Kyungpook Mathematical Journal 2022; 62(2): 363-388
Published online June 30, 2022
Copyright © Kyungpook Mathematical Journal.
Finite, Fiber-preserving Group Actions on Elliptic 3-manifolds
Department of Mathematics, St. Martin’s University Lacey, WA 98503, United States
e-mail : email@example.com
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
The main results in those papers established that firstly:
Theorem 1.1. Let
for a collection of fibers
Where an extended product action is intuitively a product action on an orientable surface with boundary cross
Secondly, it was shown that:
Corollary 1.2. Suppose that
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
A Seifert bundle is a Seifert manifold
We use the normalized notation
The Euler class of a Seifert manifold with normalized Seifert manifold is given by
We say a
Given a finite action
If we have a manifold
Given that the first homology group (equivalently the first fundamental group) of a torus is
We denote this matrix as
We say that a
Given an action
Suppose that we now have a fibering product structure
We consider two particular types of 3-orbifold. We define the solid torus with exceptional core
Figure 1. A solid torus
V(k)and Conway ball B(k)
2. Preliminary Results
We begin with some preliminary results that we will use in the next section regarding orientation-reversing diffeomorphisms.
Lemma 2.1. Let
Corollary 2.2. Let
If now we suppose that the diffeomorphism reverses the orientation of the fibers. Then the projected diffeomorphism on
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.
We begin with normalized invariants for
We then take a regular fiber γ with
We can then proceed as in  to yield a manifold
We can also now define a restricted map
According to the given product structures (with positively oriented restrictions on the boundary) we then have the following homological diagram:
Then according to the framings on
But this implies that
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 . 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.
We now consider the first homology group of
So we must have:
For some integer
Here the sign is the same for each
Hence the obstruction is nonzero. We then consider the diagram:
But by Theorem 1.1 of ,
Let the fillings of
Now, in order for
As the fibration on both
That is, for those
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.
4. Manifolds Fibering over
We apply the results of  in the case where the base space of the fibration on the Seifert manifold
4.1. Finite group actions on
By , the possible branching data of a quotient space of
The notation here is such that
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
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 .
4.2. Manifolds fibering over
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
Proposition 4.3. Let
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
We now proceed to consider the individual cases for the number of critical fibers. For each proof the construction set out in  provides the converse.
4.3. One critical fiber
We now consider the case where there is only one critical fiber.
Corollary 4.4. Let
4.4. Two critical fibers
Now consider two critical fibers. Firstly, when the respective normalized fillings are not equal.
Corollary 4.5. Let
Now consider when the fillings of the two critical fibers are equal.
Corollary 4.6. Let
Remark 4.7. Note that
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
Hence, by Proposition 4.1,
Corollary 4.9. Let
Corollary 4.10. Let
4.6. No critical fibers
In the case where there are no critical fibers, we note that there are no restrictions on
4.7. Manifolds fibering over
We here apply the results of  to yield the following result:
Corollary 4.11. Let
Now note that the induced action
So now apply the results of  to note that there is a restricted action
5. Elliptic 3-manifolds
Recall that elliptic 3-manifolds are Seifert manifolds where
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
5.1. Base space
These manifolds are lens spaces fibered without critical fibers. By , these are of the form
By Remark 2, we can only certainly work with even obstruction condition and in which case the lens space is constructed by two (
Thus we have the lens spaces
So now we apply Section 4 to state that the group
5.2. Base space
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
We can now apply Corollary 4.2. to find that the group
5.3. Base space
We first consider
5.4. Base space
In this case we apply Corollary 4.7 to yield that
In this case we instead apply Corollary 4.6 to yield that
5.5. Base space
These manifolds are again prism manifolds but fibered meridianally.
We apply Corollary 4.8 to yield that the group
5.6. Base space
In this case,
In this case we apply Corollary 4.6 to yield that
In this case we instead apply Corollary 4.5 to yield that
5.7. Base space
In both of these cases
6. Quotient Spaces
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
The main part is to establish what form the quotients of
Formally, for a representation
We then let
We then have the diagram:
We hence need to find the following:
We first consider actions constructed via the method of  that are fiber-orientation-preserving. For this section we consider
Lemma 6.1. Let
So now, we have
Remark 6.2. As
We now allow the fibers to be reversed.
Lemma 6.3. Let
To see that
To see that it is a product, we note that if it does not preserve the product structure
The result therefore follows.
Corollary 6.4. Let
Example 6.5. Consider
Example 6.6. Now consider again
Figure 2. Quotient under the action of Example 6.6
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  the action of the stabilizer on
Lemma 6.7. The quotient of a solid torus under a
Lemma 6.8. The quotient of a solid torus under a
To prove this, we first note that:
Also, if we solve
Now, there is an induced map
It follows that
So finally, the order of the exceptional core of quotient space of the whole action is:
We now consider an action of the stabilizer that reverses the orientation of the fibers. By  the action will be a
Lemma 6.9. The quotient of a solid torus under a
We again begin by assuming that the action preserves the orientation of the fibers. So now
By using product structures
Here note that we again allow
We then we have that:
Note that this is well-defined as if
Now, for any torus (either on the boundary of
So we consider the diagram:
We begin with the cyclic case. We can then choose the product structure such that
Lemma 6.10. If
So then take
We proceed with
Lemma 6.11. If
Hence we take
So now we have from above that
Example 6.12. We consider a
Here we parameterize
We calculate first
Figure 3. Quotient space
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
So then by Lemma 6.6, the exceptional set will have order:
So then again using Lemma 6.6, the exceptional set will have order:
We now compute the projection maps. By section 6.1, the projection map from both
The projection map from
The projection map from
We now calculate the projected filling of
We now calculate the projected filling of
This fully characterizes the quotient space. We visualize in Figure 4:
Figure 4. Full quotient space
7. Summary of Results and Future Work
In this paper we have studied the group actions on Seifert fibered elliptic manifolds using the results of  and . 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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