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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)

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    $\begingroup$ The second question is utterly hopeless. Just think about the case when the surface is the sphere: Take a closed connected 3-manifold and remove an open regular 3-ball from it. $\endgroup$ Dec 9 '21 at 2:17
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With regard to the first question: There are a couple of versions of unique decompositions for 3-manifolds with boundary, with respect to boundary connected sum. See Gross, Jonathan L. The decomposition of 3-manifolds with several boundary components. Trans. Amer. Math. Soc. 147 (1970), 561–572, and Swarup, G. Ananda Some properties of 3-manifolds with boundary. Quart. J. Math. Oxford Ser. (2) 21 (1970), 1–23.

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    $\begingroup$ The Bonahon paper on geometric structures offers a useful survey, as well. It phrases things as: (1) compression body decomposition (2) connect-sum decomposition (3) torus decomposition, then geometrization. $\endgroup$ Dec 9 '21 at 6:47

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