Encyclopedia > Irreducible

  Article Content

Irreducible

In Abstract algebra, irreducible is an abbreviation of irreducible element.

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 connect 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.

A theorem of 3-manifold theory is: every compact, connected 3-manifold 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 piecewise-linear category]



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Mayenne

...     Contents Mayenne Mayenne is a French département, number 53, named after the Mayenne River[?]. Préfecture (capital): ...

 
 
 
This page was created in 59.4 ms