The multiple fibration problem for seifert 3-Orbifolds

ABSTRACT

We conclude the multiple fibration problem for closed orientable Seifert three-orbifolds, namely, the determination of all the inequivalent fibrations that such an orbifold may admit. We treat here geometric orbifolds with geometries |$\mathbb R^3$| and |$\mathbb S^2\times\mathbb R$| and bad orbifolds (hence non-geometric), since the only other geometry for which the multiple fibration phenomenon occurs, namely, |$\mathbb S^3$|⁠, has been treated before by the second and third authors. For the geometry |$\mathbb R^3$| we recover, by direct and geometric arguments, the computer-assisted results obtained by Conway, Delgado-Friedrichs, Huson and Thurston.

1. INTRODUCTION

Smooth orbifolds are topological spaces that are locally homeomorphic to quotients of |$\mathbb{R}^n$| by the action of a finite group G, and they are therefore a natural generalization of smooth manifolds, allowing the presence of singular points corresponding to the fixed points of the action of G. Smooth orbifolds can be naturally constructed as quotients of a smooth manifold by a properly discontinuous action. An orbifold is called good if it can be obtained as such a global quotient, and bad otherwise.

In dimension three, a smooth orbifold is called geometric if it is locally modelled on one of the eight Thurston’s geometries |$\mathbb H^3$|⁠, |$\mathbb{R}^3$|⁠, |$\mathbb S^3$|⁠, |$\mathbb H^2\times\mathbb{R}$|⁠, |$\mathbb S^2\times \mathbb{R}$|⁠, Nil, Sol and |$\widetilde{SL_2}$|⁠. Closed geometric orbifolds are good orbifolds. They played a fundamental role, among many things, in the Orbifold Geometrization Theorem proved in [1] (see also [6] and [2]).

1.1. The multiple fibration problem

The topology of closed geometric orbifolds, whose geometry is one among |$\mathbb{R}^3$|⁠, |$\mathbb S^3$|⁠, |$\mathbb H^2\times\mathbb{R}$|⁠, |$\mathbb S^2\times \mathbb{R}$|⁠, Nil or |$\widetilde{SL_2}$|⁠, is studied very effectively via the notion of Seifert fibration for orbifold, a generalization of the classical definition of Seifert fibration for manifolds, where the fibres are allowed to be either circles or intervals. Fibres homeomorphic to circles are entirely contained either in the singular locus of the orbifold or in its complement, the regular locus. Those homeomorphic to intervals, instead, are in the regular locus except for the end points, which always lie in the singular locus.

Seifert fibred orbifolds are uniquely determined, up to orientation-preserving and fibration-preserving diffeomorphisms, by a collection of invariants, which consists of a base (a 2-dimensional orbifold |$\mathcal{B}$|⁠, whose underlying topological space is a 2-manifold with boundary) and several rational numbers: the local invariants associated with every cone and corner point of |$\mathcal{B}$|⁠, the boundary invariants associated with every boundary component of the underlying topological space of |$\mathcal{B}$| and a global Euler number.

From this perspective, closed geometric orbifolds can be classified via the collection of invariants of their Seifert fibration, with a single major difficulty: the same smooth orbifold may admit several Seifert fibrations which are not equivalent, as in the following definition.  

Equivalently, they have the same collection of invariants. Examples of inequivalent fibrations of the same orbifold are illustrated in Figures 1, 2 and 3.

In this paper we provide a conclusion of what we call the multiple fibration problem, namely, determining which closed orientable geometric orbifolds admit several inequivalent Seifert fibrations, and what are those fibrations. Here we will only analyse orbifolds with geometry |$\mathbb{R}^3$| and |$\mathbb S^2\times\mathbb{R}$| and bad orbifolds. Let us explain the reason. A reduction of the problem is provided by the following result proved in [2].  

A closed orientable good 3-orbifold has a finite fundamental group if and only if it is geometric with geometry |$\mathbb S^3$|⁠. Moreover, by Bieberbach Theorem, closed flat 3-orbifolds are covered by |$T^3$|⁠. As a consequence, for closed orbifolds, Fact 1 implies the following statement:  

For spherical orbifolds the multiple fibration problem has been solved by the second and third authors in [12], partially relying on previous results achieved in [10, 11]. This is what motivates the analysis, which is carried out in this work, of the remaining geometries |$\mathbb{R}^3$| and |$\mathbb S^2\times\mathbb{R}$| (and of the case of bad orbifolds).

1.2. Geometry |$\mathbb{R}^3$|

Closed orbifolds with geometry |$\mathbb{R}^3$| (which we will call flat in the following) are obtained as the quotients |$\mathbb{R}^3/\Gamma$|⁠, where Γ is a space group, that is, a crystallographic group of dimension three. The study of the Seifert fibrations of closed flat three-orbifolds has been tackled in [4] (including the non-orientable case, which is not treated here).

In particular, for closed flat orbifolds Conway, Delgado-Friedrichs, Huson and Thurston solved the multiple fibration problem, which is called the alias problem in their work, since a compact notation (a ‘name’) is used to denote Seifert fibrations, and a given orbifold may have several ‘names’. However, in [4] a computer-assisted method is used to solve the problem. That method is based on a consequence of Bieberbach Theorem, namely, the fact that two closed flat orbifolds are diffeomorphic if and only if their fundamental groups are isomorphic; hence an algorithm can be used to determine whether two orbifolds, expressed in terms of their Seifert fibrations, have isomorphic fundamental groups.

The first main theorem that we prove, Theorem A, is the solution of the multiple fibration problem for flat Seifert 3-orbifolds. This recovers the results of [4] by a direct proof, based on geometric and topological arguments. (See Section 1.5 for some ideas of the proof.)

Before stating Theorem A, note that in this Section 1 we will state all the theorems by using the classical notation for (orbifold) Seifert fibrations, which we now briefly recall. As discussed earlier, a Seifert fibration for a 3-orbifold is encoded by the data of a base orbifold |$\mathcal{B}$|⁠, the local invariants, the boundary invariants and the Euler number. We will therefore denote a Seifert fibered 3-orbifold by a string, whose first element is the base orbifold |$\mathcal{B}$|⁠, expressed as |$S(n_1,\dots,n_h;n_{11},\dots,n_{1h_1};\dots;n_{b1},\dots,n_{bh_b})$|⁠, where |$S$| is the underlying manifold (surface) with boundary, |$n_1,\ldots,n_h$| are the labels of cone points and |$n_{i1},\ldots,n_{ih_i}$| are the labels of the corner points lying on the i-th boundary component of |$S$|⁠. Then the string continues with several rational numbers, denoting the local invariants of every cone point of |$\mathcal{B}$| (with denominator the labels |$n_1,\ldots,n_h$| and separated by semicolons) of the corner points of each boundary component of |$S$| (again with denominators equal to the labels |$n_{1i},\dots,n_{1h_i}$|⁠). These are followed by the boundary invariants of every boundary component of |$S$| and finally by the Euler number. See Example 1.2 for an illustration of this notation.

We are now ready to provide the statement of Theorem A.  

In the table, |$Mb$| is the Möbius band and |$Kb$| is the Klein bottle. The multiple fibrations appearing in the first, second and fourth line are pictured in Figures 2, 3 and 1, respectively. In order to further explain the notation, let us make an example.

 

However, throughout the paper, we will use Conway’s notation for Seifert fibrations, which is a little less intuitive, but of very effective use. We explain this notation in Section 3.3, and all the results in the paper will then be expressed in that notation.

1.3. Geometry |$\mathbb S^2\times\mathbb{R}$|

Let us now move on to the geometry |$\mathbb S^2\times\mathbb{R}$|⁠. The following theorem provides the classification of multiple fibrations of closed orientable orbifolds with geometry |$\mathbb S^2\times\mathbb{R}$|⁠:  

Let us explain how to visualize the two inequivalent fibrations of the first line in the following example.  

1.4. Bad orbifolds

Finally, the following theorem explains the situation for bad orbifolds.  

We remark that the behaviour of the orbifolds that appear in Theorems B and C is very similar. They are treated in a unified way in Lemma 6.1. We can repeat the considerations of Example 1.3; in particular the orbifolds in the first line have underlying space |$S^2\times S^1$|⁠, and the two core fibres have singularity indices |$c$| and |$d$|.

1.5. Outline of the tools

Let us now pause to briefly discuss some of the ideas in the proofs of Theorems A and B.

First, we develop a common theory that we apply for both geometries |$\mathbb{R}^3$| and |$\mathbb S^2\times\mathbb{R}$|⁠. We prove (Proposition 4.5) that a general method to construct Seifert fibration on an orbifold with geometry |$\mathbb{R}^3$| and |$\mathbb S^2\times\mathbb{R}$| is the following. Consider a discrete subgroup of |$\mathrm{Isom}(M)\times\mathrm{Isom}(\mathbb{R})$|⁠, where M is either |$\mathbb{R}^2$| or |$\mathbb S^2$|⁠, with compact quotient |$\mathcal{O}:=(M\times\mathbb{R})/\Gamma$|⁠. Exactly like in Example 1.3, the fibration of |$M\times\mathbb{R}$| given by the parallel lines |$\{pt\}\times\mathbb{R}$| then induces a Seifert fibration of |$\mathcal{O}$|⁠. By construction, the base 2-orbifold of this fibration of |$\mathcal{O}$| is a quotient of |$M$| (hence it is flat if |$M=\mathbb{R}^2$| and spherical if |$M=\mathbb S^2$|⁠) and the Euler number vanishes. These are actually known to be the necessary condition: every Seifert fibration of a closed orientable orbifold with geometry |$\mathbb{R}^3$| (resp. |$\mathbb S^2\times\mathbb{R}$|⁠) has flat (resp. spherical) base and vanishing Euler number.

More importantly, we prove that the converse holds true (Corollary 4.7): every Seifert fibration of a closed orientable orbifold with vanishing Euler number and flat or spherical base orbifold is equivalent to the one obtained by the above construction.

Nonetheless, the situation for the geometries |$\mathbb{R}^3$| and |$\mathbb S^2\times\mathbb{R}$| is quite different, and the proofs of Theorems A and B, although partly relying on the general results (Proposition 4.5 and Corollary 4.7) described above, follow totally independent arguments. On the one hand, by Bieberbach Theorem, two flat orbifolds |$\mathbb{R}^3/\Gamma_1$| and |$\mathbb{R}^3/\Gamma_2$| are diffeomorphic if and only if the space groups |${\Gamma}_1$| and |${\Gamma}_2$| are conjugate by an affine transformation, which in particular sends families of parallel lines to families of parallel lines. Since by Corollary 4.7 every Seifert fibration of a closed flat orientable orbifold |$\mathcal{O}$| is induced by a family of parallel lines of |$\mathbb{R}^3$|⁠, the proof of Theorem A essentially consists in a careful analysis of the different families of parallel lines of |$\mathbb{R}^3$| that a space group |$\Gamma\lt\mathrm{Isom}(\mathbb{R}^3)$| may preserve.

On the other hand, unlike in |$\mathbb{R}^3$|⁠, in the geometry |$\mathbb S^2\times\mathbb{R}$| there is no notion of ‘affine transformation’, and there is a ‘privileged’ direction, namely, the vertical direction, which is preserved by the isometry group. One might be tempted to conjecture, in analogy with Bieberbach Theorem, that |$(\mathbb S^2\times\mathbb{R})/\Gamma_1$| and |$(\mathbb S^2\times\mathbb{R})/\Gamma_2$| are diffeomorphic (for |$\Gamma_i\lt\mathrm{Isom}(\mathbb S^2)\times\mathrm{Isom}(\mathbb{R})$|⁠) if and only if |${\Gamma}_1$| and |${\Gamma}_2$| are conjugate by a transformation acting by isometries on |$\mathbb S^2$| and by affine transformations on |$\mathbb{R}$|⁠. This statement is false: indeed, it would imply the uniqueness of the Seifert fibration for geometry |$\mathbb S^2\times\mathbb{R}$|⁠. In a certain sense, Theorem B describes the failure of an analogue of Bieberbach Theorem for |$\mathbb S^2\times\mathbb{R}$|⁠. Its proof shows that the orbifold |$\mathcal{O}$| in Example 1.3, together with a 2-to-1 quotient of |$\mathcal{O}$| itself, is the only situation where two discrete groups of isometries induce the same diffeomorphism type in the quotient, but the vertical fibration of |$\mathbb S^2\times\mathbb{R}$| gives rise to inequivalent fibrations in the quotient.

1.6. Some consequences

Finally, let us discuss some consequences of our results.

A particular consequence of our Theorem A is that closed orbifolds with geometry |$\mathbb{R}^3$| admit at most three inequivalent fibrations. From Theorem B (and Theorem C), the same statement does not hold for geometry |$\mathbb S^2\times\mathbb{R}$| (nor for bad orbifolds), since the closed orbifolds for which the fibration is not unique (namely, those in the table of Theorem B) admit infinitely many fibrations. For spherical geometry, in [12] the second and third authors proved that a closed spherical orbifold may admit either infinitely many fibrations or up to three fibrations. By combining these results, an immediate corollary is the following.  

Second, we provide a characterization of those closed 3-orbifolds admitting infinitely many inequivalent Seifert fibrations. Before that, we need to introduce some definitions.

A lens space is a 3-manifold obtained by glueing two solid tori along their boundaries by an orientation-reversing diffeomorphism. If we allow cores of tori to be singular curves, the glueing gives an orbifold whose underlying topological space is a lens space and the singular set is a clopen subset (possibly empty) of the union of the two cores. We call these orbifolds lens space orbifolds. Moreover, we call a Montesinos graph a trivalent graph in S³, which consists of a Montesinos link labelled 2, plus possibly one ‘strut’ for every rational tangle, namely, an interval (with any possible label) whose end points lie on the two connected components of the rational tangle. See [9, Section 4] for a detailed description.  

1.7. Organization of the paper

In Section 2, we introduce smooth and geometric orbifolds, study their relations with crystallographic groups and provide a detailed description of orbifolds of dimension two. In Section 3 we study three-dimensional orbifolds, in particular using Seifert fibrations, and we discuss the standard and the Conway notation for Seifert fibrations. In Section 4 we give some general results (in particular, Proposition 4.5 and Corollary 4.7) on the Seifert fibrations of closed orbifolds with geometries |$\mathbb{R}^3$| or |$\mathbb S^2\times\mathbb{R}$|⁠, equivalently, on the Seifert fibrations with flat or spherical base orbifold and with vanishing Euler number. In Section 5 we prove Theorem A. In Section 6 we prove Theorems B and C and Corollary E.

2. GEOMETRIC ORBIFOLDS

Let us start by introducing some basic notions on smooth and geometric orbifolds in any dimension. Additional details can be found in [2], [13] or [7].

2.1. First definitions

 

The topological space X is called the underlying topological space of |$\mathcal{O}$| and will be denoted by |$|\mathcal{O}|$|⁠. Throughout the paper, all our orbifolds will be connected. We call |$\mathcal{O}$| a closed orbifold if |$|\mathcal{O}|$| is compact.

Observe that in general the underlying topological space |$|\mathcal{O}|$| of a closed orbifold might not be a manifold. In the situations treated in this work, |$|\mathcal{O}|$| will always be a manifold, but often (in particular in dimension two, see Section 2.3) with boundary.

To every point |$x\in \mathcal{O}$| one can associate a local group, which is the smallest group |$\Gamma_x$| that gives a local chart |$\varphi:U\to\widetilde U/\Gamma_x$| around |$x$|⁠.  

If the local group |$\Gamma_x$| is the trivial group, then |$x$| is called a regular point. Otherwise x is called a singular point. The subset of |$\mathcal{O}$| consisting of regular points is, by definition, a smooth manifold. In particular, every manifold is an orbifold all of whose points are regular.

Oriented orbifolds

Of course, one can put additional structures on orbifolds. For example, the orbifold |$\mathcal{O}$| is oriented if, in Definition 2.1, each |$\widetilde U_i$| is endowed with an orientation which is preserved by the action of |$\Gamma_i$|⁠, and the lifts |$\widetilde \varphi_{ij}:\widetilde U_i\to\widetilde U_j$| of the compositions |$\varphi_j\circ\varphi_i^{-1}$| preserve such orientation. The orbifold |$\mathcal{O}$| is called orientable if such a consistent choice of orientations for |$\widetilde U_i$| exists.

Geometric orbifolds

In a similar spirit, let us introduce geometric orbifolds. Let (M, g) be a Riemannian manifold, which we will assume to be simply connected without loss of generality, and let |$\mathrm{Isom}(M,g)$| be its group of isometries. An orbifold |$\mathcal{O}$| is called geometric with geometry (M, g) if, in Definition 2.1, the open subsets |$\widetilde U_i\subset\mathbb{R}^n$| are replaced by open subsets of M, the elements of the groups |$\Gamma_i$| act on |$\widetilde U_i$| as the restrictions of elements in |$\mathrm{Isom}(M,g)$| and similarly the lifts |$\widetilde \varphi_{ij}:\widetilde U_i\to\widetilde U_j$| are the restrictions of isometries in |$\mathrm{Isom}(M,g)$|⁠.

The fundamental examples of (M, g) that the reader is advised to keep in mind for the present paper are as follows:

• the Euclidean space |$\mathbb{R}^n$| (the orbifolds with geometry |$\mathbb{R}^n$| are called flat)

• the sphere |$\mathbb S^n$| (the orbifolds with geometry |$\mathbb S^n$| are called spherical)

• products of the above two items, in particular the three-manifold |$\mathbb S^2\times\mathbb{R}$|⁠.

In dimension three, there are particular cases of the celebrated eight Thurston’s geometries used in the Geometrization Program, namely, |$\mathbb H^3$|⁠, |$\mathbb{R}^3$|⁠, |$\mathbb S^3$|⁠, |$\mathbb H^2\times\mathbb{R}$|⁠, |$\mathbb S^2\times \mathbb{R}$|⁠, Nil, Sol and |$\widetilde{SL_2}$|⁠.

Diffeomorphisms and isometries

A diffeomorphism between orbifolds |$\mathcal O$| and |$\mathcal O^{\prime}$| is a homeomorphism |$f:|\mathcal{O}|\to |\mathcal{O}^{\prime}|$| such that each composition |$\varphi^{\prime}_{j^{\prime}}\circ f|_{U_i}\circ \varphi_i^{-1}$|⁠, when it is defined, lifts to a diffeomorphism of |$\widetilde U_i$| onto its image in |$\widetilde U^{\prime}_{j^{\prime}}$|⁠.

When |$\mathcal O$| and |$\mathcal O^{\prime}$| are oriented, then f is called orientation-preserving if the lifts as above preserve the orientations chosen on |$\widetilde U_i$| and |$\widetilde U^{\prime}_{j^{\prime}}$|⁠. When |$\mathcal O$| and |$\mathcal O^{\prime}$| are geometric (with the same geometry (M, g)), f is called isometry if the lifts are restrictions of elements in |$\mathrm{Isom}(M,g)$|⁠.

Good orbifolds

Orbifolds naturally arise as the quotients |$\mathcal O=M/\Gamma$|⁠, for M a manifold and Γ a group acting smoothly and properly discontinuously on M. In this situation, the local group of a point |$[x]\in M/\Gamma$| is precisely the stabilizer of x (which is finite since the action is properly discontinuous). If M is endowed with a Riemannian metric g and Γ acts moreover by isometries on (M, g), then the quotient |$M/\Gamma$| is geometric with geometry (M, g). Similarly, if Γ preserves an orientation on M, then |$M/\Gamma$| is oriented.

An orbifold is good if it is diffeomorphic to a quotient |$M/G$| as above; otherwise it is called bad. By a standard argument, one can show that closed good geometric orbifolds with geometry (M, g) are isometric to the quotient |$M/\Gamma$|⁠, where Γ is a subgroup of |$\mathrm{Isom}(M,g)$| acting properly discontinuously on M.

We mention here that there is a notion of orbifold fundamental group. We will not introduce the formal definition; for our purpose, it will be sufficient to observe that for a good orbifold |$\mathcal{O}\cong M/\Gamma$| where M is a simply connected manifold, the orbifold fundamental group |$\pi_1(\mathcal{O})$| is isomorphic to Γ.

2.2. Flat orbifolds and crystallographic groups

As a special case of the previous paragraphs, closed good flat orbifolds are isometric to |$\mathbb{R}^n/\Gamma$|⁠, where Γ is a discrete group acting isometrically on |$\mathbb{R}^n$|⁠. Hence the following definition will be of fundamental importance:  

Let us provide some more detailed properties of crystallographic groups. First, it is well-known that every |$g\in \mathrm{Isom}(\mathbb{R}^n)$| is of the form |$g(x)=Ax+t$| for some |$A\in O(n)$| and |$t\in\mathbb{R}^n$|⁠. To simplify the notation, we will denote g by the pair (A, t). In particular |$\mathrm{Isom}(\mathbb{R}^n)\cong O(n)\ltimes T(n)$|⁠, where |$T(n)\cong\mathbb{R}^n$| is the translation subgroup, and the following is a short exact sequence:

where |$\rho(g)=A$|⁠.

  

• the point group of Γ is the image |$\rho(\Gamma)\subset O(n)$|⁠;

• the translation subgroup of Γ is |$T(\Gamma)=\mathrm{Ker}(\rho|_{\Gamma})=\{g\in\Gamma:\rho(g)=id\}$|⁠.

From the definitions, we thus have the following short exact sequence:

The following is a very classical result on crystallographic groups. See for instance [17].

(Bieberbach)

 

• the translation subgroup |$T(\Gamma)$| is isomorphic to |$\mathbb{Z}^n$|⁠;

• the point group |$\rho(\Gamma)$| is a finite group.

Two crystallographic groups are abstractly isomorphic if and only if they are conjugate by an affine transformation of |$\mathbb{R}^n$|⁠.

 

• the infinite cyclic group generated by the translation of 1;

• the infinite dihedral group generated by the reflections fixing the points 0 and |$1/2$|⁠.

Clearly the former is an index two subgroup of the latter.

We take advantage of this one-dimensional example to fix some useful notation. Any |$f\in\mathrm{Isom}(\mathbb{R})$| is described by a pair (A, t), where |$A\in\{\pm1\}$|⁠; we will then use the notation |$f=t\pm$|⁠. In this notation the two crystallographic groups of Example 2.7 are |$\lt1+\gt$| and |$\lt0-,1-\gt$|⁠.

Moreover, any isometry of |$\mathbb{R}$| induces an isometry on |$\mathbb{R}/\mathbb{Z}=\mathbb S^1$|⁠, and all the isometries of |$\mathbb S^1$| are obtained in this way. Therefore, if g of |$O(2)\cong\mathrm{Isom}\,\,(\mathbb S^1)$|is induced by an isometry |$f=t\pm$| of |$\mathbb{R}$|⁠, by a little abuse of notation, we will denote |$g$| by |$t\pm$|⁠. Hence when writing |$g\in O(2)$| as |$g=t\pm$|⁠, t is only defined modulo 1, that is, in |$\mathbb{R}/\mathbb{Z}$|⁠. For example, the rotation of angle |$2\pi\alpha$| is denoted as |$\alpha+$|⁠, and the reflection in the line with slope πβ is denoted as |$\beta-$|⁠.

 

This very simple rule for the composition turns out to be extremely practical in this paper, see, for example, the computations that appear in the proof of Theorem 4.2.

2.3. Two-dimensional orbifolds

Let us now consider orbifolds of dimension 2. In this section, we will denote all 2-orbifolds by the symbol |$\mathcal{B}$| since 2-orbifolds will represent, as explained in Section 3.2, the base orbifold of Seifert fibrations. In the following, the symbol |$\mathcal{O}$| will be thus used mostly for 3-orbifolds.

The local models

In a 2-orbifold |$\mathcal{B}$|⁠, every point has a neighbourhood that is modelled on |$D^2/\Gamma_0$| where, using Remark 2.2, Γ₀ can be assumed to be a finite subgroup of O(2). There are then four possibilities (see Figure 4):

• If Γ₀ is the trivial group, then x is a regular point;

• If Γ₀ is a cyclic group of rotations, then x is called a cone point and is labelled with the order of Γ₀;

• If Γ₀ is a group of order 2 generated by a reflection, then x is called a mirror point;

• If Γ₀ is a dihedral group, then x is called a corner point and is labelled with the order of the index two rotation subgroup of Γ₀.

The labels are also called singularity indices. Observe that, if x is a mirror point, then, in the singular locus of |$\mathcal{B}$|⁠, x has a neighbourhood that consists entirely of mirror points. We will call a connected subset of the singular locus consisting of mirror points a mirror reflector.

In particular, the underlying topological space |$|\mathcal{B}|$| is a manifold with boundary. The points on the boundary of |$|\mathcal{B}|$| are precisely mirror points and corner points.

Standard and Conway notation

The diffeomorphism type of a closed (connected) 2-orbifold |$\mathcal{B}$| is uniquely determined by the underlying (compact) manifold with boundary |$|\mathcal{B}|$|⁠, by the number of cone points together with their labels and by the corner points, together with their labels and their order up to a cyclic permutation, on each boundary component of |$|\mathcal{B}|$|⁠.

In the standard notation, |$\mathcal{B}$| is therefore denoted by the symbol

where S is the underlying manifold (surface) with boundary, |$n_1,\ldots,n_h$| are the labels of cone points and |$n_{i1},\ldots,n_{ih_i}$| are the labels of the corner points lying on the i-th boundary component of S.

It will be extremely useful to adopt Conway’s notation, which is a little less intuitive, but very practical to work with (see [5]). First, let us introduce Conway’s notation for surfaces (that is, 2-manifolds). The surface

obtained as the connected sum of t tori and p projective planes with b boundary components will be denoted by

In other words, ‘ring’ symbols represent connected sums of T², ‘cross’ symbols represent connected sums of |$\mathbb{R} P^2$| and ‘kaleidoscope’ symbols represent boundary components.

When |$\mathcal{B}$| is a 2-orbifold, we can insert the singularity indices in Conway’s notation (3) for the underlying manifold, as follows. The orbifold (1), where S is as in (2), is represented by

The reader can compare the two notations through many examples by looking at Table 1.

Fundamental group

We summarize here the standard presentation for the fundamental group of closed 2-orbifolds. For the details of the proof, which essentially relies on Van Kampen theorem for orbifolds ([2, Corollary 2.3]), see [4, Appendix III] or [5].

 

• For each |$\circ$| symbol, there are two generators |$x_s,y_s$| (for |$s=1,\dots,t$|⁠);

• For each × symbol, there is a generator z_r (for |$r=1,\dots,p$|⁠);

• For each cone point of order n_k (for |$k=1,\dots,h$|⁠), there is a generator γ_k, satisfying the relation |$\gamma_k^{n_k}=1$|⁠;

• For each boundary component of |$|\mathcal{B}|$|⁠, corresponding to a string |$\ast n_{i1},\dots,n_{ih_i}$| (for |$i=1,\dots,b$|⁠), there are |$h_i+2$| generators |$\delta_i,\rho_{i0},\rho_{i1},\dots,\rho_{ih_i}$|⁠, satisfying the relations

  $$\begin{aligned} &\delta_i^2=\rho_{i0}^2=\dots=\rho_{ih_i}^2=1 \\ &(\rho_{i0}\cdot\rho_{i1})^{n_{i1}}=(\rho_{i1}\cdot\rho_{i2})^{n_{i2}}=\dots=(\rho_{i,h_i-1}\cdot\rho_{ih_i})^{n_{ih_i}}=\delta_i\cdot\rho_{ih_i}\cdot\delta_i^{-1}\cdot\rho_{i0}=1\ . \end{aligned}$$

Moreover, there is a global relation:

The geometric interpretation of this presentation is the following, as illustrated in Figure 5.

For each cone point with label n_k, γ_k represents a loop that winds around the cone point; the relation |$\gamma_k^{n_k}$| corresponds to the fact that the corresponding local group acts as a rotation of order n_k. For boundary components of the underlying manifold, let us first consider the case where there are no corner points. In this case the above presentation gives two generators (say δ and ρ) with the relation ρ² = 1 and |$[\delta,\rho]=\delta\cdot\rho\cdot\delta^{-1}\cdot\rho=1$|⁠. The subgroup generated by δ and ρ is the fundamental group of a neighbourhood of the boundary component, where δ represents a loop around the boundary component and ρ represents a path that reaches a mirror point and goes back. It has order two since the corresponding action is a reflection. When the boundary component has corner points, ρ_ij represents the paths that reach mirror reflectors enclosed between two corner points; the relations naturally come from the fact that the action inducing a corner point is dihedral.

Euler characteristic

Let us recall the definition of Euler characteristic for orbifolds.

 

Flat and spherical 2-orbifolds

Let us briefly provide the classification of flat and spherical closed 2-orbifolds. Closed flat 2-orbifolds are good and are precisely the quotients of |$\mathbb{R}^2$| by a wallpaper group, and similarly closed spherical 2-orbifolds are the quotients of |$\mathbb S^2$| by a finite subgroup of |$\mathrm{Isom}(\mathbb S^2)$|⁠.

 

The following table contains the list of the closed flat and spherical 2-orbifolds, both in standard and in Conway notation. In the flat case there are 17 orbifolds in the list; the spherical orbifolds are infinitely many but are collected in families possibly depending on an integer parameter.

  

Some examples

Let us briefly see some examples of flat and spherical orbifolds, putting in practice Remarks 2.12 and 2.13 and the computation of fundamental groups.

  

After a little simplification we obtain

3. THREE-DIMENSIONAL SEIFERT ORBIFOLDS

Let us now move on to dimension three. In this paper we only consider orientable 3-orbifolds.

3.1. Local models

By Remark 2.2, any point admits a local chart of the form |$D^3/\Gamma_0$| for Γ₀ a finite subgroup of SO(3) and the local model is thus the cone over the spherical orientable 2-orbifold |$\mathbb S^2/\Gamma_0$|⁠. In the right column of Table 1, only the first six lines include orientable spherical orbifolds. Hence the local models (apart from the trivial model D³ with no singularity) are as shown in Figure 6.

As a consequence, the underlying topological space of an orientable 3-orbifold is a manifold and the singular set is a trivalent graph; the local group is cyclic except at the vertices of the graph. The edges of the graph are usually labelled with the order of the local (cyclic) group.

3.2. Seifert fibrations

We now focus on the topological description of orientable 3-orbifolds via Seifert fibrations. First, we recall the basic definition of diagonal action.

   

• an orbifold chart |$\varphi:U\cong \widetilde U/\Gamma$| for |$\mathcal B$| around x;

• an action of Γ on S¹;

• an orbifold diffeomorphism |$\phi:f^{-1}(U)\to (\widetilde U\times S^1)/\Gamma$|⁠, where the action of Γ on |$\widetilde U\times S^1$| is diagonal and preserves the orientation;

such that the following diagram commutes:

where the map |$p:(\widetilde{U}\times S^1)/ \Gamma\to \widetilde{U}/ \Gamma$| is induced by the projection |$pr_1:\widetilde{U}\times S^1\to \widetilde{U}$| onto the first factor.

 

Fibres that project to a regular point of |$\mathcal B$| are called generic; the others are called exceptional. A fibration-preserving diffeomorphism is an orbifold diffeomorphism |$\mathcal{O}\to\mathcal{O}^{\prime}$|⁠, which induces an orbifold diffeomorphism |$\mathcal{B}\to\mathcal{B}^{\prime}$| of the base 2-orbifolds via the fibrations |$f:\mathcal{O}\to\mathcal{B}$| and |$f^{\prime}:\mathcal{O}^{\prime}\to\mathcal{B}^{\prime}$|⁠.

Fundamental group

We collect here a useful description of the orbifold fundamental group of a Seifert 3-orbifold.

 

For the proof see [2, Proposition 2.12].

Local models and invariants

Let us discuss the local models for orientable Seifert 3-orbifolds. That is, we describe the possible fibred neighbourhoods of a fibre |$\pi^{-1}(x)$|⁠, for |$x\in\mathcal{B}$|⁠, up to fibration-preserving diffeomorphisms. Relevant references are [3, Section 2] or [8, Section 2.1.2]. We split the analysis in several cases, according to the type of singularity of the point |$x\in \mathcal{B}$|⁠. When x is a cone or corner point, we will associate with x a (numeric) local invariant, which identifies the fibred neighbourhood up to orientation-preserving, fibration-preserving diffeomorphisms.

• If x is a regular point (that is, the local group |$\Gamma_x$| is trivial, and thus the fibre |$\pi^{-1}(x)$| is generic), then there exists a small neighbourhood U of x such that |$\pi^{-1}(U)$| is a trivial circle bundle.

• If x is a cone point labelled by n, the local group |$\Gamma_x$| is a cyclic group of order n acting diagonally, by rotations both on |$\widetilde U$| and on S¹. Assume that a generator of Γ acts on |$\widetilde U$| by rotation of angle |${2\pi}/{n}$| and on S¹ by rotation of |$-{2\pi m}/{n}$|⁠. Then |$\pi^{-1}(x)$| has a neighbourhood which is the quotient of |$\widetilde U\times S^1$| by this diagonal action and hence homeomorphic to a solid torus. The fibration coincides with Seifert fibrations in the usual sense for manifolds, except that the ‘central’ fibre |$\pi^{-1}(x)$| might be contained in the singular locus (see Figure 7). Indeed, in the orbifold context m and n are not necessarily coprime and the fibre |$\pi^{-1}(x)$| has singularity index equal to |$\gcd(m,n)$| if |$\gcd(m,n)\gt1$|⁠; otherwise, the fibre is regular. We then define the local invariant of |$\pi^{-1}(x)$| as the ratio |$m/n\in\mathbb{Q}/\mathbb{Z}$|⁠. Our definition coincides with the one given in [3], but notice that in [8] the local invariant is defined with the opposite sign, that is, as |$-m/n\in\mathbb{Q}/\mathbb{Z}$|⁠.

• If x is a mirror point, the generator of the local group |$\Gamma_x\cong\mathbb{Z}_2$| acts diagonally, by reflection both on |$\widetilde U$| and on S¹. Hence |$\mathbb Z_2$| acts by an hyperelliptic involution on the solid torus |$\widetilde U\times S^1$|⁠. The quotient is topologically a 3-ball, which contains two singular arcs of index 2. The fibre |$\pi^{-1}(x)$| is an interval with each end point contained in a singular arc. See Figure 8.

• If x is a corner point, |$\Gamma_x$| is a dihedral group. It has an index 2 cyclic subgroup, whose action on |$\widetilde U\times S^1$| is exactly the one described in the case of cone points above. The non-central involutions in |$\Gamma_x$| act by reflection both on |$\widetilde U$| and on S¹ and hence by an hyperelliptic involution as in the case of mirror points above. The quotient |$(\widetilde U\times S^1)/\Gamma_x$| is again a topological 3-ball (called the solid pillow) with a singular set composed of two arcs and possibly a singular interval connecting them. This can be visualized by first taking the quotient of |$\widetilde U\times S^1$| by the index 2 cyclic subgroup |$\Gamma^c_x$| as in the case of cone points above, thus obtaining a solid torus and then taking the further quotient of this solid torus by the hyperelliptic action of |$\Gamma_x/\Gamma_x^c\cong\mathbb{Z}_ 2$|⁠, exactly as in the case of mirror points.

  As a consequence of this discussion, the fibre |$\pi^{-1}(x)$| is an interval, and it is singular if and only if |$\gcd(m,n)\gt1$|⁠, with singularity index |$\gcd(m,n)$|⁠. The local invariant |$m/n\in\mathbb{Q}/\mathbb{Z}$| associated with |$\pi^{-1}(x)$| is defined as the local invariant of the index 2 cyclic subgroup |$\Gamma_x^c$|⁠. Observe that the fibre |$\pi^{-1}(y)$| for y ≠ x is an interval with end points in the singular locus if y is a mirror point or a circle if y is a regular point. However, these circles are twisted around the singular locus, according to the local invariant over x. See Figure 9.

  Nevertheless, there is a canonical diffeomorphism that ‘untwists’ the regular fibres, so as to make them wind simply around the singular locus. This identification is developed in detail in [8, Section 2.2], and the drawback of this transformation is that the singular locus becomes knotted. See Figure 10. We call standard local model this version of the preimage |$\pi^{-1}(U)$| for U a neighbourhood of x.

Boundary invariants and Euler number

Besides the local invariants described earlier, there are two additional invariants that must be considered in order to classify Seifert fibred orbifolds.

An invariant |$\xi\in\{0,1\}$| is associated with each boundary component of the underlying manifold |$|\mathcal{O}|$|⁠. It is defined by glueing together the standard local models over the h corner points of the boundary component. To close up the solid torus, one can perform a number of twists. Up to composing with Dehn twists, the diffeomorphism type then only depends on the oddity of the number of twists. Hence we define ξ = 0 if the solid torus is obtained by applying no twist and ξ = 1 if one twist is applied. This is described in detail in [8, Section 2.3]. See Figure 11.

 

Finally, the Euler number |$e(f)\in\mathbb{Q}$| is a rational number associated with the fibration |$f:\mathcal{O}\to\mathcal{B}$|⁠. See [3, Section 3] or [8, Section 2.1.3]. It is preserved by orientation-preserving, fibration-preserving orbifold diffeomorphisms and changes sign if one changes the orientation of the Seifert orbifold.

We have already observed that the local invariants, the boundary component invariants and the Euler number are preserved by orientation-preserving, fibration-preserving orbifold diffeomorphisms. In fact, the base orbifold together with the local invariants, the boundary components invariants and the Euler number completely classify oriented Seifert fibred orbifolds in the following sense.

 

It is also important to remark that the local invariants, boundary component invariants and Euler number are not unconstrained, but must satisfy an important identity, which we state in the following theorem. Given a local invariant |$m/n\in\mathbb{Q}/\mathbb{Z}$| over a corner point, we say that the normalized local invariant is the unique representative in |$\mathbb{Q}$| that belongs to the interval |$(-1/2,1/2]$|⁠.

([3, Section 4, Page 47] [8, Theorem 2.7])

 

Notice that for this formula to hold, it is essential that the local invariants |$m_{ij}/n_{ij}$| over corner point are normalized, namely, |$-1/2\lt m_{ij}/n_{ij}\leq 1/2$|⁠.

Finally, the topology of the base orbifold |$\mathcal{B}$| and the Euler number of the fibration determine the geometry of the Seifert orbifolds.

 

The proof of Theorem 3.9 is presented in [9]; another important reference is [15]. In particular, for the geometries |$\mathbb{R}^3$| and |$\mathbb S^2\times \mathbb{R}$| of interest in this work, we get the following characterization:

 

• |$\mathcal{O}$| is flat if and only if |$\mathcal{B}$| is a good flat orbifold and the Euler number e(f) vanishes.

• |$\mathcal{O}$| is geometric with geometry |$\mathbb S^2\times \mathbb{R}$| if and only if |$\mathcal{B}$| is a good spherical orbifold and the Euler number e(f) vanishes.

3.3. Standard and Conway notation

We conclude these preliminaries by introducing compact notations to denote Seifert fibred 3-orbifolds. In light of Corollary 3.10, we will only consider Seifert fibred 3-orbifolds with vanishing Euler number. Hence we will always omit the Euler number in the notation introduced below; we will implicitly intend that |$e(f)=0$|⁠.

The standard notation is the following. Suppose that the base orbifold |$\mathcal{B}$| is as in Equation (1). Then for a Seifert fibration |$f:\mathcal{O}\to\mathcal{B}$| we write

where all the local invariants are in the same notation as above; see, for instance, the statement of Theorem 3.8.

In Conway’s notation, we simply use the expression (4) for the base orbifold, to which we add in subscript m_k (resp. m_ij) corresponding to the labels n_k (resp. n_ij) of cone points (resp. corner points), and we add ξ_i again in subscript to the symbol |$\ast$| representing the boundary component. The fibration (6) is thus represented by

   

4. VANISHING EULER NUMBER

In this section we discuss Seifert fibred orbifolds with vanishing Euler number, with particular attention to the geometries |$\mathbb{R}^3$| or |$\mathbb S^2\times\mathbb{R}$|⁠. More concretely, we will prove Theorem 4.2 and Corollary 4.7, which give a direct way to construct, given the Seifert invariants of a geometric orientable orbifold |$\mathcal{O}$| with geometry |$\mathbb{R}^3$| (resp. |$\mathbb S^2\times\mathbb{R}$|⁠), a discrete group of isometries of |$\mathbb{R}^3$| (resp. |$\mathbb S^2\times\mathbb{R}$|⁠) whose quotient is diffeomorphic to |$\mathcal{O}$|⁠.

4.1. Diagonal actions

Let us start by showing that Seifert orbifolds with geometry |$\mathbb{R}^3$| or |$\mathbb S^2\times\mathbb{R}$| and vanishing Euler number can be obtained as quotients by a diagonal action. Recall that the definition of diagonal action is given in Definition 3.1.

  

where π_ΔΓ and |$\pi_\Gamma$| simply denote the quotient maps, and we use the notation ΔΓ to remind that the action of Γ on |$M\times \mathbb S^1$| is diagonal. Since |$\pi_\Gamma,\pi_{\Delta \Gamma}$| are quotient maps, f is a continuous function.

Step 1: f is a Seifert fibration.

If |$x\in \mathcal{B}$| then by definition there exists a chart |$\varphi:U\to\widetilde U/\Gamma_{\tilde x}$|⁠, where |$\tilde x\in\widetilde U\subset M$|⁠, and we can assume that |$\Gamma_{\tilde x}$| is the stabilizer of |$\tilde x$| in Γ. As already observed, the projection |$(pr_1)|_{\widetilde U\times\mathbb S^1}:\widetilde U\times\mathbb S^1\to \widetilde U$| induces a map |$p:(\widetilde U\times\mathbb S^1)/\Gamma_{\tilde x}\to \widetilde U/\Gamma_{\tilde x}$|⁠, and we have the following commutative diagram:

as in Definition 3.3.

Step 2:  |$\mathcal{O}$|  is closed.

Clearly |$\mathcal{O}$| is connected since it is the quotient of |$M\times \mathbb S^1$|⁠, which is connected. Since |$M/\Gamma$| is compact, there exists a compact K in M such that |$\pi_\Gamma(K)=M/\Gamma$|⁠. Therefore |$\mathcal{O}$| is the image through |$\pi_{{\Delta}G}$| of |$K\times \mathbb S^1$|⁠, which is compact, and therefore |$\mathcal{O}$| is compact.

Step 3:  |$e(f)=0$|⁠.

We claim that there exists a Seifert fibred orbifold |$f^{\prime}:\mathcal{O}^{\prime}\to\mathcal{B}^{\prime}$| of vanishing Euler number that finitely covers the fibration f. This means that there is a quotient map |$c:\mathcal{O}^{\prime}\to \mathcal{O}$| that induces a map |$c^{\prime}:\mathcal{B}^{\prime}\to\mathcal{B}$| between the base orbifolds and implies by the covering formula that appears in [3, Section 4] that |$e(f)=d/d^{\prime}\cdot e(f^{\prime})$|⁠, where dʹ is the degree of cʹ and d is the degree of c, as a map from |$S^1\to S^1$|⁠, when restricted to a generic fibre. Hence in our case since |$e(f^{\prime})=0$|⁠, we will obtain |$e(f)=0$|⁠.

If |$M=\mathbb S^2$|⁠, then the projection on the first factor |$pr_1:\mathbb S^2\times \mathbb S^1\to \mathbb S^2$| is clearly a Seifert fibration with vanishing Euler number that covers the fibration f.

If |$M=\mathbb{R}^2$|⁠, by Theorem 2.6 the translation subgroup |$T(\Gamma)$| is a normal subgroup of finite index in Γ, and therefore |$\mathcal{O}=(\mathbb{R}^2\times \mathbb S^1)/\Gamma$| is the quotient of |$(\mathbb{R}^2\times \mathbb S^1)/T(\Gamma)$| by the action of |$\Gamma/T(\Gamma)$|⁠. Observe moreover that |$\mathbb{R}^2/T(\Gamma)\cong T^2$|⁠. It remains to show that |$(\mathbb{R}^2\times \mathbb S^1)/T(\Gamma)$| is diffeomorphic to |$T^2\times S^1$| via a diffeomorphism that maps the fibration of |$(\mathbb{R}^2\times \mathbb S^1)/T(\Gamma)$|⁠, whose fibres are induced by the vertical fibres |$\{pt\}\times \mathbb S^1$|⁠, to the standard vertical fibration of |$T^2\times S^1$| which has vanishing Euler number.

By Theorem 2.6, up to conjugating |$T(\Gamma)$| by an affine transformation (and leaving the action on |$\mathbb S^1$| unchanged) we can assume that |$T(\Gamma)$| is the standard lattice |$\mathbb{Z}^2\lt\mathbb{R}^2$|⁠. Let a and b be its standard generators, namely, |$a(x,y)=(x+1,y)$| and |$b(x,y)=(x,y+1)$|⁠.

Recalling that the action of Γ is diagonal and orientation-preserving, every element of Γ is associated with an element of |$\mathrm{Isom}(\mathbb{R}^2)\times\mathrm{Isom}(\mathbb S^1)$|⁠, where the projections to the factors |$\mathrm{Isom}(\mathbb{R}^2)$| and |$\mathrm{Isom}(\mathbb S^1)$| are either both orientation-preserving or both orientation-reversing. |$\alpha+$| and |$\beta+$| denote the projections to the second factor |$\mathrm{Isom}(\mathbb S^1)$| relative to the actions of a and b, which are thus rotations on |$\mathbb S^1$|⁠. It is easy to check that the diffeomorphism |$h:\mathbb{R}^2\times \mathbb S^1\to\mathbb{R}^2\times \mathbb S^1$| defined by

conjugates the diagonal action of |$\mathbb{Z}^2$| to the action on |$\mathbb{R}^2\times \mathbb S^1$| which is given by the standard action on |$\mathbb{R}^2$| and the identity on |$\mathbb S^1$|⁠. Indeed,

and similarly |$h\circ (b,\beta+) \circ h^{-1}=(b,\mathrm{id}_{S^1})$|⁠. This concludes the proof.

Second, we want to show that every Seifert fibration with vanishing Euler number and closed base orbifold of the form |$M/\Gamma$| with |$M=\mathbb{R}^2$| or |$\mathbb S^2$| can be obtained by the construction as in Proposition 4.1. Recall that, in this situation, the fundamental group of |$\mathcal{B}$| is isomorphic to Γ. If |$\pi_1(\mathcal{B})$| admits a representation |$\psi:\pi_1(\mathcal{B})\to \mathrm{Isom}(\mathbb S^1)$|⁠, we can construct a diagonal action of |$\pi_1(\mathcal{B})$| on |$(M\times \mathbb S^1)/\pi_1(\mathcal{B})$|⁠, see Remark 3.2; the quotient |$(M\times \mathbb S^1)/\pi_1(\mathcal{B})$| can then be endowed by a Seifert fibration induced by the standard vertical fibration of |$M\times \mathbb S^1$| given by the vertical fibre |$\{pt\}\times \mathbb S^1$|⁠.

  

Step 1: Construction of ψ.

Let us define |$\psi:\pi_1(\mathcal{B})\to \mathrm{Isom}(\mathbb S^1)$|⁠. We will use the presentation of the fundamental group described in Proposition 2.9. We will define ψ for each generator and check that the relations are satisfied. We will use the notation introduced at the end of Section 2.2 for elements of |$\mathrm{Isom}(\mathbb S^1)$|⁠.

Let us send the generators x_s and y_s associated with a symbol |$\circ$| to 0+ (that is, the identity), and each generator z_r associated with a symbol × to 0− (that is, the reflection in the line of slope 0, namely, in the horizontal line). Each generator γ_k corresponding to a cone point with local invariant |$m_k/n_k$| is sent to |$(-m_k/n_k)+$|⁠. The relations |$\psi(\gamma_k)^{n_k}=\mathrm{id}$| are clearly satisfied.

Let us now consider boundary components of the underlying manifold, namely, symbols |$\ast$|⁠. We define, using the same notation for the generators as in Proposition 2.9 and for the local invariants as in Section 3.3, |$\psi(\rho_{i0})=0-$| and |$\psi(\rho_{i,j-1}\cdot \rho_{ij})=(-m_{ij}/n_{ij})+$| for |$j=1,\dots,h_i$|⁠. This defines |$\psi(\rho_{ij})$| uniquely for all i. Finally, define |$\psi(\delta_i)=(-(\sigma_i+\xi_i)/2)+$|⁠, where we set

The relations |$\psi(\rho_{ij})^2=1$| are clearly satisfied since |$\psi(\rho_{ij})$| is a reflection and clearly also |$\psi(\rho_{i,j-1}\cdot \rho_{ij})^{n_{ij}}=1$| for |$j=1,\dots,h_i$|⁠. To check the relation |$\psi(\delta_i)\cdot\psi(\rho_{ih_i})\cdot\psi(\delta_i^{-1})\cdot\psi(\rho_{i0})=1$|⁠, observe that |$\rho_{ih_i}=\rho_{i0}^2\cdot \rho_{i1}^2\cdot\dots\cdot \rho_{i,h_i-1}^2\cdot \rho_{ih_i}=\rho_{i0}\cdot (\rho_{i0}\cdot \rho_{i1})\dots (\rho_{i,h_i-1}\cdot\rho_{i,h_i})$|⁠, hence (using Remark 2.8 to compute the compositions)

Then

since ξ_i is an integer number.

Finally, it remains to check that the global relation, induced by (5), holds. This follows immediately from the observation that |$[\psi(x_s),\psi(y_s))]=\psi(z_r)^2=1$| and from Theorem 3.8.

By construction, the diagonal action defined by the representation ψ is orientation-preserving. Define |$\mathcal{O}^{\prime}=(M\times \mathbb S^1)/\pi_1(\mathcal{B})$|⁠, which is a connected orientable orbifold.

Step 2:  |$\mathcal{O}^{\prime}$|  is a closed Seifert fibred orbifold with  |$\mathcal{B}$|  as base space and e = 0.

Using Proposition 4.1 with |$\Gamma=\pi_1(\mathcal{B})$|⁠, we obtain that |$\mathcal{O}^{\prime}$| has a Seifert fibration over |$M/\Gamma\cong \mathcal{B}$| and vanishing Euler number. The fibration is induced by the partition in circles of |$M\times \mathbb S^1$|⁠. Call |$f^{\prime}:\mathcal{O}^{\prime}\to \mathcal{B}$| this fibration. We will conclude the proof by showing that |$\mathcal{O}^{\prime}$| and |$ \mathcal{O}$| are orientation-preserving and fibration-preserving diffeomorphic. By Theorem 3.7, it will be sufficient to check that all Seifert invariants of |$\mathcal{O}^{\prime}$| and |$ \mathcal{O}$| are the same.

Step 3: The case of only one boundary component.

By construction of ψ, the local invariants of fʹ over cone (or corner) points of |$\mathcal{B}$| are the same as the ones of the fibration f, and by Step 2 the Euler number for the fibration fʹ is 0. If |$\mathcal{B}$| has a unique boundary component, the boundary invariant is determined by Theorem 3.8, hence in this case all the invariants match automatically.

Step 4: The case of several boundary components.

From Table 1, the only situation with several boundary invariants occurs for |$\mathcal{B}=\ast\ast=S^1\times I$|⁠. Its fundamental group has the following group presentation:

which is thus isomorphic to |$(D_2\ast D_2)\times \mathbb{Z}$|⁠. It can be realized (see Remark 2.12) as the wallpaper group generated by two reflections |$x\mapsto Ax$| and |$x\mapsto Ax+(1,0)$| in parallel lines, where |$A=\mathrm{diag}(-1,1)$| and a translation |$x\mapsto x+(0,1)$|⁠. Recalling that |$\psi(\rho_1)=0-$|⁠, the fixed point set of the action of |$(\rho_1,\psi(\rho_1))$| is |$L=\{0\}\times \mathbb{R}\times \{(\pm1,0)\}$|⁠; its image in the quotient |$\mathcal{O}^{\prime}$| is contained in the singular locus and in the preimage of one boundary component of |$|\mathcal{B}|$|⁠.

If the boundary invariant ξ₁ of |$\mathcal{O}$| equals 0, then by construction |$\psi(\delta_1)=0+$|⁠, and hence |$(\delta_1,\psi(\delta_1))$| preserves each connected component of L. This means that the image of L in |$\mathcal{O}^{\prime}$| has two components, and thus the invariant |$\xi_1^{\prime}$| is again equal to 0 by Remark 3.6. Similarly, if ξ₁ = 1, then |$\psi(\delta_1)=(1/2)+$|⁠, and hence the action of |$(\delta_1,\psi(\delta_1))$| switches the two connected components of L. In this case the image of L in |$\mathcal{O}^{\prime}$| is connected and thus |$\xi_1^{\prime}=\xi_1=1$|⁠.

We could repeat the same argument for the second boundary invariant ξ₂; however this is not necessary since it is determined by all the others by Theorem 3.8. This shows that |$f:\mathcal{O}\to \mathcal{B}$| and |$f^{\prime}:\mathcal{O}^{\prime}\to \mathcal{B}$| have the same Seifert invariants and therefore concludes the proof.

 

4.2. Horizontal part and vertical translations

In Proposition 4.1, we considered quotients of |$M\times \mathbb S^1$| via diagonal actions of a discrete group of isometries of M with compact quotient, and in Theorem 4.2 we showed that all Seifert orbifolds with flat or spherical base and vanishing Euler number are obtained in this way. However, in this work we aim to study quotients of |$M\times\mathbb{R}$|⁠. We thus need some additional results that ensure that it is not restrictive to only consider quotients of |$M\times \mathbb S^1$| as above.

 

The following proposition must be compared with Proposition 4.1, now for subgroups of |$\mathrm{Isom}(M)\times \mathrm{Isom}(\mathbb{R})$| instead of |$\mathrm{Isom}(M)\times \mathrm{Isom}(\mathbb S^1)$|⁠.

  

Since |$\pi_G,\pi_{{\Delta}G}$| are quotient maps, f is a continuous function, and |$\mathcal{B}=f(\mathcal{O})$| is compact.

Let |$G=\{(\mathrm{id}_M,h)\in\Gamma\}$|⁠. By discreteness, G is generated by a single element, which we can assume (up to applying an affine transformation in the |$\mathbb{R}$| factor) to be |$(\mathrm{id}_M,1+)$|⁠. Observe moreover that G is normal in |$\mathrm{Isom}(M)\times \mathrm{Isom}(\mathbb{R})$|⁠. Given |$g\in\Gamma_H$|⁠, by definition there exists |$h\in \mathrm{Isom}(\mathbb{R})$| such that |$(g,h)\in\Gamma$|⁠. Then we can define |$\psi(g)$| to be the element in |$\mathrm{Isom}(\mathbb S^1)$| induced by h. It is easily checked that this does not depend on the choice of h and therefore defines a homomorphism |$\psi:\Gamma_H\to \mathrm{Isom}(\mathbb S^1)$|⁠. Since G is normal in Γ, we can factorize the diagram as follows:

where |$\Gamma_H$| is acting on |$\mathbb S^1$| via the homomorphism ψ and |$\Gamma/G$| has been identified with ΔΓ_H. Therefore we can use Proposition 4.1 to complete the proof.

   

5. FLAT SEIFERT 3-ORBIFOLDS

The main result of this section is Theorem A of the Section 1, which we rewrite here using Conway notation.

 

Two Seifert fibred orbifolds in the table are orientation-preserving diffeomorphic if and only if they appear in the same line. In particular, seven flat 3-orbifolds admit several inequivalent fibrations; six of those have exactly two inequivalent fibrations and one has three.

5.1. Point groups

Let Γ be a crystallographic group. Recall that its point group |$\rho(\Gamma)$| was defined in Definition 2.5, and it is discrete by Theorem 2.6.

 

By a little abuse of notation, if |$\mathcal{O}\cong \mathbb{R}^{n+1}/\Gamma$| is a good flat orbifold, then we define the point orbifold of |$\mathcal{O}$| to be the point orbifold of Γ, which is well-defined up to diffeomorphism.

 

This is equivalent to the condition that Γ preserves the partition of |$\mathbb{R}^{n+1}$| into the lines parallel to |$\mathrm{Span}(v)$|⁠.

 

Let us now focus on dimension three. If a direction |$[v]\in\mathbb{R} P^2$| is preserved by a space group |$\Gamma\lt\mathrm{Isom}(\mathbb{R}^3)$|⁠, then Γ is contained in the subgroup |$\mathrm{Isom}(v^\perp)\times\mathrm{Isom}(\mathrm{Span}(v))\lt\mathrm{Isom}(\mathbb{R}^3)$|⁠. Hence we will extend Definition 4.4 in this context, by defining

Clearly |$\Gamma_H^{[v]}$| is identified with a subgroup of |$\mathrm{Isom}(\mathbb{R}^2)$|⁠, which is well-defined up to conjugacy.

In this setting, Proposition 4.5 tells us that, if Γ admits a translation parallel to v, then |$\mathbb{R}^3/\Gamma$| has a Seifert fibration induced by the partition of |$\mathbb{R}^3$| in lines parallel to |$\mathrm{Span}(v)$|⁠, with base |$v^\perp/\Gamma_H^{[v]}$|⁠. The following result tells us that the converse is true, namely, all Seifert fibration of closed flat 3-orbifolds is obtained by this construction.

  

A very similar result in a more general context and with a different approach is obtained in [14, Theorem 7].

In [15] it is showed that if a space group Γ admits an invariant direction |$[v]$|⁠, then it also admits an invariant direction |$[v^{\prime}]$|⁠, possibly different, such that Γ admits a non-trivial translation parallel to vʹ. Therefore if a closed flat orbifold |$\mathbb{R}^3/\Gamma$| is not Seifert fibred, then it does not admit any invariant direction. Furthermore Γ has an element of order 3, see [15, Lemma 4.2]. Notice also that if Γ has an element of order 3, then this element permutes the vertices of an equilateral triangle, and therefore its barycentre is a fixed point.

An immediate consequence is the following, since for flat manifolds Γ acts freely on |$\mathbb{R}^3$|⁠:

  

On the road towards the proof of Theorem A, we will classify the point groups of space groups that admit several invariant directions. We first give here a restricted list of possible candidates.

 

• If  Γ admits several invariant directions in |$\mathbb{R} P^2$|⁠, then |$\mathbb S^2/\rho(\Gamma)$| is diffeomorphic to |$1,\times , \ast, 22,2\ast,\ast22,222,\ast222,2\times$| or |$2\!\ast\!2$|⁠.

• If moreover Γ is orientation-preserving, then |$\mathbb S^2/\rho(\Gamma)$| is diffeomorphic to |$1,22$| or 222.

By ‘1’ here we simply mean S², that is, |$\rho(\Gamma)$| is the trivial group.

  

5.2. Multiple fibrations for flat orientable orbifolds.

By Proposition 5.7, to study orientation-preserving space groups Γ whose quotients have several fibrations, it suffices to consider those having point orbifold in |$\{1,22,222\}$|⁠.

The following statement will be useful to further reduce the problem.  

  

Point orbifold is diffeomorphic to 1.

First, let us prove the following lemma.

  

Point orbifold diffeomorphic to 22.

Next, we prove the following lemma.

  

The invariant directions are the equivalence classes in |$\mathbb{R} P^2$| of the following set:

Therefore, we have two possibilities:

1. If the invariant direction is |$[e_3]$|⁠, we get

   $$\rho(\Gamma_H^{[e_3]})=\langle\begin{bmatrix} -1 & 0\\ 0 & -1 \end{bmatrix}\rangle \cong \mathbb{Z}_2.$$

   In this case |$\mathbb{R}^2/\Gamma_H^{[e_3]}$| is orientable and has at least one cone point, each cone point has singularity index equal to 2. From Table 1, |$\mathbb{R}^2/\Gamma_H^{[e_3]}$| is diffeomorphic to orbifold 2222.

2. If the invariant direction is |$[v]$|⁠, where |$v\in \mathbb S^1\times\{0\}$|⁠, we get

   $$\rho(\Gamma_H^{[v]})=\langle\begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix}\rangle\cong D_2.$$

   In this case |$\mathbb{R}^2/\Gamma_H^{[v]}$| is non-orientable and does not have any cone or corner point. From Table 1, |$\mathbb{R}^2/\Gamma_H^{[v]}$| is diffeomorphic to one of the following orbifolds: |$\ast\ast$|⁠, |$\ast\times$| or |$\times\times$|⁠.

Step 2: All possible Seifert invariants for a fibration of  |$\mathcal{O}$|  are listed in the table.

By a direct computation using the fact that the Euler number is zero and the identity in Theorem 3.8, one checks that all possible Seifert fibrations are listed in the statement.

Step 3: The orbifolds  |$(2_02_02_02_0),(2_02_02_12_1)$|  and  |$(2_12_12_12_1)$|  are not diffeomorphic.

Each cone point in the base orbifold with Seifert invariant 2₀ gives a singular fibre, while the fibres associated with the cone points of invariant 2₁ are not singular. By counting the components of the singular set we can distinguish the three orbifolds.

Step 4: Seifert orbifolds in the same line are orientation-preserving diffeomorphic.

We first claim that a Seifert fibred orbifold with base space |$\mathcal{B}\in\{\ast\ast,\ast\times,\times\times\}$| from the table in the statement also admits a Seifert fibration with base space orbifold 2222.

To see this, let Γ be a space group, |$\mathbb{R}^3/\Gamma$| be one of those Seifert fibred orbifolds and |$f:\mathbb{R}^3/\Gamma\to \mathcal{B}$| be the associated Seifert fibration. As above, up to a conjugation with an orthogonal map we can assume that |$\rho(\Gamma)$| is generated by the matrix |$\mathrm{diag}(-1,-1,1)$|⁠, since the point orbifold is diffeomorphic to 22.

Since |$\mathcal{B}=\mathbb{R}^2/G$| is diffeomorphic to |$\ast\ast,\ast\times$| or |$\times\times$|⁠, one can show (see Remark 2.12) that in all three cases G has two linearly independent translations whose directions are preserved by G. By Proposition 5.9 the group Γ has three linearly independent translations whose directions are preserved by Γ. Therefore, by Proposition 4.5, |$\mathbb{R}^3/\Gamma$| admits Seifert fibrations induced by those directions. By Step 1, all invariant directions lie in a plane with a single exception, the direction |$[e_3]$|⁠. As shown in Step 1, the base orbifold induced by the direction |$[e_3]$| is diffeomorphic to |$\mathbb{R}^2/\Gamma_H^{[e_3]}\cong 2222$|⁠.

Finally, we can identify the correct flat orbifold by looking at the number of components of the singular locus.

This concludes the proof.

5.2. Point orbifold is diffeomorphic to 222.

Finally, let us consider the remaining case.  

  

5.3. Conclusion of the proof of Theorem A

We are now ready to conclude the main result of the section.  

6. GEOMETRY |$\mathbb S^2\times\mathbb{R}$| AND BAD SEIFERT 3-ORBIFOLDS

The goal of this section is to prove Theorems B and C of Section 1, which we rewrite here using Conway notation.  

  

6.1. Explicit orbifold diffeomorphisms

First, in the following lemma we prove some explicit diffeomorphisms between Seifert fibred 3-orbifolds. We will show later that these are actually the only diffeomorphisms between inequivalent fibrations of Seifert fibred 3-orbifolds which are bad or have geometry |$\mathbb S^2\times\mathbb{R}$|⁠.  

     

Based on Lemma 6.1, we can immediately prove Theorem C.  

We remark that Lemma 6.1 also includes orbifolds with geometry |$\mathbb S^2\times \mathbb{R}$|⁠: in fact, if the base 2-orbifold is bad then we have a bad Seifert 3-orbifold, otherwise we get an orbifold with geometry |$\mathbb S^2\times \mathbb{R}$|⁠. We now move on to the situation for the geometry |$\mathbb S^2\times \mathbb{R}$|⁠.

6.2. All Seifert fibrations

In this section we enumerate all Seifert fibrations of a closed orientable Seifert fibred 3-orbifold with geometry |$S^2\times \mathbb{R}$|⁠. By Theorem 3.9 we know that those are the fibrations with base space a closed spherical 2-orbifold, with the Euler number equal to 0 and with compatible Seifert invariants (as in Theorem 3.8).

Therefore if |$f:(\mathbb S^2\times\mathbb{R})/\Gamma\to \mathcal{B}$| is a Seifert fibration, then |$\mathcal{B}$| is diffeomorphic to an orbifold in the right column of Table 1. Using Theorem 3.8 to find all compatible Seifert invariants, all possible fibrations are computed in Table 3. In the same table we report the underlying topological space of each fibred orbifold, which we computed using the following facts:

1. We can remove a cone or corner point that has a local invariant equal to |$0/p$| without affecting the topology of the underlying space.

2. We can substitute a cone or corner point that has invariant |$m/n$| with local invariant |$(m/\mathrm{gcd}(m,n))/(n/\mathrm{gcd}(m,n))$| without changing the underlying space.

3. A fibration with base space a disc without cone points has S³ as underlying topological space (see [8, Proposition 2.11]).

4. The fibration |$(n_mn_{(n-m)})$| has |$S^2\times S^1$| as underlying space (by the proof of Lemma 6.1).

5. The fibration |$(2_1\ast_1)$| has |$\mathbb{R} P^3$| as underlying space (see [8, Proposition 2.11]).

6. The fibration |$(n_0\overline \times)=(\mathbb{R} P^2;0/n;;0)$| has |$\mathbb{R} P^3\# \mathbb{R} P^3$| as underlying space (we can reduce to the manifold case by using Fact 2 and for manifolds, namely, to the case n = 1, where the conclusion is obtained in [15, pp. 457–459]).

In particular, all closed orientable Seifert orbifolds with geometry |$\mathbb S^2\times\mathbb{R}$| have as underlying space one of the following four manifolds (which are pairwise non-diffeomorphic):

6.3. Structure of the fundamental group

A short exact sequence involving the fundamental group of a Seifert fibred orbifold |$f:\mathcal{O}\to\mathcal{B}$| is described in Proposition 3.5: the generic fibre generates a normal cyclic subgroup that we denote by C, and we have

By the construction in the proof of Corollary 4.7, if |$\mathcal{O}$| has geometry |$\mathbb S^2\times\mathbb{R}$|⁠, then |$\mathcal{O}\cong(\mathbb S^2\times\mathbb{R})/\Gamma$| and C is identified to the subgroup of Γ consisting of vertical translations. The elements of |$\pi_1(\mathcal{B})\cong \Gamma_H$| can be seen as elements acting on |$\mathbb S^2$|⁠; in this sense we will distinguish between orientation-preserving and orientation-reversing elements of |$\pi_1(\mathcal{B})$|⁠.

  

We will use the symbol |$\pi_1^+(\mathcal{B})$| to denote the subgroup of |$\pi_1(\mathcal{B})$| of orientation-preserving elements. It has index either one or two.

  

Finally, to compute the index of a maximal abelian normal subgroup |$N=(f_*)^{-1}(M)$|⁠, we denote |$\pi_1^+(\mathcal{O}):=(f_*)^{-1}(\pi_1^+(\mathcal{B}))$|⁠. Observe that the index of |$\pi_1^+(\mathcal{O})$| in |$\pi_1(\mathcal{O})$| is equal to the index of |$\pi_1^+(\mathcal{B})$| in |$\pi_1(\mathcal{B})$| (and they are both equal either to one or to two), so it suffices to check that |$|\pi_1^+(\mathcal{O}):(f_*)^{-1}(M)|=|\pi_1^+(\mathcal{B}):M|$|⁠. Now, we have |$\pi_1^+(\mathcal{O})/N=(\pi_1^+(\mathcal{O})/C)/(N/C)=\pi_1^+(\mathcal{B})/M$|⁠.

 

6.4. Proof of Theorem B

We will now proceed to the proof of Theorem B. As a preliminary step, in Table 4 we provide a table that computes some invariants that will be used in the proof in order to distinguish different diffeomorphism classes of orbifolds. However, these invariants will not always be sufficient to distinguish non-diffeomorphic orbifolds. Hence we need an additional lemma, which will be used several times in the proof of Theorem B.

  

We are now ready to provide the proof.

Proof of Theorem B

 

• We have Group 9 with n = 2, Group 10 with n = 2, Group 12 and Group 13 where S³ appears as underlying topological space. Counting the number of connected components and the number of vertices of the singular locus, one checks that these are pairwise non-diffeomorphic except for the case of |$(\ast_12_12_12_0)$| and |$(2_0\ast_0 2_0)$|⁠. These are distinguished by Lemma 6.6 with n = 2.

• The two orbifolds with underlying space |$\mathbb{R} P^3$| are distinguished by the number of vertices.

• The only remaining possibility is the orbifold within Group 8, having |$S^2\times S^1$| as the underlying topological case, and also in this case the number of connected components distinguishes them.

We can conclude that in this case two fibred orbifolds can be diffeomorphic only if they are included in Group 12; these diffeomorphisms are treated in Lemma 6.1.

Step 5:  |$|\pi_1(\mathcal{O}):N|=4$|⁠.

Index 4 is achieved by the orbifold with a fibration included in the following groups: Group 9 and Group 10 with |$n\not\in\{2,4\}$|⁠, see Table 4. Diffeomorphisms between fibred orbifolds of the same group cannot occur by the same argument used in the previous step. Counting the number of vertices, we only have to exclude a diffeomorphism between |$(\ast_12_12_1n_0)$| and |$(2_0\ast_0n_0)$|⁠. This is impossible by Lemma 6.6. This concludes the proof.

Finally, we conclude by discussing Corollaries D and E. The proof of Corollary D follows immediately from the results of this paper and of [12], as explained in the Section 1. Let us provide the proof of Corollary E.

Proof of Corollary E

  

Acknowledgements

The third author has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01) funded by the French program Investissements d’avenir. All three authors are members of the national research group GNSAGA.

References