Classification of generalized Seifert fiber spaces
Fernando Galaz-Garcia Jesús Núñez-Zimbrón







Generalized Seifert Spaces]Classification of generalized Seifert fiber spaces


F. Galaz-García]Fernando Galaz-García
[Galaz-García]Department of Mathematical Sciences, Durham University, United Kingdom
fernando.galaz-garcia@durham.ac.uk

J. Núñez-Zimbrón]Jesús Núñez-Zimbrón
[Núñez-Zimbrón]Facultad de Ciencias, UNAM, Mexico
nunez-zimbron@ciencias.unam.mx






§ ABSTRACT 
We provide a symbolic classification of generalized Sei­fert fiber spaces, which were introduced by Mitsuishi and Yamaguchi in the classification of collapsing Alexandrov 3-spaces. Additionally, we show that the canonical double branched cover of a non-manifold generalized Seifert fiber space is a Seifert manifold and compute its symbolic invariants in terms of those of the original  space.  






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§ MAIN RESULTS




Generalized Seifert fiber spaces first appeared in the work of Mitsuishi and Yamaguchi on the classification of collapsing Alexandrov 3-spaces <cit.> and generalize Seifert 3-manifolds, sharing a similar structure: they decompose into fibers over a 2-orbifold, with the fibers being either circles or closed intervals. As in the manifold case, small tubular neighborhoods of the circle fibers are standard fibered tori. In contrast, small neighborhoods of the interval fibers are homeomorphic to , a fundamental non-negatively curved Alexandrov 3-space that serves as a “singular” fibered torus and can be written as the boundary connected sum of two cones over P^2 (see Section <ref> for a detailed definition).


A closed (i.e., compact and without boundary) Alexandrov 3-space is homeomorphic to either a 3-manifold or to a non-orientable 3-manifold with finitely many boundary components homeomorphic to P^2, the real projective plane, that have been capped off with cones over P^2 <cit.>; the latter spaces have also appeared in the literature under the name singular 3-manifolds <cit.>. Conversely, any 3-manifold or singular 3-manifold is homeomorphic to some Alexandrov space  <cit.>. A consequence of the surgery procedure introduced in <cit.>, which removes spaces homeomorphic to  from non-manifold Alexandrov 3-spaces, combined with the geometrization theorem for Alexandrov 3-spaces <cit.>, is that every closed Alexandrov 3-space can be decomposed into hyperbolic pieces and generalized Seifert fiber spaces. Therefore, it is natural to consider a classification of generalized Seifert fiber spaces, as they constitute fundamental components of all Alexandrov 3-spaces and play a central role in their collapse theory <cit.>. Our first main result is such a classification, extending the classical one originally due to Seifert <cit.>.







Let p X→ B be a generalized Seifert fibration  with X closed and connected. Then the generalized Seifert fiber space X is determined up to fiber-preserving homeomorphism by the set of invariants

    { b,ε,g, ι, { (α_i, β_i) }_i=1^n }.



For concision, the precise definitions of the symbols in Theorem <ref> are stated in Section <ref>. 
Some of these invariants coincide with those appearing in the classification of local circle actions on Alexandrov 3-spaces <cit.>. However, Theorem <ref> is not a special case of the classification of local circle actions on Alexandrov 3-spaces <cit.>, as the canonical “fibration” of the space  does not arise from any local circle action (see the discussion after <cit.>). Our strategy to prove Theorem <ref> involves excising small neighborhoods of the interval fibers and gluing appropriate “blocks” to obtain an associated space that does admit a local circle action, and use the corresponding classification. 



Every closed non-manifold Alexandrov 3-space X is the base of a two-fold branched cover πX̃→ X whose total space X̃ is a closed orientable manifold 
and whose branching set is the set of non-manifold points of X, i.e., points which have neighborhooods homeomorphic to a cone over P^2 <cit.>. In particular, there is an
orientation reversing (piecewise linear) involution ιX̃→X̃ with only isolated fixed points such that X
is homeomorphic to the quotient X̃ /ι <cit.>. The manifold  X̃ is unique up to homeomorphism and we refer to it as the double branched cover of X.   

In Theorem <ref>, we show that the canonical double branched cover of a non-manifold generalized Seifert fiber space X is a Seifert manifold X̃ where the Seifert fibration commutes with the double branched cover construction. We also compute the symbolic invariants of X̃ in terms of those of the original space X, finding, roughly speaking, that the invariants of X are doubled in X̃.  



Let p X→ B be a generalized Seifert fibration where X is closed, connected, and not a manifold, 
and let πX̃→ X be the canonical double branched cover of X. Then there exists a Seifert fibration p̃X̃→B̃ commuting with the double branched cover. Moreover, if X is determined by the invariants

    { b,ε,g, ι, { (α_i, β_i) }_i=1^n },

then X̃ is determined by the invariants 

    {b , ε, 2g, 0, { (α̃_i, β̃_i) }_i=1^2n},

where (α̃_2k,β̃_2k)=(α_k, β_k) and (α̃_2k-1,β̃_2k-1)=(α_k, β_k) for each k=1,2,…,n.





Our article is organized as follows. In Section <ref>, we recall basic facts on generalized Seifert fibered spaces and their topology, and collect basic facts on the classification of Alexandrov 3-spaces with local circle actions that we will use in the proofs of the main theorems. We prove Theorems <ref> and <ref> in Sections <ref> and <ref>, respectively.



JNZ acknowledges support from PAPIIT-UNAM project number IN101322. JNZ wishes to thank Diego Corro and Luis Jorge Sánchez Saldaña for useful conversations.








§ PRELIMINARIES




To keep our presentation concise, throughout the article we assume the reader to be familiar with the basic theory of Seifert manifolds and Alexandrov spaces of curvature bounded below, particularly in the three-dimensional case. For an introduction to  Alexandrov spaces, we refer to the standard references <cit.>. For basic results in dimension three, we refer the reader to <cit.>. For further results on Alexandrov 3-spaces, see  <cit.>.  Our main reference for the basic theory of Seifert fiber manifolds will be <cit.>.

In this section, we only recall the key aspects necessary for our discussion. 




 §.§ Alexandrov 3-spaces
 Let X be a closed Alexandrov 3-space. The space of directions Σ_x X at each point x in X is a closed Alexandrov 2-space with curvature bounded below by 1 and,  by the Bonnet–Myers Theorem <cit.>, the fundamental group of Σ_x X is finite. Thus, Σ_x X is homeomorphic to a 2-sphere S^2 or a real projective plane P^2. A point in X whose space of directions is homeomorphic to S^2 is a topologically regular point. Points in X whose space of directions is homeomorphic to P^2 are topologically singular.  We will denote the set of topologically singular points of X by sing(X). The set of topologically regular points is open and dense in X. By Perelman's Conical Neighborhood Theorem <cit.>, every point x in X has a neighborhood that is pointed-homeomorphic to the cone over Σ_x X. Therefore, the set of topologically singular points of X is finite, and X is homeomorphic to a compact 3-manifold with a finite number of P^2-boundary components to which one glues in cones over P^2. If X contains topologically singular points, it is homeomorphic to the quotient of a closed, orientable, topological 3-manifold X by an orientation-reversing involution ιX̃→X̃ with only isolated fixed points. The 3-manifold X̃ is the orientable double branched cover of X (see, for example, <cit.>). One may lift the metric on X to X̃ so that X̃ becomes an Alexandrov space with the same lower curvature bound as X and ι is an isometry with respect to the lifted metric (see <cit.> and <cit.>) and compare with <cit.>). In particular, ι is equivalent to a smooth involution on X̃ considered as a smooth 3-manifold. 







 §.§ Generalized Seifert fiber spaces


Let us recall the definition of generalized Seifert spaces, introduced by Mitsuishi and Yamaguchi in <cit.>.
We begin with the definition of the space . Consider the isometric involution (with respect to the flat metric) 

    α S^1× D^2    →    S^1× D^2
    
    (e^iθ,x)   ↦    (e^-iθ,-x).

The space  is then defined as (S^1× D^2)/⟨α⟩, and we denote the projection by π S^1× D^2 →. There is a natural projection 

    p B(pt) → K_1(S^1)

induced by (e^iθ,x)↦ x. 




    Observe that the restriction of α to S^1× S^1 has as quotient the flat Klein bottle K^2, viewed as the non-orientable circle bundle over S^1 (see, for example, <cit.>). Consequently, the fibration on K^2 induced by the restriction p K^2 → S^1 is fiberwise equivalent (i.e., there is a fiberwise homeomorphism) to the fibration induced by the free local circle action (see next section for the definition of a local circle action) on K^2 given by rotating one of the circle factors.  


A generalized Seifert fibration of a (topological) 3-orbifold X over a (topological) 2-orbifold B (possibly with non-empty boundaries) is a map f X→ B whose fibers are homeomorphic to circles or bounded closed intervals, and which satisfies the following conditions. For every x∈ B, there is a neighborhood U_x homeomorphic to a 2-disk satisfying the following conditions:

    
  (i) If f^-1(x) is homeomorphic to a circle, then there is a fiber-preserving homeomorphism of f^-1(U_x) to a Seifert fibered solid torus in the usual sense. In this case, we say that f^-1(x) is a C-fiber.
    
  (ii) If f^-1(x) is homeomorphic to an interval, then there exists a fiber-preserving homeomorphism of f^-1(U_x) to the space 
    B(pt), with respect to the fibration
    
    ( B(pt), p^-1(o)) →( K_1(S^1), o).

    In this case, we say that f^-1(x) is an I-fiber.

Moreover, for any compact component C of the boundary of B, we require the existence of a collar neighborhood N of C in B such that f|_f^-1(N) is a circle bundle over N. In this context, we say that X is a generalized Seifert fiber space with base B and use the notation X=(B). The base spaces of two generalized Seifert fibered spaces X=(B) and Y=(C) are isomorphic if there exists a homeomorphism between B and C that preserves the fiber type.

As noted in <cit.>, when there are I-fibers, the fibration f is not induced by any local circle action.





 §.§ Local circle actions



We briefly summarize the classification of local circle actions on Alexandrov 3-spaces obtained in <cit.>, which will play a key role in our discussion (see <cit.> for the manifold case). 

A local circle action on a closed Alexandrov 3-space X is a decomposition of X into (possibly one-point) disjoint, simple, closed curves (the fibers), having a tubular neighborhood which admits an effective circle action whose orbits are the curves of the decomposition. The local circle action is isometric if the circle actions on each tubular neighborhood of the fibers are isometric with respect to the restricted metric.





 §.§ Fiber types
 The fiber types of a local circle action are in correspondence with the isotropy groups of the fibers viewed as orbits of the isometric circle action on a small tubular neighborhood around them. These types are the following:

    
  * F-fibers are topologically regular fixed-point fibers.
    
  * SF-fibers are topologically singular fixed-point fibers.
    
  * E-fibers are those that correspond to ℤ_k isotropy, acting in such a way that local orientation is preserved.
    
  * SE-fibers correspond to ℤ_2 isotropy, reversing the local orientation.
    
  * R-fibers are fibers that are not F-, SF-, E- or SE-fibers.
  

The sets of points in F-, SF-, E-, SE- and R-fibers are denoted by F, SF, E, SE and R, respectively. The fiber space X^* is a 2-orbifold. Its boundary consists of the images of F-, SF- and SE-fibers under the natural projection map, while the interior consists of R-fibers and a finite number of E-fibers.



 §.§ Building blocks
 A closed Alexandrov 3-space X with a local isometric circle action can be decomposed into building blocks of types F, SF, E and SE, defined by considering small tubular neighborhoods of connected components of fibers of the corresponding type. A building block is simple if its boundary is homeomorphic to a torus, and twisted if its boundary is homeomorphic to a Klein bottle. Note that the stratum of R-fibers is an S^1-fiber bundle with structure group O(2).




Let X be a closed, connected Alexandrov 3-space with 2r≥ 0 topologically singular points. Then, the set of isometric local circle actions (up to equivariant equivalence) is in one-to-one correspondence with the set of unordered tuples

    {b; ε, g, (f,k_1), (t,k_2), (s,k_3); { (α_i, β_i) }_i=1^n; (r_1,r_2, …, r_s-k_3); (q_1, q_2, …, q_k_3)}.



The definition of the symbols appearing in this result is as follows:

    
  * (ε,k) is the pair that classifies the S^1-bundle of R-fibers according to <cit.>.
    
  * g≥ 0 is the genus of X^*.
    
  * The symbols f, t, k_1, k_2 are non-negative integers  such that k_1 ≤ f and k_2≤ t, where k_1 is the number of twisted F-blocks and k_2 is the number of twisted SE-blocks (therefore f-k_1 is the number of simple F-blocks and t-k_2 is the number of simple SE-blocks).
    
  * n is the number of E-fibers and we let { (α_i, β_i)}_i=1^n be the corresponding Seifert invariants (see <cit.> for the definition).
    
  * b is an integer or an integer mod 2 with the following conditions: b=0 if f+t>0 or if ε∈{o_2,n_1,n_3,n_4} and some α_i=2 (see <cit.> for the precise definitions of the o_i, n_j and α_l); b∈{0,1} if f+t=0 and ε∈{ o_2,n_1,n_3,n_4} and all α_i≠ 2. In the remaining cases, b is an arbitrary integer;
    
  * We let s, k_3 be non-negative integers, where  k_3≤ s is the number of twisted SF-blocks (thus s-k_3 is the number of simple SF-blocks).
    
  * (r_1, r_2, …, r_s-k_3) and (q_1, q_2, …, q_k_3 ) are (s-k_3)- and k_3-tuples of non-negative even integers corresponding to the number of topologically singular points in each  simple and twisted SF-block, respectively.
    
  * The numbers k, k_1, k_2, and k_3 satisfy k_1 + k_2 + k_3 = k.






§ INVARIANTS AND CLASSIFICATION




In this section, we define the equivariant and topological symbols that appear in Theorem <ref> and prove this result. 





 §.§ Invariants


Let X=(B) be a closed, generalized Seifert fiber space and associate the following information to X:



    
  * The subset X_0 of X obtained by removing all I-fibers and C-fibers with non-trivial isotropy is an S^1-bundle with structure group O(2), so we associate to X the classifying pair (ε,k). In fact, it will follow from the proof of Theorem <ref> that k=ι (where ι is defined below).
    
  * The integer (or integer mod 2 depending on the orientability of B) b is the obstruction to extending a cross-section of the bundle X_0 to X. 
    
  * The symbol g stands for the genus of B.
    
  * The symbol ι is the (necessarily finite) number of I-fibers.
    
  * Finally, the ordered pairs (α_i,β_i) are the Seifert invariants of the C-fibers with non-trivial isotropy.


All of these symbols, except ι, are inspired by those assigned to a space with a local circle action as can readily be seen. In turn, they are subject to the same restrictions on its values as those indicated in Theorem <ref> (except obviously ι, which is new); however, not all values are possible under the assumption that X is a generalized Seifert space, 
as will follow from the proof of Theorem <ref>.
 
To summarize, to each generalized Seifert bundle we associate a set of invariants of the form 

    { b,(ε,k),g, ι, { (α_i, β_i) }_i=1^n }.




 §.§ Classification


If there are no I-fibers, Theorem <ref> reduces to the classification of usual Seifert manifolds by letting ι=0, and the fact that the classical classification of Seifert (see <cit.>) is contained in the classification of local circle actions on 3-manifolds of Fintushel, Orlik and Raymond (see <cit.>). Therefore, to prove Theorem <ref>,  it is sufficient to consider the case in which the set of I-fibers is non-empty. Note that this implies that sing(X), the set of topologically singular points of X, is non-empty.


To establish Theorem <ref>, we need to prove existence and uniqueness statements as follows: that given a list of symbols as in <ref>, one is able to construct a generalized Seifert fiber space realizing these symbols, and that if two generalized Seifert fiber spaces share the same set of invariants, then there exists a fiber-preserving homeomorphism  between them.




 §.§ Proof of Theorem <ref>
 
We first prove the uniqueness statement. Let X_1=(B_1) and X_2=(B_2) be closed and connected generalized Seifert spaces. It is clear that if X_1 is fiberwise homeomorphic to X_2, then B_1 is isomorphic to B_2. We now prove that if there is an isomorphism of fiber spaces φ B_1 → B_2, then X_1 is equivalent to X_2.


Let I^∗_j={x^∗_j1, x^∗_j2, …, x^∗_jι}⊆ B_j for j=1,2. We now choose disjoint closed disk neighborhoods D_1i of each x^∗_1i such that π_1^-1(D_1i)≅. Without loss of generality, as φ is fiber-type preserving, we can assume φ(x^∗_1i)=x^∗_2i and denote D_2i= φ(D_1i), i=1,…, ι. Furthermore, we can assume that D_1i is small enough so that π_2^-1(D_2i)≅ for all i=1,…,ι. 

Now, observe that φ restricts to a fiber-type preserving homeomorphism B_1 ∖⋃_i=1^ιint(D_1i) → B_2 ∖⋃_i=1^ιint(D_2i) and consider for j=1,2

    Z_j:= X_j∖⋃_i=1^ιπ_j^-1( int(D_ji) ).

Then ∂ Z_j is a disjoint union of ι copies of Klein bottles K^2. Observe moreover that Z_j is a Seifert manifold with boundary. 

We let V_ji be twisted F-blocks for j=1,2, i=1,…, ι and consider the spaces Y_j obtained by gluing each V_ji to each boundary component of Z_j via fiber-preserving homeomorphisms. Note that the spaces Y_j satisfy that sing(Y_j)=∅ as all topologically singular points of X_j must be on I-fibers and not on C-fibers. Thus, Y_j is a closed topological 3-manifold, j=1,2 and it carries a local circle action. Therefore, by the work of Orlik and Raymond <cit.> and Fintushel <cit.>, the Y_j are determined up to fiber-preserving homeomorphism by the same set of invariants, namely 

    {b,ε,g, (f=ι,k_1=ι), (t=0,k_2=0), {(α_j,β_j) }_j=1^n},


and there is a fiber-preserving homeomorphism Ψ Y_1 → Y_2 which is determined by cross-sections to the local actions away from the exceptional orbits. 

Observe now that Ψ(V_1i)⊂ Y_2 is a twisted F-block so that, without loss of generality, we can assume Ψ(V_1i)=V_2i, for each i=1,…,ι. Thus, by restricting Ψ we obtain a fiber-preserving homeomorphism φ Z_1 → Z_2 determined by cross-sections ρ_j∂ ( B_j ∖⋃_i=1^ιint(D_ji))→∂ Z_j, that is, such that 

    Ψ( ρ_1( ∂ ( B_1 ∖⋃_i=1^ιint(D_1i)))) = ρ_2( ∂ ( B_2 ∖⋃_i=1^ιint(D_2i))).

Furthermore, as the local circle action restricted to ∂ Z_j is equivalent to the “free” local circle action, we can parametrize each component of ∂ Z_J, homeomorphic to the Klein bottle, seen as the non trivial bundle S^1×S^1 as an equivalence class (with respect to the involution α as in (<ref>)) of points [e^iθ, e^iη], where e^iθ∈ρ_1( ∂ ( B_1 ∖⋃_i=1^ιint(D_1i))) ≅ S^1 and e^iη parametrizes the fiber direction. Hence, Ψ|_∂ Z_1 is of the form 

    Ψ|_∂ Z_1([e^iθ, e^iη]) = [Ψ_1(e^iθ), Ψ_2,θ(e^iη)],

where Ψ_1 and Ψ_θ,2 are self-homeomorphisms of S^1 for each θ∈ [0,2π). 

We now conclude by observing that Ψ can clearly be extended to a fiber-preserving homeomorphism X_1→ X_2 by parametrizing each  block by classes of points  [e^iθ, re^iη] where r∈ [0,1], and defining 

    Ψ|_([e^iθ, re^iη]):= [Ψ_1(e^iθ), rΨ_2,θ(e^iη)].


We now prove the existence part of the theorem. By the classification of spaces with local circle actions <cit.>, there exists a space Y with a local circle action with the invariants

    {b,ε,g, (f=ι,k_1=ι), (t=0,k_2=0), {(α_j,β_j) }_j=1^n},

and an orbifold Riemannian metric of curvature bounded below. 
In a similar fashion to the uniqueness part of the proof, we take out all ι twisted F-blocks from Y, obtaining a Riemannian orbifold with boundary Z with curvature bounded below and with a local circle action. We now take ι copies V_i of  with the orbifold metric induced by the flat metric on S^1× D^2. By the collar theorem, the orbifold metrics on Z and V_i are product metrics and thus, we can glue the V_i to Z by ι copies of the cylinder I× K^2 in such a way that the metric on {0}× K^2 coincides with the metric on the i-th boundary component of Z and the metric on {1}× K^2 coincides with the metric on the boundary of V_i. Moreover, this can be done so that the glued space X is a smooth Riemannian orbifold which by compactness has curvature bounded below, and in particular, X is an Alexandrov space admitting a generalized Seifert fiber space structure with the invariants

    { b,(ε,k),g, ι, { (α_i, β_i) }_i=1^n }.

As we pointed out, k=ι so that we remove the superfluous invariant k and uniquely associate (up to fiber-preserving homeomorphism) the list 

    { b,ε,g, ι, { (α_i, β_i) }_i=1^n }.

to each generalized Seifert fiber space, as stated in Theorem <ref>.



§ COMPATIBILITY WITH THE DOUBLE BRANCHED COVER



The canonical double branched cover of an Alexandrov space X with sing(X)≠∅ is a fundamental construction which, in dimension 3, de-singularizes the space <cit.>. In this section, we show that the double branched cover of a generalized Seifert fiber space is a Seifert manifold and compute its invariants from the invariants of the original space, thus proving Theorem <ref>.
 

The following lemma follows directly from the fact that the orientation double cover of a non-orientable manifold can be characterized in the following way: Let M be a non-orientable manifold and pM̃→ M its orientation double cover. If N is an oriented manifold and π N→ M is a double cover with an orientation-reversing non-trivial deck transformation, then (N,π) is isomorphic to (M̃,p), (see for example <cit.>).



The canonical double branched cover of  is isomorphic to the projection used in the construction of , that is S^1× D^2 →, given by (e^iθ, x)↦ [e^iθ, x]. 


With this observation in hand, we may prove our second main theorem.



 §.§ Proof of Theorem <ref>


By Lemma <ref>, the preimage in S^1× D^2 of the fibers of the standard generalized Seifert fibration on  (with base K_1(S^1)) define a (usual) Seifert fibration p̃ S^1× D^2 → D^2, where we have an action of ℤ_2 on the D^2-base by rotations of angle π, with quotient K_1(S^1) (we call the quotient map π̃). In other words the following diagram commutes

    S^1× D^2 [d]_π[r]^p̃     D^2 [d]^π̃
    [r]_p     K_1(S^1)


In all other points away from the branching locus, the action of the canonical involution of the double branched cover is free. Therefore, slightly abusing the notation and denoting πX̃→ X the canonical double branched cover, and p X→ B the generalized Seifert fibration, we have that there exists a 2-orbifold B̃ and a (usual) Seifert fibration ρ̃X̃→B̃ such that the following diagram commutes

    X̃[d]_π[r]^p̃   B̃[d]^π̃
    
    X [r]_p     B


where π̃ is the obvious map. Thus we have shown that for any closed Alexandrov 3-space X with non-empty set of topologically singular points, its canonical double branched cover X is a Seifert 3-manifold.

Let us denote the symbolic invariants associated to X=(B) as in (<ref>) and those of X̃=(B̃) by

    {b̃,ε̃,g̃, ι̃, { (α̃_i, β̃_i) }_i=1^ñ}.


It is then immediately clear that, since sing(X̃)=∅, then ι̃=0. Let us now point out that near the exceptional C-orbits, the canonical involution of X̃ acts freely and therefore the preimage in X̃ of each exceptional orbit on X consists of two copies of itself. Thus ñ=2n and we can enumerate the Seifert invariants of X̃, for example as (α̃_2k,β̃_2k)=(α_k, β_k) and (α̃_2k-1,β̃_2k-1)=(α_k, β_k) for each k=1,2,…,n. Because the local actions of ℤ_2 on B̃ that give rise to B are orientation preserving, it follows that B̃ is orientable if and only if B is orientable. Therefore, ε̃ = ε. Moreover, it is also clear from the definition of B̃ that g̃=2g. Finally, any section B→ X determines a section B̃→X̃ and vice versa, so that b̃=b. 


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