi\colon\partial X\to\partial Y$ be a gluing map, and let $M=X\cup_\varphi Y$.…

Let $X$ and $Y$ be 3-manifolds, each with a single incompressible torus boundary component, and let $M$ be obtained by gluing them along their boundaries. There is a complexity on…