In mathematics, the term **irreducible** is used in several ways. - In representation theory (group theory), an
**irreducible representation** is a nontrivial representation with no nontrivial subrepresentations. Similarly, an **irreducible module** is another name for a simple module. - In the theory of manifolds, an n-manifold is
**irreducible** if 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 n-manifold is called prime, if it cannot be written as a connected sum of two n-manifolds (neither of which is an n-sphere). 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 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 S^1 and the twisted 2-sphere bundle over S^1. |