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  "body": "\n## Statement\n\nA closed connected $3$-manifold $P$ is *prime* if $P$ is not homeomorphic to $S^3$ and the following holds.\nWhenever $P$ is homeomorphic to a [[connected sum|Y0K0K5NGWW]] $A \\# B$ of closed $3$-manifolds, one of $A$ and $B$ is homeomorphic to $S^3$.\n\nLet $M$ be a closed connected [[orientable|702SNFQG0W]] [[3-manifold|R20TB2KEQQ]] not homeomorphic to the [[3-sphere|T62369RFH6]] $S^3$.\nThen:\n\n1. $M$ is homeomorphic to a connected sum $P_1 \\# \\cdots \\# P_k$ of finitely many prime $3$-manifolds.\n2. The summands are unique.\n   If $M$ is also homeomorphic to $Q_1 \\# \\cdots \\# Q_l$ with each $Q_j$ prime, then $k = l$, and after reordering $P_i$ is homeomorphic to $Q_i$ for each $i$.\n\n## Remarks\n\nExistence is due to Kneser and uniqueness to Milnor.\nFor 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.\nUniqueness comes from an innermost-disk argument that makes two systems of splitting spheres disjoint.\nA 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.\nFor example, $\\mathbb{RP}^3 \\# \\mathbb{RP}^3$ has the two prime summands $\\mathbb{RP}^3$, and no other decomposition.\nWithout 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.\n\n## Explanations\n\n### L1 human\n\nEvery closed orientable $3$-manifold breaks up, in exactly one way, into basic pieces glued together by connected sums.\nIt is the analogue of factoring an integer into primes.\nTo understand all $3$-manifolds, it is enough to understand the prime ones.\n\n### L2 human\n\nThe theorem reduces the classification of closed orientable $3$-manifolds to irreducible ones and $S^2 \\times S^1$.\nFor irreducible manifolds, the next step is the JSJ decomposition along incompressible tori.\nThese two decompositions are the first stages of the geometrization theorem.\nOn the algebra side, the fundamental group of a connected sum is the free product of the fundamental groups of the summands.\n",
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