Prime decomposition (3-manifold)

The prime decomposition theorem for 3-manifolds, proved by Hellmuth Kneser in 1929 and refined by John Milnor, states that every compact orientable 3-manifold can be uniquely decomposed as a connected sum of prime manifolds — manifolds that cannot be written as a non-trivial connected sum. This decomposition is the first and coarsest level of the structural hierarchy that culminates in the geometrization of 3-manifolds, splitting arbitrary complexity into irreducible building blocks. The theorem's uniqueness mirrors the fundamental theorem of arithmetic: just as integers factor uniquely into primes, 3-manifolds factor uniquely into prime manifolds, suggesting that topological complexity obeys a law of discrete atomicity at its foundation.