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.
Consider specifically 3-manifolds.
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.
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} ).
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.
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.
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.
An irreducible manifold M {\displaystyle M} is prime. Indeed, if we express M {\displaystyle M} as a connected sum M = N 1 # # --> N 2 , {\displaystyle M=N_{1}\#N_{2},} then M {\displaystyle M} is obtained by removing a ball each from N 1 {\displaystyle N_{1}} and from N 2 , {\displaystyle N_{2},} and then gluing the two resulting 2-spheres together. These two (now united) 2-spheres form a 2-sphere in M . {\displaystyle M.} The fact that M {\displaystyle M} is irreducible means that this 2-sphere must bound a ball. Undoing the gluing operation, either N 1 {\displaystyle N_{1}} or N 2 {\displaystyle N_{2}} is obtained by gluing that ball to the previously removed ball on their borders. This operation though simply gives a 3-sphere. This means that one of the two factors N 1 {\displaystyle N_{1}} or N 2 {\displaystyle N_{2}} was in fact a (trivial) 3-sphere, and M {\displaystyle M} is thus prime.
Let M {\displaystyle M} be a prime 3-manifold, and let S {\displaystyle S} be a 2-sphere embedded in it. Cutting on S {\displaystyle S} one may obtain just one manifold N {\displaystyle N} or perhaps one can only obtain two manifolds M 1 {\displaystyle M_{1}} and M 2 . {\displaystyle M_{2}.} In the latter case, gluing balls onto the newly created spherical boundaries of these two manifolds gives two manifolds N 1 {\displaystyle N_{1}} and N 2 {\displaystyle N_{2}} such that M = N 1 # # --> N 2 . {\displaystyle M=N_{1}\#N_{2}.} Since M {\displaystyle M} is prime, one of these two, say N 1 , {\displaystyle N_{1},} is S 3 . {\displaystyle S^{3}.} This means M 1 {\displaystyle M_{1}} is S 3 {\displaystyle S^{3}} minus a ball, and is therefore a ball itself. The sphere S {\displaystyle S} is thus the border of a ball, and since we are looking at the case where only this possibility exists (two manifolds created) the manifold M {\displaystyle M} is irreducible.
It remains to consider the case where it is possible to cut M {\displaystyle M} along S {\displaystyle S} and obtain just one piece, N . {\displaystyle N.} In that case there exists a closed simple curve γ γ --> {\displaystyle \gamma } in M {\displaystyle M} intersecting S {\displaystyle S} at a single point. Let R {\displaystyle R} be the union of the two tubular neighborhoods of S {\displaystyle S} and γ γ --> . {\displaystyle \gamma .} The boundary ∂ ∂ --> R {\displaystyle \partial R} turns out to be a 2-sphere that cuts M {\displaystyle M} into two pieces, R {\displaystyle R} and the complement of R . {\displaystyle R.} Since M {\displaystyle M} is prime and R {\displaystyle R} is not a ball, the complement must be a ball. The manifold M {\displaystyle M} that results from this fact is almost determined, and a careful analysis shows that it is either S 2 × × --> S 1 {\displaystyle S^{2}\times S^{1}} or else the other, non-orientable, fiber bundle of S 2 {\displaystyle S^{2}} over S 1 . {\displaystyle S^{1}.}
Плесенська сільська громадаОсновні даніКраїна УкраїнаОбласть Хмельницька областьРайон Шепетівський районУтворена 14 серпня 2019 рокуАдмін. центр ПлеснаТериторія та населенняНаселені пунктиМіста 0Смт 0Села 3Селища 0Органи владиРада Плесенська сільська радаАдреса с. П
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