Prime manifold explained

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

S ^{2} x S ^{1} and the non-orientable fiber bundle of the 2-sphere over the circle 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.

Definitions

Consider specifically 3-manifolds.

Irreducible manifold

A 3-manifold is if every smooth sphere bounds a ball. More rigorously, a differentiable connected 3-manifold

M is irreducible if every differentiable submanifold S homeomorphic to a sphere bounds a subset D (that is, S = \partial D ) which is homeomorphic to the closed ball$$D^3\; =\; \backslash .$$The assumption of differentiability of 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 .

Prime manifolds

A connected 3-manifold

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

Examples

Euclidean space

\R ^{3} is irreducible: all smooth 2-spheres in it bound balls.

On the other hand, Alexander's horned sphere is a non-smooth sphere in

\R ^{3} that does not bound a ball. Thus the stipulation that the sphere be smooth is necessary.

Sphere, lens spaces

S ^{3} is irreducible. The product space S ^{2} x S ^{1} is not irreducible, since any 2-sphere S ^{2} x \{pt\} (where pt is some point of S ^{1} ) has a connected complement which is not a ball (it is the product of the 2-sphere and a line).L(p,q) with p ≠ 0 (and thus not the same as S ^{2} x S ^{1} ) is irreducible.

Prime manifolds and irreducible manifolds

A 3-manifold is irreducible if and only if it is prime, except for two cases: the product

S ^{2} x S ^{1} and the non-orientable fiber bundle of the 2-sphere over the circle S ^{1} are both prime but not irreducible.

From irreducible to prime

An irreducible manifold

M is prime. Indeed, if we express M as a connected sum$$M=N\_1\backslash \#N\_2,$$then M is obtained by removing a ball each from N _{1} and from N _{2,} and then gluing the two resulting 2-spheres together. These two (now united) 2-spheres form a 2-sphere in M. The fact that M is irreducible means that this 2-sphere must bound a ball. Undoing the gluing operation, either N _{1} or N _{2} is obtained by gluing that ball to the previously removed ball on their borders. This operation though simply gives a 3-sphere. This means that one of the two factors N _{1} or N _{2} was in fact a (trivial) 3-sphere, and M is thus prime.

From prime to irreducible

Let

M be a prime 3-manifold, and let S be a 2-sphere embedded in it. Cutting on S one may obtain just one manifold N or perhaps one can only obtain two manifolds M _{1} and M _{2.} In the latter case, gluing balls onto the newly created spherical boundaries of these two manifolds gives two manifolds N _{1} and N _{2} such that$$M\; =\; N\_1\backslash \#N\_2.$$Since M is prime, one of these two, say N _{1,} is S ^{3.} This means M _{1} is S ^{3} minus a ball, and is therefore a ball itself. The sphere S is thus the border of a ball, and since we are looking at the case where only this possibility exists (two manifolds created) the manifold M is irreducible.

It remains to consider the case where it is possible to cut

M along S and obtain just one piece, N. In that case there exists a closed simple curve \gamma in M intersecting S at a single point. Let R be the union of the two tubular neighborhoods of S and \gamma. The boundary \partial R turns out to be a 2-sphere that cuts M into two pieces, R and the complement of R. Since M is prime and R is not a ball, the complement must be a ball. The manifold M that results from this fact is almost determined, and a careful analysis shows that it is either S ^{2} x S ^{1} or else the other, non-orientable, fiber bundle of S ^{2} over S ^{1.}

References

• Book: William Jaco. William Jaco. Lectures on 3-manifold topology. 0-8218-1693-4.

See also

• 3-manifold
• Connected sum
• Prime decomposition (3-manifold)