Prime decomposition of a closed orientable three-manifold

Every closed connected orientable three-manifold is a finite connected sum of prime three-manifolds, unique up to order and the insertion of three-sphere factors. The corresponding fundamental group is the free product of the summand groups. A prime orientable three-manifold is either irreducible or $S^2\times S^1$; this separates aspherical infinite-group cases from the infinite cyclic case.