This is in some sense a generalization of the question I asked some time ago. I am very sorry if this question is too basic for MathOverflow, but I just started learning about some more detailed things of the topology of 3-manifolds quite recently for some project and hence, I am still missing some concepts.

**My general question is the following:**

I am interested in the class of all compact, orientable and connected 3-dimensional topological manifolds $\mathcal{M}$ with boundary $\partial\mathcal{M}\cong_{\mathrm{homeo.}}\Sigma_{g}$ where $g\in\mathbb{N}_{0}$ and $\Sigma_{g}$ denotes the orientable genus-g surface. Is there a way to "divide" this class into different types?

Obviously, I am not asking for a classification in the strict sense, but just a way to distinguish such manifolds by their construction. As an example, in Moishe Kohan's great answer to the question I linked above, he explained that all 3-manifolds (with the properties as above) with $\partial\mathcal{M}=T^{2}$, where $T^{2}=S^{1}\times S^{1}$ denotes the 2-torus, is of one (and only one) of the following types:

- $\mathcal{M}$ is homeomorphic to the solid torus $\overline{T}^{2}=D^{2}\times S^{1}$.
- $\mathcal{M}$ is homeomorphic to $\overline{T}^{2}\#\mathcal{N}$, where $\mathcal{N}$ is some closed, connected and orientable 3-manifold, which is not $S^{3}$ and where $\#$ denotes the (internal, oriented) connected sum.
- $\mathcal{M}$ has incompressible boundary.

Is a similar trichotomy also true for the more general case with $\partial\mathcal{M}\cong_{\mathrm{homeo.}}\Sigma_{g}$?