- SEIFERT FIBERED MANIFOLDS AND GRAPH MANIFOLDS 193
- Seifert fibered manifolds and graph manifolds
In this section we give some topological definitions related to collapsing se-
quences of compact 3 -ma nifolds, which we shall study in the next chapter.
DEFINITION 31.52 (Foliation). A smooth k-dimensional foliation of a n n-
dimensional differentia ble m anifold Mn is a n atlas { (Ua , Xa)} of M such that for
any a, (3, the transition function
r.p 13 o r.p;;;^1 : 'Pa (Ua n U 13 ) --t r.p 13 (Ua n U 13 )
h as the first n -k dep endent va riables dep ending only on the first n -k independent
variables. That i s , if we let xo = (x^1 , ... , xn-k) and x • = (xn-k+l, ... , xn) , thenr.p 13 o r.p;;;^1 (xo, x•) = ( (xo), w (xo, x•))
for some functions and w mapping into JRn-k and JRk, res pectively.
The submanifolds{ x E Ua : (r.p;, ... , r.p~-k) = const},
where 'Pa ~ ( r.p~,. .. , r.p~), are called plaques. Note that each plaque is a k-dimensional submanifold of M. A leaf is a k-dimensional submanifold of M which
is a maximal union of plaques.DEFINITION 31.53 (Seifert fibered manifold). A Seifert fibered manifold is
a compact 3-manifold which admits a foliation whose leaves are circles.
Examples of Seifert fibered ma nifolds are a product of a surface with the circle,
the unit tangent bundle of a Riemannian surface, and a 3-manifold with a circle
action which has no global fixed points. The simplest exa mple of a Seifert fibered
ma nifold is the solid torus B^2 x 51 with the product folia tion whose leaves are
{p} x 51. It turns out that the solid torus B x 51 h as many different 51 foliations
F p ,q , where p , q E Z are relatively prime integers. These folia tions on the solid
torus are the building blocks for constructing all Seifert fibered manifolds.
Here is the description of the foliation F p ,q· Consider the solid torus B x 51
as the quotient of B x JR by the action of Z whose generator T sends (z, t) to
(e^2 Prri/qz, t + 1). The diffeo morphism T preserves the product foliation {p} x JR on
B x JR. Furthermore, each leaf {p} x JR b ecomes a circle in the quotient solid torus.
Thus the quotient foliation, denoted by F p,q, is a foliation whose leaves a re circles.
A Seifert fibered 3-manifold is said to h ave an orientable foliation if the circle
fibers can b e coherently oriented. A simple topological argument shows that each
Seifert fibered 3-manifold either has an orientable foliation or has a two-fold cover
which has an orientable foliation. All Seifert fibered 3 -manifolds with orientable
fibrations are obtained as follows. Take a compact surface X with nonempty bound-
ary and consider the product foliation on Xx 51. Now glue solid tori with foliations
F p , ,q, to the boundary components of X x 51 by diffeomorphisms preserving the
51 fibers.
Seifert fibered 3-manifolds have been classified up to diffeomorphism. The
classification is given by the invariants (pi, qi ), the topology of the surface X , and
the Euler characteristic class of the circle fibration. It can b e shown that each Seifert
fibered 3 -ma nifold h as a finite cover which is either the product space X x 51 for