Prime decomposition

The Prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold can be uniquely expressed as a Connected sum of prime 3-manifolds — those that cannot be decomposed nontrivially as a connected sum of two other 3-manifolds. Proved by Kneser and refined by Milnor, the theorem is the three-dimensional analogue of the fundamental theorem of arithmetic: just as every integer factors uniquely into primes, every 3-manifold factors uniquely into prime 3-manifolds. The theorem reduces the wild diversity of all 3-manifolds to the study of their irreducible building blocks, and it is the essential first step in the classification program that culminated in the Geometrization conjecture.