Prime three-manifold

A closed connected orientable smooth manifold of dimension three is prime when a decomposition as a connected sum forces at least one summand to be the three-sphere. Unlike an irreducible three-manifold, it may contain a nonseparating embedded two-sphere.

Table of contents
- Prime decomposition of a closed orientable three-manifold Prime three-manifold
- Nonseparating sphere gives a sphere-circle summand Prime three-manifold

Prime decomposition of a closed orientable three-manifold

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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.

Nonseparating sphere gives a sphere-circle summand ( $M ≅ N # (S^{2} \times S^{1})$ )

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In an orientable three-manifold, an embedded nonseparating two-sphere has a product neighborhood. Join its two boundary spheres by an arc outside that neighborhood and take a regular neighborhood of the union. This region is

S^{2} \times S^{1}

with an open ball removed. Its boundary sphere exhibits the displayed connected sum. Consequently a prime but nonirreducible orientable three-manifold is