This is in some sense a generalization of the [question](https://mathoverflow.net/questions/410387/decomposition-of-manifolds-with-toroidal-boundary) 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:


1. $\mathcal{M}$ is homeomorphic to the solid torus $\overline{T}^{2}=D^{2}\times S^{1}$.
2. $\mathcal{M}$ 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.
3. $\mathcal{M}$ has incompressible boundary.


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



"Classification" of (orientable) 3-manifolds with genus-g-surface as their boundary
You can find the discussion of the characteristic compression body in Section 3.3 of [Bonahon’s survey.](https://www.math.csi.cuny.edu/abhijit/gt3m/bonahon-gs3m.pdf)  See Theorem 3.7 for the irreducible case, which together with the uniqueness of connect sum in Theorem 3.1 I think gives the sort of generalization of the trichotomy you are seeking. Either the manifold has incompressible boundary, or it has compressible boundary and it is a handlebody, or it has compressible boundary and is obtained from a compression body (with genus g compresible boundary) attached to some manifolds with incompressible boundary. I think this gives a reference for John Pardon’s comment.


