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.
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
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.
A 3-manifold is irreducible if every smooth sphere bounds a ball.
is irreducible if every differentiable submanifold
homeomorphic to a sphere bounds a subset
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.
is prime if it cannot be expressed as a connected sum
Three-dimensional Euclidean space
is irreducible: all smooth 2-spheres in it bound balls.
Thus the stipulation that the sphere be smooth is necessary.
) has a connected complement which is not a ball (it is the product of the 2-sphere and a line).
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
is obtained by removing a ball each from
is irreducible means that this 2-sphere must bound a ball.
Undoing the gluing operation, either
In the latter case, gluing balls onto the newly created spherical boundaries of these two manifolds gives two manifolds
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
It remains to consider the case where it is possible to cut
In that case there exists a closed simple curve
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