Kneser's theorem

Statement

Every compact connected orientable manifold $M$ of dimension 3 is expressible as a connected sum of finitely many prime 3-manifolds as follows:

$\!M=P_{1}\#P_{2}\#\dots \#P_{n}$

where the choice of $P_{i}$ with multiplicity is unique up to rearrangement and the addition/deletion of copies of the 3-sphere $S^{3}$ (if we remove all copies of $S^{3}$, it is unique up to rearrangement).

Note

This does not completely reduce the problem of classifying compact connected orientable 3-manifolds to the problem of classifying prime 3-manifolds, because homotopy type of connected sum depends on choice of gluing map, and thus, in some cases, the same pair of 3-manifolds may have different possible choices of connected sum. However, there are at most two different choices for each connected sum (based on the orientation of identification of spheres) and thus, knowing the decomposition into prime 3-manifolds reduces the number of alternatives to a finite list.