In topology (a mathematical discipline) 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.
Contents
- Definitions
- Irreducible manifold
- Prime manifolds
- Euclidean space
- Sphere lens spaces
- Prime manifolds and irreducible manifolds
- From irreducible to prime
- From prime to irreducible
- References
The notions of irreducibility in algebra and manifold theory are related. An irreducible manifold is 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 the circle S1 and the twisted 2-sphere bundle over S1.
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.
Definitions
Let us consider specifically 3-manifolds.
Irreducible manifold
A 3-manifold is irreducible if any smooth sphere bounds a ball. More rigorously, a differentiable connected 3-manifold
The assumption of differentiability of
A 3-manifold that is not irreducible is reducible.
Prime manifolds
A connected 3-manifold
Euclidean space
Three-dimensional Euclidean space
On the other hand, Alexander's horned sphere is a non-smooth sphere in
Sphere, lens spaces
The 3-sphere
A lens space
Prime manifolds and irreducible manifolds
A 3-manifold is irreducible if and only if it is prime, except for two cases: the product
From irreducible to prime
An irreducible manifold
then
From prime to irreducible
Let
Since
It remains to consider the case where it is possible to cut