Horizontal vector fields and Seifert fiberings
Andy Hammerlindl


Diagonalwise Refactorization: An Efficient
 Training Method for Depthwise Convolutions

    
Zheng Qin, Zhaoning Zhang^*^*Corresponding author., Dongsheng Li, Yiming Zhang, Yuxing Peng
Science and Technology on Parallel and Distributed Laboratory 

National University of Defense Technology

Changsha, China

qinzheng12@nudt.edu.cn; zzningxp@gmail.com; lds1201@163.com; sdiris@gmail.com; pengyuxing@nudt.edu.cn

    
===================================================================================================================================================================================================================================================================================================================================





    This paper gives a classification of the topology of
    vector fields which are
    nowhere tangent to the fibers of a Seifert fibering.





§ INTRODUCTION
 

This paper gives a classification of
the topology of horizontal vector fields on
Seifert fiber spaces.
Here, horizontal means that the
vector field is nowhere tangent to the Seifert fibering.
This work was inspired in part by the classification of horizontal foliations
on Seifert fiber spaces
<cit.>.
Depending on the geometry of the 3-manifold,
the condition of having a horizontal vector field can
be either more restrictive or less restrictive
than having a horizontal foliation.

The key observation is that a horizontal
vector field corresponds to a fiber-preserving map from the
Seifert fiber space to the unit tangent bundle of the base orbifold.

We give definitions and review the theory for
orbifolds and Seifert fiberings in sections <ref>
and <ref> respectively,
and develop a general theory for  in <ref>.
In this introductory section, however,

we first list
out the possibilities based on the geometry of the base orbifold.

First and most interesting are Seifert fiber spaces
where the base orbifold is a bad orbifold.
These are lens spaces and each lens space L(p,q) supports
infinitely many distinct Seifert fiberings.



 
    Suppose M is diffeomorphic to the lens space L(p,q) with p ≥ 0.
    
        
        
        
        
  * 
        If p = 1 or p = 2,
        then every Seifert fibering on
        M has a .
        
  * If p ≥ 3 and q ≡± 1  p,
        then M has infinitely many Seifert fiberings
        which support  and infinitely many which do not.
        
  * If p ≥ 8, p is divisible by 4,
        and q ≡12p ± 1  p,
        then M has exactly one Seifert fibering
        which supports a  and all others do not.
        
  * For all other cases of p and q,
        no Seifert fibering on M has a .
         
These cases are explained in detail in <ref>.
To prove <ref>, we introduce the notion of a “marked lens space”
which carries additional information on the orientation of the two solid
tori which are glued together to construct the lens space.
Under such a notion, L(p, q) and L(-p, -q) are distinct marked lens spaces
even though they are diffeomorphic. Further, except for a few special
cases,
every Seifert fibering of L(-p, -1) has a  while no Seifert fibering
of L(p, 1) has a .

We next consider Seifert fiberings where the base orbifold is elliptic;
that is, the orbifold is finitely covered by the 2-sphere.
Some of these spaces, such as the 3-sphere, are also lens spaces and are
handled by the previous theorem.

 
    Suppose M is a Seifert fiber space with elliptic base orbifold
    and M is not diffeomorphic to a lens space.
    Then M has a  if and only if
    M, up to orientation, is
    the unit tangent bundle of the base orbifold.

Next consider Seifert fiber spaces over parabolic orbifolds.
Such an orbifold is covered by the Euclidean plane and corresponds to a wallpaper
group with only rotational symmetry.

 
    Suppose M is a Seifert fiber space with parabolic base orbifold.
    Then M has a  if and only if either
    
        
  * M, up to orientation, is
        the unit tangent bundle of the base orbifold, or
        
  * the base orbifold is a surface (either the 2-torus or the Klein
        bottle).
     
<Ref> explains both of the above cases in detail.
Note that the two cases overlap when the base orbifold is a surface.

Finally, the most general case is for hyperbolic orbifolds.

 
    Suppose M is a Seifert fiber space with hyperbolic base orbifold.
    Then the following are equivalent:
    
        
  * M has a ,
        
  * M finitely covers the unit tangent bundle of the base orbifold,
        
  * M supports an Anosov flow.
     
The equivalence (2) ⇔ (3) in <ref>
was proved by Barbot, extending a result of Ghys on circle bundles
<cit.>.
The proof of (1) ⇔ (2) is discussed in <ref>.

<Ref> was part of the original motivation
for exploring the properties of these vector fields.
In joint work with Mario Shannon and Rafael Potrie,
we analyze partially hyperbolic dynamical systems
on Seifert fiber spaces
<cit.>.
Most of that paper deals only with the case of dynamical systems
defined on circle bundles.
However, <cit.> employs <ref>
above to extend these results to the case of a general Seifert fiber
space. See the cited paper for further details.

<Ref> studies the homotopy classes of ,
and
<ref> looks at the case of Seifert fiberings on manifolds with
boundary.







§ ORBIFOLDS
  

Intuitively,
an orbifold is an object that locally resembles
a quotient of ^n by a finite group.
In this paper, we will only consider those orbifolds
that appear as the base space of a Seifert fibering,
and so we give a simpler definition restricted to this case.

Let D^2 = { z ∈ : |z| ≤ 1 } be the closed unit disk.
For an integer a ≥ 1,
define r_a : D^2 → D^2, z → e^2 π i/a z.
This is a rotation of order a.
Define
D_a as the topological quotient space
D_a = D^2 / r_a.
If a > 1, then D_a is homeomorphic to a disk,
but is not a smooth surface.
By a slight abuse of notation,
we write 0 ∈ D_a for the image of 0 ∈ D^2 under the quotient.
If U is an open subset of D_a,
then U is a smooth manifold if and only if 0 ∉ U.
Let (D_a) denote the interior of D_a.

An orbifold  is a closed topological surface
equipped with an atlas of charts such that

    
  * each chart is of the form
    
        (φ, U, a)
    
    where a ≥ 1 is an integer,
    U is open, and
    φ : U →(D_a) is a homeomorphism;
    
  * the charts cover ; and
    
  * if (φ, U, a) and (ψ, V, â)
    are distinct charts in the atlas,
    then φ(U ∩ V) and ψ(U ∩ V)
    are smooth manifolds and
    
        ψ∘φ: φ(U ∩V) →ψ(U ∩V)
    
    is a diffeomorphism.

The final item above implies
in particular
that if a > 1 then φ(0) ∉ V and
if â > 1 then ψ(0) ∉ U.
We call x ∈ a cone point if the atlas has a chart
(φ, U, a) such that φ(x) = 0 and a > 1.
Here, a is the order of the cone point.
In this version of the definition of an orbifold,
each cone point is contained in exactly one chart.
Since the orbifold is compact, it can have at most finitely many
cone points.

In this paper,
we refer to orbifolds using the orbifold notation of Thurston and Conley
<cit.>.
In this notation, the starting surface is a sphere
and orbifolds are constructed from this by surgery.
A positive integer a in the notation corresponds to excising a disk
and gluing in D_a in its place.
For example, the 237 orbifold is a sphere with cone points
of orders 2, 3, and 7 added.
We allow ones in the notation in order to make some results easier to state.
For instance, consider the family of pp orbifolds with p ≥ 1.
This is an infinite family of orbifolds that contains the 11 orbifold,
i.e, the 2-sphere, as a member of the family.
The symbol “” corresponds to replacing a disk with a cross cap,
and so
the 22 orbifold is the sphere with two cones points of order two
and one cross-cap added.
The symbol “o” corresponds to adding a handle, and so
the 23o o orbifold is
a genus-two surface with points of orders 2 and 3 added.
We do not consider orbifolds with silvered edges or corner reflectors
and so the “*” symbol will not occur in any of the notation here.

For an orbifold , the underlying topological surface _0
is just the orbifold itself treated as a topological surface
and forgetting any of the additional information given by the charts
(φ, U, a).



For a smooth surface  equipped with a Riemannian metric,
one can define the unit tangent bundle 
consisting of all tangent vectors of length v  = 1.
This is a 3-manifold 
with a natural circle fibering coming from the projection
→.


We now extend this notion of a unit tangent bundle
to the case of an orbifold.

First, consider a choice of Riemannian metric g on the disk D^2
such that with respect to this metric the rotation r_a : D^2 → D^2
is an isometry.
If U is an open subset of D_a with 0 ∉ U,
then g descends to a Riemannian metric on U.
By a slight abuse of notation, we call such a g a
Riemannian metric on D_a.
Define T D_a as the quotient of T D^2 by the
tangent map T r_a : T D^2 → T D^2.
The tangent bundle T D^2
has the structure of a smooth 4-manifold with boundary,
but if a > 1, then the quotient T D_a does not
have such structure.
Since r_a is an isometry,
its tangent map restricts to a diffeomorphism
of the unit tangent bundle of D^2.
This diffeomorphism and its iterates
T r_a^i|_U T D^2 for 1 ≤ i < a
have no fixed points.
From this property, one can see that the quotient
U T D_a = U T D^2 / T r_a
is a smooth 3-manifold with boundary.
In fact, it is a solid torus.

For an orbifold ,
we define the tangent bundle using using equivalence classes of tangent
vectors in charts.


That is, consider the set X of all tuples
(p, (ϕ, U, a), v)
where p is a point in , (ϕ, U, a) is a chart containing p,
and v a vector in T D_a based at ϕ(p).
The equivalence relation is defined by

    (p, (ϕ, U, a), v) ∼ (q, (ψ, V, â), w)

if and only if

    p = q
        
        T(ψ∘ϕ)(v) = w.

The tangent bundle T of the orbifold
is defined as the set of equivalence classes of X.

A Riemannian metric on an orbifold 
is a choice for every chart (ϕ, U, a)
of Riemannian metric on D_a
such that every transition map
ψ∘ϕ : ϕ(U ∩ V) →ψ(U ∩ V)
is an isometry.
Suppose  is an orbifold equipped with a Riemannian metric.
As the transition maps are isometries,
every element v ∈ T has a well-defined length,
even those vectors based at cone points,
and we may define the unit tangent bundle

    = { v ∈T :  v  = 1 }.


A key observation is that even though  is an orbifold,
its unit tangent bundle is a smooth 3-dimensional manifold.
To see this,
note that  is covered by open sets of the form
U T V where V is given by a chart (V, ϕ, a)
and T ϕ : U T V → U T D_a
gives a smooth structure to U T V.
Distinct choices of metric on  will yield
unit tangent bundles which are distinct as subsets of T ,
but which are nonetheless diffeomorphic to each other.
Because of this,
in what follows we will at times discuss “the”
unit tangent bundle of an orbifold without giving a specific choice of metric.

The unit tangent bundle of an orbifold is oriented
if even the orbifold itself is not.
This is because at a point x ∈,
a choice of orientation of the tangent space T_x
induces an orientation on the circle of unit tangent vectors.
Regardless of the choice of orientation on T_x ,
the resulting orientation on the 3-manifold is the same.



One way to produce an orbifold is to quotient a surface by a discrete group of
isometries.
We call an orbifold  a good orbifold
if there is a Riemannian surface S of constant curvature
and a finite group of isometries G acting on S
such that = S / G with the orbifold structure of  
coming from the smooth structure of S.
We do not make a distinction between “good” and “very good”
orbifolds as the notions are equivalent in the 2-dimensional setting.
An orbifold is bad if it is not good.
Proofs of the following two results are given
in <cit.>.
See also the discussion in <cit.>.


    An orbifold  is bad if and only if
    it is either
    
        
  * a sphere with a single cone point of order p > 1 added, or
        
  * a sphere with two cone points of orders p  q added.
         
For an orbifold  with
underlying topological surface _0
and cone points of order _1, …, _n,
define the Euler characteristic of 
as

    χ() = χ(_0) - ∑_i ( 1 - 1/_i ).

One way to think of this definition is that it is the
formula χ() = V - E + F
for vertices, edges, and faces, but with a cone point counting
as a fraction 1/_i of a vertex.


    If = S / G is a good orbifold, then
    χ() = 1/|G|χ(S).
    

We call a good orbifold , elliptic if χ() > 0,
parabolic if χ() = 0,
and
hyperbolic if χ() < 0.
These cases correspond to the orbifold being covered
by the 2-sphere, 2-torus, or a hyperbolic higher-genus surface
respectively.



We may define vector fields on orbifolds analogously to vector fields
on manifolds.
If  is an orbifold, a vector field on 
is a continuous function v : → T such that π(v(x)) = x
for all x ∈ where π is the canonical projection π : T →.

 
    If v is a vector field on an orbifold ,
    then v is zero at every cone point on .
    In particular, if v is a non-zero vector field,
    then  is a surface and is either the 2-torus or Klein bottle.


    First, consider the case of a vector field defined on D_a with a > 1.
    Such a vector field lifts to a vector field v : D^2 → T D^2
    which is invariant under the rotation r_a;
    that is, T r_a(v(x)) = v(r_a(x)) for all x ∈ D^2.
    If x = r_a(x) is the center of the disk, this is only possible
    if the vector is zero at this point.
    This shows that a vector field must be zero at the cone point in D_a.
    Since orbifolds are locally modelled on D_a, any vector field
    on an orbifold must be zero at all cone points.

    If an orbifold  has a non-zero vector field, then  must have
    no cone points and is a true surface. We are only considering
    the case of surfaces without boundary and the Poincaré-Hopf theorem
    implies that χ() = 0.



§ SEIFERT FIBERINGS
  

We now discuss the definition and properties of Seifert fiberings.

Recall that D^2 is a closed unit disk and r_a is a rotation of order a
on D^2.
Let S^1 =  D^2 and note that r_a restricts to S^1.
For coprime integers a,b with a ≥ 1,
define a diffeomorphism  of the solid torus D^2  S^1
by

    : D^2  S^1 → D^2  S^1,    (z,w) ↦ (r_a^b(z), r_a(w)).

This generates a cyclic group of order a acting freely on D^2  S^1
and the quotient

    := D^2  S^1 /

is again a solid torus.
Consider the projection Π : D^2  S^1 → D^2 onto the first coordinate.
The fibers of this projection are circles and this fibering is invariant
under . Hence it defines a fibering of  by circles.
The solid torus  along with this fibering by circles is called
is called a standard fibered torus.
The fibering may also be viewed directly as the fibers of a map
from  to D_a = D / r_a.

In the special case that a = 1,
 is the identity map
and = D^2  S^1 has a trivial fibering. We call this a
trivial fibered torus.



Let M be a closed 3-manifold along with a decomposition of M into a disjoint
union of circles.
We say that M is a Seifert fiber space
if for every x ∈ M there is a solid torus U embedded in M
with x in its interior such that U is a union of circles
and these circles give U the structure of a standard fibered torus.
That is, there are integers a and b and a diffeomorphism
from  to U which takes circles to circles.

Let  be the topological space defined by quotienting M along these
circles.
Then any standard fibered torus U embedded in M
quotients down to a subset of 
with the same topological and smooth structure
as the quotiented disk D_a.
From this, one may show that the Seifert fibering on M
induces the structure of an orbifold on .
The projection π : M → determines the fibering of circles on M,
and so we will often refer to a Seifert fibering as given
by a map M →
where M is a closed 3-manifold and  is an orbifold.
If x is a cone point in , then the fiber π(x) is called
an exceptional fiber.
If x is not a cone point, then π(x) is called a regular fiber.



We now outline the standard method of construction of a Seifert fiber space
by gluing solid tori into a trivial circle bundle with boundary.
This largely follows the exposition in <cit.>.

Let g ≥ 0 be an integer and let (_1, _1), …, (_n, _n)
be such that for each i, _i and _i are coprime integers and _i ≥ 1.
Let F be a closed surface of genus g and remove n disjoint open disks from F
to produce
F_0 = F  (D_1^2 ∪…∪ D_n^2).
Let M_0 = F_0  S^1 be the trivial circle bundle over F_0
and write

    M_0 = (S_1^1  S^1) ∪…∪ (S_n^1  S^1).

Define

    R
               = F_0 {1},
        
    
        H_i
               = {1} S^1  S^1_i  S^1, 
    
        C_i
               = R ∩ (S^1_i  S^1) = S^1_i {1}.


Here,
H_i is a vertical fiber on the boundary of M_0
and C_i is a horizontal circle around the excised disk.
We assume C_i has the same orientation as the boundary of the excised disk.
This means that the boundary components of R have the opposite orientation as
the circles C_i.

Let T_i = D^2  S^1 be a solid torus.
It has a meridian M_i = S^1 {1} T_i which bounds a disk inside
T_i and a longitude L_i = {1} S^1  T_i.
Note that M_i and L_i form a basis for the first homology group
of T_i
and that C_i and H_i form a basis for the first homology group of
S^1_i  S^1  M_0.
At times, we consider rational linear combinations of elements of this
homology,
and so we use homology with rational coefficients throughout.

To glue T_i into M_0, we choose a linear map identifying the two
tori and such that M_i ∼_i C_i + _i H_i in homology.
That is, we choose integers '_i and '_i such that the matrix

    [  _i  _i; '_i '_i ]

lies in SL(2, ) and use this as the gluing map.
By a slight abuse of notation, we write

    [ M_i; L_i ]
        =
        [  _i  _i; '_i '_i ][ C_i; H_i ]

to concisely express how the circles are related in homology.
Once T_i is glued into the manifold, there is a unique way to extend
the fibering to T_i in such a way that T_i is a standard fibered torus.

Note that the longitude of the solid torus is not well defined
and we could replace L_i with a curve homologous to L_i + k M_i
for any integer k.
The gluing matrix is not unique
and we could replace '_i and '_i by '_i + k _i
and '_i + k _i.
These, in fact, correspond to the same ambiguity
and they have no impact on the topology of the resulting manifold.

After gluing in all of the tori, the result is
a closed oriented 3-manifold.
We denote this manifold equipped with this Seifert fibering as

    M(g; (_1, _1),  …,  (_n, _n)).



We now consider the case where g < 0,
which denotes a Seifert fibering over a non-orientable orbifold.
Readers only interested in Seifert fiberings over
oriented base orbifolds may safely skip over this paragraph.
Let g < 0 and let F be a closed non-orientable surface of genus |g|.
Let F_0 = F  (D_1^2 ∪…∪ D_n^2).
Let G_0 be the orientable double cover of F_0
and let τ : G_0 → G_0 be an involution such that
F_0 = G_0 / τ.
Define an involution ϕ on G_0  S^1 by
ϕ(g, z) = (τ(g), z̅) where complex conjugation z ↦z̅
is an involution of the circle.
Then define a twisted circle bundle over F_0 by M_0 = (G_0  S^1) / ϕ.
Define R as the image of G_0 {1} under this quotient.
Note that while M_0 is orientable,
the fibers are neither orientable nor transversely orientable
on all of M_0.
However, we can find a subset M_1 of M_0 which is a union of
fibers, contains the boundary of M_0, and on which the fibers are
both orientable and transversely orientable.
This subset M_1 may be identified with F_1  S^1
where F_1 is an orientable subset of F_0
and such that

    R ∩M_1 = F_1 {1}.

Under this identification, the construction of the Seifert fibering
by gluing proceeds exactly as before.
Because of the existence of this subset M_1,
we also freely assume in the proofs in the remainder of this section
that the fibering is always orientable.



Two Seifert fiberings M_1 →_1 and M_2 →_2 are isomorphic
if there is a diffeomorphism from M_1 to M_2 which takes fibers to fibers.
Every Seifert fibering M → of an oriented 3-manifold 
is isomorphic to one constructed by gluing in tori as above,
and so we define the Seifert invariant
of the fibering as

    (g; (_1, _1),  …,  (_n, _n)).

Two invariants define the same fibering if and only if
one can be transformed into the other by changes of the following form:

    
  * altering each 
    _i/_i
    by an integer, but keeping ∑_i/_i fixed,
    
  * re-ordering the pairs (_i,_i), and
    
  * inserting or removing pairs of the form (1,0).

For proofs of the above assertions,
see <cit.> or <cit.>.

The Euler number of a fibering is defined as

    e(M →) = - ∑_i=0^n _i/_i.

One can see that none of the above transformations to the Seifert invariant
will affect the Euler number.

It is possible to extend the notation to define Seifert invariants
for fiberings on non-orientable 3-manifolds.
However, this complicates the notation and is not useful for the current
paper, so we only consider invariants for oriented manifolds.

For most Seifert fiber spaces, the topology of the 3-manifold
uniquely determines the fibering up to isomorphism.
However, there are a few important exceptions, such as for lens spaces.
We discuss this non-uniqueness in detail in <ref>.



Recall that the unit tangent bundle  of an orbifold 
is a closed 3-manifold.
Further,
the unit tangent bundle of a quotiented disk D_a = D^2 / r_a
is a standard fibered torus where the fibers are given
by the projection T D_a → D_a.
From this, one can see that the projection →
gives  a canonical Seifert fibering.

 
    For an orbifold  with
    underlying topological surface _0 of genus g
    and cone points of order _1, …, _n,
    the unit tangent bundle has Seifert invariant
    
    (g;   (1, n - χ(_0)),   (_1, -1), …, (_n, -1)).
 
This fact is stated several times in the literature.
See, for instance, <cit.>.
However, we know of no place that gives a complete proof
including the case of bad orbifolds,
and so
we give a full proof here.


    We will first consider the sphere and then certain quotients
    of the sphere before handling general surfaces and orbifolds.

    For the unit tangent bundle of S^2,
    travelling once around a circle in the fibering corresponds to
    staying in one place on the globe,
    starting facing a certain direction,
    and then rotating once around
    counter clockwise (CCW).
    Consider a unit vector field on the 2-sphere defined everywhere
    except on small disks centered at the north and south poles.
    This unit vector field always points north
    and plays the role of R in the construction of a Seifert fibering above.

    Looking down on the north pole in a small chart,
    we may trivialize the unit tangent bundle
    in this chart.
    For instance, we may take a section of the unit tangent bundle
    to be a vector field
    which always points “up” inside this chart.
    If we consider a small disk around the north pole,
    then its unit tangent bundle is a solid torus
    and the meridian M_i consists of vectors pointing “up”
    everywhere on the circle which bounds the disk.
    If we walk CCW around this circle once,
    then a north-pointing compass needle will make a complete CCW rotation.
    From this, one can see that C_i is homologous to M_i + H_i.
    Similarly at the south pole, if we walk CCW once
    around the circle, then the compass needle will rotate once CCW.
    and so C_i ∼ M_i + H_i for this disk as well.
    Together, these show that the Seifert invariant of S^2
    is
    
        (0; (1, -1), (1, -1)).
    
    

    Now let > 1 be an integer and the consider
    a rotation of order  of the sphere such that the
    quotient by this rotation is an orbifold 
    and each of the north and south
    poles of the sphere quotients down to a cone point of order .
    This quotient defines a map π : U T S^2 → U T
    and one can verify that this a smooth covering map
    between manifolds.
    In what follows, we use uppercase letters
    R, C_i, H_i, M_i, L_i
    for curves
    and surfaces in U T S^2 and lowercase letters
    r, c_i, h_i, m_i, ℓ_i
    for corresponding objects in U T .

    The north-pointing vector field is invariant under the rotation,
    and so it quotients down a vector field defined
    everywhere on the orbifold except for small disks
    about each of the cone point.
    The section R defined in the previous case
    covers a section r
    defined everywhere except on disks about the two cone points
    and the covering R → r is of degree .

    Let c_i denote a boundary circle of r,
    but with opposite orientation.
    It is -fold covered by C_i.
    Let h_i denote a regular fiber which is the image of H_i
    under the quotient.
    
    The north pole's meridional disk on S^2 maps down injectively
    to a meridional disk on , and
    the boundary m_i of this disk
    wraps  times around the cone point in .
    Then
    
    π(C_i) =  c_i,   π(H_i) = h_i, π(M_i) = m_i

    in homology.
    It follows from C_i ∼ H_i + M_i that c_i is homologous to h_i + m_i.
    The exact same relations hold at the south pole.
    Hence, m_i ∼ c_i - h_i at both poles
    and the Seifert invariant for  is
    
        (0; (, -1), (, -1)).
    
    

    Now let  be a surface of any genus.
    Put a vector field on  which is non-zero
    everywhere except a finite set consisting
    of sinks, sources, and saddles.
    We can define small disks about each
    of the critical points and each disk yields a solid torus
    in the unit tangent bundle.
    As with the case of the sphere,
    each of the sinks and sources has a gluing
    with M_i ∼ C_i + H_i.
    A “compass” following the vector field around
    a saddle will rotate once clockwise, and so
    one can show that a saddle corresponds to a gluing
    with M_i ∼ C_i - H_i
    
    Hence, the circle bundle has an Euler number
    which is the number of sinks and sources minus the number of saddles.
    The Poincaré-Hopf theorem tells us that this is exactly
    the Euler characteristic of the surface.

    

    Finally, consider the general case of an orbifold .
    Consider a vector field on the underlying surface _0 
    which has finitely many sinks, sources, and saddles.
    Further assume that every cone point in 
    corresponds to a point in _0 which is a sink or source for the flow.
    Such a vector field may be produced, for instance,
    by considering a gradient flow where all of the cone points 
    correspond to local minima or maxima.
    
    The previous case tells us that _0
    has a Seifert invariant which may be written as
    
    (g;   (1, n - χ(_0)),  (1, -1),  …,   (1, -1)_n  times).

    Based on our above analysis of the 2-sphere and its quotient,
    one sees that
    the effect of removing a point in a surface
    and replacing it by a cone point of order _i
    is to remove a pair (1, -1) and replace it
    by (_i, -1) in the Seifert invariant.
    This then proves the desired result.









 
    The equality e(→) = χ()
    holds for any orbifold.




    This follows directly from <ref>
    and the definitions of Euler number and Euler characteristic.


Suppose M_1 → and M_2 →
are two Seifert fiberings
over the same orbifold.
We call a covering map u:M_1 → M_2 a fiberwise covering
if it quotients down to the identity map on .
That is, the following diagram commutes:

    M_1uM_2𝕀.




As described in <cit.>,
a Seifert fibering M →
may be viewed as the action of S^1 on the manifold
where the orbits of the action are fibers of the fibering.
One may also think of the fibers as being equipped
with a metric such that every regular fiber has length exactly one.
The metric varies continuously and so an exceptional
fiber will then have a length of 1/_i.
The S^1 action then corresponds to moving along the fibers at constant speed.

A discrete subgroup of S^1 will be a cyclic group
/ d and we may consider the quotient of M by / d.

 
    Suppose M is a Seifert fiber space with invariant
    
    (g; (_1, _1), …, (_n, _n))

    and d is a positive integer coprime to _i
    for all i.
    Then the quotient M / ( / d) is a Seifert fiber space
    with invariant
    
    (g; (_1, d _1), …, (_n, d _n)).

    Moreover, any fiberwise covering of degree d ≥ 1 
    between oriented
    Seifert fiber spaces is of this form.


    A very similar result is stated as
    <cit.>,
    but without any distinction between the topological and smooth
    categories of manifolds.
    There, they allow the case where d and _i
    share a common factor.
    In this case, / d does not act freely on M.
    Further, the quotient destroys the smooth structure of M
    and so M / ( / d) is not a smooth manifold.
    As we are only concerned with smooth fiberwise coverings in the current
    paper, we do not allow such quotients.

To prove <ref>,
we first consider the effect of such quotienting
on a standard fibered torus.
For an integer d > 1, define

    Q_d : D^2  S^1 → D^2  S^1,    (z,w) ↦ (z, r_d(w))

and note that this map commutes with .

 
    Consider a standard fibered torus = D^2  S^1 / with a > 1
    and the map Q_d for some d > 1.
    
        
  * If a and d are coprime,
        then the group ⟨, Q_d ⟩
        acts freely on D^2  S^1 and the quotient
        is a standard fibered torus.
        
  * If a and d are not coprime,
        then the group ⟨, Q_d ⟩
        does not act freely on D^2  S^1 and the quotient
        does not have the structure of
        a smooth manifold with boundary.
     

    First, suppose a and d are coprime.
    
    Then, there are integers n and m such that
    
        n d + m a = 1.
    
    Define F as the composition ^n ∘ Q_d^m.
    Using
    
    n ( b/a, 1/a)
            +
            m ( 0, 1/d)
            =
            ( n b/a, 1/a d)
            ∈^2,




















    one can show that
    
    F(z,w) = (r_a^n b(z), r_a d(w)) = R_âb̂(z, w)

    where â = a d and b̂ = n b d.
    Further, F^a = Q_d and F^d = ,
    and so
    ⟨, Q_d ⟩ is a cyclic group generated
    by F
    and D^2  S^1 / F is a standard fibered torus.







    

    Now suppose a and d are not coprime.
    There is 0 < n < a such that n d is a multiple of a,
    and further there is m such that
    
        n d + m a = 0.
    
    In this case,
    
    ^n ∘ Q_d^m (z,w) = (r_a^n(z), w)

    and if z = 0, then (z,w) will be a fixed point.
    This shows that ⟨, Q_d ⟩
    does not act freely on the solid torus
    and that the quotient by this group
    does not have the structure of a smooth manifold with boundary.


    Suppose M is a Seifert fiber space
    as in the statement of the proposition.
    By <ref> and the fact that d is coprime to _i
    for all i,
    it follows that every standard fibered torus
    embedded in M quotients down to a standard fibered torus
    in M / ( / d).
    Hence, the latter is also a Seifert fiber space.

    Let R, C_i, H_i, M_i, and L_i
    be as in the construction of a Seifert fibering
    by gluing as described earlier
    and use r, c_i, h_i, m_i, ℓ_i
    for surfaces and curves in the covered manifold M / ( / d).
    
    Let π : M → M / ( / d)
    be the quotient map.
    One can see that
    π(R) = r,
    π(C_i) = c_i, and
    π(H_i) = d h_i
    in homology.
    For a solid torus neighbourhood of an exceptional fiber in M,
    a meridional disk maps injectively down to a meridional disk
    in M / ( / d)
    and so π(M_i) = m_i.
    A longitude ℓ_i in the quotient is d-fold covered by
    a longitude of a solid torus in M.
    However, we do not know precisely which longitude
    covers ℓ_i
    and so the most we can say
    is that π(L_i + k_i M_i) = d ℓ_i for some integer k_i.
    
    Then
    
    π[ M_i; L_i ]   =
            [  _i  _i; '_i '_i ] π[ C_i; H_i ]  ⇒  
    [    1    0; -k_i    d ][ m_i; ℓ_i ]   =
            [  _i  _i; '_i '_i ][ 1 0; 0 d ][ c_i; h_i ]  ⇒  
    [ m_i; ℓ_i ]   =
            [               _i             d _i; 1d('_i + k_i _i)     '_i + k_i _i ][ c_i; h_i ]
            .

    This formula would make sense whether or not
    the entries are integers
    as we are considering first homology with rational coefficients.
    However, since we already know M / ( / d)
    is a well-defined Seifert fiber space,
    it must be that
    1d('_i + k_i _i) 
    is an integer.
    This shows that the covered manifold has a Seifert invariant
    as stated in the proposition.

    To prove the final statement,
    suppose M_1 → M_2 is a fiberwise covering of degree d.
    If we equip the fibers of M_2 with a metric
    such that every regular fiber has length exactly 1 / d,
    this lifts by the covering to a metric on M_1 where
    every regular fiber has length exactly one.
    The quotient from M_1 to M_1 / ( / d)
    is then exactly the same as the quotient from M_1 to M_2.

We also wish to consider fiberwise coverings of negative degree.
First, we consider the case of d = -1.

 
    If M has Seifert invariant
    
        (g; (_1, _1), …, (_n, _n)),
    
    then -M with the same fibering has invariant
    
        (g; (_1, -_1), …, (_n, -_n)).
     

    Let M → -M be the orientation-reversing identity map.
    To give -M the opposite orientation as M,
    we may assume that the orientation along the fibers has been reversed,
    but the orientation transverse to the fibers remains the same.
    Then, keeping the same notation as in the previous proof,
    π(R) = r,
    π(C_i) = c_i,
    π(H_i) = -h_i,
    π(M_i) = m_i, and
    π(L_i + k_i M_i) = -ℓ_i for some integer k_i.
    One can then calculate the Seifert invariant of -M
    much as in the previous proof.


 
    Suppose M is a Seifert fiber space with invariant
    
    (g; (_1, _1), …, (_n, _n))

    and d is a non-zero integer coprime to _i
    for all i.
    Then M fiberwise covers a Seifert fiber space
    with invariant
    
    (g; (_1, d _1), …, (_n, d _n)).

    Moreover, any fiberwise covering of degree d
    between oriented
    Seifert fiber spaces is of this form.


    This follows from the previous two propositions.

 
    If M_1 → and M_2 →
    are Seifert fiberings and there is a fiberwise covering of degree d
    from M_1 to M_2,
    then
     
        d ·e(M_1 →) = e(M_2 →).
     




§ GENERAL PROPERTIES
  

We now consider the general properties of 
on Seifert fiber spaces.
Suppose v is a vector field on a Seifert fiber space M →.
That is, v is a continuous map v : M → TM such that
v(x) lies in T_x M for every x.
We say that v is horizontal if for all x, the vector
v(x) is not tangent to the fiber through x.

 
    A Seifert fibering M → has a 
    if and only if there is a continuous map u: M →
    such that
    
    Mu𝕀

    commutes.
    
To prove this, we first show the equivalence inside a
standard fibered torus.
Recall the definitions of a standard fibered torus

    = D^2 S^1 / 
and a quotiented disk

    D_a = D^2 / r_a.

Both  and the unit tangent bundles D_a
have fiberings given by canonical maps π : → D_a
and π_a :  D_a → D_a.

 
    Any horizontal vector field on a standard fibered torus
     induces a continuous fiber-preserving map from  to
    the unit tangent bundle of   D_a.


    A vector field v on  may be considered as
    a continuous function
    v : → T .
    Lifting the vector field to D^2  S^1,
    we have a function
    v_1 : D^2  S^1 → T(D^2  S^1)
    which satisfies
    
    v_1 ∘ = T ∘ v_1.

    The lifted vector field is horizontal
    and for a point y = (z, w) ∈ D^2  S^1
    the vector v_1(y) is not tangent to the fiber
    {z} S^1.
    In particular, if Π : D^2  S^1 → D^2
    is the projection onto D^2,
    then 
    T Π (v_1(y)) ∈ T_z D^2
    is non-zero.
    Define a map
    
    u_2 : D^2  S^1 → D^2,
              
            y ↦ T Π(v_1(y)) .

    One may show, using the definition of ,
    that u_2 satisfies the symmetry relation
    
        T r^b_a ∘u_2 = u_2 ∘T .
    
    Hence,
    u_2 quotients to a map u from
    = D^2  S^1 /
    to
    D_a =  D^2 / T r_a.
    Moreover, as u_2 takes fibers to fibers,
    so does u.

Note that the above proof works for any choice of Riemannian metric
on D_a.

 
    Let  be a standard fibered torus,
    P a plane field on  transverse to the fibering,
    and
    u : → D_a
    a continuous function for which 
    
    u D_aππ_a𝕀

    commutes.
    Then there exists a unique  v : → T
    such that for all x ∈,
    v(x) ∈ P(x)
    and
    T π(v(x)) = u(x).


    Each of  and D_a is a solid torus
    with an infinite cyclic fundamental group.
    Since both covering maps D^2  S^1 →
    and D^2 → D_a
    are of the same degree a,
    one may verify that the lifting criterion
    <cit.>
    applies here and u : → D_a
    lifts to a map u_1 : D^2  S^1 → D^2.
    Moreover, we may choose the lift in such a way that
    
    D^2  S^1u_1 D^2D^2𝕀D^2

    commutes.
    To see this, first choose a lift so that the above
    diagram commutes for at least one point y ∈ D^2  S^1,
    and then use that D^2  S^1 is connected
    to show that it commutes for all points.
    The lifted map satisfies
    
        u_1 ∘= T r^n_a ∘u_1
    
    for some integer n, and the above commutative diagram
    shows that n ≡ b  a.
    Without loss of generality, assume n = b.

    The transverse plane field P on 
    lifts to a transverse plane field P_1 on D^2  S^1
    which is invariant under T .
    Let Π : D^2  S^1 → D^2 be the canonical projection.
    By transversality, for each point y = (z, w) ∈ D^2  S^1
    the tangent map
    
    T_y Π : T_y (D^2  S^1) → T_z D^2

    restricts to an isomorphism from the 2-dimensional subspace
    P(y) onto T_z D^2.
    Define u_2(y) as the unique vector in P(y)
    which satisfies T Π (v_2(y)) = u_1(y) ∈ T_z D^2.
    Doing this at all points in D^2  S^1 defines a vector field
    v_2 : D^2  S^1 → T (D^2  S^1).

    Using P((y)) = T (P(y))
    and
    u_1((y)) = T r^b_a u_1(y),
    one can see that
    v_2((y)) = T  v_2(y)).
    Hence,
    u_2 quotients down to a vector field u : → T
    and this is the desired vector field.
    Since u_2 is uniquely determined, so is u.


    Suppose v : M → TM is a .
    Define a function
    
    u : M →,    x ↦ T π(v(x))

    where π : M → defines the fibering.
    As v is horizontal, u is well defined.
    Any point x ∈ M lies inside of a standard fibered torus,
    where the definition of u agrees with the map given in
    <ref>.
    Hence, u is continuous and is the desired map.

    For the converse direction, suppose u : M →
    is a map as in the statement of the proposition.
    Choose a plane field P transverse to the fibering.
    Such a plane field always exists;
    for instance, one can equip M with a Riemannian metric
    and consider the planes which are orthogonal to the fibers.
    
    Then define v : M → TM as the unique vector field
    tangent to P
    which satisfies T π(v(x)) = u(x) for all x ∈ M.
    By <ref>,
    such a v exists and is continuous.
    Moreover, v is horizontal by construction.


Fiber-preserving maps must be of a specific form.

 
    Suppose M_1 and M_2 are Seifert fiber spaces over the same
    base orbifold 
    and
    u:M_1 → M_2 is a fiber-preserving map such that
    
    M_1uM_2𝕀

    commutes.
    Then either
    
        
  * 
        u is homotopic to a composition of the form
        M_1 →→ M_2, or
        
  * 
        u is homotopic to a fiberwise covering M_1 → M_2.
     

    In the first case, the map M_1 → is the projection
    defining the fibering and the second map → M_2
    is a continuous section for the fibering M_2 →.
    In either case, the homotopy is along the fibers of M_2,
    as may be seen from the proof below.


    First assume that M_1 and M_2 and their fibers are orientable.
    Because u projects down to the identity map
    on , the degree of u is equal to the degree of the restriction
    of u to any fiber of M_1.
    Further, since the fiberings of M_1 and M_2 are smooth,
    we may perturb the continuous map u slightly to produce a smooth function
    which also projects to the identity on .
    This smooth perturbation is homotopic to u,
    and so without loss of generality, we assume u itself is smooth.

    Equip each of M_1 and M_2 with a metric so that every regular
    fiber has length exactly one.
    The exceptional fibers will then have lengths 1/_i
    for integers _i>1.
    Consider a point x ∈ M_1 and
    let L_1 be the fiber through x
    and L_2 the fiber through u(x).
    Define length-preserving covering maps
    π_1: → L_1 and
    π_2: → L_2
    and choose a lift ũ: →
    such that π_2 ∘ũ = u ∘π_1.
    Define û: → by
    
    û(t) = ∫_t-12^t+12ũ(s)   ds
                so that    d û/d t
            = ũ(t+12)
            - ũ(t-12) = (u).

    There is then a unique function h:L_1 → L_2 such that
    h ∘π_1 = π_2 ∘û
    and h is independent of the choice of lift ũ.
    When defined fiber by fiber on all of M_1,
    this gives a smooth map h:M_1 → M_2.
    By shrinking the interval [t-1/2,t+1/2] down to a point,
    one can show that h is homotopic to u
    and that the homotopy is through fiber-preserving maps.

    If u = 0, then h is constant on every fiber of M_1,
    and so it factors as a composition M_1 →→ M_2.
    If u  0, then h is a covering map on each fiber.
    Since h projects to the identity on ,
    one can see that it is a global covering map M_1 → M_2.

    If any of M_1, M_2, or the fiberings is not orientable,
    the steps above still work.
    However, the degree of u is only defined up to a sign.
    The interval [t-1/2,t+1/2] was used above in order to
    avoid having to choose an orientation on a fiber.

Because of <ref>,
we primarily consider <ref> in the case where
M_2 is a unit tangent bundle.
When is the case M →→ possible?
This requires a horizontal section →;
that is, a vector field on  which by <ref>
is possible only if  is the 2-torus or Klein bottle.
This situation is discussed in further detail in <ref>.
Outside of this case, we have the following equivalences.

 
    Let M → be a Seifert fibering over
    a base orbifold  which is neither the 2-torus
    nor the Klein bottle.
    Then the following are equivalent:
    
        
  * M has a ;
        
  * there is a continuous map u: M →
        such that
        
    Mu𝕀

        commutes;
        
  * there is a fiberwise covering from M to
        the unit tangent bundle ;
        
  * M is orientable, the fibering has Seifert invariant
        
    (g; (_1, _1), …, (_k, _k))

        and there is a non-zero integer d
        such that
        
            d ·e(M →) = χ()
        
        and
        
    d _i/_i ≡ -1/_i

        for all i = 1, …, k.
     

    The equivalence
    (1) ⇔ (2)
    is given by <ref>
    and
    (3) ⇔ (4)
    is a combination of
    propositions <ref> and <ref>.
    Note that as  is oriented,
    any manifold M covering  is orientable.
    The direction (2) ⇐ (3)
    follows immediately from the definition of a fiberwise covering.
    The only place we use that  is not the 2-torus or Klein bottle
    is to invoke <ref> to ensure we are in the second case
    of <ref>.
    This gives (2) ⇒ (3).

The integer d in item (4) above has topological significance
in the other three items as well.
In (2) and (3), d is the degree of the map u : M →.
In (1), d is the number of turns that the vector field makes
while going around a regular fiber.

In the next four sections,
we consider the consequences of <ref>
for the cases of bad, elliptic, parabolic, and hyperbolic
base orbifolds respectively.




§ LENS SPACES
  

We now consider Seifert fiberings on lens spaces.
To aid in the study, we introduce in this paper
the notion of a “marked” lens space.

For a solid torus, D^2  S^1,
the core of a solid torus
is the curve {0} S^1
where 0 is the center of the disk D^2.

A lens space
is a closed oriented 3-manifold L constructed
by gluing together together two solid tori
U_1 and U_2
by a linear map A : ^2 →^2
which identifies the two boundaries.

A marked lens space
is a lens space along with a choice of orientation
for each of the cores of the two solid tori.

A fibered marked lens space
is a marked lens space equipped with a Seifert fibering
such that each of U_1 and U_2 are standard fibered tori.



Later in this section,
we show that there are two families of Seifert fiberings on lens spaces.
One of these is the fibered marked lens spaces.
The other is produced by taking a single standard fibered torus and,
by identifying points on the boundary,
closing it up into a closed 3-manifold.
We first fully treat the fibered marked lens spaces before handling
this second case.



 §.§ Fibered marked lens spaces


Now consider a marked lens space L.
For the boundaries of the two solid tori,
the meridians M_i are well defined
(at least up to homology)
as they bound solid disks.
The longitudes, however,
are not well defined.
If L_i is a longitude on the boundary of U_i,
then for any integer k_i,
there is a diffeomorphism from U_i to itself
which maps L_i to a curve homologous
to L_i + k_i M_i.
As we are assuming the core of U_i has a
specified orientation,
we only consider longitudes with a compatible orientation.

Viewing the gluing map A as two-by-two matrix,
we have

    [ M_1; L_1 ]
        =
         A [ M_2; L_2 ]

in homology.
In order for the lens space L to be an oriented 3-manifold
the gluing map must reverse the orientation of the boundary,
and so this matrix has a determinant equal to -1.
In this section, we write this gluing matrix as

    A
        =
        [ -q_2    p;       q_1 ]

with a * for the lower-left entry.
We omit this entry as it is uniquely determined
by (A) = -1 and the other three entries,
and whereas the other entries have topological
significance, this lower-left entry
is usually given by a relatively complicated formula
with no particular significance.

The matrix depends on the choice of longitudes L_i + k_i M_i
and a different choice of longitudes would yield a matrix

    [   1   0; k_1   1 ][ -q_2    p;       q_1 ][   1   0; k_2   1 ]
        =
        [  -q_2-p k_2           p;             q_1 + p k_1 ]
        .

This shows that the integer p
is well defined
and that q_1 and q_2 are well defined modulo p.
If we relabelled U_1 and U_2
as U_2 and U_1 respectively,
then the gluing map would be replaced by its inverse,
and the matrix would then be

    [ -q_2    p;       q_1 ]
        =
        [ -q_1    p;       q_2 ]
        .

Note also that the condition (A) = -1
implies that q_1 q_2 ≡ -1  p.

For coprime integers p and q, define L(p,q)
as the marked lens space
constructed by gluing together two solid tori
by a gluing map with a matrix of the form

    [ -q_2    p;       q_1 ]

where q_1 = q and the first column is chosen so the determinant is -1.
As the choice of the first column
of the matrix corresponds only to a choice
of longitude on U_1, the marked lens space L(p,q)
is well defined.
Further, for any integer k,
L(p, q) and  L(p, q + p k) are the same marked lens space.
If 
q_1 q_2 ≡ -1  p.
then
L(p, q_1) and L(p, q_2)
are the same marked lens spaces,
differing only by the choice of labelling
of the two solid tori as U_1 and U_2.

Now suppose we take a marked lens space
and produce a new marked lens space by
reversing the orientation of the core of exactly
one of the two solid tori,
say U_2.
This reverses the orientation of the longitude L_2 and,
in order to keep the original orientation of the 3-manifold
unchanged,
we must reverse the orientation of the meridian M_2 as well.
With respect to these new orientations,
the gluing matrix is now given by

    [ -q_2    p;       q_1 ][ -1  0;  0 -1 ]
        =
        [  q_2   -p;      -q_1 ]
        .

This leads to the following key observation:

    L(p,q) and L(-p,-q)
    are diffeomorphic as oriented 3-manifolds,
    but they are not the same marked lens space.


We can reverse the orientation of a lens space,
say by leaving the orientations of both cores unchanged,
but reversing the orientations of both meridians.
The new gluing matrix is

    [  1  0;  0 -1 ][ -q_2    p;       q_1 ][  1  0;  0 -1 ]
        =
        [ -q_2   -p;       q_1 ]

from which one sees that -L(p,q) = L(-p,q).



Lens spaces give examples of manifolds which are homotopy equivalent,
but not homeomorphic <cit.>.
In fact, it is a highly non-trivial result
that the only homeomorphisms between lens spaces
are the “obvious” ones explained above.

 
    Lens spaces L(p_1, q_1) and L(p_2, q_2)
    are homeomorphic
    if and only if |p_1| = |p_2|
    and q_1 ≡± q_2^± 1 p.

Now consider a Seifert fiber space of the
form M(0; (_1, _1), (_2, _2)).
This is constructed by gluing two standard fibered tori
into a manifold with boundary M_0 = F_0  S^1
where F_0 is a sphere with two disks removed and so is an annulus.
As M_0 is an I-bundle,
this is effectively the same as gluing the boundaries of the
two solid tori together by a linear map that respects the fiberings,
and we may think of the Seifert fiber space as a fibered marked lens space.

 
    The Seifert invariant (0; (_1, _1), (_2, _2)) yields 
    a fibered marked lens space with gluing matrix
    
    [ -_1 '_2 - '_2 _1    _1 _2 + _2 _1;                   '_1 _2 + _2 '_1 ]  where  [  _i  _i; '_i '_i ]∈
            SL(2, ).


    In particular, the marked lens space is L(p,q)
    with
    
    p = _1 _2 + _2 _1 
            q = '_1 _2 + _2 '_1.
 






    Recall in the construction of a Seifert fiber space by gluing
    that
    
    [ M_i; L_i ]
            =
            [  _i  _i; '_i '_i ][ C_i; H_i ]

    where H_i is a vertical fiber,
    C_i is a horizontal circle,
    and '_i and '_i are chosen to yield so that
    the matrix lies in SL(2, ).

    Note that we are not free to replace L_i with -L_i here
    without changing the gluing.
    Hence, the pair (_i, _i) in the Seifert invariant
    determines an orientation of the longitude L_i
    and so also determines an orientation of the core of the solid torus.
    This means we are in fact constructing a marked lens space.

    As the two circles C_1 and C_2 are the boundaries of two disks
    punched out of a sphere, C_1 ∼ -C_2 in homology.
    As regular vertical fibers, H_1 ∼ H_2 in homology.
    Hence,
    
    [ C_1; H_1 ]
            =   [ -1  0;  0  1 ][ C_2; H_2 ]  ⇒  
    [  _1  _1; '_1 '_1 ][ M_1; L_1 ]
            =   [ -1  0;  0  1 ][  _2  _2; '_2 '_2 ][ M_2; L_2 ]  ⇒  
    [ M_1; L_1 ]
            =   [  _1  _1; '_1 '_1 ][ -1  0;  0  1 ][ -'_2   _2; -'_2   _2 ][ M_2; L_2 ]  ⇒  
    [ M_1; L_1 ]
            =   [ -_1 '_2 - '_2 _1    _1 _2 + _2 _1;                   '_1 _2 + _2 '_1 ][ M_2; L_2 ]
            .
 






Using this, we may calculate which lens spaces arise as unit tangent bundles.

 
    If  is a sphere with 0, 1, or 2 cone points added,
    then  is a marked lens space of the form L(p, -1)
    with p ≤ -2.


    By <ref>, we may write the fibering as
    
        
        M(0; (_1, -1), (_2, -1))
    
    where _1 = _2 = 1 if = S^2 is the two sphere
    and _1 = 1 if  has a single cone point.
    One can then apply <ref> with '_i = 1 and '_i = 0
    to get the result.
    
    Alternatively, it is fairly easy
    to calculate the gluing matrix directly.
    Adapting the proof of <ref>, the matrix is given by
    
    [ _2 -1;  1  0 ][ -1  0;  0  1 ][ _1 -1;  1  0 ]
            =
            [      1 -_1-_2;            -1 ]
            .

    Since _1, _2 ≥ 1, it follows that p = -_1-_2 ≤ -2.

 
    Let L̂ and L be fibered marked lens spaces
    and let π : L̂→ L be a fiberwise
    covering of degree d.
    Then there are integers p and q such that
    L̂ = L(p, q)
    and
    L = L(d p, q).


    We use the same notation as in the proofs of
    propositions <ref> and <ref>.
    It follows from those proofs that
    π(M_i) = m_i
    and π(L_i + k_i M_i) = d ℓ_i for some integer k_i.
    Then,
    
    π[ M_1; L_1 ]
            =   [ -q_2    p;       q_1 ] π[ M_2; L_2 ]  ⇒  
    [    1    0; -k_1    d ][ m_1; ℓ_1 ]
            =   [ -q_2    p;       q_1 ][    1    0; -k_2    d ][ m_2; ℓ_2 ]  ⇒  
    [ m_1; ℓ_1 ]
            =   1/d[   d   0; k_1   1 ][ -q_2    p;       q_1 ][    1    0; -k_2    d ][ m_2; ℓ_2 ]  ⇒  
    [ m_1; ℓ_1 ]
            =   [ -q_2 - k_2 p          d p;               q_1 + k_1 p ][ m_2; ℓ_2 ]

    Hence, the claim holds with q = q_1 + k_1 p.

Note that q  d p uniquely determines q  p,
and so the covered lens space uniquely determines the
covering lens space.


 
    A fibered marked lens space L(p,q) 
    has a  if and only if
    p  0 and q ≡ -1  p.


    First assume L(p,q) has a .
    Then by <ref> and <ref>,
    it fiberwise covers L(d p, -1) for some integer d
    with the further property that d p ≤ -2.
    This implies that p must be non-zero.
    By <ref>, L(p,q) = L(p, -1) as a marked lens space
    and so q ≡ -1  p.

    Conversely, assume p  0 and q ≡ -1  p.
    As explained earlier in this section,
    we may make choices of longitudes,
    or equivalently choices of '_i and '_i,
    such that the gluing matrix is
    
    [  _1  _1; '_1 '_1 ][ -1  0;  0  1 ][  _2  _2; '_2 '_2 ]
            =
            [  1  p;    -1 ]
            .

    Then _1 '_1 - '_1 _1 = 1
    as it is the determinant of a matrix in SL(2, ).
    <Ref> implies that
    '_1 _2 + _2 '_1 = -1.
    These equations may be rewritten as
    
    '_1 ( -_1/_1) = 1/_1 - '_1
            '_1 ( -_2/_2) = 1/_2 + '_1.

    One may then show that <ref> holds with d = '_1.
    Indeed,
    d(-_i/_i) ≡ 1/_i
    since '_1 is an integer, and summing the above equations gives
    
    d · e(M →)
            =
            d ( - _1/_1 - _2/_2)
            =
            1/_1 + 1/_2
            =
            χ().
 
Before proceeding, we attempt here to give a more intuitive
and topological explanation of why q ≡ -1  p
is a necessary and sufficient condition for a 
and why d = '_1 in the proof above.
Since the results have already been rigorously proved above,
we do not give complete details in the following explanation.

First, suppose that we have Seifert fibering on a marked lens space such that
the gluing torus is a union of fibers and that there is a .
For any closed curve on the gluing torus,
we can measure how many turns the  makes as we travel around this curve.
This may be done, say, by counting how many times the vector field
crosses the gluing torus and changes from pointing into one solid torus
to pointing into the other.
This counting of turns induces a group homomorphism
ϕ : H_1(T) →
where H_1(T) is the first homology group of the gluing torus.
Since M_i bounds a meridional disk and the  projects
to a non-zero vector field on the disk, the Poincaré-Hopf theorem
implies that the vector field must make exactly one clockwise turn
when going around M_i. Hence ϕ(M_i) = -1.
In fact, this is only restriction on ϕ. One can check that
for any  defined on the boundary of a standard fibered torus,
that so long as ϕ(M_i) = -1 it is possible to extend the  to all of the
solid torus.

The relation

    [ M_1; L_1 ]
        =
        [ -q_2    p;       q_1 ][ M_2; L_2 ]

implies that M_1 ∼ -q_2 M_2 + p L_1
and so

    -1 = ϕ(M_1) = -q_2  ϕ(M_2) + p  ϕ(L_1)
        = q_2 + p  ϕ(L_1).

That is, q_2 ≡ -1  p and the marked lens space is L(p, -1).

Conversely, suppose we have a Seifert fibering on L(p, -1)
and we wish to construct a .
It is enough to find a homomorphism ϕ : H_1(T) →
such that ϕ(M_1) = ϕ(M_2) = -1.
Up to choosing different longitudes on the solid tori,
we may assume that the gluing matrix is of the form

    [ M_1; L_1 ]
        =
        [  1  p;  0 -1 ][ M_2; L_2 ]
        .

Under such a choice, we may then define ϕ by ϕ(M_i) = -1
and ϕ(L_i) = 0 for i = 1,2 and see that this agrees with the gluing.
In the construction of a Seifert fibering by gluing,
the cycles in homology satisfy

    [ C_i; H_i ]  =  [  _i  _i; '_i '_i ][ M_i; L_i ]  =  [  '_i  -_i; -'_i   _i ][ M_i; L_i ]

and so ϕ(H_i) = -'_i ϕ(M_i) + _i ϕ(L_i) = '_i.
Since H_i is a regular fiber,
ϕ(H_i) measures the number of twists the vector field
makes around the regular fiber and so equals the degree d
of the covering M →.
This shows that d = '_1 = '_2.
This ends our discussion of the proof of <ref>.
    


We now prove a version of <ref> for fibered marked lens spaces.

 
    Suppose M is diffeomorphic to the lens space L(p,q) with p ≥ 0
    and consider only the Seifert fiberings on M which
    have an invariant of the form (0; (_1, _1), (_2, _2)).
    
        
        
        
        
  * 
        If p = 1 or p = 2,
        then every such fibering on
        M has a .
        
  * If p ≥ 3 and q ≡± 1  p,
        then M has infinitely many fiberings of this form
        which support  and infinitely many which do not.
        
  * For all other cases of p and q,
        no fibering of this form on M has a .
         

    We prove this using <ref> and <ref>.
    If p = 0, then the only marked lens space diffeomorphic to M is
    L(0,1) = S^2  S^1 and <ref> implies that no
    fibering has a .

    If p = 1 or p = 2, 
    the fact that q is coprime to p implies that q ≡ -1  p.
    That is, the only marked lens space diffeomorphic to M
    is L(p, -1) and so every fibering has a .

    If M is diffeomorphic to L(p,q) with
    |p| > 3 and q ≡ -1  p,
    then M is diffeomorphic to both L(p,-1) and L(p,+1).
    The former has infinitely many fiberings,
    all of which have a ;
    the latter has infinitely many fiberings,
    none of which has a .

    Finally,
    if q ≢± 1  p,
    then <ref> shows that M is not diffeomorphic to L(p, -1)
    and so no fibering of the form (0; (_1, _1), (_2, _2))
    will have a .
    



 §.§ Other fiberings of lens spaces


To complete the proof of <ref>,
we must consider all possible Seifert fiberings of lens spaces.
To do this, we first list out all closed 3-manifolds
which have multiple possible Seifert fiberings.

 
    A Seifert fibering of an oriented 3-manifold (without boundary)
    is unique up to isomorphism
    with the following exceptions:
    
        
  * Every lens space has infinitely many different Seifert fiberings,
        and
        any Seifert fibering of a lens space
        is either of the form
        
    (0; (_1, _1), (_2, _2))
                
                (-1; (, ± 1)).

        
  * For a non-zero rational number p/q written in lowest terms,
        write _1 = |p|,
        _3 = |q|,
        and define _1 and _3 such that
        _3/_3 = _1/_1 = p/q,
        then the invariants
        
    (0; (2, 1), (2, -1), (_3, _3))
                
                (-1; (_1, _1))

        give fiberings of the same manifold.
        This manifold is a lens space if and only if _3 = 1.
        
  * The unit tangent bundles of the 2222 orbifold
        and the Klein bottle
        are diffeomorphic as manifolds.
     

    This is a re-statement of a known result on the uniqueness of Seifert fiberings.
    As a reference, we give the lecture notes of Hatcher
    <cit.>.
    The theorem as stated there has five cases which we quote here using
    the notation conventions from those notes:
    
        
  * M(0,1; /), the various model Seifert fiberings of S^1  D^2.
        
  * M(0,1; 1/2, 1/2) = M(-1,1;), two fiberings of
        
            S^1 
            S^1 
            I. 
        
  * M(0,0; _1/_1, _2/_2),
        various fiberings of S^3, S^1  S^2 and lens spaces.
        
  * M(0,0; 1/2, -1/2, /) = M(-1, 0; /) for , 0.
        
  * M(0,0; 1/2, 1/2, -1/2, -1/2) = M(-2, 0;),
        two fiberings of 
        
            S^1 
            S^1 
            S^1.
         
    Cases (a) and (b) are for manifolds with boundary
    and can be ignored here.
    We have used <ref> above to rephrase case (e)
    in terms of unit tangent bundles.
    Since the manifold in case (e) is finitely covered by the 3-torus,
    one can show that
    it does not overlap with cases (c) or (d).
    
    Assuming for simplicity that  and  are positive,
    the invariants in case (d) may be written in our notation as
    
    (0; (2, 1), (2, -1), (_3, _3))
            
            (-1; (_1, _1))
        
    where / = _1/_1 = _3/_3 and _i > 0.
    
    Note that if _3 = 1, then 
    
    M(0; (2, 1), (2, -1), (1, _3))
            =
            M(0; (2, -1), (2, 2 _3 - 1))

    is a fibered marked lens space,
    and so there is actually some overlap
    between cases (c) and (d).
    
    Using the formula in <cit.>,
    the fundamental group of the manifold
    M(-1; (_1, _1)) is given by
    
    ⟨ a, q, h  : 
             a  h a = h ,
             [h,q] = 1,
             q^_1 h^_1 = 1,
             q a^2 = 1
            ⟩

    or equivalently
    
    ⟨ a, h  : 
             a  h a = h ,
             a^2 _1 = h^_1⟩.

    If |_1| > 1, this group is non-cyclic.
    and the manifold is not a lens space.
    (In fact, the manifold is a prism manifold.)


Hence, the only fiberings on lens spaces left to study
are those of the form (-1; (, ± 1)).
Up to a choice of orientation, these are all unit tangent bundles.

 
    Let M be diffeomorphic to a lens space and suppose M is Seifert
    fibered, but is not a fibered marked lens space.
    Then (up to a change of orientation) there is ≥ 1
    such that M is unit tangent bundle of the  orbifold,
    M is diffeomorphic to L(4 , 2 ± 1),
    and the fibering has a .


    Recall from the orbifold notation introduced in <ref>
    that the  orbifold is P^2 with a cone point of order 
    added. If = 1, this orbifold is P^2 itself.


    By <ref>, the fibering must have Seifert invariant
    (-1; (, ± 1)).
    By <ref>, up to a possible change of orientation
    we may assume that the invariant is (-1; (, - 1)).
    <Ref> then implies that M is the unit tangent
    bundle of the  orbifold
    and <ref> shows that it has a .
    By <ref>, this manifold is diffeomorphic
    to
    
    M(0; (2, 1), (2, -1), (1, ))
            =
            M(0; (2, 1), (2, 2  - 1)).

    Using <ref> or directly from the computation
    
    [      2 2  - 1;      1        ][ -1  0;  0  1 ][ 2 1; 1 1 ]
            =
            [ -(2  + 1)         4;              2  + 1 ]

    one may show that this lens space is
    L(4 , 2  + 1).
    By <ref>,
    L(p, q) is diffeomorphic to M
    if and only if |p| = 4 and q ≡ 2 ± 1  p.

This shows that beyond the fibered marked lens spaces,
there is exactly one additional Seifert fibering on
L(4 , 2 ± 1) for each ≥ 1.
<Ref> now follows as a combination of
propositions <ref> and <ref>.



§ ELLIPTIC ORBIFOLDS
  

We now consider Seifert fiberings over elliptic orbifolds.
We start by listing out all of the possible orbifolds.

 
    An elliptic orbifold is of one of the following forms:

2
    
        
  * the pp orbifold with p ≥ 1;
        
  * the 2 2 p orbifold with p ≥ 2;
        
  * the 2 3 q orbifold with 3 ≤ q ≤ 5;
        
  * the p orbifold with p ≥ 1.
    


Note that the 2-sphere and real projective plane occur
as special cases with p = 1.
For further details and a proof of this result,
see the section of <cit.> entitled
“The geometry of S^2”
and the references therein.

 
    Let M → be a Seifert fibering over
    an elliptic base orbifold.
    Then M is diffeomorphic to a lens space if and only if either
    
        
  *  is the pp orbifold with p ≥ 1; or
        
  * M = ± where  is the  p orbifold with p ≥ 1.
     

    If  is the pp orbifold, the Seifert invariant may be written
    as
    
    (g; (_1, _1), (_1, _2))

    with _1 = _2 = p
    and so M is a fibered marked lens space.
    
    If  is the p orbifold, then
    propositions <ref> and <ref>
    imply that M is diffeomorphic to a lens space if and only if
    M = ±.
    
    The other elliptic orbifolds listed in <ref>
    may be ruled out by <ref>.


Now, consider Seifert fiber spaces M_1 and M_2 over an orbifold 
and suppose there is a fiberwise covering M_1 → M_2 of degree d > 1.
<Ref> states that d · e(M_1 →) = e(M_2 →).
Roughly speaking,
the Euler number measures the amount of “twisting” of the
fibering, and the effect of quotienting M_1 down to M_2
is to increase this twisting by a factor of d.
Conversely, if we consider M_2 first,
finding a fiberwise covering map M_1 → M_2 
is in some sense equivalent to finding a way
to reduce the twisting of the fibering by a factor of d.
If we know, however, that the fibering M_2 →
already has the minimum possible amount of (non-zero)
twisting for a fibering over , then it cannot be fiberwise
covered by another fibering.

This is the technique we will use to prove <ref>.
We will show for all but one of the cases
that e(→) is the smallest possible positive Euler number
for any Seifert fibering over .
In the exceptional case, a degree two covering is possible,
but M must be a lens space.

 
    Suppose  is an elliptic orbifold and
    M → is a fiberwise covering
    of positive degree d.
    Then the degree d is at most two.
    
    Moreover, the d = 2 case occurs exactly when M is of the form
    
        M(0;  (, _1),  (, _2))
    
    where ≥ 1 is odd,
    
    _1  =  2 -  12_2  =  -2 -  12.
 
The two-fold covering above with = 1
corresponds to the so-called “belt trick” or “plate trick”
where S^3 double-covers S^2.


    We proceed through the cases in <ref>
    in an order which makes the overall proof easiest to follow.

    
    
    [3]

    Case 1.  is the 23q orbifold for 3 ≤ q ≤ 5.
    
    
    
    
    

    We give a full proof for the 235 orbifold.
    Here e(→) = χ() = 1/30
    and M has a Seifert invariant which may be written as
    
        (0; (2, _1), (3, _2), (5, _3))
    
    for some values of _i.
    The Euler number of M → is a linear combination
    of 1/2, 1/3 and 1/5 with integer coefficients
    and hence must be an integer multiple of 1/30.
    Say k is such that e(M →) = k/30.
    Then d · e(M →) = e(→)
    implies that d k = 1.
    As d and k are integers and d is positive,
    the only possibility is that d = k = 1.
    The 233 and 234 orbifolds have similar proofs
    with 1/6 and 1/12 replacing 1/30 above.
    
    
    
    
    
    
    
    
    

    

    Case 2.  is the 22p orbifold where p is even.

    Here e(→) = χ() = 1/p
    and the Euler number of M → is a linear combination
    of 1/2 and 1/p with integer coefficients.
    Hence, e(M →) = k/p for some integer k
    and d k = 1 implies that d = k = 1.

    

    Case 3.  is the 22p orbifold where p is odd.

    As in the previous case,
    e(→) = χ() = 1/p
    and the Euler number of M → is a linear combination
    of 1/2 and 1/p with integer coefficients.
    However, as p is odd, we may only conclude that
    e(M →) = k/2p for some integer k
    and so d k = 2.
    As shown by <ref>,
    when quotienting along fibers to produce a d-fold
    cover, the integer d must be coprime to the orders
    of all of the cone points.
    As  has cone points of order 2, d cannot be even
    and so d = 1 is the only possibility.

    

    Case 4.  is the p orbifold.

    Here e(→) = χ() = 1/p
    and M has a Seifert invariant which may be written as
    
        (-1; (p, _1))
    
    for some value of _1.
    Hence, e(M →) = k/p for some integer k
    and d k = 1 implies that d = k = 1.

    

    Case 5.  is the pp orbifold where p is even.

    Here e(→) = χ() = 2/p
    and e(M →) = k/p for some integer k.
    Hence, d k = 2.
    By <ref>,
    d is coprime to p and so must be odd.
    This is only possible if d = 1.

    

    Case 6.  is the pp orbifold where p is odd.

    As in the previous case,
    d k = 2 where d is coprime to p.
    As p is odd, both d = 1 and d = 2 are possible.
    Assume d = 2 for the rest of the proof.
    
    Case (4) of <ref> holds with d = 2 and = _1 = _2 = p,
    showing that
    
    d · e(M →)
            = - 2 (_1 + _2)
            = 2
            = χ()
            
            2 _i / ≡ -1 / .

    As  is odd, it follows that
    
        _i ≡12 (- 1) .
    
    Using the allowed manipulations on Seifert invariants,
    we may adjust the ratios _i/_i by integer amounts
    and assume with loss of generality
    that
     
        _1  =  2  -  12.
            
    Then _1 + _2 = -1 implies that
    
        _2  =  -2  -  12.
         
Proving <ref> is now a matter of combining the above results.


    By <ref>, having a  is equivalent to having
    a fiberwise covering M → of some degree d.
    Up to changing the orientation of M,
    we may assume d ≥ 1.
    Since we are assuming M is not diffeomorphic to a lens space,
    propositions <ref> and <ref>
    imply that d = 1 and so M = .

As a final observation,
note that for each p > 1, the unit tangent bundle of the 22p orbifold
has two fiberings

    M(0; (2,1), (2,-1), (p, -1)
        
        M(-1; (1, -p))

given by item (2) of <ref>.
The first of these fibering is the standard fibering of the unit tangent
bundle, and so has a .
The second is a fibering over the real projective plane
and since p > χ( P^2) = 1, this fibering does not have a .



§ PARABOLIC ORBIFOLDS
  

We now consider Seifert fiberings over elliptic orbifolds.
There are exactly seven orbifolds of this type.

 
    The parabolic orbifolds are as follows:

2
    
        
  * the 2-torus ^2;
        
  * the Klein bottle K;
        
  * the 236 orbifold;
        
  * the 244 orbifold;
        
  * the 333 orbifold;
        
  * the 2222 orbifold;
        
  * the 22 orbifold.
    


For further details and a proof of this result,
see the section of <cit.> entitled
“The geometry of E^3”
and the references therein.
See also 2.1 of <cit.>.
It is actually a somewhat enjoyable exercise to prove the result directly
by starting with a surface _0 with χ(_0) ≥ 0
and seeing which combinations of cone points may be added to get
χ() down to exactly zero.



 §.§ Self-coverings

For each of the parabolic orbifolds,
 is an oriented Seifert fiber space
and <ref> implies that e(→) = χ() = 0.
It turns out that, up to orientation,
these unit tangent bundles
are the only Seifert fiber spaces of this form.

 
    There are exactly ten distinct Seifert fiberings M →
    for which M is oriented and e(M →) = χ() = 0.
    These are all of the form M = ±.
    Moreover, if  is the 236, 244, or 333 orbifold,
    then -.
    For the other parabolic orbifolds, = -.


    A list of these Seifert invariants appears in a number of places.
    For instance, 8.2 of <cit.>
    and the sections of <cit.> and <cit.>
    mentioned above.
    As the result is not difficult,
    we outline the proofs for two of the cases here and leave the others,
    which are similar, to the reader.

    The 236 orbifold.     Consider a Seifert fibering M →
    having invariant
    
    (0; (2, _1), (3, _2), (6, _3))

    and
    e(M →) = 0.
    As _2 is coprime to 3,
    we have either _2 ≡ 1 or _2 ≡ -1  3.
    By reversing the orientation of the manifold if necessary
    and using <ref>,
    we may assume the latter holds.
    By doing valid operations on the Seifert invariant,
    adding an integer to one ratio _i/_i and subtracting it to another
    we may reduce to the case where the invariant
    is (0; (2, -1), (3, -1), (6, _3).
    Then e(M →) = 0 implies that _3 = 5 and one may verify
    M = .

    The 333 orbifold.     Consider a Seifert fibering M →
    having invariant
    
    (0; (3, _1), (3, _2), (3, _3))

    and
    e(M →) = 0.
    Each _i is coprime to 3.
    One can show that _1 ≡_2 ≡_3  3
    as otherwise e(M →) would not be an integer.
    Applying <ref> if necessary,
    we may assume _1 ≡_2 ≡_3 ≡ -1.
    Similar to above,
    we may reduce to the case where the invariant
    is (0; (3, -1), (3, -1), (3, _3).
    Then e(M →) = 0 implies that _3 = 2 and
    M = .

    Using <ref>, one can determine whether or not
     and - have the same Seifert invariants.
    
    

 
    Suppose  is an elliptic orbifold
    with a (possibly empty) set of cone points
    of orders _1, …, _n.
    If M → is a fiberwise cover, then M = ±.
    Moreover, such a fiberwise cover of degree d
    exists if and only if d is coprime to all of the _i.
    

    First, suppose M_1 → is a fiberwise covering.
    Then, <ref> implies that 
    d · e(M_1 →) = χ() = 0
    and so
    e(M_1 →) = 0.
    As M_1 has the same base orbifold as ,
    <ref> applies and M_1 = ±.

    Conversely, starting from an integer d coprime to all _i,
    <ref> gives a fiberwise covering → M_2.
    <Ref> implies that
    
    e(M_2 →) = 0
    and then <ref> implies that M_2 = ±.
    Hence, we have constructed a covering →±.
    Up to a possible change of orientation,
    this gives a covering ±→ as desired.

Using the above propositions, it is a straightforward task to list
out all of the possible coverings:

    
  * If = ^2, then 

    ≅ - and
    has a d-fold self-covering for all d  0.
    
  * If = K is the Klein bottle, then 

    ≅ - and
    has a d-fold self-covering for all d  0.
    
  * 2366
    
  * 2444
    
  * 3333
    
  * If  is the 2222 orbifold, then 

    ≅ - and
    d-fold covers itself if
    d ≡ 1 mod 2.
    
  * If  is the 22 orbifold, then 

    ≅ - and
    d-fold covers itself if
    d ≡ 1 mod 2.



We now attempt to impart some intuition
about the structure of these self-coverings and their associated .
Every parabolic orbifold is the quotient of ^2
by a wallpaper group
consisting of affine isometries.
Consider the unit tangent bundle ^2
of the Euclidean plane.
An element of

    ^2 ≅^2 ^1

may be represented by a pair (x, θ) where
x ∈^2 and θ is an angle.
Then ^2 has self-covering maps of the form

    (x, θ) ↦(x, d θ)

for d ≥ 1.

If A : ^2 →^2 is an orientation-preserving isometry,
then its derivative is of the form

    (x, θ) ↦(A(x),   θ+ θ_0)

for some constant θ_0.
In some cases, this will commute with the d-fold self-covering;
for instance, when A is a rotation by an angle θ_0 = 2π/k
and d ≡ 1 mod k.
If the self-covering commutes with every element of the wallpaper group,
then this defines a self-covering map on unit tangent bundle of the orbifold.

As a concrete example, consider the wallpaper group associated
to the 236 orbifold.
This group is generated by three rotations of orders 2, 3, and 6
as depicted in <ref>.

If d ≡ 1  6, the self-covering on ^2 will quotient
to the unit tangent bundle .




 §.§ Vector fields on surfaces

To completely handle Seifert fiberings over parabolic orbifolds,
we must also consider case (1) of <ref>.

 
    Let  be the 2-torus or Klein bottle.
    Then any Seifert fibering M →
    has a  (whether or not M is orientable).


    On such a surface, one can construct
    a unit vector field v : →.
    Define u : → as the composition of v with
    the projection π : M → defining the Seifert fibering.
    Then, a  exists by <ref>.

For each of ^2 and K,
there are infinitely many oriented Seifert fiberings,
one for each integer e(M →).
These manifolds have Euclidean geometry if e(M →) = 0
and Nil geometry otherwise
<cit.>.
There are exactly four non-orientable 3-manifolds with Euclidean
geometry.
(See 8.2 of <cit.>.)
Each of these has a Seifert fibering over K
and therefore has a .
Two of these manifolds also have Seifert fiberings over ^2
which therefore also have .








    If  is a surface with χ() = 0,
    then <ref> establishes the theorem,
    so suppose  is not a surface.
    <Ref> and <ref> then
    imply that
    
    M has a  ⇔ M fiberwise covers  ⇔  
            M = ±.
 



§ HYPERBOLIC ORBIFOLDS
  

The final class of orbifolds to consider are those with hyperbolic geometry.
Even though this is the most general case,
there is the least to say here.
In this setting,
it is relatively easy to find fiberwise coverings M → of degree d > 1.
For circle bundles,
if  is a surface of genus g ≥ 2, then there is a cover of degree d
for every factor d of χ(g) = 2 - 2g.
Even for orbifolds without handles, non-trivial covers are possible.
For instance, using <ref>, one may show that

    M(0; (1,-1), (5,2), (5,2), (5,2))

double covers the unit tangent bundle of the 555 orbifold.

For certain choices of cone points,
there may be no non-trivial covers.
For instance, if  is the 237 orbifold,
one may show that only d = ± 1 is possible.
The proof is of the same form as Case 1 of the proof of <ref>.




If  is a hyperbolic orbifold,
the geodesic flow on  is an Anosov flow
and the flow is generated by a vector field which is horizontal.
This flow lifts to any finite cover and is still Anosov on the cover.
Ghys and Barbot showed that (up to orbit equivalency)
every Anosov flow on a Seifert fiber space is of this form
<cit.>.
This establishes 
the equivalence (2) ⇔ (3) in <ref>.
The equivalence (1) ⇔ (2)
is a re-statement of the results in <ref>.













































§ HOMOTOPIES OF 
  

For a given Seifert fibering M →,
consider the space of all .
What are the connected components of this space?
That is, when can one vector field be deformed into another
along a path of ?
For simplicity, we only consider this in the case where
the base orbifold  is oriented.

By <ref>, the question reduces
to studying homotopy classes of maps of the form

    Mu𝕀.



Consider two such maps u, v : M →
and suppose they have the same degree d.
(Otherwise, they are clearly not homotopic.)
By the averaging method used in the proof of <ref>,
we may assume there are metrics on the fibers of 
M and  and that u and v have the same constant speed on all fibers.
As  is oriented, the fibers of  are oriented.
Using the metric, there is then a well-defined difference map

    u - v : M →^1

measuring the angle between vectors.
This map is constant on fibers
and therefore quotients to a map g : →^1.
The map u is homotopic to v
if and only if
g is homotopic to a constant map.
Since we are only concerned with the homotopy class of g,
the smooth orbifold structure of  is unimportant
and we may consider g as a map from _0 to ^1
where _0 is the underlying topological surface.

Note that these steps are reversible.
Given a continuous map g : _0 →^1,
we may, up to homotopy, assume g is smooth everywhere
and constant on a neighbourhood of the cone points.
If π : M → is the Seifert fibering,
then g ∘π : M →^1 is a smooth map.
Given u as above, we may construct v by v = u + g ∘π
where the plus sign denotes addition of angles.
If we assume u is fixed, then this gives a bijection
between homotopy classes of maps v : M →
of the same degree as u and homotopy classes
of maps g : _0 →^1.

There are canonical isomorphisms identifying

    
  * the homotopy classes of maps from _0 to ^1,
    
  * homomorphisms from π_1(_0) to = π_1(^1),
    
  * homomorphisms from the first homology group H_1(_0) to , and
    
  * elements of the first cohomology group H^1(_0,).

For _0 = ^2, this follows because all of the above are trivial.
For other oriented surfaces, it follows because both _0 and ^1
are K(π,1).
(See also Theorem 4.57 and the discussion on page 198 of <cit.>.)


Each connected component of the space of 
is therefore uniquely determined by the degree d
(which is also the number of turns that the vector field
makes around any fiber)
and an element of H^1(_0,).

For a Seifert fibering M → of an oriented manifold,
let  denote the “allowable degrees”
of a .
That is d ∈ if and only if
there is a degree-d map u : M → of the form given
in <ref>.
The above reasoning may then be used to show the following.

 
    For a Seifert fibering M → over an oriented orbifold,
    there is a bijection between the connected components
    of the space of  and pairs of the form
    
        (d, φ) ∈H^1(_0,).
     
    
If χ()  0, then <ref> shows that
the degree d is uniquely determined by the fibering.
In particular,
for a bad, elliptic, or hyperbolic orbifold with _0 = ^2,
<ref> implies that
the Seifert fibering uniquely determines the  up to homotopy.
Conversely if H^1(_0,) is non-trivial,
then many homotopy classes are possible.
For instance,
consider the 3-torus ^3 with fibers tangent to
the vertical z-direction.
Then U T ^2 may be identified with ^3 and
the classes of  correspond to

maps of the form

    ^3 →^3,    (x,y,z) ↦ (x,y, a x + b y + d z)

for all (a,b,d) ∈^3.














§ MANIFOLDS WITH BOUNDARY
  

In this final section, we consider Seifert fiberings on
manifolds with boundary.
First, consider the case where there are boundary conditions
for the vector field.
We could require that the vector field be either tangent or transverse
to the boundary and the resulting restrictions on M will be the same.

 
    For a Seifert fibering M → on a manifold with boundary,
    the following are equivalent:
    
        
  * M has a  everywhere tangent to the boundary,
        
  * M has a  everywhere transverse to the boundary,
        
  * the base orbifold  is either the annulus or the Möbius band.
     

    We show (2) ⇔ (3).
    The proof of (1) ⇔ (3)
    is similar and left to the reader.
    Note that <ref> holds
    for manifolds with boundary using the same proof.
    Suppose M supports a  which is everywhere transverse
    to M and consider the associated map u : M →.
    For a point x ∈,
    the restriction of u to the fiber over x
    cannot be surjective as its range omits
    the two unit vectors at x which are tangent to .
    This implies that u has degree zero
    and <ref> shows that u is homotopic
    to a composition M →→.
    In the proof of the proposition,
    the vector field → is
    defined by averaging
    and one can verify that is it transverse to .
    Further, the same argument as in <ref>
    shows that  has no cone points and is therefore
    a surface with boundary.
    The Poincaré–Hopf theorem then implies that
     is either the annulus or Möbius band.

    Conversely, for any circle bundle
    over an annulus or Möbius band,
    we may compose the projection M →
    with a vector field → transverse to 
    and produce a  on M transverse to M.

For the remainder of the section,
we consider  with no boundary conditions.

 
    Let M → be a Seifert fibering on a manifold with boundary.
    Then M has a  if and only if
    one or both of the following hold:
    
        
  * the base orbifold  is a surface with boundary, or
        
  * M fiberwise covers the unit tangent bundle of .
     
To prove this, we use the following.

 
    An orbifold with boundary supports a non-zero vector field
    if and only if it has no cone points
    (i.e., it is a surface with boundary).


    As shown in the proof of <ref>,
    it is impossible to define a non-zero vector field in the neighbourhood
    of a cone point
    and so the existence of such a vector field implies that  is a surface with boundary.
    Conversely,
    if we have no boundary conditions,
    we may construct a non-zero vector field on any surface with boundary.
    For instance,
    we may take a generic vector field on a closed surface.
    This is zero at finitely many points <cit.>,
    and we can excise one or more topological disks to remove all of these
    points.

Using this lemma and checking that the results of <ref>
extend to the case of a manifold with boundary,
one may then prove <ref>.

For an oriented manifold with boundary, write the Seifert invariant as

    (g, n; (_1, _1), …, (_k, _k))

where n > 0 is the number of boundary components.
We can reorder the pairs, add or remove pairs of the form (1,0),
and (specifically for M ∅) add an integer to
any of the ratios _i/_i without changing the fibering.
Note here that integer changes to one ratio _i/_i do
not need to be offset with a change to another ratio  _j/_j.
Because of this, the Euler number e(M →) is not well defined.
See 2 of <cit.> for more details.

By adapting arguments in <ref>
one can prove the following analogue of <ref>.
Note that the condition

    d ·e(M →) = χ()

has been removed.


 
    Let M be a manifold with boundary 
    and let M → be a Seifert fibering
    such that  has one or more cone points.
    Then the following are equivalent:
    
        
  * M has a ;
        
  * there is a continuous map u: M →
        such that
        
    Mu𝕀

        commutes;
        
  * there is a fiberwise covering from M to
        the unit tangent bundle ;
        
  * M is orientable, the fibering has Seifert invariant
        
    (g, n; (_1, _1), …, (_k, _k))

        and there is a non-zero integer d
        such that
        
    d _i/_i ≡ -1/_i

        for all i = 1, …, k.
     

Even with the Euler number condition removed,
it may not be possible to find a .
Consider, for instance (g,n; (3,1), (3,2)).
No integer d satisfies

    13 d ≡23 d ≡-13    
and so no  exists.


If a Seifert fibering satisfies item (4) of <ref>
for some integer d,
then it also satisfies the condition when
d is replaced by d + m ℓ
where m is any integer and
ℓ is the least common multiple of {_1, …, _k}.
Using this and adapting the results in <ref>,
one may show that if a Seifert fibering on a manifold with boundary
has one , then it has infinitely many homotopy classes
of such vector fields.












































































































































































































Acknowledgements.
The author thanks
Jonathan Bowden,
Heiko Dietrich,
Rafael Potrie,
Jessica Purcell,
Mario Shannon, and
Stephan Tillmann
for helpful conversations.
We also thank the reviewer for helpful comments.
This research was partially funded by the Australian Research Council.






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