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.

A 3-manifold is irreducible if and only if it is prime, except for two cases: the product ${\displaystyle S^{2}\times S^{1}}$ and the non-orientable fiber bundle of the 2-sphere over the circle ${\displaystyle S^{1}}$ are both prime but not irreducible. 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.

Consider specifically 3-manifolds.

A 3-manifold is irreducible if every smooth sphere bounds a ball. More rigorously, a differentiable connected 3-manifold ${\displaystyle M}$ is irreducible if every differentiable submanifold ${\displaystyle S}$ homeomorphic to a sphere bounds a subset ${\displaystyle D}$ (that is, ${\displaystyle S=\partial D}$) which is homeomorphic to the closed ball $${\displaystyle D^{3}=\{x\in \mathbb {R} ^{3}\ |\ |x|\leq 1\}.}$$ The assumption of differentiability of ${\displaystyle M}$ 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.

A connected 3-manifold ${\displaystyle M}$ is prime if it cannot be expressed as a connected sum ${\displaystyle N_{1}\#N_{2}}$ of two manifolds neither of which is the 3-sphere ${\displaystyle S^{3}}$ (or, equivalently, neither of which is homeomorphic to ${\displaystyle M}$).

Three-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ is irreducible: all smooth 2-spheres in it bound balls.