My question is the following:
Can all compact, connected and orientable 3-dimensional topological manifolds $\mathcal{M}$ with connected boundary $\partial\mathcal{M}\cong S^{2}$ be obtained by removing a closed ball inside some arbitrary closed, connected and orientable 3-manifold $\mathcal{N}$?
Furthermore, I know that there is also the notion of manifolds with "incompressible" boundary. However, I am far from beeing an expert on this, so I am not sure if this can happen in this case...