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

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

• Probably you want to take a maximal compression of the boundary to get an incompressible surface $F$. So then $M$ is obtained from a three-manifold $M_0$ with incompressible boundary $F$ by attaching some $1$-handles. Dec 19, 2021 at 14:52