Finite, fiber- and orientation-preserving group actions on orientable Seifert manifolds with orientable base space

In this paper we consider the finite groups that act fiber- and orientation-preservingly on orientable Seifert manifolds that fiber over an orientable base space without boundary. We establish a method of constructing such group actions and then show that if an action satisfies a condition on the obstruction term of the Seifert manifold, it can be derived from the given construction. The obstruction condition is refined and the general structure of the finite groups that act via the construction is provided.

Given that the first homology group (equivalently the first fundamental group) of a torus is Z × Z 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 : T 1 → T 2 and product structures k i : S 1 × S 1 → T i , then we can express the induced map on the first homology groups by a matrix that uses bases for H 1 (T i ) derived from the meridian-longitude framings that arise from k i : S 1 × S 1 → T i . We denote this matrix as a 11 a 12 a 21 a 22 If we have a manifoldM with torus boundary components and each of those boundary tori T i have a product structure k i : S 1 × S 1 → T i , then we say a G-action ϕ : G → Dif f (M ) respects the product structures on the boundary tori if k −1 j • ϕ(g) • k i : S 1 × S 1 → S 1 × S 1 can be expressed as (ϕ 1 (g), ϕ 2 (g)) where ϕ 1 : G → Dif f (S 1 ) and ϕ 2 : G → Dif f (S 1 ). These again are not necessarily injections.
Suppose that we now have a fibering product structure k : S 1 × F → M . 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.

Dehn fillings and Seifert manifolds.
We first establish some background work on Dehn fillings and Seifert manifolds.
This section broadly follows the construction from [4]. We use the notation for a closed orientable Seifert manifold M with orientable base space: (g, o 1 |(q 1 , p 1 ), . . . , (q n , p n )), q i > 0 This notation implies that M is a manifold that can be decomposed into a manifoldM ∼ = S 1 × F that is trivially fibered with boundary ∂M = T 1 ∪ . . . ∪ T n , and X = V 1 ∪ . . . ∪ V n , a disjoint collection of fibered solid tori (the notation specifies the fibration). M is reobtained by a gluing map d : ∂X → ∂M . This is defined as follows: Take a given fibering product structure kM : S 1 × F →M onM , and some particular product structure k X : S 1 × (D 1 ∪ . . . ∪ D n ) → X where each D i is a disk. Then define product structures k ∂Vi : S 1 × S 1 → ∂V i and k Ti : S 1 × S 1 → T i by parameterizing each component of ∂F and ∂D i with a a positive orientation by some diffeomorphisms ρ i : S 1 → (∂F ) i and σ i : S 1 → ∂D i , and then taking k ∂Vi (u, v) = k X (u, σ i (v)) and k Ti (u, v) = kM (u, σ i (v)).
d : ∂X → ∂M is then a diffeomorphism such that d(∂V i ) = T i and Where x i q i − y i p i = −1 and |y i | < q i .
We note therefore that the induced fibration on each solid torus V i , is a (−q i , y i ) fibration (according to k ∂Vi ).
We now quote Theorem 1.1. from [5] regarding Seifert invariants: Theorem 2.1. Let M and M be two orientable Seifert manifolds with associated Seifert invariants (g, o 1 |(α 1 , β 1 ), . . . , (α s , β s )) and (g, o 1 |(α 1 , β 1 ), . . . , (α t , β t )) respectively. Then M and M are orientationpreservingly diffeomorphic by a fiber-preserving diffeomorphism if and only if, after reindexing the Seifert pairs if necessary, there exists an n such that: The consequence of this theorem is that we can perform the following "moves" on the Seifert invariants: From this we yield the Corollary: Corollary 2.2. Let M and M be two orientable Seifert manifolds with associated Seifert invariants (g, o 1 |(α 1 , β 1 ), . . . , (α s , β s )) and (g, o 1 |(α 1 , β 1 + m 1 α 1 ), . . . , (α s , β s + m s α s )) respectively. Then M and M are orientation-preservingly diffeomorphic by a fiber-preserving diffeomorphism if and only if Proof. By Theorem 2.1, we need only consider the third condition. The first two conditions hold trivially. So, the two manifolds are diffeomorphic if and only if: Hence, if and only if We can now define normalized Seifert invariants so that any orientable Seifert manifold over an orientable base space can be expressed as: Where 0 < p i < q i and b is some integer called the obstruction term.
The constant: is known as the Euler class of the Seifert bundle and is zero if and only if the Seifert bundle is covered by the trivial bundle. Alternatively, it is zero if the manifold M has the geometry of either S 2 × R, H 2 × R, or E 3 . For more details, refer to [6].

Construction of a finite, orientation and fiber-preserving action
We now present a construction for a finite, orientation and fiber-preserving action on a Seifert manifold M = (g, o 1 |(q 1 , p 1 ), . . . , (q n , p n )). Here the Seifert invariants are not necessarily normalized.
We first note that according to Section 2, we can decompose M intoM and X whereM ∼ = S 1 × F is trivially fibered and X is a disjoint union of n solid tori. We then have a gluing map d : ∂X → ∂M , so that for a fibering product structure kM : S 1 × F →M , there is some k X : S 1 × (D 1 ∪ . . . ∪ D n ) → X and restricted positively oriented product structures k ∂Vi : S 1 × S 1 → ∂V i and k Ti : 3.1 Constructing a finite, fiber-preserving action onM .
We pick a finite, fiber-preserving group action onM by first choosing some (not-necessarily effective) group action ϕ 1 : G → Dif f (S 1 ). This will necessarily be of the form: Here θ 1 : G → S 1 and α : G → {−1, 1}. The precise nature of these maps is shown in Section 3.5.
Then we define our group action ϕ : G → Dif f (M ) by: So now we can fully express ϕ : G → Dif f (M ) on the boundary ofM by: We note here that (according to the set framing of each boundary torus), each element g ∈ G acts on a boundary tori T i by mapping it to T β(g)(i) with: • a rotation by θ 1 (g) in the longitudinal direction.
• a rotation by θ 2 (i, g) in the meridianal direction.
• a reflection in the meridian and longitude if α(g) = −1.
3.2 Inducing a finite, fiber-preserving action on ∂X.
So we can now induce an action on ∂X by: This we can fully express (after simplification) as: Therefore (according to the set framing of each boundary torus), each element g ∈ G acts on a ∂V i by mapping it to ∂V β(g)(i) with: • a rotation by θ 1 (g) −qi θ 2 (i, g) pi in the longitudinal direction.
• a reflection in the meridian and longitude if α(g) = −1.
Alternatively, we could view this action by each element g ∈ G mapping ∂V i to ∂V β(g)(i) with: • a rotation by θ 1 (g) along (−q j , y j ) curves (along the fibers).
• a reflection in the meridian and longitude if α(g) = −1.

Extending the induced action to X.
We here note that: Where D is the unit disc. Hence the action ψ : G → Dif f (X) straightforwardly extends by coning inwards. This works as the product structure on X is such that the fibration is normalized. Hence, the extended action is fiber-preserving.

The final action.
So now we have defined finite, fiber-preserving actions onM and X such that they agree under the gluing map d : ∂X → ∂M . This completes the construction. We here establish some necessary and sufficient conditions in the construction of ϕ 1 : G → Dif f (S 1 ) and ϕ 2 : G → Dif f (F ).

Actions onM
In order to find out to what extent finite, fiber-preserving actions can be derived from the construction set out in Section 3, we first need to establish a result regarding actions onM . In this section we always take F to be an orientable surface with boundary andM to be the fibered manifold that has boundary made up of tori described earlier.
The main result we prove in this section is an adaptation of Theorem 2.3 in [7]. It will state that ifM has a product structure, then provided the restricted product structures on each boundary component are respected by the action, then there is another product structure onM that is left invariant by the group action. Moreover, the two product structures foliate the boundary tori identically.
We first state some preliminary results.
Then F contains a ϕ-equivariant essential simple arc.
Proof. F/ϕ is a 2-orbifold. We can then pick an essential simple arc in the underlying space of F/ϕ that doesn't intersect the exceptional points and then lift this to a ϕ-equivariant essential simple arc in F .
be a finite group action on a Seifert fibered torus. Suppose that there exists a fibering product structure k : is equivalent to a fiber-preserving group action that leaves the product structure k : S 1 × S 1 → T invariant. Moreover, the conjugating map is fiber-preserving and isotopic to the identity.
We then note that by [2], the only possible quotient types are a torus or S 2 (2, 2, 2, 2). By [8] these refer respectively to actions of groups Z m × Z n and Dih(Z m × Z n ) where Z m × Z n acts by preserving the orientation of the fibers and the dihedral Z 2 subgroup of Dih(Z m × Z n ) acts by reversing the orientation of the fibers.
We first consider the torus case. This will receive an induced fibration from T . We then can pick a fibering product structure on T /ψ . This product structure can be lifted to an invariant fibering product structure k : According to this product structure, the group acts as rotations along the fibers or along loops k ({u} × S 1 ). As such, it preserves any fibration up to isotopy. So we can assume that k : This is a product. It also follows that f is fiber-preserving and isotopic to the identity.
If the action has quotient of S 2 (2, 2, 2, 2), then we note that as the fiber orientation-preserving subgroup Z m × Z n is a normal subgroup, we can consider the induced Z 2 -action on the quotient of the Z m × Z n -action. This is necessarily a "spin" action by [8] and we can pick a fibering product structure on T /(Z m × Z n ) as above but that is further left invariant under the "spin" action. Lemma 4.3. Let k : S 1 × F →M and k : S 1 × F →M be fibering product structures so that they foliate the boundary tori identically. Then So now by composing with the diffeomorphism l : are freely isotopic by isotoping along the fibers.
The proof of the theorem follows that of [7] in an adapted and expanded form.
Theorem 4.4. Let k : S 1 × F →M be a fibering product structure such that the finite group action ϕ : G → Dif f f p + (M ) respects the restricted product structures on each boundary torus. Then there exists an isotopic fibering product structure k : S 1 ×F →M such that the group action ψ : G → Dif f (S 1 ×F ) given by ψ(g) = k −1 •ϕ(g)•k for each g ∈ G is a product action and foliates the boundary identically to k.
Proof. We proceed by induction on the Euler Characteristic of F .
We therefore haveM as a trivially fibered solid torus with k : S 1 × F →M , a fibering product structure. By the product structure on the boundary, we have a foliation by meridianal circles that each bound a disc and the usual longitudinal Seifert fibration by circles. So any of the meridianal circles are necessarily ϕ-equivariant. Then taking such a circle, we apply the equivariant Dehn's Lemma to yield a ϕ-equivariant disc D whose boundary agrees with the product structure on the boundary of the solid torus. We now decompose along Orb(D) = {D 1 , . . . , D s } to yield a collection B 1 , . . . , B s of balls, each which are homeomorphic to I × D and fibered by arcs.
So starting with B 1 we have the action ϕ 1 : Note that the quotient orbifold B 1 /ϕ 1 necessarily has boundary either S 2 (n, n) or S 2 (2, 2, n). This follows from [9], that show that these are the only orientable quotients of S 2 where the action fixes one point or exchanges two points (corresponding to the two discs D 1 , D 2 ).
We here use the proof of the Smith conjecture (see ball orbifolds in [10]) to see that B 1 /ϕ 1 has the following possible forms with induced (orbifold) foliations on part of the boundary shown by Figure 1. This first can then clearly be foliated by discs that agree with the foliation by circles on the boundary. The second can be foliated by discs with a cone point of order n with the discs agreeing with the foliation by circles on the boundary.
The third can be foliated by discs with cone points order n -with the discs having boundaries given by the circles -and a 2-orbifold of the form shown in Figure 2.
Each of these can be taken to hit each induced orbifold I-fiber once and will lift to an invariant foliation of B 1 by discs that each hit each I-fiber once. We therefore have a product structure k 1 : I × F → B 1 left invariant by the action ϕ 1 : Stab(B 1 ) → Dif f (B 1 ) whose foliation (by arcs and circles) on the part of its boundary that intersects with the boundary ofM is equal to the restricted foliation from k : We can then define product structures k i : Note that as each ϕ(g i ) leaves the original product structure k : S 1 × F →M invariant on the boundary ofM then each k i : I × F → B i foliates B i (by arcs and circles) on the part of its' boundary that intersects with the boundary ofM the same way as the restricted foliation from k : This is a product by above. So now we have a collection of product structures on each B 1 , . . . , B s that are left invariant under the action. We view these now as invariant foliations by arcs and discs. By construction, we yield invariant foliations ofM by circles and discs. This is possible as each of the invariant foliations of B i are equal to the restricted foliation from k : S 1 × F →M on the part of its' boundary that intersects with the boundary ofM .
These invariant foliations give our required k : S 1 × F →M .

Inductive
Step: We suppose the result holds for χ(F ) > c and now consider χ(F ) = c.
We induce the action ϕ F : G F → Dif f (F ) on the base space of the fibration and then apply Lemma 4.1 to yield a ϕ F -equivariant essential simple arc in F . We call this arc λ and define A 1 to be the annulus made up of fibers that project to λ. As ϕ : G → Dif f (M ) is fiber-preserving, this is necessarily ϕ-equivariant.
Cutting along the collection of annuli Orb(A 1 ) will yield a disjoint collection {M 1 , . . . ,M n } of manifolds with boundary which fiber over surfaces {F 1 , . . . , F n }. Necessarily, each of these have greater Euler number than F . Now pickM 1 . Then pick any boundary torus T ofM 1 that contains A 1 . This consists of annuli that were originally contained in a boundary tori ofM before being cut open -we refer to these as A 1 , . . . , A m -or some annuli in the collection Orb(A 1 ) -we refer to these as A 1 , . . . , A m . Note that there must be an equal number of each type of annulus. Each of A 1 , . . . , A m inherit product structures k A i : S 1 × I → A i that are respected under the restricted action of Stab(T ). Now consider T /Stab(T ). This will necessarily be either another torus consisting of two glued annuli -one referring to the projection of A 1 and the other referring to the projection of A 1 -or an S 2 (2, 2, 2, 2) consisting of two glued together D(2, 2) -again, one referring to the projection of A 1 and the other referring to the projection of A 1 . This follows from [2].
The annulus covered by A 1 has an induced Seifert fibration and foliation by arcs. The annulus covered by A 1 has an induced Seifert fibration and can by foliated by arcs so that T /Stab(T ) is foliated by circles that cross each fiber once.
The D(2, 2) covered by A 1 has an induced orbifold Seifert fibration and orbifold foliation as shown below in Figure 3. The D(2, 2) covered by A 1 has an induced orbifold Seifert fibration and can be orbifold foliated so that T /Stab(T ) is orbifold foliated so that each leaf crosses each fiber once. Moreover these orbifold foliations can be chosen so that they lift to give T a foliation that is invariant under Stab(T ); agrees with the foliation by arcs given by k A i : S 1 × I → A i ; and is isotopic to the induced foliation of T from the original k : S 1 × F →M . This follows from Lemma 4.2.
This then defines a product structure k T : S 1 × S 1 → T invariant under the action of Stab(T ) which restricts to a product structure k A1 : We then define product structures k Ti : So now for any g ∈ G with ϕ(g)(T i ) = T j for some i, j, we have that g = g j g g −1 i for some g ∈ Stab(T 1 ). So then Hence it is a product and the product structures on each of the tori T i are respected under Stab(M 1 ).
We do this for each orbit of boundary components ofM 1 to yield product structures on each boundary tori that are respected under Stab(M 1 ) and that agree with the inherited product structure from the original boundary ofM .
We can now translate these product structures to the boundaries of eachM i . Now we can reconstructM and can assume that we have respected product structures on each of the connected components of the union of ∂M and Orb(A 1 ). Pick the first connected component C that yielded T when we cut as shown in Figure 4. The product structure on this connected component is necessarily isotopic to the original product structure by construction. Suppose that the product structure on some other connected component C was defined by translating by ϕ(g). We now note that k : S 1 × F →M and ϕ(g) • k : S 1 × F →M satisfy the requirements of Lemma 4.3. Hence applying the lemma, we yield that the restricted product structure on C from ϕ(g) • k : S 1 × F →M is isotopic to the original product structure k : S 1 × F →M .
Hence, in regular neighborhoods of each of the connected components, we adjust the product structure k : S 1 × F →M to equal the invariant product structures on the connected components.
It then follows that the respected product structures on each of the boundary tori ofM 1 extend within.
We can then apply the inductive hypothesis to assume that kM 1 : S 1 × F 1 →M 1 is in fact left invariant under the action of Stab(M 1 ).
We translate this product structure to eachM i to yield the required invariant product structure.

Remark 2.
We remark here that it is not sufficient simply that there are product structures on the boundary tori that are respected by the action. It is required also that the product structures can be extended within. We give the following example to illustrate this: Example 4.1. We let F be an annulus and k : S 1 × F →M be a fibering product structure. Let G = Z m act onM by simply rotating by 2π m along the fibers. This action will preserve any fibering product structure (up to isotopy) on each boundary torus.
So now pick meridians on the first torus to be the loops that are (0, 1) curves according to k : S 1 × F →M and meridians on the second torus to be loops that are (1, 1) curves according to k : S 1 × F →M . These are both left invariant, but there is no product structure onM that restricts to these on the boundary.

Main Result.
We now prove the main result, whic states that given a condition on the obstruction term, any finite, orientation and fiber-preserving action on an orientable Seifert 3-manifold that fibers over an orientable base space can be derived via the presented construction in Section 3.
To prove this, we first state Theorem 2.8. be an admissible diffeomorphism, and suppose that for some regular fiber γ in M 1 , f (γ) is homotopic in M 2 to a regular fiber. Then f is admissibly isotopic to a fiber-preserving diffeomorphism. If f is already fiber-preserving on some union U of elements of m 1 , then the isotopy may be chosen to be relative to U .
This then leads us to what we will require: Lemma 5.2. Let W be a Seifert fibered torus and let h : T → T be a fiber-preserving diffeomorphism with h * = id. Then h : T → T can be extended to a fiber-preserving diffeomorphism h : T × I → T × I with h(x, 1) = (h(x), 1), h(x, 0) = (x, 0). Here T × I is fibered as a unique extended fibration.
Proof. We note first that an isotopy to the identity exists. We then need only check that such an isotopy can be taken to fiber-preserving.
So there exists a diffeomorphism H : W × I → T , such that H(x, 1) = h(x) and H(x, 0) = x with H t : T → T a diffeomorphism for each t ∈ I.
We now define the diffeomorphismH : T × I → T × I byH(x, t) = (H(x, t), t). This diffeomorphism is fiber-preserving on the boundary of T × I.
We now give T × I the boundary pattern consisting of the union of its' two boundary tori. Then certainlyH is an admissible diffeomorphism and moreover it is the identity on one boundary component, so the image of a fiber being homotopic to a fiber condition is trivially satisfied.
We now prove the main result: Proof. We let M be the Seifert 3-manifold with normalized invariants: p 1 ), . . . , (q n , p n ), (1, b)) Firstly, without loss of generality, we can assume that the orbits of each {α 1 , . . . , α m } are distinct. If α i , α j were in the same orbit, then we note that b i · #Orb ϕ (α i ) + b j · #Orb ϕ (α j ) = (b i + b j ) · #Orb ϕ (α i ) so that we do not require α j for the property to still hold.
Secondly, we can suppose without loss of generality that the first t of the fibers {α 1 , . . . , α t } are regular and each critical fiber {γ 1 , . . . , γ n } is in the orbit of one of {α t+1 , . . . , α m }. If one is not, it can be added into the collection with a coefficient of zero. This will not change the sum.

Now let
Then rewrite the Seifert invariants as: Take closed, fibered regular neighborhoods N (α 1 ), . . . , N (α m ) and then define: For convenience, denote: p 1 ), . . . , (q n , p n ), (q n+1 , p n+1 ), . . . , (q n+A , p n+A ) From Section 2, this gives us a fibering product structureM : S 1 × F →M and a product structure k X : S 1 × (D 1 ∪ . . . ∪ D n+A ) → X so that according to it, each V i in X has a normalized fibration. We then have that for the nontrivially fibered solid tori according to these product structures. So now the fibrations on these V i are a (−q i , y i ) fibration. The action can only send some V i to a V j if they have the same fibration. Hence (−q i , y i ) = (−q j , y j ).
We now show that the action can only send some V i to a V j if they have the same associated fillings. Now, we have x i q i − y i (p i + h(i)q i ) = −1 and x j q i − y i (p j + h(i)q i ) = −1. Hence: We can henceforth assume that if the action sends some V i to a V j , then the fillings must be the same. Note that this is true also for the fillings of trivially fibered tori by construction.
So thenM is a Seifert fibered 3-manifold with boundary such that there is a fiber-preserving restricted action given by: We now proceed to show that there is a product structure onM such thatφ respects the restricted product structures on the boundary tori. We do so to employ Theorem 4.4.
Now take T i arbitrarily and consider the action given byφ(g)| Ti for each g ∈ Stab(T i ).
By restricting kM : S 1 × F →M and k X : S 1 × (D 1 ∪ . . . ∪ D n+A ) → X as in Section 2 to k Ti : S 1 × S 1 → T i and k ∂Vi : S 1 × S 1 → ∂V i we have the following homological diagram: Now, as the action extends into V i and is finite, we must have that (d| −1 ∂Vi •φ(g)| Ti • d| ∂Vi ) * = ±id. Hence (φ(g)| Ti ) * = ±id for all g ∈ Stab(T i ).
We can then apply Lemma 4.2 to get f i : T i → T i such that f i is fiber-preserving, isotopic to the identity, and k −1 We translate the conjugating map f i : where: ifφ(g j ) reverses the orientation of the fibers Each f j is certainly fiber-preserving, we check that they are isotopic to the identity.
Note that we have the diagram: So that necessarilyφ(g j ) * = ±id depending on whether the orientation on the fibers are reversed or not. Consequently, f j is isotopic to the identity.
Then for any g ∈ G, g = g j2 hg −1 j1 , for some h ∈ Stab(T i ) and some T j1 , T j2 ∈ Orb(T i ). We calculate: k −1 is also a product map, and the product structures f j • k Tj : We can now do this for each of the distinct orbits of boundary tori.
As each f j is isotopic to the identity and fiber-preserving, we can employ Lemma 5.2. to define f ∈ Dif f f p + (M ) so that f | Tj = f j and f is the identity outside of a regular neighborhood of each boundary torus. f is necessarily isotopic to the identity.
So now, the product structure f • kM : S 1 × F →M is such that f • k Tj : S 1 × S 1 → T j for each T j are respected underφ and moreover is isotopic to kM .
Then we have what we require to employ Theorem 4.4: a product structure onM such thatφ respects the restricted product structures on the boundary tori. So we yield a product structure k M : Therefore, we must have that on each boundary component T i : But now α 1 (g) = α 2 (g) as the action is orientation-preserving.
It remains to show that we can pick a product structure on X that is left invariant. We know that there is a product structure k X : S 1 × (D 1 ∪ . . . ∪ D l ) → X so that according to the product structure k M : S 1 × F →M we have: If we let ϕ X be the action restricted to X, we have that according to this product structure, the action on the boundary of X looks like: That is, it respects the restricted product structures. Hence we can consider Stab(V i ) for each V i to apply Theorem 4.4 and translate in a similar way to above and in the proof of Theorem 4.4.
This completes the proof.
We now state some of the immediate corollaries of Theorem 5.3: Corollary 5.4. Let M be an orientable Seifert 3-manifold that fibers over an orientable base space. Let ϕ : G → Dif f f p + (M ) be a finite group action on M such that a fiber is left invariant. Then ϕ can be derived via the construction set out in Section 3.
Corollary 5.6. Let M be an orientable Seifert 3-manifold that fibers over an orientable base space. Let ϕ : G → Dif f f p + (M ) be a finite group action on M so that there are two fibers α, β with #Orb ϕ (α), #Orb ϕ (β) coprime. Then ϕ can be derived via the construction set out in Section 3.
These corollaries give some simple situations under which the conditions of Theorem 5.1 are satisfied. We give an example of the use of these corollaries: Example 5.1. We consider a Seifert manifold M which fibers over an orientable base space B which has the cone points 2, 2, 3, 3, 3. Now any action on B would necessarily only be able to exchange the two cone points of order 2 and permute the cone points of order 3. Hence a critical fiber α referring to one of the cone points of order 2, must have that #Orb ϕ (α) is 1 or 2. Similarly, there is a critical fiber β referring to one of the cone points of order 3, that must have either #Orb ϕ (β) as 1 or 3. If either #Orb ϕ (α) or #Orb ϕ (β) is 1, then we can apply Corollary 5.4. If #Orb ϕ (α) = 2 and #Orb ϕ (β) = 3, then we can apply Corollary 5.6.
In all cases any finite, orientation and fiber-preserving action on M must be derived via the construction set out in Section 3. This is regardless of the obstruction term.
Remark 3. We note that the obstruction condition is not always satisfied. We give the following example: Example 6.1. Construct by a Seifert 3-manifold M fibering over an even genus g surface with no exceptional fibers and odd obstruction b by taking two trivially fibered manifolds M 1 = S 1 × F 1 and M 2 = S 1 × F 2 where F 1 , F 2 are genus g 2 surfaces with a disc removed, and then gluing according to the map d( Define the rotation rot 2 : F i → F i to be an order 2 rotation that leaves the boundary invariant. Define an orientation-preserving finite and fiber-preserving action on M 1 and M 1 by: f 1 (u 1 , x 1 ) = (u 1 , rot 2 (x 1 )), f 1 (u 2 , x 2 ) = (−u 2 , rot 2 (x 2 )) and f 2 (u 1 , x 1 ) = (u 2 , x 2 ), f 2 (u 2 , x 2 ) = (u 1 , x 1 ) It can be checked that these agree over the gluing torus.
So then the projected action on the genus g surface is a Z 2 × Z 2 -action and all orbit numbers are even. Hence, it cannot be that We now proceed to refine the obstruction condition. First, two lemmas are established and then a proposition which provides a convenient equivalent statement for the obstruction condition that can be used to apply our results. Lemma 6.1. Let ϕ : G → Dif f (S) be a finite group action on a surface S. Suppose that the orbifold S/ϕ has data set (n 1 , . . . , n k ; m 1 , . . . , m l ). Then the possible orbit numbers under ϕ are |G|/n 1 , . . . , |G|/n k , |G|/2m 1 , . . . , |G|/2m l and |G|.
Proof. S is an order |G| orbifold cover of S/ϕ. Therefore any regular point of S/ϕ lifts to |G| points of S, any of these points have orbit number |G|. Any neighborhood of a cone point of order n i is covered by a collection of discs in S, each disc is an n i -fold cover of the neighborhood. Hence the number of discs that cover the neighborhood is |G| ni . Thus the center of each disc has orbit number |G| ni . Any neighborhood of a corner reflector of order m i is covered by a collection of discs in S, each disc is an 2m i -fold cover of the neighborhood. Hence the number of discs that cover the neighborhood is |G| 2mi . Thus the center of each disc has orbit number |G| 2mi . Proof. We work by induction. For the base case we use the result that gcd(x, y)lcm(x, y) = xy for any integers x, y. This implies: gcd( N n 1 , N n 2 )lcm(n 1 , n 2 ) = N 2 lcm(n 1 , n 2 ) n 1 n 2 lcm( N n1 , N n2 ) = N 2 lcm(n 1 , n 2 ) lcm(n 2 N, n 1 N ) = N 2 lcm(n 1 , n 2 ) N lcm(n 2 , n 1 ) = N For the inductive step, we work in a similar fashion: Without loss of generality, we can assume that the orbit numbers of all the x i are different, that s = k + l (set b i = 0 if necessary), and that the branching data of each x i is n i for i = 1, . . . , k and 2m i for i = k + 1, . . . , l.
This result then allows us to quickly establish whether the obstruction condition is satisfied based on the order of the induced action on the base space and the least common multiple of the data from the orbifold quotient of the induced action. This is a convenient way to establish results based on possible quotient types.

Group structures.
We now establish the possible structures of the groups that can act orientation-preservingly on a Seifert manifold (satisfying the obstruction condition).
Proposition 7.1. Suppose that ϕ : G → Dif f (S 1 ) × Dif f (F ) is a finite group action with ϕ(g)(u, x) = (ϕ S 1 (g)(u), ϕ F (g)(x)) such that ϕ S 1 (g) is orientation-preserving if and only if ϕ F (g) is orientation-preserving. Suppose that there exists g − ∈ G such that ϕ S 1 (g − ) is orientation-reversing and g 2 − = 1. Then G is isomorphic to a subgroup of a semidirect product of Z n × ϕ F (G) + and Z 2 .
Proof. First let ϕ(G) f op be the subgroup of ϕ(G) where each element is orientation-preserving on both components.
We now consider the structure of ϕ(G) f op . We note that ϕ(G) f op is a finite subgroup of ϕ S 1 (G) + × ϕ F (G) + . Now ϕ S 1 (G) + ∼ = Z n for some n and so ϕ(G) f op is a finite subgroup of Z n × ϕ F (G) + .
This splits if there is an element in ϕ(G) of order 2 that is not in ϕ(G) f op . By assumption, ϕ(g − ) is such an element. The result then follows.

Summary
We have shown that provided that the obstruction term is satisfied, then a finite, fiber-and orientation-preserving action can be constructed via our method. The final section above gives some form to the kinds of finite groups that act this way. We note that there is the restriction that G contains an order 2 element that reverses the orientation of the fibers and therefore reverses the orientation on the base space. In the particular case of the base space being S 2 this is not a restriction as any finite group that acts is a subgroup of a finite group that has this property. [9] In particular, we will establish in a future paper that the finite groups that act fiber-and orientation-preservingly on Seifert manifolds fibering over S 2 (and satisfiying the obstruction condition) are of the form (Z n × H) • −1 Z 2 where Z 2 acts by anticommuting with each element of Z n × H and H is a finite group that acts orientation-preservingly on S 2 .