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.
Let us consider specifically 3-manifolds.
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.
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 ).
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
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 .