Prime manifold

In topology, a branch of mathematics, 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 , 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. Consider specifically 3-manifolds. A 3-manifold is if any smooth sphere bounds a ball. More rigorously, a differentiable connected 3-manifold is irreducible if every differentiable submanifold homeomorphic to a sphere bounds a subset (that is, ) which is homeomorphic to the closed ball The assumption of differentiability of is not important, because every topological 3-manifold has a unique differentiable structure. The assumption that the sphere is smooth (that is, that it is a differentiable submanifold) is however important: indeed the sphere must have a tubular neighborhood. A 3-manifold that is not irreducible is called . A connected 3-manifold is prime if it cannot be expressed as a connected sum of two manifolds neither of which is the 3-sphere (or, equivalently, neither of which is homeomorphic to ). Three-dimensional Euclidean space is irreducible: all smooth 2-spheres in it bound balls.