A note on Schwartzman–Fried–Sullivan Theory, with an application

We prove a theorem on the existence of global surfaces of section with prescribed spanning orbits and homology class. This result is a modification and a refinement of a result due to Fried, recast in terms of invariant measures instead of homology directions.


Introduction
Throughout this paper we fix a smooth flow φ t on a smooth closed oriented connected 3-manifold M . The interplay between the topology of M and the dynamics of φ t can be studied by following a program outlined in the 1950s by Schwartzman [15]. If μ is a φ t -invariant Borel probability measure, then μalmost every point is recurrent and its trajectory almost closes up infinitely often. One may close such "almost periodic" long trajectories with short paths, and average in time the value obtained by hitting with a degree one cohomology class y. With the help of Ergodic Theory one can show that this procedure defines a μ-integrable function; its integral will be denoted by μ · y ∈ R and called an intersection number for geometric reasons which will soon become clear. See Sect. 2 for precise definitions. This construction can be localized to invariant open subsets. In some sense the values of μ · y, for all possible μ and y, give a complete portrait of the flow. This claim will be made precise in the context of a particular problem, namely that of deciding when a finite collection of periodic orbits bounds a global surface of section.
The notion of a global surface of section goes back to Poincaré's work on the 3-body problem. It is intimately connected to the development of Symplectic Topology: the discovery of Poincaré's annulus in the context of Celestial Mechanics led to the statement of his last geometric theorem [13], nowadays known as the Poincaré-Birkhoff theorem, which in turn led Arnold [1] to make his conjectures on the number of fixed points of Hamiltonian diffeomorphisms and of Lagrangian intersections. The Arnold conjectures led to the creation of Floer theory, see for instance [5]. Definition 1.1. A global surface of section for φ t is a compact embedded surface Σ → M such that (i) ∂Σ is a finite collection of periodic orbits, and φ t is transverse to Σ\∂Σ.
Remark 1.2. The case ∂Σ is the empty set is not excluded. All our global surfaces of section are oriented by the ambient orientation and the co-orientation induced by the flow: intersection points of trajectories (oriented by the flow) with Σ\∂Σ count +1.
Using Σ one may study φ t in terms of the first return map, defined by following a point in Σ\∂Σ until it hits Σ in the future. This point of view opens the door to methods in two-dimensional dynamics.
From now on L ⊂ M is a fixed null-homologous link consisting of periodic orbits of φ t , or the empty set. The task at hand is to look for qualitative information that will tell us when L is the boundary of a global surface of section. Besides intersection numbers, other important players are the rotation numbers of components of L relative to some y ∈ H 1 (M \L; R). These are roughly described as follows. The linearized flow along a periodic orbit γ ⊂ L induces an orientation preserving diffeomorphism on the circle of oriented rays issuing from a point of γ. A class y determines (up to homotopy) an isotopy from the identity to this diffeomorphism. We end up with a real-valued rotation number, denoted by ρ y (γ). See Sect. 2 for a precise discussion.
We denote by P φ (M \L) the set of φ t -invariant Borel probability measures on M \L. If b ∈ H 2 (M, L; Z), then y b denotes its dual class, seen in H 1 (M \L; R). Recall that, by definition, a Seifert surface is a compact connected orientable embedded surface. We say that L spans the Seifert surface if it is equal to its boundary. Two conventions are useful: the empty set is said to bind an open book decomposition if the ambient manifold fibers over the circle, and a Seifert surface spanned by the empty set is just a closed connected orientable embedded surface.
The case where L = ∅ and the homology class b is not prescribed is sketched by Ghys in [8]. In fact, the reader will immediately realize that these notes are much inspired by, and owe a lot to, the exposition in [8].
The case L = ∅ of Theorem 1.3 is close to work of Fried. The definition of a global surface of section in [7, section 5] is more restrictive than Definition 1.1. The difference is that in [7] one asks for the first return time function to be bounded and the angle between the surface and the vector field not to vanish to first order as one approaches the boundary; let us call these strong global surfaces of section. It is not hard to construct by hand global surfaces section which are not strong, and the reader is invited to check that our arguments will produce strong ones from assumption (iii). In [7, section 5] one also finds necessary and sufficient conditions for the existence of strong global surfaces of section, but these are stated in terms of assumptions on homology directions in a certain "blown up" manifold; such conditions are hard to work with. Our conditions are stated in terms of rotation numbers on the ambient manifold, which are more directly related to the flow and comfortable to work with. The presentation of this new set of sufficient conditions is our first contribution. Another novelty is the identification of a rather simple C ∞ -generic assumption that guarantees that the sufficient conditions are also necessary, it reads: For every periodic orbit γ ⊂ L, if the rotation number of γ with respect to y b vanishes then γ is hyperbolic. The proof can be found in Sect. 3.4. The strength of Theorem 1.3 is its generality; it deals with any flow on any oriented 3-manifold. Its weakness comes from the fact that it might not be easy to check (iii) in concrete examples. However, there are restrictive but still very interesting classes of flows for which more applicable existence results can be proved. The basic example is the following: There is an important point to be made here: one should work to make the general results from Schwartzman-Fried-Sullivan theory more applicable, and to make restrictive geometric results such as Birkhoff's theorem more general. Pseudo-holomorphic curve theory in symplectizations and symplectic cobordisms, as introduced by Hofer [9], implements this program within the class of Reeb flows. The following remarkable theorem is our guide. We refer to [11] for an overview of results obtained in this direction. In [12] new results which can be used to prove Theorem 1.4 quite directly will appear. The reader can find in the book [6] by Frauenfelder and van Koert a nice introduction to global surfaces of section in the context of Symplectic Dynamics, with emphasis in Celestial Mechanics.

Intersection numbers
Let R φ denote the set of recurrent points of φ t .
there exists a function f μ,y ∈ L 1 (μ) with the following property: For μ-almost every point Here k(T n , p) denotes any loop obtained by concatenating to φ [0,Tn] (p) a path from φ Tn (p) to p inside V .
Definition 2.2. The above lemma allows one to use the integral as a definition of the intersection number. 1 We now work towards the proof of Lemma 2.1. Let Ω 1 L ⊂ Ω 1 (M \L) be the subspace defined as follows. Choose a connected component γ ⊂ L and an orientation preserving diffeomorphism defined on some neighborhood N of γ, satisfying Ψ(φ t (p 0 )) = (t, 0) for some p 0 ∈ γ. Here D ⊂ C is the closed unit disk oriented by the complex orientation of (C, i), T > 0 is the primitive period of γ, R/T Z is oriented by the canonical orientation of the real line, and R/T Z × D is oriented as a product. On N \γ we have coordinates via the identification Ψ −1 (t, re iθ ) (t, r, θ), which will be referred to as tubular polar coordinates for γ. Define Ω 1 L to be the set of smooth 1-forms on M \L that can be represented as Adt + Bdr + Cdθ with bounded coefficients A, B, C with respect to tubular polar coordinates (t, r, θ) around each connected component of L. Proof of Lemma 2.1. LetS μ,y be the set of points p ∈ R φ \L with the following property: There exists some open contractible neighborhood V ⊂ M \L of p such that for every sequence T n → +∞ satisfying φ Tn (p) → p the limit exists. Here k(T n , p) denotes, for n large enough, any loop obtained by concatenating to φ [0,Tn] (p) a path from φ Tn (p) to p inside V . By contractibility of V , the above limits do not depend on choice of closing paths, but they could in principle depend on the sequence T n . Clearly the existence and the values of the above limits do not depend on V . By Lemma 2.5 we can choose a closed representative β ∈ Ω 1 L of y. Then β(X) is bounded, by Lemma 2.4. By the ergodic theorem there exists a Borel set S ⊂ M \L with the following properties: μ(S) = 1, and for every p ∈ S the limit exists, and defines a μ-integrable function satisfying Let p ∈ S ∩ R φ , choose any contractible open neighborhood V ⊂ M \L of p, and let T n → +∞ be any sequence satisfying φ Tn (p) → p. Fix any auxiliary Riemannian metric g and, for n large enough, consider the loop k (T n , p) obtained by concatenating to φ [0,Tn] (p) a short unit speed g-geodesic φ Tn (p) to p inside V . Since the g-norm of β is bounded near p, we get Since V is contractible, the limit obtained by replacing the loop k (T n , p) on the right-hand side by any k(T n , p) exists and is again equal to lim T →∞ The above argument proves two facts. The first is that S ∩ R φ ⊂S μ,y . The second is that for every p ∈ S ∩ R φ the values of limits as in (5) do not depend on a particular sequence T n , and they define a function f μ,y ∈ L 1 (μ) satisfying Corollary 2.6 (Of the proof). If β ∈ Ω 1 L represents y ∈ H 1 (M \L; R) and We end with proofs of Lemmas 2.4 and 2.5.  (3). Let re iθ = x + iy be polar coordinates on D.
as r → 0. The conclusion follows since, by assumption, Proof of Lemma 2.5. Choose any closed 1-form β 0 in M \L representing y.
In tubular polar coordinates (t, r, θ) around a connected component γ of L we can write β 0 in the frame {dt, dr, dθ} as pdt + qdθ + df where p, q are constants depending only on y. Consider a smooth function ρ : Consider the smooth closed 1-form agreeing with β 0 far from γ, and with pdt + qdθ + d(ρf ) near γ. It still represents y since it differs from β 0 by d((1 − ρ)f ). Repeating this process near each connected component of L we obtain the desired representative in Ω 1 L .

Rotation numbers
Fix a connected component γ ⊂ L, and consider the trivial vector bundle E γ = T M| γ /T γ over γ. Let T > 0 be the primitive period of γ. Coordinates (t, x + iy = re iθ ), with t ∈ R/T Z and x + iy ∈ D, defined on a neighborhood of γ by a diffeomorphism as in (3) induce a trivializing frame {∂ x , ∂ y } on E γ , and a vector bundle isomorphism E γ R/T Z × R 2 . This frame induces an angular fiber coordinate on the circle bundle (E γ \0)/R + still denoted by θ ∈ R/2πZ with no fear of ambiguity. We end up with a bundle isomorphism (E γ \0)/R + R/T Z × R/2πZ, with coordinates (t, θ). The linearized flow dφ t on (E γ \0)/R + gets represented as the flow of a vector field of the form on this torus. Seeing b(t, θ) as a T Z × 2πZ periodic function on R 2 , this flow lifts to a flow on Definition 2.7. If y is cohomologous to pdt + qdθ near γ, then we define Remark 2.8. The argument in [2, pp. 104-105] proves that the limit in (8) exists, does not depend on (t 0 , θ 0 ), and the convergence is uniform in (t 0 , θ 0 ). The task now is to show that ρ y (γ) is independent of choice of coordinates. Another such choice induces new coordinates (t , θ ) ∈ R/T Z × R/2πZ on the circle bundle (E γ \0)/R + and, moreover, t = t. The degree of is independent of θ and denoted by m ∈ Z. Hence the class y is cohomologous to (p − 2πmq/T )dt + qdθ near γ and With these new choices we would then have defined ρ y (γ) to be

Proof of Theorem 1.3
Note first that (ii) ⇒ (i) follows from definitions.

Blowing periodic orbits up
Enumerate the components γ 1 , . . . , γ h of L, and denote their primitive periods by T j > 0. For each j ∈ {1, . . . , h} choose a neighborhood N j of γ j and an orientation preserving diffeomorphism as in (3) which is aligned with b, i.e. the loop t ∈ R/T j Z → Ψ −1 j (t, 1) ∈ M \L has zero algebraic intersection number with b, and for some (hence any) t 0 ∈ R/T j Z the algebraic intersection number of the loop θ ∈ R/2πZ → Ψ −1 j (t 0 , e iθ ) ∈ M \L belongs to {1, −1}. Remark 3.1. Such a choice is only possible since it is assumed in Theorem 1.3 that b comes from a Seifert surface for L. This is an absolutely crucial choice which the reader must keep in mind. It will not be used in this paragraph, but will play an important role in the proof of (iii) ⇒ (ii) in Theorem 1.3.
The Ψ j induce tubular polar coordinates (t, r, θ) ∈ R/T j Z × (0, 1] × R/2πZ around the γ j , as explained in 2.1. A smooth 3-manifold M L can be constructed by blowing L up: more precisely it is defined as where, for each j ∈ {1, . . . , h}, the point (t, r, θ) From now on we fix j, work on N j , and consider Z = (Ψ j ) * X = Z(t, x+ iy). Then Z is a vector field on R/T j Z × D ⊂ R/T j Z × C, and as such it can be seen as smooth function Z : R/T j Z × D → R × C. Write x + iy = re iθ and consider the smooth map Then Φ −1 is well-defined and smooth on R/T j Z × D\0, and is a smooth vector field on R/T j Z×(0, 1]×R/2πZ. We claim that there exists a smooth extension is smooth (the second component of Z is a complex number). Moreover, Hence, if we see the second component as an element of R 2 , we get which, as the reader will immediately see, is smooth all the way up to r = 0. Hence it has a smooth extension W to R/T Z × (−∞, 1] × R/2πZ, and at r = 0 this vector field has no component in ∂ r . In fact, we see from the above formula that It follows that X| M \L can be smoothly extended to a vector field tangent to the tori to have no zeros on M L \D L . We will also denote the flow of X L by φ t with no fear of ambiguity. Remark 3.2. The linearized flow dφ t | (0,0) (s 0 , u 0 ) along γ j in coordinates (t, x+ iy) induced by Ψ j is the solution of the initial value problem From (12) we see that u(t) satisfies the linear equationu(t) = A 2 (t, 0)u(t).
With the aid of Ψ j the coordinates x + iy are precisely induced by the frame which is used to write down the vector field (7). This shows that the matrix M (t) representing the linearized flow on E γj satisfiesṀ (t) = A 2 (t, 0)M (t). Thus, M (t)u 0 = u(t), and in polar coordinates u(t) = r(t)e iθ(t) one easily computesθ This gives a concrete formula for the function b(t, θ) appearing in (7).

Schwartzman cycles and structure currents
The manifold M L (10) was obtained by blowing L up, and the smooth domain D L ⊂ M L was defined to be the closure of M \L in M L . We see that has boundary equal to ∂D L = h j=1 Σ j where the Σ j are the tori (16). The smooth vector field X L (15) restricts to D L as the unique continuous extension of X from M \L to D L , and it is tangent to ∂D L .
In [4] de Rham equips Ω 1 (M L ) with the C ∞ loc -topology and defines a 1-current with compact support as an element of its topological dual C 1 = Ω 1 (M L ) . The space C 1 is equipped with its weak* topology. It is a useful fact, proved in [4, §17], that the map Ω 1 (M L ) → C 1 (topological dual of C 1 with its weak* topology) given by ω → ·, ω is a linear homeomorphism; in other words Ω 1 (M L ) is reflexive.
More generally, one may consider the space C p of p-currents with compact support, defined as the topological dual of Ω p (M L ) equipped with the C ∞ loc -topology. As before C p is equipped with its weak* topology. The boundary operator be the subset of those which are X L -invariant probability measures supported on D L . Any μ ∈ P defines a 1-current c μ ∈ C 1 by the formula We follow Sullivan's notation and write c μ = ML X L dμ. A simple calculation shows that if μ ∈ P is supported in D L , then c μ is a cycle if, and only if, μ is X L -invariant. The elements of the set will be called Schwartzman cycles.
In [16] C is called the cone of structure currents in D L .

Compactness
Lemma 3.3. The following hold: Proof of (I). We first claim that for every neighborhood O of 0 in C 1 there exists δ > 0 such that if p 1 , . . . , p N ∈ D L and a 1 , . . . , a N > 0 satisfy i a i < δ, then i a i δ pi ∈ O. Here N ≥ 1 is arbitrary. In fact, by the definition of weak* topology we find η 1 , . . . , η J ∈ Ω 1 (M L ) and > 0 such that Fix R > max j η j (X L ) L ∞ (DL) and set δ = /R. Consider a finite sum i a i δ pi where the p i are points of D L and the a i > 0 satisfy i a i < δ. Then We claim that c, ω > 0 for every c ∈ C \{0}. In fact, fix any c ∈ C \{0} arbitrarily. Since and a i > 0 for all i, then i a i ≥ δ and we can conclude that By the definition of C we get c, ω ≥ δd > 0, as desired. Proof of (II). Define the convex set K = {c ∈ C | c, ω = 1} where ω is a 1-form satisfying (I). Note that ω(X) is pointwise strictly positive over D L since for all p ∈ D L we have δ p ∈ C \{0}. The set K is closed since it is the intersection of two closed sets. If we can show that K is contained on a compact subset of C 1 , then it will follow that K is compact. We claim that the set { c, η | c ∈ K} ⊂ R is bounded, for every η ∈ Ω 1 (M L ). By the definition of C one needs only to consider the case where c ∈ K is a finite linear combination of Dirac currents at points of D L with positive coefficients. There exists a constant A > 0, depending on η and ω such that |η(X L )| ≤ Aω(X L ) holds pointwise on the compact set D L .
as desired. We apply the Banach-Steinhaus Theorem [14, 2.5] to conclude that K is an equicontinuous set of linear functionals, namely there exists a neighborhood V of 0 in Ω 1 (M L ) such that Now the Banach-Alaoglu theorem [14, 3.15] asserts that K V is compact. Compactness of K follows. The map f → c, f dx 1 defines a distribution on U of order 0; here f stands for a test function on U . This means that c, f n dx 1 → 0 holds for any sequence f n of test functions on U supported on a common compact subset F ⊂ U , and satisfying f n → 0 uniformly. This is true since it is clearly true for finite combinations of Dirac measures, and hence also for currents in their closure. Similarly, we conclude that c, f dx 1 ≥ 0 holds for any non-negative test function on U .

Representation by measures
We have checked that we can apply the Riesz representation theorem to find a unique positive Borel measure μ on U , finite on compact subsets of U , such that c, f dx 1 = U f dμ for all compactly supported continuous functions f : U → R. Pushing forward to V we get a unique Borel measure on V , supported on V ∩ D L and still denoted by μ, such that c, ω = V ω(X L ) dμ holds for all ω ∈ Ω 1 (M L ) which is compactly supported on V . The uniqueness property allows us to patch such local constructions to obtain a unique Borel measure μ on M L , supported on D L , satisfying for all ω ∈ Ω 1 (M L ). The total μ-measure of M L is finite since compact subsets have finite measure and μ is supported in D L .

Proof of (iii) ⇒ (ii)
We only deal with the case L = ∅; the case L = ∅ is easier and left to the reader. Recall the chosen tubular neighborhoods N j of the connected components γ j ⊂ L, used to blow L up and obtain the manifold M L (10) as explained in 3.1. These were equipped with tubular polar coordinates (t, r, θ) in R/T j Z × (0, 1] × R/2πZ on N j \γ j in such a way that the boundary component of D L corresponding to γ j is the torus Σ j = R/T j Z × 0 × R/2πZ. Assume that b satisfies (iii) in Theorem 1.3. Since the Ψ j are aligned with b as explained in 3.1, we can choose a closed 1-form β ∈ Ω 1 (M L ) that represents the class y b in M \L and is written as on the end R/T j Z × (−∞, 0] × R/2πZ corresponding to γ j , for some j ∈ {1, −1}. In particular, the restriction of β to M \L belongs to Ω 1 L . For simplicity we may just write y instead of y b . Remark 3.5. If we fix t 0 ∈ R/T j Z and r 0 > 0 small enough then j is the algebraic intersection number of the loop θ ∈ [0, 2π] → (t 0 , r 0 , θ) with the class b.
Denote by P XL (D L ) the set of Borel probability measures on M L supported in D L which are invariant by the flow of X L . We claim that Proof. As in (14) the vector field X L | Σj is written as ∂ t + b(t, θ)∂ θ . We lift it as a periodic vector field on the universal covering R 2 , where t and θ lift to real-valued coordinates. Let θ(t; t 0 , θ 0 ) be the unique solution toθ = b(t, θ) with value θ 0 at time t = t 0 . The flow of X L on Σ j is φ t (t 0 , θ 0 ) = (t + t 0 , θ(t + t 0 ; t 0 , θ 0 )), modulo T j Z × 2πZ. It follows from the argument in [2, pp. 104-105], and from the definition of ρ y (γ j ), that the sequence of functions converges to ρ y (γ j ) uniformly in (t 0 , θ 0 ) ∈ R 2 as n → ∞. By periodicity of b, the g n descend to functions on Σ j R/T Z × {0} × R/2πZ → R. We compute Thenμ and the μ j are X L -invariant Borel measures, and μ =μ + j μ j . We have because ·, η > 0 on C \{0}. Consider inclusions ι : D L → M L and ι j : Σ j → M L . The periods of ι * η are integers since y comes from an integral class on M \L and the inclusion map M \L → D L induces isomorphism in cohomology. Using coordinates (t, θ) in R/T j Z × 0 × R/2πZ = Σ j , then {dt, dθ} is a basis of H 1 (Σ j ; R), and ι * j η is homologous to j dθ/2π. The set of periods of ι * η is equal to the set of all integers Z. This is so because the class y pulls back to ±dθ/2π near each Σ j . Choose p 0 ∈ M \L and define a map pr : D L → R/Z by setting pr(p) to be the integral of ι * η along any path from p 0 to p modulo Z The map pr : D L → R/Z is a smooth surjective submersion in view of (24). An important property that follows from this construction is that if c : The preimages pr −1 (x) are the leaves of a foliation of D L obtained by integrating ker ι * η. Each pr −1 (x) is a compact embedded submanifold of D L that intersects the boundary ∂D L cleanly, since it is transverse to X L and X L is tangent to ∂D L . It follows that pr −1 (x) ⊂ D L is a smooth embedded surface transverse to X L with boundary equal to pr −1 (x) ∩ ∂D L .
Each leaf pr −1 (x) can be co-oriented by the vector field X L and can also be co-oriented by pulling back the canonical orientation of R/Z via the map pr. These co-orientations coincide, basically by construction. Equation (25) implies that each pr −1 (x) induces the class b seen in H 2 (M, L; R). Later we will be in position to show that the same is true with Z coefficients.
We need to show that each pr −1 (x) ∩ M \L is a properly embedded surface spanned by L which is a global surface of section. First, note that all trajectories in D L will hit all leaves pr −1 (x) in finite time both in the future and in the past: this follows from compactness of D L and from (24). In particular, the return time is finite for all trajectories in M \L.
The next step is to study the structure of the boundary of the leaves. We claim that for all j and x the submanifold pr −1 (x) ∩ Σ j is a circle dual to j dt/T j . Let α be a connected component of pr −1 (x) ∩ Σ j oriented as a piece of the boundary of pr −1 (x). Then α is an embedded circle in Σ j . It is non-trivial in H 1 (Σ j ; Z) since otherwise it would bound a disk D ⊂ Σ j with X L ∂D, thus forcing a singularity of X L on Σ j . Let {e 1 , e 2 } be a basis in H 1 (Σ j ; Z) dual to {dt/T j , dθ/2π} and write α = n 1 e 1 + n 2 e 2 in homology. Since α is tangent to the kernel of ι * j η we get 0 = α ι * j η = j n 2 ⇒ n 1 = 0 and n 2 = 0. If |n 1 | > 1, then α would have self-intersections. Hence n 1 = ±1. One can use Remark 3.5 to conclude that n 1 = j . So far we have proved that if pr −1 (x)∩Σ j is non-empty, then its components are embedded circles dual to j dt/T j . Hence to conclude our study of the boundary we need to show that pr −1 (x)∩Σ j is connected and non-empty. It has to be non-empty since Σ j is an invariant torus; hence if pr −1 (x) did not touch Σ j then trajectories in Σ j would not hit pr −1 (x), impossible. Let s be the number of connected components of pr −1 (x) ∩ Σ j , and consider the loop c : R/2πZ → Σ j defined by c(θ) = (t 0 , 0, θ) in polar coordinates near Σ j . Then the degree d of pr • c satisfies |d| = s. Using (25) together with the fact that y j dθ/2π near Σ j we compute d = y, c * [R/2πZ] = j dθ/2π, c * [R/2πZ] = j ⇒ s = 1.
It only remains to be shown that leaves are connected. Fix x ∈ R/Z, denote the first return map pr −1 (x) → pr −1 (x) by ψ, and denote by the map defined by requiring Ψ(p) to be the first point in the future trajectory of p ∈ D L \pr −1 (x) that belongs to pr −1 (x). In other words, Ψ(p) is the first hitting point in pr −1 (x). By transversality of X L to pr −1 (x), ψ is a smooth diffeomorphism and Ψ is a smooth submersion. For each connected component Y of pr −1 (x) we set In other words, C Y is the set of points p ∈ D L which will first hit pr −1 (x) at a point of Y in its future trajectory φ (0,+∞) (p). We claim that pr −1 (x) has no boundaryless connected components. Our argument is indirect. Suppose that some connected component has no boundary. By compactness of pr −1 (x), we find finitely many boundaryless connected components Y 1 , . . . , Y N of pr −1 (x) such that Y j = ψ(Y j−1 ) for all 2 ≤ j ≤ N and Y 1 = ψ(Y N ). It follows that C Y1 ∪ · · · ∪ C YN is open and closed in D L ; hence, it is equal to D L since D L is connected. But this would force pr −1 (x) to be equal to Y 1 ∪· · ·∪Y N and have no boundary, contradicting our standing assumption L = ∅.
The previously explained analysis of the boundary of pr −1 (x) told us that each Σ j contains exactly one connected component of the boundary of pr −1 (x), which is a circle dual to j dt/T j . It follows that if p is a boundary point of pr −1 (x), then ψ(p) is in the same connected component of the boundary as p. Since we already proved that all connected components of pr −1 (x) have non-empty boundary, we can conclude that ψ preserves each connected component of pr −1 (x). It follows that C Y is open and closed in D L for each connected component Y ⊂ pr −1 (x). By connectedness of D L , each such C Y is either empty or equal to D L . Hence pr −1 (x) is connected.
Finally, we need show that the projection of pr −1 (x) to M induces the class b ∈ H 2 (M, L; Z). We already know this to be true with R-coefficients. We argue as follows: Consider an embedded compact oriented surfaceB ⊂ D L that intersects ∂D L cleanly, satisfies ∂B =B ∩ ∂D L , and projects to M as a