It is well-known that every closed, connected and orientable 3-manifold $\mathcal{M}$ can uniquely be decomposed as

$$\mathcal{M}=P_{1}\#\dots\# P_{n}$$

where $P_{i}$ are prime manifolds, i.e. manifolds which can not be written as a non-trivial connected sum of two 3-manifolds.

My main question is the following:Is there are similar result for 3-dimensional compact, connected and orientable manifolds with connected boundary?

I know that in this case there are two types of connected sums, namely:

- the
*internal*connected sum, which is defined as the connected sum of manifolds without boundary by choosing balls, which are purely in the interior. This also includes the hybrid case, i.e. the connected sum of a manifold with boundary with a manifold without boundary and - the
*boundary*connected sum, where one choses balls on the boundaries and glues them together.

In any case, can any 3-manifold with boundary (with the properties written above) be decomposed in some kind of connected sum?

As a second, very related question:If I fix the boundary to be some compact, oriented surface of arbitrary genus, is there some kind of classification/decomposition result classifying all manifolds having a boundary homeomorphic to our chosen boundary surface?

(This question was previously posted to MathStackExchange)