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### Article

Kyungpook Mathematical Journal 2022; 62(2): 363-388

Published online June 30, 2022

### Finite, Fiber-preserving Group Actions on Elliptic 3-manifolds

Benjamin Peet

Department of Mathematics, St. Martin’s University Lacey, WA 98503, United States
e-mail : bpeet@stmartin.edu

Received: May 16, 2020; Accepted: December 8, 2021

In two previous papers the author presented a general construction of finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds. In this paper we restrict our attention to elliptic 3-manifolds. For illustration of our methods a constructive proof is given that orientation-reversing and fiber-preserving diffeomorphisms of Seifert manifolds do not exist for nonzero Euler class, in particular elliptic 3-manifolds. Each type of elliptic 3-manifold is then considered and the possible group actions that fit the given construction. This is shown to be all but a few cases that have been considered elsewhere. Finally, a presentation for the quotient space under such an action is constructed and a specific example is generated.

Keywords: geometry, topology, 3-manifolds, finite group actions, Seifert fiberings, elliptic

### 1.1. Discussion of results

In previous papers [8] and [9] we considered orientable Seifert manifolds and the possible finite groups that can act fiber- and orientation-preservingly.

The main results in those papers established that firstly:

Theorem 1.1. Let M be a closed, compact, and orientable Seifert 3-manifold that fibers over an orientable base space. Let φ:GDiff+fp(M) be a finite group action on M such that the obstruction class can expressed as

b= i=1m(bi#Orbφ(αi))

for a collection of fibers {α1,,αm} and integers {b1,,bm}. Then φ is an extended product action.

Where an extended product action is intuitively a product action on an orientable surface with boundary cross S1 extended across Dehn fillings of the boundary tori.

Secondly, it was shown that:

Corollary 1.2. Suppose that φ:GDiff(M) is a finite group action on an orientable Seifert manifold with a non-orientable base space. Then provided that the unique lifted group action φ˜:GDiff(M˜) satisfies the obstruction condition, G is isomorphic to a subgroup of 2×H where H is a finite group that acts orientation-preservingly on the orientable base space of M˜.

These two results will allow us to consider the elliptic 3-manifolds in particular and present the possible finite, fiber- and orientation-preserving groups that can act on them.

We then present a proof that all finite, fiber-preserving actions on Seifert manifolds with non-zero Euler class must be orientation-preserving by using our particular construction and in particular apply this to elliptic manifolds.

Finally, we consider the quotient orbifolds that will arise under the given actions and present a thorough example of one such action.

### 1.2. Preliminary definitions

Throughout we consider M to be an oriented smooth manifold of dimension 3 and without boundary. G is always assumed to be a finite group. We denote Diff(M) as the group of self-diffeomorphisms of M, and then define a G-action on M as an injection φ:GDiff(M). We use the notation Diff+(M) for the group of orientation-preserving self-diffeomorphisms of M.

M is further assumed to be an orientable Seifert-fibered manifold. That is, M can be decomposed into disjoint fibers where each fiber is a simple closed curve and each fiber has a fibered neighborhood which can be mapped under a fiber-preserving map onto a solid fibered torus.

A Seifert bundle is a Seifert manifold M along with a continuous map p:MB where p identifies each fiber to a point. For clarity, we denote the underlying space of B as BU.

We use the normalized notation (g,ϵ|(q1,p1),,(qn,pn),(1,b)) to indicate an orientable Seifert manifold with normalized Seifert invariants (q1,p1),,(qn,pn), obstruction class b, and ϵ=o1 if the base space is orientable and ϵ=n2 is the base space is not orientable.

The Euler class of a Seifert manifold with normalized Seifert manifold is given by e=(b+ i=1npiqi) and an elliptic Seifert manifold is such that e=0 and the base orbifold has positive Euler characteristic.

We say a G-action is fiber-preserving if for any fiber γ and any gG,φ(g)(γ) is some fiber of M. We use the notation Difffp(M) for the group of fiber-preserving self-diffeomorphisms of M (given some Seifert fibration). Given a fiber-preserving G-action, there is an induced action φBU:GBUDiff(BU) on the underlying space BU of the base space B.

Given a finite action φ:G Difffp(M), we define the orbit number of a fiber γ under the action to be #Orbφ(γ)=#{α|φ(g)(γ)=αforsomegG}.

If we have a manifold M, then a product structure on M is a diffeomorphism k:A×BM for some manifolds A and B. [6] If a Seifert-fibered manifold M has a product structure k:S1×FM for some surface F and k(S1×{x}) are the fibers of M for each x∈ F, then we say that k:S1×FM is a fibering product structure of M.

Given that the first homology group (equivalently the first fundamental group) of a torus is × generated by two elements represented by any two nontrivial loops that cross at a single point, we can use the meridian-longitude framing from a product structure as representatives of two generators. If we have a diffeomorphism f:T1T2 and product structures ki:S1×S1Ti, then we can express the induced map on the first homology groups by a matrix that uses bases for H1(Ti) derived from the meridian-longitude framings that arise from ki:S1×S1Ti.

We denote this matrix as a11a12a21a22k2k1.

We say that a G-action φ:GDiff(A×B) is a product action if for each g∈ G, the diffeomorphism φ(g):A×BA×B can be expressed as (φ1(g),φ2(g)) where φ1(g):AA and φ2(g):BB. Here φ1:GDiff(A) and φ2:GDiff(B) are not necessarily injections.

Given an action φ:GDiff(M) and a product structure k:A×BM, we say that φ leaves the product structure k:A×BM invariant if ψ(g)=k1φ(g)k defines a product action ψ:GDiff(A×B).

Suppose that we now have a fibering product structure k:S1×FM. We then say that each boundary torus is positively oriented if the fibers are given an arbitrary orientation and then each boundary component of k({u}×F) is oriented by taking the normal vector to the surface according the orientation of the fibers.

We consider two particular types of 3-orbifold. We define the solid torus with exceptional core V(k) to be a solid torus with an exceptional set of order k running along the core loop of the solid torus. We define the Conway ball B(k) to be a ball with exceptional set consisting of two arcs of order two joined by an arc of order k according to Figure 1 below:

Figure 1. A solid torus V(k) and Conway ball B(k)

We begin with some preliminary results that we will use in the next section regarding orientation-reversing diffeomorphisms.

Lemma 2.1. Let F be an orientable surface with boundary. Let the boundary be positively oriented according to some orientation of F and f:FF be a diffeomorphism. Then f is orientation-preserving on F if and only if f is orientation-preserving between some pair of boundary components.

Proof. In a regular neighborhood of two exchanged boundary components (they may be the same), the diffeomorphism is either a reflection or a rotation (given parameterizations of the annuli). If it is a reflection, the orientation on the boundary is reversed and and the orientation on F is reversed. If it is a rotation, the orientation on the boundary is preserved and and the orientation on F is preserved.

Corollary 2.2. Let M^ be an oriented trivially Seifert fibered 3-manifold with positively oriented boundary M^=T1Tn. Then a fiber-preserving diffeomorphism f:M^M^ is orientation-preserving if and only if f is orientation-preserving between some pair of boundary tori.

Proof. Firstly, there is a fibering product structure k:S1×FM^. Suppose that the diffeomorphism preserves the orientation of the fibers. Then the projected diffeomorphism on F must be orientation-preserving. By Lemma 2.1, this is if and only if it is orientation-preserving between some pair of boundary components. As the diffeomorphism preserves the orientation of a fiber, this is equivalent to f being orientation-preserving between some pair of boundary tori.

If now we suppose that the diffeomorphism reverses the orientation of the fibers. Then the projected diffeomorphism on F must be orientation-reversing. By Lemma 2.1, this is if and only if it is orientation-reversing between some pair of boundary components. As the diffeomorphism reverses the orientation of a fiber, this is equivalent to f being orientation-preserving between some pair of boundary tori.

### 3. Conditions for an Orientation-reversing Action

We now use the previous section to establish some results about the conditions under which an orientation-reversing action is possible.

Firstly, a condition on the order of critical fibers:

Proposition 3.1. All finite, fiber-preserving actions on an orientable Seifert 3-manifold fibering over an orientable base space with at least one critical fiber of order greater than two are orientation-preserving.

Proof. Suppose for contradiction that there exists a periodic, fiber-preserving and orientation-reversing diffeomorphism f:MM.

We begin with normalized invariants for M=(g,o1|(q1,p1),,(qn,pn),(1,b)).

We then take a regular fiber γ with #Orbf(γ)=l for some l. Then adjust the invariants to yield M=(g,o1|(q1,p1),,(qn,pn),(1,b1),,(1,bl)) where each (1,bi) refer to a fiber in Orbf(γ). Necessarily, i=1lbi=b.

We can then proceed as in [8] to yield a manifold M^ with fibering product structure kM^:S1×FM^ and a collection of solid tori X with product structure kX:S1×(D1Dn+l)X.

We can also now define a restricted map f^Diff(M^). Suppose that the filling of Ti yields a critical fiber of order greater than 2.

Suppose that f^(Ti)=Tj. It could be that i=j.

According to the given product structures (with positively oriented restrictions on the boundary) we then have the following homological diagram:

(d|Vi)*H1(Ti)H1(Vi)f^*(d|Vj1° f ^ °d|Vi)*H1(Tj)H1(Vj)(d|Vj)*

Now, f:MM is orientation-reversing and extends into the solid tori Vi,Vj, hence (d|Vj1°f^°d|Vi)*=±10a1kVjkVi or ±10a1kVjkVi. By Corollary 2.2, we must have the second case.

Then according to the framings on Vi,Vj, the fibrations are given by a (qi,yi)=(qj,yj) curve where qi=qj>2. f^ must preserve the fibration hence:

±10a1qiyi=±qiyi

But this implies that aqi+yi=yi, and so aqi=2yi. This further implies that qi divides 2 which is a contradiction.

Secondly, we establish that if the Euler class of the manifold is non-zero, then there are no orientation-reversing actions. This can be seen by noting that fiber-preserving diffeomorphisms induce isomorphisms on Euler class. In particular, orientation-preserving maps induce the identity and orientation-reversing induce the negative identity map. For more details, see Theorem 2.4 in [1]. The following result then follows straightforwardly, but we present a constructive proof to further illustrate our methods:

Proposition 3.2. All finite, fiber-preserving actions on an orientable Seifert 3-manifold fibering over an orientable base space with nonzero Euler class are orientation-preserving.

Proof. Again suppose for contradiction that there exists a periodic, fiber-preserving and orientation-reversing diffeomorphism f:MM. We proceed as in the previous proposition to yield a manifold M^ with fibering product structure kM^:S1×FM^, a collection of solid tori X with product structure kX:S1×(D1Dn+l)X, and a restricted diffeomorphism f^:M^M^.

We now consider the first homology group of M^. We have the presentation:

H1(M^)=α1,,αn+l,a1,b1,,ag,bg,t|α1αn+l=1,allcommute

Where t represents an oriented fiber and α1,,αn+l represent positively oriented loops kTi({u}×S1) on each boundary torus.

So we must have:

f^*(αi)=αj(i)±1tci

For some integer ci and some permutation jperm{1,n+l}.

Here the sign is the same for each αi. So then:

1=f^*(α1αl)=t i=1 n+lci

Hence,

i=1 n+lci=0

Case 1: There are no critical fibers. That is, n=0.

Hence the obstruction is nonzero. We then consider the diagram:

(d|Vi)*H1(Ti)H1(Vi)(f^|Ti)*(d|Vj(i)1f^|Tid|Vi)*H1(Tj(i))H1(Vj(i))(d|Vj(i))*

So now (d|Vi)*=1 bi 01kTikVi and (d|Vj(i)1f^|Tid|Vi)*=±1001kVj(i)kVi This is as the diffeomorphism extends, is fiber-preserving, and orientation-reversing as well as each Vi being trivially fibered. Here we again use Corollary 2.2.

Hence: (f^|Ti)*=±1(bi +bj(i) )01kTj(i)kTi . So then from above, ci=(bi+bj). Hence, we have:

i=1l(bi+bj(i))=2 i=1lbi=0

But by Theorem 1.1 of [7], i=1lbi is the obstruction term and by assumption is nonzero. Hence there can be no such f.

Case 2: There are critical fibers.

Let the fillings of T1,,Tn be by nontrivially fibered solid tori and the fillings of Tn+1,,Tl be by trivially fibered solid tori.

Firstly, for T1,,Tn we have the diagram:

(d|Vi)*H1(Ti)H1(Vi)(f^|Ti)*(d|Vj(i)1°f^|Ti°d|Vi)*H1(Tj(i))H1(Vj(i))(d|Vj(i))*

Now, in order for (d|Vj(i)1f^|Tid|Vi)* to extend into the solid torus, preserve a nontrivial fibration, and be orientation-reversing, according to Corollary 2.2 we must have:

(d|Vj(i)1f^|Tid|Vi)*=±1011kVj(i)kVi

As the fibration on both Vi and Vj(i) is a (-2,1) fibration by Proposition 3.1. Hence, we have:

(d|Vi)*=0112kTikVi

and

(d|Vj(i))*=0112kTikVj(i)

So that:

(f^|Ti)*=±1101kTj(i)kTi

That is, for those Vi that are nontrivially fibered, ci=∓1. Here again the sign is the same for all.

For Tn+1,,Tl we proceed as in Case 1, to yield ci=(bi+bj(i)) for i=n+1,,l.

So now,

0= i=1 n+lci= i=1n1+ i=n+1l(bi+bj(i))=n2b=±2e

This is twice the Euler class of the bundle which is nonzero. This yields our contradiction

This proposition establishes the fact that there are no orientation-reversing actions on elliptic manifolds as these have nonzero Euler class.

We apply the results of [8] in the case where the base space of the fibration on the Seifert manifold M has underlying space S2. Recall for an action φ:GDifffp(M), there is an induced action φS2:GS2Diff(S2). We first consider these possible actions.

### 4.1. Finite group actions on S2

By [11], the possible branching data of a quotient space of S2 acted on by a finite group is given by Table 10.1.1. The semidirect product ° is defined so that for H2, the 2 generator anti-commutes with each element of H. Indeed, throughout, this will be the only semidirect product used. If H happens to be abelian, we use Dih(H) instead.

The notation here is such that rotnx is a rotation of order n about the x-axis when S2 is embedded about the origin in 3, similarly with rotny,rotnz. Then refxy is a reflection in the x-y plane, and again similarly with other reflections. Lastly rotL1,rotL2,rotL3 refer to rotations about lines regarding the rotational symmetry of a tetrahedron, an octahedron, and an icosahedron when inscribed inside S2. For more details see [3]. Note that the groups may be given by different names in other sources. For example, A42 is really S4, but we write as a semidirect product for convenience.

These groups form partially ordered sets. We do not expressly show these, but they can be worked out by referring to the generators given.

Remark 4.1. By reference to the generators, it is clear is that any finite group that acts on S2 is a subgroup of a finite group that is a semidirect product of a group of orientation-preserving diffeomorphisms and a 2 generated by an orientation-reversing element. Again, the semidirect product is such that the 2 generator anti-commutes with all elements of the group of orientation-preserving diffeomorphisms.

This leads us to consider which of these will satisfy the obstruction condition in Table 1:

Remark 4.2. Note that for all actions with induced actions as above, the obstruction condition will be satisfied if the obstruction term is even, but there could actions that will not satisfy the obstruction condition if the obstruction term is odd. One such action is exhibited in [8].

### 4.2. Manifolds fibering over S2

We now prove a general result that will set up the group structure for the groups acting on manifolds fibering over an orbifold with underlying space S2. Throughout this section we assume normalized form for Seifert invariants.

Proposition 4.3. Let M=(0,o1|(q1,p1),,(qn,pn),(1,b)) and φ:GDiff+fp(M) be a finite action that satisfies the obstruction condition. Then G is isomorphic to a subgroup of (m×GS2+)2 for some m and GS2+ is the orientation-preserving subgroup of the induced action φS2:GS2Diff(S2).

Proof. As φ:GDiff+fp(M) satisfies the obstruction condition, we can restrict to the action φ^:GDiff(S1)×Diff(F).

Now, consider φ^S1(G), the projection onto Diff(S1). So then φ^S1(G) is a subgroup of Dih(m)m2 for some m.

Also, φ^F(G) the projection onto Diff(F) will be a subgroup of φ^F(G)+2 by the remark above, where φ^F+(G)+ is the orientation-preserving subgroup.

So now, φ^(G) φ ^ S1(G)× φ ^ F(G)(m2)×( φ ^ F(G)+2).

But, φ^(G) is orientation-preserving. Hence, we consider the orientation-preserving subgroup of (m2)×(φ^F(G)+2).

Note that g((m2)×(φ^F(G)°2))+ if and only if g=(g1,g2) or g=(g1z1,g2z2) for (g1,g2)m×φ^F(G)+ and z1,z2 are respective generators of the two 2 components. It therefore follows that ((m2)×(φ^F(G)+2))+=(m×φ^F(G)+)2 where the 2 is generated by z=(z1,z2), and the semidirect product is defined by z(g1,g2)z1=(g11,g21).

Now, φ^F(G)+GS2+ and φ^(G)G so that G(m×GS2+)2.

This result essentially states that we need only check that the obstruction condition is satisfied and calculate the possible orientation-preserving subgroup of the induced action φS2:GS2Diff(S2). This we can do by reference to the Table 1 and 2.

Orbit numbers of finite group actions on S2.

NumberUnderlying SpaceGDataGenerators
1S2Trivial()id
2n(n,n)rotnz
3Dih(2n)(2,2,2n)rot2nz,rot2y
4Dih(2n+1)(2,2,2n+1)rot2n+1z,rot2y
5A4(2,3,3)rot2z,rot3L1
6S4(2,3,4)rot2z,rot3L2
7A5(2,3,5)rot2z,rot3L3
8D2(;)refxy
92n×2(2n;)rot2nz,refxy
104n+2(2n+1;)rot2n+1zrefxy
11Dih(n)(;n,n)rotnz,refyz
12Dih(2n)2(;2,2,2n)rot2nz,rot2y,refyz
13Dih(2n+1)2(;2,2,2n+1)rot2n+1z,rot2y,refyz
14A4°2(;2,3,3)rot2z,rot3L1,refyz
15S4×2(;2,3,4)rot2z,rot3L2,refxy
16A5×2(;2,3,5)rot2z,rot3L3,refxy
17Dih(2n)2(2;2n)rot2nz,rot2y,refxz
18Dih(2n+1)2(2;2n+1)rot2n+1z,rot2y,refxz
19A4×2(3;2)rot2z,rot3L1,refxy
2022()rot2zrefxy
212n(n)rot2nzrefxy

Orbit numbers of finite group actions on S2..

NumberGOrbit Numbers of non-regular pointsLCM|G|/LCMOC Satisfied?
1Trivial111all b
2n1, 1n1all b
3Dih(2n)2, 2n, 2n2n2b even
4Dih(2n+1)2, 2n + 1, 2n + 14n + 21all b
5A44, 4, 662b even
6S46, 8, 12122b even
7A56, 10, 15301all b
82121all b
92n×22, n2n2b even
104n+22, 2n + 14n + 21all b
11Dih(n)1, 12n1all b
12Dih(2n)22, 2n, 2n4n2b even
13Dih(2n+1)22, 2n + 1, 2n + 18n + 41all b
14A424, 4, 6122b even
15S4×26, 8, 12242b even
16A5×26, 10, 15602b even
17Dih(2n)22, 4n4n2b even
18Dih(2n+1)22, 4n + 24n + 22b even
19A4×26, 8122b even
20212b even
212n2n2b even

We now proceed to consider the individual cases for the number of critical fibers. For each proof the construction set out in [8] provides the converse.

### 4.3. One critical fiber

We now consider the case where there is only one critical fiber.

Corollary 4.4. Let M=(0,o1|(q,p),(1,b)). There exists a finite action φ:GDiff+fp(M) if and only if G is isomorphic to a subgroup of Dih(m×n) for some m,n.

Proof. Note that the induced action φS2:GS2Diff(S2) must fix a point. By Table 1 and 2, we can assume that this is of the form of action 11. This action satisfies the obstruction condition for any b. Hence, GS2+n for some n. Then by Proposition 4.1, G is isomorphic to a subgroup of (m×n)2=Dih(m×n).

### 4.4. Two critical fibers

Now consider two critical fibers. Firstly, when the respective normalized fillings are not equal.

Corollary 4.5. Let M=(0,o1|(q1,p1),(q2,p2),(1,b)) with (q1,p1)(q2,p2). There exists a finite action φ:GDiff+fp(M) if and only if G is isomorphic to a subgroup of Dih(m×n) for some m,n.

Proof. Note that the induced action φS2:GS2Diff(S2) must fix two points. By Table 1 and 2, we again assume the form of action 11. This action satisfies the obstruction condition for any b. Hence, GS2+n for some n. Then by Proposition 4.1, G is isomorphic to a subgroup of (m×n)2=Dih(m×n).

Now consider when the fillings of the two critical fibers are equal.

Corollary 4.6. Let M=(0,o1|(q,p),(q,p),(1,b)) with b even. There exists a finite action φ:GDiff+fp(M) if and only if G is isomorphic to a subgroup of (m×Dih(n))2 for some m,n.

Proof. We assume that the induced action φS2:GS2Diff(S2) exchanges two points referring to the critical fibers. Otherwise, we apply Corollary 4.3. Given that two points are exchanged, we consult the Tables to note that we can assume that φS2 is in the form of actions 12/13 or 17/18. The obstruction condition will be satisfied for each of these as we assume that b is even. Then in either case, GS2+Dih(n) and by Proposition 4.1, G is isomorphic to a subgroup of (m×GS2+)2(m×Dih(n))2.

Remark 4.7. Note that M=(0,o1|(q,p),(q,p),(1,b)) with b even is simply S2×S1. M=(0,o1|(q,p),(q,p),(1,b)) with b odd is a Lens space and so as an exception to our results, we refer the reader to [4] for a classification of finite actions on these manifolds.

### 4.5. Three critical fibers

We now move on to having three critical fibers and break into the three possible scenarios: that they all have different fillings; that two have the same fillings; and that they all have the same filling.

Corollary 4.8. Let M=(0,o1|(q1,p1),(q2,p2),(q3,p3),(1,b)) with (qi,pi) all different. There exists a finite action φ:GDiff+fp(M) if and only if G is isomorphic to a subgroup of Dih(m) for some m.

Proof. Note that the induced action φS2:GS2Diff(S2) must fix three points. By Table 1 and 2, the only such induced action is the trivial action 1, that is GS2 is the trivial group. This action trivially satisfies the obstruction condition for any b.

Hence, by Proposition 4.1, G is a subgroup of (m×GS2+)2m2=Dih(m).

Corollary 4.9. Let M=(0,o1|(q1,p1),(q,p),(q,p),(1,b)) with (q1,p1)(q,p). There exists a finite action φ:GDiff+fp(M) if and only if G is isomorphic to a subgroup of Dih(m×2) for some m.

Proof. Note that the induced action φS2:GS2Diff(S2) must fix a point and at most exchange two others. By Table 1 and 2, the only such action is of the form of action 11 with n=2. This action satisfies the obstruction condition for any b. So GS2+2. Hence by Proposition 4.1, G is a subgroup of (m×2)2=Dih(m×2).

Corollary 4.10. Let M=(0,o1|(q,p),(q,p),(q,p),(1,b)). There exists a finite action φ:GDiff+fp(M) if and only if G is isomorphic to a subgroup of (m×Dih(3))2.

Proof. We assume that φS2:GS2Diff(S2) exchanges three points, else apply Corollary 4.5. or Corollary 4.6. So now by Table 1 and 2 we can assume that φS2:GS2Diff(S2) is of the form of action 13 with n=1. This action satisfies the obstruction condition for any b and GS2+Dih(3). Hence, by Proposition 4.1, G is isomorphic to a subgroup of (m×Dih(3))2.

### 4.6. No critical fibers

In the case where there are no critical fibers, we note that there are no restrictions on φS2:GS2Diff(S2). Hence we cannot guarantee that the obstruction condition will be satisfied unless b is even. In such a case the group will be a subgroup of a group of the form (m×H)2 where H is a group from the list of groups that act orientation-preservingly on S2. Note, however that once again, these manifolds are Lens spaces of the form L(b,1) and we again refer the reader to [4].

### 4.7. Manifolds fibering over ℙ2

We here apply the results of [9] to yield the following result:

Corollary 4.11. Let M=(1,n2|(q,p),(1,b)). There exists a finite action φ:GDiff+fp(M) if and only if G is isomorphic to a subgroup of 2×Dih(n) for some n.

Proof. Let M˜=(0,o1|(q,p),(q,p),(1,2b)) be the orientable base space double cover of M. According to [9], we consider a corresponding finite action φ˜:GDiff+fop(M˜) that commutes with the covering translation τ:M˜M˜.

Now note that the induced action φ˜S2:GS2Diff(S2) can exchange two points but must be orientation-preserving as φ˜:GDiff+fop(M˜) is fiber-orientation-preserving. We can then assume that φS2 is in the form of actions 3/4. Then, GS2Dih(n) for some n. Again by Table 2, it will satisfy the obstruction condition as 2b is even.

So now apply the results of [9] to note that there is a restricted action φ˜^:GDiff( M˜^) and product structure k:S1×FM˜^ (F is in fact an annulus) so such that (k1φ˜^(g)k)(u,x)=(ϵ(g)u, φ ˜ ^2(g)(x)) for ϵ(g)=±1. We then note that φ˜^2(G)GS2Dih(n). Hence, φ˜^(G)G is isomorphic to a subgroup of φ˜^1(G)× φ˜^2(G)2×Dih(n). Again, our construction set out in [8] provides the converse.

Recall that elliptic 3-manifolds are Seifert manifolds where χorb(B)>0 and the Euler class of the Seifert bundle is nonzero. [10] By [11], the orbifolds without boundary that have positive orbifold Euler characteristic are:

S2,S2(q1),S2(q1,q2),S2(2,2,q),2(q),S2(2,3,3),S2(2,3,4),S2(2,3,5)

We note that by Proposition 3.2, all fiber-preserving actions on elliptic manifolds are orientation-preserving as the Euler class must be nonzero. Hence we can break down the possible base spaces and apply the results of the previous sections. In each subsection, suppose that we have a finite action φ:GDifffp(M).

### 5.1. Base space S2

These manifolds are lens spaces fibered without critical fibers. By [4], these are of the form L(p,q) where q=±1(modp).

By Remark 2, we can only certainly work with even obstruction condition and in which case the lens space is constructed by two (b,1) fillings of S1×A. We then calculate:

1b0110011b01=12b01

Thus we have the lens spaces L(2b,1) for nonzero b.

So now we apply Section 4 to state that the group G will be a subgroup of a group of the form (m×H)2 where H is a group from the list of groups that act orientation-preservingly on S2 and m.

### 5.2. Base space S2(q)

These manifolds are again lens spaces, but fibered with one critical fiber. All lens spaces can be given such a fibration except those of the form L(p,q) where q=±1(modp) mentioned above. This follows from fibering one solid torus side of the Heegaard torus trivially and inducing a fibration on the other side.

We can now apply Corollary 4.2. to find that the group G is a subgroup of Dih(m×n) for m,n.

### 5.3. Base space S2(q1,q2)

So M=(0,o1|(p1,q1),(p2,q2),(1,b)). Once again, these manifolds are lens spaces, but now fibered with two critical fibers. All lens spaces can be fibered in this way.

We first consider (p1,q1)(p2,q2). Then G is a subgroup of Dih(m×n) for m,n by Corollary 4.3.

If (p1,q1)=(p2,q2) then our results only apply in the case where b is even, in which case the manifold is not elliptic by Remark 3.

### 5.4. Base space S2(2,2,q)

So M=(0,o1|(q,p),(2,1),(2,1),(1,b)). These manifolds are now prism manifolds fibered longitudinally. We split into the two cases:

Case 1: (q,p)=(2,1)

In this case we apply Corollary 4.7 to yield that G is a subgroup of (m×Dih(3))2 for m.

Case 2: (q,p)(2,1)

In this case we instead apply Corollary 4.6 to yield that G is a subgroup of Dih(m×2) for m.

### 5.5. Base space ℙ2(p)

These manifolds are again prism manifolds but fibered meridianally.

We apply Corollary 4.8 to yield that the group G is a subgroup of 2×Dih(n) for some n.

### 5.6. Base space S2(2,3,3)

In this case, M=(0,o1|(2,1),(3,p1),(3,p2),(1,b)) for p1=1,2 and p2=1,2. We hence break into the two possible cases:

Case 1: p1=p2

In this case we apply Corollary 4.6 to yield that G is a subgroup of (m×Dih(3))2 for m.

Case 2: p1p2

In this case we instead apply Corollary 4.5 to yield that G is a subgroup of Dih(m) for some m.

### 5.7. Base space S2(2,3,4) and S2(2,3,5)

In both of these cases M=(0,o1|(2,1),(3,p1),(q2,p2),(1,b)) for q22,3. Hence we apply apply Corollary 4.7 to yield that G is a subgroup of Dih(m) for some m.

We now consider the quotient spaces under these constructed actions.

### 6.1. General outline of construction

We first note that an orientation and fiber-preserving action on a fibered torus will have quotient type either a torus or a S2(2,2,2,2). This follows from [10] and the fact that S2(2,3,6), S2(3,3,3), and S2(2,4,4) cannot be Seifert fibered. We then consider the quotient of M^S1×F under a product action and the stabilizers of the boundary tori. Here, F will be a disc with holes. There will be then glued in either a solid torus with exceptional core or a Conway ball.

The main part is to establish what form the quotients of M^ and each Vi will be, and then how the gluing maps look under the projection.

Formally, for a representation M=(0,o2|(q1,p1),,(qn,pn)), we take an action φ:GDiff+fp(M) that restricts to an action φ^:GDiff+fp(M^) which leaves some fibering product structure k:S1×FM^ invariant and which extends over the fillings of a collection of fibered solid tori X=V1Vn. We denote φX:GDiff+fp(X) to be the restricted action on X.

We then let p^:M^M^/φ^ and \$pX:XX/φX be the quotient maps.

We then have the diagram:

d|ViTiVip^|Ti(pX)|ViTi/Stab φ ^ (Ti)Vi/Stab φX(Vi)d|Vi/Stab φX(Vi)

We hence need to find the following:

M^/φ^

Vi/StabφX(Vi)

d|Vi/StabφX(Vi)

### 6.2. M^/φ^

We first consider actions constructed via the method of [8] that are fiber-orientation-preserving. For this section we consider F in the more general setting as any orientable surface with boundary.

Lemma 6.1. Let φ^S1×F:GDiff+(S1)× Diff+(F) be a finite group action such that no element leaves an isolated fiber invariant. Then (S1×F)/φ^S1×F is a trivially fibered Seifert 3-manifold with fibering product structure S1×(F/(φ^S1×F)F).

Proof. It is clear that (S1×F)/φ^S1×F is a trivially fibered Seifert 3-manifold. It remains to show that it has the fibering product structure S1×(F/(φ^S1×F)F). We examine the diagram:

projFS1×FFpφS1×F(S1×F)/φ^S1×F=S1×F Fproj F

Now, pφS1×F:S1×FS1×F can be chosen so that pφS1×F(u,x)=(p1(u,x),p2(x)).

So now, we have projF(p(u,x))=p2(x) and p2 is the covering map for (φ^S1×F)F. Hence F=F/(φ^S1×F)F.

Remark 6.2. As φ^S1×F:GDiff+(S1)×Diff+(F) will be extended over some fillings, if it does happen to leave some isolated fibers invariant, then we can simply drill out these fibers and restrict to φ^ S1 ×F:GDiff+(S1)×Diff+(F) and then consider the resultant torus boundaries to be filled according to a (1,0) filling. Therefore, for our purposes, we can without loss of generality assume that our action φ^S1×F:GDiff+(S1)×Diff+(F) does not leave any isolated fibers invariant and so the previous lemma holds.

We now allow the fibers to be reversed.

Lemma 6.3. Let φ^S1×F:GDiff(S1)×Diff(F) be a finite, orientation-preserving group action so that (φ^S1×F)+:G+Diff+(S1)× Diff+(F) is such that no element leaves an isolated fiber invariant. Then any element that reverses the orientation on both components will induce some product involution f=(f1,f2) of S1×(F/((φ^S1×F)+F) that also reverses the orientation on both components. Then (S1×F)/φ^S1×F is found by taking I×(F/(φ^S1×F)+F) and identifying (i,x) with (i,f2(x)) for i=0,1 and leaving exceptional sets of order 2 as properly embedded arcs or circles according to the fixed point set of f2.

Proof. If g is an element of G, so that φ^S1×F(g) reverses the orientation on both components, then we have some f:S1×(F/((φ^S1 ×F)+F)S1×(F/((φ^S1 ×F)+F) so that p(φ^S1×F)+φ(g)=fp(φ^S1×F)+.

To see that f is an involution requires only the observation that G+ is an index two subgroup of G.

To see that it is a product, we note that if it does not preserve the product structure S1×(F/((φ^S1×F)+F), then g cannot preserve the product structure S1×F. Hence f is a product reversing the orientation on both components.

The result therefore follows.

Corollary 6.4. Let F be a genus 0 surface with boundary. Let φ^S1×F:GDiff(S1)×Diff(F) be a finite, orientation-preserving group action so that (φ^S1×F)+:G+Diff+(S1)× Diff+(F) is such that no element leaves an isolated fiber invariant. Then any element that reverses the orientation on both components will induce some product involution f=(f1,f2) of S1×(F/((φ^S1×F)+F) that also reverses the orientation on both components. Then (S1×F)/φ^S1×F is a ball B less a disjoint collection of balls and solid tori in the interior of B with exceptional sets of order 2 as properly embedded arcs or circles according to the fixed point set of f2.

Proof. From Lemma 6.2, we note that S1×(F/((φ^S1×F)+F) is simply another genus 0 surface with boundary cross I. It then follows that the boundary identification will fold S1×(F/((φ^S1×F)+F) up to a ball with removed interior balls and solid tori and exceptional sets of order 2 as properly embedded arcs or circles according to the fixed point set of f2.

Example 6.5. Consider F×S1 where F is a disc with three discs removed. Then take a Dih(2)-action on F×S1 generated by g1: an order 2 rotation on F fixing two of the boundary components and exchanging the other two with no rotation in the S1 component, and g2: the antipodal map on F and a reflection on S1. Then (F×S1)/g1 is an annulus cross S1. g2 induces an involution on this space consisting of the antipodal map on the annulus and a reflection in the S1 component. This quotients to a ball with no interior balls removed and no exceptional set.

Example 6.6. Now consider again F×S1 where F is a disc with three discs removed. This time take a Dih(2)-action on F×S1 generated by g1: an order 2 rotation in the S1 component and the identity on F and g2: a reflection on F that leaves two boundary components invariant, exchanging the other two, and a reflection in the S1 component.

Here (F×S1)/g1 is homeomorphic to F×S1 and g2 induces the same map on the quotient space. This then quotients to the following space:

Figure 2. Quotient under the action of Example 6.6

### 6.3. Vi/StabφX(Vi)

We begin by assuming that the action preserves the orientation of the fibers and note that the filling is of a fibered solid torus where the critical fiber is also an exceptional set. By [5] the action of the stabilizer on Vi will be a m×l-action where m divides l with generators φ(g1)(u,v)=(e2aπiu,e2bπiv) and φ(g2)(u,v)=(e2cπiu,e2dπiv) where a=a1a2,b=b1b2,c=c1c2,d=d1d2 are rational numbers. The quotient will then be a solid torus with an exceptional core of order k, where m|k|l. This follows again from [5].

Lemma 6.7. The quotient of a solid torus under a m-action with generator φ(g1)(u,v)=(e2aπiu,e2bπiv) is a solid torus with exceptional core of order:

k=b2gcd(a2,b2)

Proof So φ(g1)a2(u,v)=(u,e2a2bπiv) and φ(g1)a2 is an order lcm(a2,b2)a2=b2gcd(a2,b2) element that fixes the core. The quotient space then has an exceptional core of order k=b2gcd(a2,b2).

Lemma 6.8. The quotient of a solid torus under a m×l-action where m divides l with generators φ(g1)(u,v)=(e2aπiu,e2bπiv) and φ(g2)(u,v)=(e2cπiu,e2dπiv) is a solid torus with exceptional core of order:

k=b2d2gcd(a2,c2)gcd(d2gcd(a2,c2)gcd(a2,b2),a2b1c1d2z+b2c2d1)

Where z is such that a1z+1a2.

Proof We begin by noting that the quotient of the solid torus under the normal group generated by g1 is V(k') where k=b2gcd(a2,b2). We then claim the projection under the restricted action of g1, is pg1(u,v)=(ua2,uz vk ) where z=a2b1gcd(a2,b2)z for z such that a1z+a2y=1.

To prove this, we first note that:

pg1(e2aπiu,e2bπiv)=((e2aπiu)a2,(e2aπiu)z (e2bπiv)k )=(ua2,e2πi(a z +b k )uz vk )

So then:

az+bk=b1gcd(a2,b2)a1z+b1gcd(a2,b2)  =a1b1gcd(a2,b2)(1a2y)+b1gcd(a2,b2)  =a2ygcd(a2,b2)

So that:

pg1(e2aπiu,e2bπiv)=(ua2,uz vk )

Also, if we solve pg1(u,v)=(1,1), we yield ua2=1 and u zv k=1. So that there are a2k=a2b2gcd(a2,b2) possible solutions. This is the order of g1.

Now, there is an induced map φ(g2)¯ such that φ(g2)¯pg1=pg1φ(g2).

We compute:

φ(g2)¯(ua2,uz vk )=(φ(g2)¯pg1)(u,v)      =(pg1φ(g2))(u,v)      =pg1(e2cπiu,e2dπiv)      =(e2a2cπiua2,e2πi(c z +d k )uz vk )

It follows that φ(g2)¯(u,v)=(e2πa2ciu,e2π(cz+dk)iv).

So then φ(g2)¯c2gcd(a2,c2)(u,v)=(u,e2πc2gcd(a2,c2)(cz+dk)iv).

Now φ(g2)¯c2gcd(a2,c2) is an element that fixes the core of V(k') and is of order the denominator of c2gcd(a2,c2)(cz+dk) when in reduced form. We calculate:

c2gcd(a2,c2)(cz+dk)=c1d2z+d1c2kd2gcd(a2,c2)

Hence φ(g2)¯c2gcd(a2,c2) has order d2gcd(a2,c2)gcd(d2gcd(a2,c2),c1d2z+c2d1k).

So finally, the order of the exceptional core of quotient space of the whole action is:

k=kd2gcd(a2,c2)gcd(d2gcd(a2,c2),c1d2z+c2d1k)=b2gcd(a2,b2)d2gcd(a2,c2)gcd(d2gcd(a2,c2),c1d2z+c2d1k)=b2d2gcd(a2,c2)gcd(d2gcd(a2,b2)gcd(a2,c2),c1d2a2b1z+c2d1b2)

We now consider an action of the stabilizer that reverses the orientation of the fibers. By [5] the action will be a Dih(m×l)-action where m divides l with generators φ(g1)(u,v)=(e2aπiu,e2bπiv), φ(g2)(u,v)=(e2cπiu,e2dπiv), and φ(g3)(u,v)=(u1,v1). We here note that similar to the proof of Lemma 6.2. we can consider the quotient of the m×l-action and then the induced involution upon it. The following lemma then holds:

Lemma 6.9. The quotient of a solid torus under a Dih(m×l)-action where m divides l with generators φ(g1)(u,v)=(e2aπiu,e2bπiv), φ(g2)(u,v)=(e2cπiu,e2dπiv), and φ(g3)(u,v)=(u1,v1) is a Conway ball with exceptional core of order:

k=b2d2gcd(a2,c2)gcd(d2gcd(a2,c2)gcd(a2,b2),a2b1c1d2z+b2c2d1)

Where z is such that a1z+1a2.

Proof. This follows from considering an orientation-preserving involution on V(k) the quotient of the m×l-action generated by φ(g1)(u,v)=(e2aπiu,e2bπiv) and φ(g2)(u,v)=(e2cπiu,e2dπiv).

### 6.4. d′|∂Vi/StabφX(∂Vi)

We again begin by assuming that the action preserves the orientation of the fibers. So now M^/φ^ has a collection of boundary tori. These will be filled by solid tori with a possible exceptional core. It remains to show how the gluing map from the boundary of the solid tori into M^/φ^ will look.

By using product structures k:S1×FM^ and k:S1×F/φ^FM^/φ^ that restrict to positively oriented product structures kTi:S1×S1Ti and kTi:S1×S1Ti, we can consider:

H1(M^)=t,x1,,xs,a1,b1,,ag,bg|x1xs=1,allcommute
H1(M^/φ^)=t, x 1,, x s , a 1, b 1,, a g , b g | x 1 x s =1,allcommute

Here note that we again allow F to be more generally any orientable surface with boundary as it presents no extra complication to the calculations. t,t' represent a fiber of M^ and M^/φ^ respectively; x1,,xs represent the boundary loops of k({1}×F); and similarly x1,,xs represent the boundary loops of k({1}×F/φ^F).

We then we have that:

(pφ^)*(t)=ta,(pφ^)*(xi)=xj(i)mj(i)tlj(i)

Here j:{1,,s}{1,,s} is a surjection.

Note that this is well-defined as if j(i1)=j(i2) then (pφ^)*(xi1)=(pφ^)*(φ(g)*(xi2)) for some gG.

Now, 1=(pφ^)*(x1xs)=x1m1#j1(1)x sm s #j1(s )tl1#j1(1)++l s #j1(s ).

So that 0=l1#j1(1)++ls #j1(s) and necessarily mj(i1)#j1(i1)=mj(i2)#j1(i2) for any i1,i2{1,,s}.

Now, for any torus (either on the boundary of M^ or on the boundary of one of the solid tori) we have that Stab(T)m or m×l where m divides l. [5]

So we consider the diagram:

d|ViTiVipφ^|Tip φX|Vi T i=Ti/Stab(Ti) V id| V i

We begin with the cyclic case. We can then choose the product structure such that (kTi1φ^(g)kTi)(u,v)=(e2aπiu,e2bπiv) for some a,b and g a generator of Stab(Ti). Letting a=a1a2,b=b1b2 be fully reduced, this has order lcm(a2,b2).

Lemma 6.10. If Stab(Ti)m, then (kTi1pφ^kTi)(u,v)=(ulcm(a2,b2)b2vlj(i),vb2) where lj(i) is an integer such that a1lcm(a2,b2)a2b2+lj(i)b1b2=a1gcd(a2,b2)+lj(i)b1b2 is integer valued.

Proof. The projection pφ^|Ti will need to send a fiber to a fiber, hence (kTi1pφ^kTi)(u,v)=(urvs,vt). But now:

(urvs,vt)=(kTi1pφ^kTi)(e2aπiu,e2bπiv))=(e2(ar+bs)πiurvs,e2btπivt)

So then take t=b2.

Now consider (kTi1pφ^kTi)(u,v)=(urvs,vt)=(1,1). This should have lcm(a2,b2) solutions. So ur=1 has lcm(a2,b2)b2 solutions and r=lcm(a2,b2)b2.

Now, ar+bs=alcm(a2,b2)b2+bs. So let s=lj(i) be a solution to this. This exists as gcd(a2,b2) divides b2 by [2]. There are however an infinite number of choices depending upon the product structure kTi:S1×S1Ti.

The projection pφX|Vi will need to extend over the entire solid torus and so will need to send a meridian to a meridian. We can again choose the product structure such that (kVi1φX(g)kVi)(u,v)=(e2aπiu,e2bπiv). Hence pφX|Vi will similarly give (kVi1pφ^kVi)(u,v)(ua2,uzvlcm(a2,b2)a2). Here the choice of z will not affect the filling but depends upon the product structure kVi:S1×S1Vi.

We proceed with Stab(Ti)m×l where m divides l. Then (kTi1φ^(g1)kTi)(u,v)=(e2aπiu,e2bπiv) and (kTi1φ^(g1)kTi)(u,v)=(e2cπiu,e2dπiv) for some a,b,c,d and g1,g2 generators of Stab(Ti). Here m=lcm(a2,b2) and l=lcm(c2,d2).

Lemma 6.11. If Stab(Ti)m×l, then (kTi1pφ^kTi)(u,v)=(umllcm(b2,d2)vlj(i),vlcm(b2,d2)), where lcm(b2,d2) divides lj(i).

Proof. The projection pφ^|Ti will again need to send a fiber to a fiber. Hence it will again be of the form (kTi1pφ^kTi)(u,v)=(urvs,vt). Now:

(urvs,vt)=(kTi1pφ^kTi)(e2aπiu,e2bπiv))=(e2(ar+bs)πiurvs,e2btπivt)
(urvs,vt)=(kTi1pφ^kTi)(e2cπiu,e2dπiv))=(e2(cr+ds)πiurvs,e2dtπivt)

Hence we take t=lcm(b2,d2).

Now consider (kTi1pφ^kTi)(u,v)=(urvs,vt)=(1,1). This should have ml solutions. So ur=1 has mllcm(b2,d2) solutions and r=mllcm(b2,d2).

Finally, s is such that amllcm(b2,d2)+bs and cmllcm(b2,d2)+ds. We now calculate:

lcm(a2,b2)lcm(c2,d2)=lcm(lcm(a2,b2),lcm(c2,d2))gcd(lcm(a2,b2),lcm(c2,d2))=lcm(lcm(a2,b2),lcm(c2,d2))lcm(a2,b2)

So that:

lcm(c2,d2)=lcm(lcm(a2,b2),lcm(c2,d2))=lcm(lcm(a2,c2),lcm(b2,d2))

Then:

amllcm(b2,d2)=a1lcm(a2,b2)lcm(c2,d2)a2lcm(b2,d2)    =a1lcm(a2,b2)lcm(lcm(a2,c2),lcm(b2,d2))a2lcm(b2,d2)

Similarly, cmllcm(b2,d2). So then we require that bs,ds. Hence, b2 and d2 must divide s and we take s=lj(i) to be a multiple of lcm(b2,d2).

The projection pφX|Vi will need to extend over the entire solid torus and so will need to send a meridian to a meridian. We can again choose the product structure such that (kVi1φX(g1)kVi)(u,v)=(e2aπiu,e2bπiv) and (kVi1φX(g1)kVi)(u,v)=(e2cπiu,e2dπiv). Hence pφX|Vi will similarly give (kVi1pφ^kVi)(u,v)=(ulcm(a2,c2),uzvmllcm(a2,c2)). Here the choice of z will not again affect the filling but depends upon the product structure kVi:S1×S1Vi.

So now we have from above that 0=l1#j1(1)++ls #j1(s). Hence we have the degree of freedom to choose c1,,cs1 (according to the conditions), but then cs will be uniquely determined.

Each filling d| V i will now be determined be solving:

(pφ^| Ti)*(d|Vi)*=(d| V i )*(p φX|Vi)*

Example 6.12. We consider a Dih(6×12)-action on the lens space M=(0,o1|(3,2),(1,5)) constructed by:

f1:S1×AS1×A,f1(u,ρv)=(e2πi6u,ρe2πi3v)
f2:S1×AS1×A,f1(u,ρv)=(u,ρe2πi12v)
f2:S1×AS1×A,f1(u,ρv)=(u1,ρv1)

Here we parameterize A={ρv|1ρ2,vS1}. Note that according to the product structure S1×A one boundary torus is positively oriented and the other negatively depending on the orientation on the fiber. We take S1×S1 to be positively oriented and S1×2S1 to be negatively oriented.

We calculate first (S1×A)/Dih(6×12). This will be I×A where (0,ρv) is identified with (0,ρv1) and (0,ρv) is identified with (0,ρv1) with four arcs of order 2. It will be S2×S1 with four properly embedded arcs looking as shown in Figure 3:

Figure 3. Quotient space (S1×A)/Dih(6×12)

Next we compute the orders of the exceptional sets of the two Conway balls that fill the two boundary components. We first calculate the generators of the induced action on the solid tori V1 and V2 that correspond to the fillings (3,2) and (1,5).

Firstly, for V1, we compute:

(d|V11f1d|V1)(u,v)=(d|V11f1)(u1v2,u1v3)=d|V11(e2πi6u1v2,e2πi3u1v3)=(e2πi(36+23)u,e2πi(16+13)v)=(e2πi6u,e2πi6v)(d|V11f2d|V1)(u,v)=(d|V11f2)(u1v2,u1v3)=d|V11(u1v2,e2πi12u1v3)=(e2πi(212)u,e2πi(312)v)=(e2πi6u,e2πi4v)

So then by Lemma 6.6, the exceptional set will have order:

k=(6)(12)gcd(6,6)gcd(12gcd(6,6)gcd(6,6),(6)(1)(1)(12)z+(6)(6)(11))=432gcd(432,72z+396)

Here z is such that z+16. So take z=-1 and then k=432gcd(432,324)=432108=4.

Secondly, for V2, we compute:

(d|V21f1d|V2)(u,v)=(d|V21f1)(u1v5,v)=d|V21(e2πi6u1v5,e2πi3v)=(e2πi(1653)u,e2πi3v)=(e2πi6u,e4πi3v)(d|V21f2d|V2)(u,v)=(d|V21f2)(u1v5,v)=d|V21(u1v5,e2πi12v)=(e2πi(512)u,e2πi12v)=(e14πi12u,e10πi12v)

So then again using Lemma 6.6, the exceptional set will have order:

k=(3)(12)gcd(2,12)gcd(12gcd(2,12)gcd(2,3),(2)(2)(7)(12)z+(3)(12)(5))=72gcd(24,288z+180)

Here z is such that z+12. So take z=-1 and then:

k=72gcd(24,108)=7212=6

We now compute the projection maps. By section 6.1, the projection map from both S1×S1 and S1×2S1 will have the matrix:

mllcm(b2,d2)c0lcm(b2,d2)=6c012

Here lcm(b2,d2)=12 divides c, so we take c=12.

The projection map from V1 will have matrix:

lcm(a2,c2)0zmllcm(a2,c2)=60612

The projection map from V2 will have matrix:

lcm(a2,c2)0zmllcm(a2,c2)=12066

We now calculate the projected filling of S1×S1 with V1 by solving:

(pφ^|S1×S1)*(d|V1)*=(d| V 1)*(p φX|V1)*
6120121213=xpyq60612

This yields:

18481236=6x+6p12p6y+6q12q

So then p=4,q=3,x=7, and y'=-5.

We now calculate the projected filling of S1×2S1 with V2 by solving: (pφ^|S1×2S1)*(d|V2)*=(d| V 2)*(p φX|V2)*

6120121501=xpyq12066

This yields:

642012=12x+6p6p12y+6q6q

So then p=7,q=2,x=4, and y'=-1.

This fully characterizes the quotient space. We visualize in Figure 4:

Figure 4. Full quotient space

In this paper we have studied the group actions on Seifert fibered elliptic manifolds using the results of [8] and [9]. We have extended the results of those papers by considering when an orientation-reversing action is possible and shown this can only happen if there are no critical fibers of order greater than 2 and the Euler class is non-zero. These results allowed us to consider the possible base spaces of the Seifert manifolds and determine what the possible group actions are. As future work, Seifert manifolds that do admit orientation-reversing actions could be considered as well as a construction of such an action.

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