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 category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds. A 3-manifold is irreducible if and only if it is prime, except for two cases: the product S 2 × S 1 {\displaystyle S^{2}\times S^{1}} and the non-orientable fiber bundle of the 2-sphere over the circle S 1 {\displaystyle S^{1}} are both prime but not irreducible. This is somewhat analogous to the notion in algebraic number theory of prime ideals generalizing Irreducible elements. 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.
Definitions Consider specifically 3-manifolds.
Irreducible manifold A 3-manifold is irreducible if every smooth sphere bounds a ball. More rigorously, a differentiable connected 3-manifold M {\displaystyle M} is irreducible if every differentiable submanifold S {\displaystyle S} homeomorphic to a sphere bounds a subset D {\displaystyle D} (that is, S = ∂ D {\displaystyle S=\partial D} ) which is homeomorphic to the closed ball
D 3 = { x ∈ R 3 | | x | ≤ 1 } . {\displaystyle D^{3}=\{x\in \mathbb {R} ^{3}\ |\ |x|\leq 1\}.}
The assumption of differentiability of M {\displaystyle M} is not important, because every topological 3-manifold has a unique differentiable structure. However it is necessary to assume that the sphere is smooth (a differentiable submanifold), even having a tubular neighborhood. The differentiability assumption serves to exclude pathologies like the Alexander's horned sphere (see below). A 3-manifold that is not irreducible is called reducible.
Prime manifolds A connected 3-manifold M {\displaystyle M} is prime if it cannot be expressed as a connected sum N 1 # N 2 {\displaystyle N_{1}\#N_{2}} of two manifolds neither of which is the 3-sphere S 3 {\displaystyle S^{3}} (or, equivalently, neither of which is homeomorphic to M {\displaystyle M} ).
Examples
Euclidean space Three-dimensional Euclidean space R 3 {\displaystyle \mathbb {R} ^{3}} is irreducible: all smooth 2-spheres in it bound balls. On the other hand, Alexander's horned sphere is a non-smooth sphere in R 3 {\displaystyle \mathbb {R} ^{3}} that does not bound a ball. Thus the stipulation that the sphere be smooth is necessary.
Sphere, lens spaces The 3-sphere S 3 {\displaystyle S^{3}} is irreducible. The product space S 2 × S 1 {\displaystyle S^{2}\times S^{1}} is not irreducible, since any 2-sphere S 2 × { p t } {\displaystyle S^{2}\times \{pt\}} (where p t {\displaystyle pt} is some point of S 1 {\displaystyle S^{1}} ) has a connected complement which is not a ball (it is the product of the 2-sphere and a line). A lens space L ( p , q ) {\displaystyle L(p,q)} with p ≠ 0 {\displaystyle p\neq 0} (and thus not the same as S 2 × S 1 {\displaystyle S^{2}\times S^{1}} ) is irreducible.
Prime manifolds and irreducible manifolds A 3-manifold is irreducible if and only if it is prime, except for two cases: the product S 2 × S 1 {\displaystyle S^{2}\times S^{1}} and the non-orientable fiber bundle of the 2-sphere over the circle S 1 {\displaystyle S^{1}} are both prime but not irreducible.
From irreducible to prime An irreducible manifold M {\displaystyle M} is prime. Indeed, if we express M {\displaystyle M} as a connected sum
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