Prime decomposition of 3-manifolds
In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-manifolds. A manifold is "prime" if it cannot be presented as a connected sum of more than one manifold, none of which is the sphere of the same dimension. This condition is necessary since for any manifold M of dimension formula_1 it is true that formula_2 (where formula_3 means the connected sum of formula_4 and formula_5). If formula_6 is a prime 3-manifold  then either it is formula_7 or the non-orientable formula_8 bundle over formula_9 or it is irreducible, which means that any embedded 2-sphere bounds a ball. So the theorem can be restated to say that there is a unique connected sum decomposition into irreducible 3-manifolds and fiber bundles of formula_8 over formula_11 The prime decomposition holds also for non-orientable 3-manifolds, but the uniqueness statement must be modified slightly: every compact, non-orientable 3-manifold is a connected sum of irreducible 3-manifolds and non-orientable formula_8 bundles over formula_11 This sum is unique as long as  we specify that each summand is either irreducible or a non-orientable formula_8 bundle over formula_11 The proof is based on normal surface techniques originated by Hellmuth Kneser. Existence was proven by Kneser, but the exact formulation and proof of the uniqueness was done more than 30 years later by John Milnor.