Prime manifold

In topology (a mathematical discipline) a prime manifold is an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds. Non-trivial means that neither of the two is an n-sphere. A similar notion is that of an irreducible n-manifold, which is one in which any embedded (n − 1)-sphere bounds an embedded n-ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds.

The notions of irreducibility in algebra and manifold theory are related. An irreducible manifold is prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over the circle S¹ and the twisted 2-sphere bundle over S¹.

According to a theorem of Hellmuth Kneser and John Milnor, every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) collection of prime 3-manifolds.

Let us consider specifically 3-manifolds.

A 3-manifold is irreducible if any smooth sphere bounds a ball. More rigorously, a differentiable connected 3-manifold ${\displaystyle M}$ is irreducible if every differentiable submanifold ${\displaystyle S}$ homeomorphic to a sphere bounds a subset ${\displaystyle D}$ (that is, ${\displaystyle S=\partial D}$) which is homeomorphic to the closed ball

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