In the theory of manifolds, an nmanifold is irreducible if any embedded (n1) sphere bounds an embedded nball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewiselinear manifolds.
The notions of irreducibility in algebra and manifold theory are related. An nmanifold is called prime, if it cannot be written as a connect sum of two nmanifolds (neither of which is an nsphere). An irreducible manifold is thus prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however, the topologist (in particular the 3manifold topologist) finds the definition above more useful. The only compact, connected 3manifolds that are prime but not irreducible are the trivial 2sphere bundle over S^1 and the twisted 2sphere bundle over S^1.
A theorem of 3manifold theory is: every compact, connected 3manifold has a prime decomposition, i.e. can be written as a connected sum with each summand being prime. This prime decomposition is also unique (up to homeomorphism of summand). [Again, we must be working in either the differentiable or piecewiselinear category]
Search Encyclopedia

Featured Article
