"Classification" of (orientable) 3-manifolds with genus-g-surface as their boundary

Question

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:

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 $\partial\mathcal{M}\cong_{\mathrm{homeo.}}\Sigma_{g}$?

Answer 1

You can find the discussion of the characteristic compression body in Section 3.3 of Bonahon’s survey. 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.