Prime decomposition of a closed orientable three-manifold
= 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.