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.
Definitions
Let us consider specifically 3-manifolds.
Irreducible manifold
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 manifolds
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 the 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 '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 manifolds
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 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
.
Bibliography
- William Jaco. Lectures on 3-manifold topology. ISBN 0-8218-1693-4.