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.

DefinitionsEdit

Consider specifically 3-manifolds.

Irreducible manifoldEdit

A 3-manifold is irreducible 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 reducible.

Prime manifoldsEdit

A connected 3-manifold   is prime if it cannot be obtained as a connected sum   of two manifolds neither of which is the 3-sphere   (or, equivalently, neither of which is homeomorphic to  ).

ExamplesEdit

Euclidean spaceEdit

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 spacesEdit

The 3-sphere   is irreducible. The product space   is not irreducible, since any 2-sphere   (where 'pt' 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 manifoldsEdit

A 3-manifold is irreducible if and only if it is prime, except for two cases: the product   and the non-orientable fiber bundle of the 2-sphere over the circle   are both prime but not irreducible.

From irreducible to primeEdit

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 irreducibleEdit

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  .

ReferencesEdit

  • William Jaco. Lectures on 3-manifold topology. ISBN 0-8218-1693-4.

See alsoEdit