Prime decomposition of 3-manifolds

Remarks

Existence is due to Kneser and uniqueness to Milnor. For existence, one cuts $M$ along embedded $2$-spheres that do not bound balls and caps the pieces with balls; Kneser's finiteness argument shows that this process stops. Uniqueness comes from an innermost-disk argument that makes two systems of splitting spheres disjoint. A closed orientable prime $3$-manifold is either $S^2 \times S^1$ or irreducible, meaning that every embedded $2$-sphere bounds an embedded $3$-ball. For example, $\mathbb{RP}^3 \# \mathbb{RP}^3$ has the two prime summands $\mathbb{RP}^3$, and no other decomposition. Without orientability, uniqueness fails in this form: for a non-orientable $P$, the sums $P \# (S^2 \times S^1)$ and $P \# (S^2 \tilde{\times} S^1)$ are homeomorphic, where $S^2 \tilde{\times} S^1$ is the non-orientable $S^2$-bundle over the circle.

Going deeper

For a student

The theorem reduces the classification of closed orientable $3$-manifolds to irreducible ones and $S^2 \times S^1$. For irreducible manifolds, the next step is the JSJ decomposition along incompressible tori. These two decompositions are the first stages of the geometrization theorem. On the algebra side, the fundamental group of a connected sum is the free product of the fundamental groups of the summands.