Seifert Fiber Space - Classification

Classification

Seifert classified all closed Seifert fibrations in terms of the following invariants. Seifert manifolds are denoted by symbols

where: is one of the 6 symbols:, (or Oo, No, NnI, On, NnII, NnIII in Seifert's original notation) meaning:

o₁ if B is orientable and M is orientable.

o₂ if B is orientable and M is not orientable.

n₁ if B is not orientable and M is not orientable and all generators of π₁(B) preserve orientation of the fiber.

n₂ if B is not orientable and M is orientable, so all generators of π₁(B) reverse orientation of the fiber.

n₃ if B is not orientable and M is not orientable and g≥ 2 and exactly one generator of π₁(B) preserves orientation of the fiber.

n₄ if B is not orientable and M is not orientable and g≥ 3 and exactly two generators of π₁(B) preserve orientation of the fiber.

g is the genus of the underlying 2-manifold of the orbit surface.

b is an integer, normalized to be 0 or 1 if M is not orientable and normalized to be 0 if in addition some a_'i is 2.

(a₁,b₁),...,(a_r,b_r) are the pairs of numbers determining the type of each of the r exceptional orbits. They are normalized so that 0<b_i<a_i when M is orientable, and 0<b_i≤a_i/2 when M is not orientable.

The Seifert fibration of the symbol

can be constructed from that of symbol

by using surgery to add fibers of types b and b_i/a_i.

If we drop the normalization conditions then the symbol can be changed as follows:

• Changing the sign of both a_i and b_i has no effect.
• Adding 1 to b and subtracting a_i from b_i has no effect. (In other words we can add integers to each of the rational numbers (b, b₁/a₁, ..., b_r/a_r provided that their sum remains constant.)
• If the manifold is not orientable, changing the sign of b_i' has no effect.
• Adding a fiber of type (1,0) has no effect.

Every symbol is equivalent under these operations to s unique normalized symbol. When working with unnormalized symbols, the integer b can be set to zero by adding a fiber of type (1, b).

Two closed Seifert oriented or non-orientable fibrations are isomorphic as oriented or non-orientable fibrations if and only if they have the same normalized symbol. However, it is sometimes possible for two Seifert manifolds to be homeomorphic even if they have different normalized symbols, because a few manifolds (such as lens spaces) can have more than one sort of Seifert fibration. Also an oriented fibration under a change of orientation becomes the Seifert fibration whose symbol has the sign of all the bs changed, which after normalization gives it the symbol

and it is homeomorphic to this as an unoriented manifold.

The sum b + Σb_i/a_i is an invariant of oriented fibrations, which is zero if and only if the fibration becomes trivial after taking a finite cover of B.

The orbifold Euler characteristic χ(B) of the orbifold B is given by

χ(B) = χ(B₀) − Σ(1−1/a_i)

where χ(B₀) is the usual Euler characteristic of the underlying topological surface B₀ of the orbifold B. The behavior of M depends largely on the sign of the orbifold Euler characteristic of B.