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 irreduciblen-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.
A 3-manifold is irreducible if any smooth sphere bounds a ball. More rigorously, a differentiableconnected 3-manifold is irreducible if every differentiable submanifoldhomeomorphic 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 reducible.
Prime manifolds
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 ).
Examples
Euclidean space
Three-dimensional Euclidean space is irreducible: all smooth 2-spheres in it bound balls.
On the other hand, Alexander's horned sphere is a non-smooth sphere in that does not bound a ball. Thus the stipulation that the sphere be smooth is necessary.
Sphere, lens spaces
The 3-sphere is irreducible. The product space is not irreducible, since any 2-sphere (where is some point of ) has a connected complement which is not a ball (it is the product of the 2-sphere and a line).
A lens space with (and thus not the same as ) 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 and the non-orientablefiber bundle of the 2-sphere over the circle are both prime but not irreducible.
From irreducible to prime
An irreducible manifold is prime. Indeed, if we express as a connected sum
then is obtained by removing a ball each from and from and then gluing the two resulting 2-spheres together. These two (now united) 2-spheres form a 2-sphere in The fact that is irreducible means that this 2-sphere must bound a ball. Undoing the gluing operation, either or 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 or was in fact a (trivial) 3-sphere, and is thus prime.
From prime to irreducible
Let be a prime 3-manifold, and let be a 2-sphere embedded in it. Cutting on one may obtain just one manifold or perhaps one can only obtain two manifolds and In the latter case, gluing balls onto the newly created spherical boundaries of these two manifolds gives two manifolds and such that
Since is prime, one of these two, say is This means is minus a ball, and is therefore a ball itself. The sphere 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 is irreducible.
It remains to consider the case where it is possible to cut along and obtain just one piece, In that case there exists a closed simple curve in intersecting at a single point. Let be the union of the two tubular neighborhoods of and The boundary turns out to be a 2-sphere that cuts into two pieces, and the complement of Since is prime and is not a ball, the complement must be a ball. The manifold that results from this fact is almost determined, and a careful analysis shows that it is either or else the other, non-orientable, fiber bundle of over
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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 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 Contents 1 Definitions 1 1 Irreducible manifold 1 2 Prime manifolds 2 Examples 2 1 Euclidean space 2 2 Sphere lens spaces 3 Prime manifolds and irreducible manifolds 3 1 From irreducible to prime 3 2 From prime to irreducible 4 References 5 See alsoDefinitions EditConsider specifically 3 manifolds Irreducible manifold Edit A 3 manifold is irreducible if any 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 ballD 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 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 reducible Prime manifolds Edit 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 EditEuclidean space Edit 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 Edit 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 EditA 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 Edit An irreducible manifold M displaystyle M is prime Indeed if we express M displaystyle M as a connected sumM 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 From prime to irreducible Edit 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 thatM 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 g 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 g 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 References EditWilliam Jaco Lectures on 3 manifold topology ISBN 0 8218 1693 4 See also Edit3 manifold Connected sum Prime decomposition 3 manifold Retrieved from https en wikipedia org w index php title Prime manifold amp oldid 1054280673, wikipedia, wiki, book, books, library,