Contact Homology of Orbit Complements and Implied Existence

JOURNALOF MODERN DYNAMICS doi: 10.3934/jmd.2011.5.409 VOLUME 5, NO. 3, 2011, 409–472

CONTACT HOMOLOGY OF ORBIT COMPLEMENTS AND IMPLIED EXISTENCE

AL MOMIN (Communicated by Leonid Polterovich)

ABSTRACT. For Reeb vector ﬁelds on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other knotting/linking properties relative to the original set. We work out a few examples on the 3-sphere to illustrate the theory, and describe an application to closed geodesics on S2 (a version of a result of Angenent in [1]).

1. INTRODUCTION Let V be a closed 3-manifold with a contact form λ and associated Reeb vector field Xλ. In this article we will be concerned with the following question about closed orbits: General Question: If one has a Reeb vector field with a known set of closed Reeb orbits L, can one deduce the existence of other closed Reeb orbits from knowledge about L? Let us call an affirmative answer to such a question an implied existence result. We will see that in some instances one can obtain an affirmative answer to this question. These results agree with affirmative answers to analogous questions for certain similar conservative systems in low dimensions studied in e.g., [1, 22, 23], but the present methods differ significantly. Specifically, to exhibit implied existence we study cylindrical contact homology on the complement of L for nondegenerate contact forms λ imposing as few conditions on the orbit set as we can manage. We will show in detail one approach to such a theory inspired by the intersection theory of [40] for the necessary compactness arguments (it has been pointed out to the author that other approaches are possible e.g., using convexity for compactness arguments, such as in [8]). We use established techniques [9,5] to compute this homology in some cases, and use these computations as a tool to affirmatively answer the above question in certain cases.

Received December 22, 2010; revised October 12, 2011. 2000 Mathematics Subject Classiﬁcation: Primary: 37J45; Secondary: 53D42. Key words and phrases: Reeb vector ﬁeld, periodic orbit, forcing, contact homology, link complement, holomorphic curves, intersection theory.

1.1. A version of cylindrical contact homology on Reeb orbit complements. In this paper we denote by V a 3-dimensional manifold. A one-form λ on V is called a contact form if λ dλ is everywhere nonzero. Such a one-form ∧ uniquely determines its Reeb vector field X by the equations λ(X ) 1 and λ λ ≡ dλ(X , ) 0. It also determines a distribution ξ kerλ, which is a contact λ · ≡ = structure, and two forms λ induce the same contact structure if and only if ± λ f λ for some nowhere vanishing function. We note that different choices + = · − of contact form which induce the same contact structure have different Reeb vector fields, and the dynamics of the Reeb vector field can vary drastically between two different choices of contact forms. The Conley–Zehnder index of a closed orbit for this vector field is a measure of the rotation of the flow around the closed orbit - see Proposition 2.4 for one characterization of the Conley– Zehnder index. The Conley–Zehnder index usually depends on a choice of a trivialization of the contact structure, but in many cases we will consider, e.g., the tight 3-sphere, there is a global trivialization which is used to define Conley–Zehnder indices independently of choices.

1.1.1. The hypotheses (E) and (PLC). We introduce two types of technical hypotheses below, which are not mutually exclusive. They simplify the construction of contact homology on V \L which we use to deduce implied existence. We will give some examples of forms on the 3-sphere later which we hope will help to clarify these hypotheses. We will consider pairs (λ,L) in which λ is a nondegenerate contact form on V and L is a link composed of closed orbits of the Reeb vector ﬁeld of λ. Sometimes it may be convenient to refer to the subset of closed Reeb orbits with image contained in L; we will abuse language and denote this subset of closed Reeb orbits by L again i.e., L refers to both a closed embedded submanifold which is tangent to the Reeb vector ﬁeld X , as well as the set of solutions x : R/(T Z) V,x˙ X (x) modulo reparametrization with image contained in L · → = (this includes multiple covers of components of L). Following standard terminology, an orbit is elliptic if the eigenvalues of its linearized Poincaré return map (a linear symplectic map on ξ) are nonreal.

DEFINITION 1.1. We will say (λ,L) satisﬁes the “ellipticity” condition (abbrevi- ated by (E)) if each orbit in L - that is the orbit L and all its multiple covers Lk - is non- • degenerate elliptic, and each contractible Reeb orbit y not in L “links” with L. To be precise, we • will assume that for any disk with boundary y, L intersects the interior of the disk.

We remark that the first condition can be determined from the linearized return map of the simply covered orbit L: it is satisfied if and only if the ei- 2πiθ genvalues of the linearized return map are e± for some irrational real number θ. We shall see the hypotheses of Definition 1.1 force the compactness of

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 411 moduli of holomorphic cylinders in V \L necessary to deﬁne cylindrical contact homology. Before we continue, let us note a simple example:

EXAMPLE 1.2. Consider R4 R2 R2 with the usual polar coordinates (r ,θ ) = × i i (i 1,2) on each R2 factor of R4 and let E be the ellipsoid determined by the = equation r 2 r 2 1 2 1 a + b = where a,b are positive constants. The one-form 2 X 1 2 λ0 ri dθi = i 1 2 = restricted to E deﬁnes a contact form which we denote λ. If the ratio a/b is irrational, then there are precisely two geometrically distinct closed Reeb orbits, P E C {0},Q E {0} C. The orbits P,Q and all their multiple covers P k ,Qk = ∩ × = ∩ × are nondegenerate and elliptic with Conley–Zehnder indices j ³ a ´k ¹ µ b ¶º CZ(P k ) 2 k 1 1, CZ(Qk ) 2 k 1 1. = + b + = + a + Since the linking numbers `(P k ,Q) k,`(P,Qk ) k, the pairs (λ,P), (λ,Q), (λ,P = = t Q) all satisfy (E). See Example 6.1 in Section6 for a more interesting class of examples. There is another way to control compactness of holomorphic cylinders in V \L. Again assume L is a link of closed orbits for λ, and let [a] be the homotopy class of a loop a in the complement.

DEFINITION 1.3. We say that (λ,L,[a]) satisﬁes the “proper link class” condition (PLC) if for any connected component x L, no representative γ [a] can be ho- • ⊂ ∈ motoped to x inside V \L i.e., there is no homotopy I : [0,1] S1 V (with × → I(0, ) γ and I(1, ) x) such that I([0,1) S1) V \L. Such a homotopy · = · = × ⊂ class [a] will be called a proper link class for L. for every disk F with boundary ∂F y a closed (nonconstant, but possibly • = multiply covered) Reeb orbit (possibly in L), there is a component x of L that intersects the interior of F . Figure 1(1) provides an example of a proper link class: the components of the link have linking number 0 with each other, but the loop a has linking number 1 with each. Thus it is impossible to homotope a to one component in the complement of the other, since the linking numbers would be different and constant along the homotopy. In contrast, Figure 1(2) is a counter-example, because one easily ﬁnds a homotopy a from a a to L a such that for s = 0 = 1 each s 1 the loop a lies in the complement V \L. 6= s The class [a] is meant to be analogous to a proper braid class studied in [22,23]. Both conditions (E) and (PLC) contain a “no contractible orbits in V \L” hypothesis. Such a hypothesis is necessary to preclude “bubbling”, which is

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 412 AL MOMIN

First component of L

The loop "a" L a

Second component of L

(1) The homotopy class of a is a proper link class (2) Here a does not repre- for the two component link L above. sent a proper link class for L.

FIGURE 1.1. Example and counter-example of proper link classes in Definition 1.3. an obstruction to defining cylindrical contact homology in general [14]. The hypotheses we place on contractible orbits is more restrictive than necessary, but suffices for many applications and is relatively simple to state and check. We remark that the “no contractible orbits” hypothesis in (PLC) is more restrictive. We point to Examples 6.1, 6.2 below for concrete examples of orbits and (Morse–Bott) contact forms on the tight S3 satisfying conditions (E) and (PLC) respectively.

1.1.2. An invariant, contact homology. We deﬁne a relation on pairs (λ,L) as above. Suppose we have two such pairs (λ ,L ). We write (λ ,L ) (λ ,L ) if ± ± + + ∼ − − ker(λ ) ker(λ ) and L L . We say (λ ,L) (λ ,L) if they are related by + = − + = − + ≥ − ∼ and the Conley–Zehnder indices of the orbits in L (including multiple covers) are always greater or equal when considered as orbits for λ than when con- + sidered as orbits for λ . If (λ0,L) (λ1,L) and (λ1,L) (λ0,L) then we write − ≥ ≥ (λ ,L) (λ ,L). 0 ≡ 1 In Section4 we associate with a pair ( λ,L) satisfying (E) and the homotopy class [a] of a loop a, or with a triple (λ,L,[a]) satisfying (PLC), an invariant CCH [a]([λ] rel L). It is invariant in the sense that it depends only on the equivalence∗ class in case (λ,L) satisfy (E) (that is, it depends on the Conley–Zehnder ≡ indices of the elliptic orbits in L). In fact, in this case it can depend on the equivalence classes of and not just the coarser equivalence class , which ≡ ∼ we will see in Example 6.1. In the case (PLC) it depends only on equivalence classes of (i.e., only on the contact structure ξ). CCH [a]([λ] rel L) will be an ∼ ∗ isomorphism class of graded vector spaces, so it makes sense to say whether or not it is zero.

1.2. A general implied existence result. From the existence of the invariant CCH [a]([λ] rel L) alone one can deduce the existence of periodic orbits in the homotopy∗ class [a] under the hypotheses required to deﬁne the invariant. These hypotheses, however, are unnecessarily restrictive. In Section5 we derive the

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 413 following implied existence result which signiﬁcantly reduces the assumptions on λ:

THEOREM 1.4. Suppose λ is a contact form with a closed orbit set L. Suppose either L is nondegenerate and elliptic, or • 1. L is such that every disk F with boundary ∂F L and [∂F ] 0 H (L) • ⊂ 6= ∈ 1 has an interior intersection with L, and 2.[ a] is a proper link class relative to L. If [a] is simple and CC H [a]([λ] rel L) 0, then there is a closed Reeb orbit in the ∗ 6= homotopy class [a]. To clarify a possible point of confusion in the statement of Theorem 1.4, [a] when we say CCH ([λ] rel L) 0 we mean that there is a λ0 λ resp. λ0 λ ∗ 6= ≡ ∼ satisfying (E) resp. (PLC) for which CCH [a] can be computed and is nonzero. ∗ REMARK 1.5. When [a] is a proper link class, Theorem 1.4 requires no nondegeneracy hypotheses whatsoever on λ, and in the ellipticity case requires only nondegeneracy of the orbits in L. These hypotheses apply in many interesting cases: for an instance of an application of Theorem 1.4 using the first hypothesis see Example 6.1, and for examples using the second hypothesis see Section 6.4. Specifically, Lemma 6.6 shows that hypothesis (1) in Theorem 1.4 is satisfied for a large class of interesting examples. 1.3. Some applications in S3. The results here are corollaries of Theorem 1.4 in Section 1.2 and the examples and computations of Section6. It is useful to recall some notions and facts from contact topology, which can be found in the text [20]. A transverse link in a contact manifold V with contact structure ξ is a link L such that at each point x L any nonzero tangent ∈ vector v T L\0 satisfies v ξ .A transverse isotopy of a transverse link L is a ∈ x ∉ x smooth isotopy Lt starting at the link L through transverse links. It is a useful fact - see Theorem 2.6.12 in [20] - that a transverse isotopy can be extended to an isotopy of (V,ξ) through contactomorphisms (diffeomorphisms preserving the contact structure). Transverse knots and links admit an invariant called the self-linking number. This invariant is an integer, and it is defined as follows. Given a transverse link L, a Seifert surface ΣL for L, and a nonvanishing section s of ξ , consider the link L0 obtained by pushing L slightly in the direction of |ΣL the vector field s along L: the self-linking number is the oriented intersection 3 number sl(L) L0 Σ . As a remark, in S with the tight contact structure this = · L number is independent of the choice of Seifert surface. It satisfies an inequality known as the Bennequin bound (see e.g., Theorem 4.6.34 in [20]) (1.1) sl(L) χ(Σ ) ≤ − L where ΣL is any Seifert surface for L. An example of a transverse link in S3 with its standard tight contact structure is the Hopf link, which is precisely the pair of knots P Q of Example 1.2. We t will also call any transverse link L in S3 which is transversely isotopic to P Q t JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 414 AL MOMIN a Hopf link. By Theorem 4.1 in [15], one can identify these Hopf links by its (topological) link type and self-linking number alone. We note that the self- linking number of P Q is sl(P Q) 0. t t = Our first application is the existence of analogs of “(p,q)-type orbits” if a certain pair of closed Reeb orbits form a transverse Hopf link of self-linking number 0 as described above and satisfy a “non-resonance” condition (namely the pair violates the “resonance” condition introduced in [2] when the differential of cylindrical contact homology in the tight S3 vanishes). In the following statement `( , ) denotes the linking number of two knots in S3. We also intro- · · duce the following notation: given two pairs of real numbers (x1, y1),(x2, y2) in the subset S {(x, y) x 0 or y 0} R2 we write (x , y ) (x , y ) if the vector = | ≥ ≥ ⊂ 1 1 < 2 2 (x2, y2) lies counter-clockwise to the vector (x1, y1) in S.

COROLLARY 1.6. Let λ be a tight contact form on the 3-sphere. Suppose that there is a pair of periodic orbits L ,L such that L L is a Hopf link with self- 1 2 1 t 2 linking 0. Suppose L1 and L2 (and all multiple covers) are nondegenerate elliptic, and let θ1,θ2 be the unique irrational numbers satisfying CZ(Lk ) 2 k (1 θ ) 1, for all k 1, 1 = b + 1 c + ≥ CZ(Lk ) 2 k (1 θ ) 1, for all k 1. 2 = b + 2 c + ≥

Then if p,q is a relatively prime pair of integers such that (p,q) S R2 and ∈ ⊂ (1,θ ) (p,q) (θ ,1), or (θ ,1) (p,q) (1,θ ), 1 < < 2 2 < < 1 there is a simple closed Reeb orbit P(p,q) such that `(P ,L ) q and `(P ,L ) p. (p,q) 1 = (p,q) 2 = REMARK 1.7. There are no hidden hypotheses on the contact form being either dynamically convex or nondegenerate (except in the explicit hypotheses on the orbits L1,L2). There are other approaches to prove such a result using the surface of section constructed by [31,27] and applying the result of [18]; however, the construction of these surfaces of section require additional hypotheses, so it is not clear that they always exist. The result above applies even when there may be no such surface of section.

Proof. By Theorem 4.1 in [15], there is a transverse isotopy from L L to the 1 t 2 link H H in Example 6.1, which extends to an ambient contact isotopy, and 1 t 2 therefore a contactomorphism taking L L to H H . By applying the given 1 t 2 1 t 2 contactomorphism we can assume (λ,L L ) (λ0, H H ) with (λ0, H H ) 1 t 2 ≡ 1 t 2 1 t 2 from Example 6.1. It follows immediately from Theorem 1.4 (each component is elliptic by hypothesis) and the computation in Example 6.1.

It is interesting to interpret this result in the case the contact form is obtained from a metric on S2. In Section 6.2 we recall Poincaré’s inverse rotation number ρ(γ) associated with a closed geodesic γ, which roughly speaking

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 415 measures the density of zeros of solutions to the Hill equation for γ on increas- ingly large intervals (see equation (6.1)). Angenent [1] proved that if one has a simple, closed geodesic for a C 2,µ-Riemannian metric on S2, then whenever Poincaré’s inverse rotation number ρ 1, for every p/q (ρ,1) (1,ρ) there is a 6= ∈ ∪ geodesic γp,q with the flat-knot type of a (p,q)-satellite geodesic. Corollary 1.6 can be applied to this problem by the following observations. Given a metric, which can be taken to be Finsler, the geodesic flow on the unit cotangent bundle is actually the Reeb flow of the restriction of the canonical one-form to the unit cotangent bundle. The tight 3-sphere double covers the unit cotangent bundle, so the geodesic flow lifts to a Reeb flow on S3 for a (tight) contact form. Moreover, any simple closed curve traversed in opposite directions at unit speed lifts to a Hopf link with self-linking number 0 in S3. These facts are well known, and are explained briefly in Section 6.2. The following corollary, obtained by applying Corollary 1.6 to the Reeb flow just described and the Hopf link formed by the lifts of a simple closed curve which is a geodesic when traversed in both directions, is a version of Angenent’s result with stronger hypotheses on the rotation number and weaker conclusions about the resulting geodesics, but allowing the metric to be Finsler.

COROLLARY 1.8. Let F be a Finsler metric on S2. Suppose γ is a simple, closed 2 curve in S with the property that the two embeddings γ1,γ2 into the unit cotangent bundle defined by F which are obtained by traversing γ in opposite directions at unit speed are geodesics for F with irrational (and possibly different) inverse rotation numbers ρ1,ρ2. Then for every pair of relatively prime integers p,q such that (1,2ρ 1) (p,q) (2ρ 1,1), or (2ρ 1,1) (p,q) (1,2ρ 1), 1 − < < 2 − 2 − < < 1 − there is a geodesic γp,q for F with the following topological property. If γ1,γ2 3 denote the lifted double covers of the geodesics γ1,γ2 to S (traversed “forwards” 3 and “backwards”), then the lift γp,q of γp,q to S has linking number q with γ1 and p with γ2. In particular, the geodesics γp,q are topologically distinguished. The derivation of Corollary 1.8 from Corollary 1.6 is explained in some detail in Section 6.2 (one can see in the argument why the quantities “2ρ 1” ap- − pear). We make two comments about Corollary 1.8. The first is that it applies to any simple closed geodesic for a reversible Finsler metric (for example, any Riemannian metric) and in this case the rotation numbers ρ ρ ρ(γ) are 1 = 2 = the same, so the interval is nonempty whenever ρ(γ) 1. The second is that it 6= is interesting to note that in the case of “resonance” 1 ρ1 1/2, ρ2 1/2, 2ρ1 1 , > > − = 2ρ 1 2 − the interval is empty and there may be no other closed geodesics: thus, for example, this must be the case in the metrics constructed by Katok [34] having precisely two closed geodesics which cover the same embedded curve in opposite directions. In Section 6.2 we briefly discuss the relationship between the geodesic γp,q found here and the satellite geodesics found by [1].

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REMARK 1.9. In subsequent work with Umberto Hryniewicz and Pedro Salomão [25], the author and collaborators managed to remove the irrationality hypotheses on the components of the Hopf link/closed geodesics appearing in Corol- laries 1.6 and 1.8. This requires a more careful analysis of the asymptotic convergence of holomorphic cylinders which is somewhat special in the particular case of the Hopf link.

We give other examples by considering fibered knots or links in the 3-sphere, in which conclusions can be drawn without nondegeneracy hypotheses. One class of examples is obtained by considering open book decompositions, which are described in Section 6.3 and play a prominent role in contact topology due to a well-known classification theorem of Giroux [21]. Let us say that a knot B in the tight 3-sphere is a tight fibered hyperbolic knot if the following conditions hold: 1. (fibered) The knot B is the binding of an open book decomposition (see Section 6.3) of S3. 2. (tight) The contact structure supported by the open book (see Section 6.3) is the tight contact structure on S3. 3. (hyperbolic) If B is the binding of an open book decomposition, it follows that the complement V \B is diffeomorphic to a mapping torus Σ(S,h) of a surface with boundary S, where h : S S is a diffeomorphism called the → monodromy map. We will say B is hyperbolic if the monodromy map h of the associated open book decomposition is pseudo-Anosov. 1 1 4. The map h∗ I : H (S;R) H (S;R) (where S,h are as described in (3)) is − → invertible. We remark that the fourth condition actually follows automatically from the first; it will be a convenient fact (see Section 6.5 where we also explain why this follows). We make some further brief remarks about this class of knots in Section 6.5. For now we merely note that there are infinitely many examples, such as the Fintushel–Stern knot which is the Pretzel knot ( 2,3,7). For tight − fibered hyperbolic knots in S3 (see Section 6.5):

COROLLARY 1.10. Suppose λ is a tight contact form on S3. If its Reeb vector field has a closed orbit which is a tight fibered hyperbolic knot realizing the Ben- nequin bound (1.1) as a transverse knot, then there are infinitely many geometrically distinct closed Reeb orbits and the number of such orbits of period at most T is bounded below by an exponential function of T .

The required calculation is essentially due to Colin–Honda [9], but turns out simpler because one does not need to consider holomorphic curves that intersect the binding.

REMARK 1.11. Again, there are no hidden nondegeneracy hypotheses, and it applies to all tight contact forms (not only dynamically convex ones). Theorem 1.4 allows one to draw conclusions about free homotopy classes which must contain closed Reeb orbits.

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Finally, cylindrical contact homology is usually thought to be only applicable to tight contact structures. However, even for an over-twisted contact structure after removing a certain orbit set L it may be possible to apply cylindrical contact homology on the complement V \L. For example, we can take the figure eight knot in the 3-sphere V S3, which satisfies properties 1, 3, 4 in the = definition of fibered hyperbolic supporting knots, but not property 2 (that is, it is fibered and hyperbolic, but does not support the tight contact structure). Hence it supports an over-twisted contact structure instead; Theorem 1.4 still applies from which one can deduce the following:

COROLLARY 1.12. Let λ be a contact form for the (over-twisted) contact structure on S3 supported by the open book decomposition with binding the ﬁgure eight knot, page diffeomorphic to a once-punctured torus, and monodromy map given by the matrix transformation ·2 1¸ 1 1 Suppose λ has a closed Reeb orbit transversely isotopic to the binding. Then the number of geometrically distinct periodic orbits of action at most N grows at least exponentially in N. We note that in the above one can describe homotopy classes that must contain a closed orbit. We will go over the details of this particular example in Section6. An analogous result will hold for any other (non-tight) ﬁbered hyperbolic knot in S3, examples of which are plentiful [12]. Finally, in Corollary 6.3 we will also give another example, related to the example used in Corollary 1.6, which does not assume nondegeneracy of the orbit set L. We postpone the statement to Sec- tion6 where it will be easier to describe the link L.

1.4. Further comments and outline.

1.4.1. Comments. It is possible to study the cylindrical contact homology of stable Hamiltonian structures on complements of elliptic Reeb orbits (or in proper link classes) in general. For example, the mapping tori of Hamiltonian diffeomorphisms of surfaces fits in this category. In this case there are clearly no contractible Reeb orbits, and bubbling of spheres is not difficult to rule out in most cases. The portion of the loops space corresponding to the first-return map are all simple homotopy classes and thus CCH is well-defined and coin- ∗ cides with Floer homology. This particular case has been carried out explicitly (at least in the case the surface is a disk) and is called “braid Floer homology” in [23].

1.4.2. Outline. In Section2 we recall the geometric and analytic context. Sec- tion3 describes an intersection theory of pseudoholomorphic curves in symplectizations [39] and derives some compactness results from this intersection theory. In Section4 we describe the chain complexes, maps and homotopies. In Section5, we describe how to draw conclusions when the contact form might

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 418 AL MOMIN be degenerate or there may be contractible (in V \L) closed Reeb orbits. In Sec- tion6 some explicit examples on the 3-sphere are worked out in detail.

2. REVIEWOFGEOMETRICANDANALYTICCONTEXT 2.1. Symplectizations. Let (V,ξ) be a co-oriented contact manifold. Then there is a symplectic manifold associated with it, W ˙ ξ⊥\0 (the annihilator of ξ in ξ= T ∗V minus the zero section) which is a subbundle and symplectic submanifold of T ∗V with respect to the exterior derivative of the Liouville form, dθcan. The symplectic manifold (W ,ω dθ ) is called the symplectization of ξ ξ = can|Wξ (V,ξ). Because ξ is co-oriented there are two connected components, Wξ, . The ± co-orientation distinguishes the connected component Wξ, with the property + that any contact form inducing the chosen co-orientation is a section of Wξ, . + From now on when we write Wξ we mean this connected component only. There is a natural R action (a,λ) a λ ea λ given by scalar multiplication in 7→ ∗ = · the fibers, and a natural projection to V by restricting the projection T ∗V V → to Wξ. Any positively co-oriented contact form λ for ξ defines a global section of Wξ which gives an identification µ ¶ λ0 ψλ : Wξ R V, λ0 a ln ,x Π(λ0) , =∼ × 7→ = λ =

(where Π denotes the restriction to W of the projection T ∗V V ) in which ξ → the symplectic form for W is d(ea λ). ξ · Given such a λ we denote by X the Reeb vector field, defined by X λ 1, X dλ 0. ¬ ≡ ¬ ≡ We will later assume that λ is nondegenerate, a generic condition meaning that no closed periodic orbit has a Floquet multiplier equal to 1; for the moment this is not necessary. Given a contact form λ, we define a splitting of TWξ: 1 1 T W R ∂ R d(ψ− ) (0, X ) d(ψ− ) (0 ξ) λ0(x) ξ = · a ⊕ · λ ψ(λ0) ⊕ λ ψ(λ0) × : R ∂a R Xb ξb. = · ⊕ · ⊕ Though the subspaces R Xb and ξb both depend on the choice of λ, we might · abuse notation and use X to denote Xb and ξ for ξb.

2.1.1. Almost-complex structures. Given a contact form λ for ξ, consider the symplectic vector bundle (ξ,dλ) over V and denote by J (ξ) the set of dλ- compatible almost-complex structures. This deﬁnes a set of almost-complex structures on ξb via 1 Jb(λ0) (dψ− )ψ(λ ) ξ J (dψλ)λ . = λ 0 | ◦ ◦ 0 |ξb

Finally this extends to all of Tλ0 Wξ by

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We denote the set of almost-complex structures on TWξ that arise in this way J (λ); we can identify this with J (ξ). The almost-complex structures in J (λ) are called cylindrical almost-complex structures. 2.1.2. Cylindrical ends. To compare the contact homology between different choices of λ, one studies the symplectization and almost-complex structure with cylindrical ends. We define an order relation on the fibers of π: Wξ V 1 → as follows: given λ ,λ π− (x), we say λ λ (resp. λ λ ) if λ /λ 1 0 1 ∈ 0 ≺ 1 0 ¹ 1 1 0 > (resp. λ /λ 1). Given two contact forms λ ,λ for ξ, we write λ λ if 1 0 ≥ 0 1 0 ≺ 1 λ (x) λ (x) on each fiber, or equivalently when we write λ r λ we have 0 ≺ 1 1 = 0 r 1 pointwise. If λ λ , then we set > − ≺ + © ª W (λ ,λ ) λ Wξ λ (π(λ)) λ λ (π(λ)) . − + = ∈ | − ¹ ¹ + This is an exact symplectic cobordism between (V,λ ),(V,λ ). Let © − ª + W −(λ ) λ Wξ λ λ (π(λ)) , − = ∈ | ¹ − © ª W +(λ ) λ Wξ λ (π(λ)) λ , + = ∈ | + ¹ so (by ∂+ we mean ∂a is outward pointing and by ∂− we mean ∂a is inward pointing) [ [ Wξ W −(λ ) W (λ ,λ ) W +(λ ). = − − + + ∂+W −(λ ) ∂+W (λ ,λ ) − = − + ∂−W (λ ,λ ) ∂−W +(λ ) − + = + An almost-complex structure with cylindrical ends is then an almost-complex structure J such that

J agrees with Jb cJ (λ ) on (a neighborhood of) W ±. • ± = ± ± J is ω -compatible on all of W . • ξ Denote the set of such almost-complex structures by J (Jb , Jb ). A well-known − + argument shows that this is a nonempty contractible set. For J J (Jb , Jb ) the ∈ + − almost-complex manifold (W, J) is said to have cylindrical ends W ±. It is also necessary to consider families of almost-complex structures; we will denote by Jτ(Jb , Jb ) the space of smooth paths [0,1] J (Jb , Jb ), τ Jτ. − + → − + 7→ 2.1.3. Splitting almost-complex structures. Suppose we have λ λ0 λ . Con- − ≺ ≺ + sider cylindrical almost-complex structures Jb , Jb0, Jb and almost-complex struc- − + tures J1 J (Jb , Jb0), J2 J (Jb0, Jb ). Then there is a smooth family of almost- ∈ − ∈ + complex structures J 0 on W for R 0 defined by (using the coordinates ψ ): R ξ ≥ λ0 ½ J1(a R,x) if a R J 0 (a,x) + ≤ R = J (a R,x) if a R 2 − ≥ − which fits together smoothly because Jb0 is R-translation-invariant. Let δ 0 be an arbitrarily small but fixed number. Choose diffeomorphisms > g − : R ( ,0), g + : R (0, ) such that g −(a) a δ if a 0 and g +(a) a δ δ → −∞ δ → ∞ δ = − ≤ δ = + if a 0. Choose a smooth family of diffeomorphisms g (δ,R) : R R (for R δ) ≥ → ≥ with the properties g (δ,δ)(a) a, • = JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 420 AL MOMIN

½a R δ if a R g (δ,R)(a) − + ≥ • = a R δ if a R (δ,R) (+δ,R)− ≤ − g ( R),g ( R) converge to g +,g − in C ∞ (R,R) as R . • · + · − δ δ loc ↑ ∞ Then we have diffeomorphisms G(δ,R)(a,x) (g (δ,R)(a),x) (using the coordi- = (δ,R) nates ψλ0 for Wξ), and almost-complex structures JR G JR0 . = ∗ We may concatenate the matching Hamiltonian structured ends of (Wξ, J1), (W , J ) to get (W , J ) (W , J ) (see Section 3.1). Then we have a diffeomor- ξ 2 ξ 1 ¯ ξ 2 phism to the concatenation G : W (W, J ) (W, J ) deﬁned by ξ → 1 ¯ 2 g −(a) a 1 e δ λ0(x) e λ0(x) ψ− (a,x), on W −(λ0)\λ0(V ) Wξ. • · 7→ = λ0 ⊂ g +(a) a 1 e δ λ0(x) e λ0(x) ψ− (a,x), on W +(λ0)\λ0(V ) Wξ. • · 7→ = λ0 ⊂ e0 λ (x) ( ,x) ( ,x) { } V { } V , on λ (V ) W . • · 0 7→ ∞ ∼ −∞ ∈ +∞ × ∼ −∞ × 0 ⊂ ξ It follows from the construction of JR that G JR converges to the concatenated ∗ almost-complex structure J J uniformly on compact subsets of W \λ (V ). 1 ¯ 2 ξ 0 In (Wξ, JR ), denote the regions W W ±(λ ), and denote by W0 the region ± = ± ( δ,δ) λ0(V ). Note that JR W cJ ,(W0, JR W0 ) ([ R,R] λ0(V ), Jb0). Let us − ∗ | ± = ± | =∼ − ∗ denote the set of such almost-complex structures JR constructed in this way from J1, J2 by J (J1, J2) [0, ). =∼ ∞

2.1.4. Holomorphic maps and finite energy. A J-holomorphic map on a punctured Riemann surface (Σ, j,Γ) is a map U : Σ\Γ W that satisfies the Cauchy– → ξ Riemann equation DU J DU j 0. + · · = We only consider solutions of this equation having finite energy; for the definition of finite energy J-holomorphic curves we refer to [3]. A crucial example of a finite energy J-holomorphic cylinder (in the particular case of a cylindrical almost-complex structure J) is the trivial cylinder over a periodic Reeb orbit x of some period T : it is a solution of the form (for (s,t) ∈ R S1) ψ U(s,t) (Ts a ,x(T t)). If a contact form is (Morse–Bott) non- × λ ◦ = + 0 · degenerate, then at a (nonremovable) puncture a finite energy J-holomorphic map is asymptotic to a half-trivial cylinder over a closed Reeb orbit in one of the ends W ±, see [29] or [3]. When considering finite energy holomorphic maps, we will always assume that all contact forms have nondegenerate orbits, so this condition is always satisfied in everything that follows. A puncture is said to be positive (resp. negative) if it is asymptotic to the positive (resp. negative) half of a trivial cylinder in W + (resp. W −). We will say that a finite energy holomorphic map is positively (resp. negatively) asymptotic to a Reeb orbit x if there is a positive (resp. negative) puncture z (perhaps already specified) at which U is asymptotic to a half-trivial cylinder over x in W + (resp. W −). When the contact form is (Morse–Bott) nondegenerate, then the asymptotic convergence to the trivial half-cylinder mentioned above is exponential in na- ture and one has an asymptotic formula [30, 40]. If one looks at the space of maps asymptotic to fixed closed Reeb orbits with a given rate of exponential

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 421 convergence, then there is a Fredholm theory for the Cauchy–Riemann operator [13, 43]. One more crucial property of finite-energy holomorphic curves is the compactness theory of [3]. In [3, Section 8], a notion of a k 1 k -level holomorphic −| | + building and convergence of a sequence of holomorphic maps to that building is given. We refer to that paper for the definitions. [3, Theorem 10.3] extends to the case where W has cylindrical ends as well, so the space of JR - holomorphic maps can be compactified for sequences with R . The com- ↑ ∞ pactification consists of a holomorphic building in (W −(λ ), J 0 ) (W −(λ ), J 0 ) 0 1 t 0 2 together with a Jb0(λ0)-holomorphic building, and all of these glue together to W −(λ0) W +(λ0) Wξ. We can think of these as k 1 k0 2 k -level holomor- ¯ = −| | | | + phic buildings, which we describe next. Let S be a decorated Riemann surface [3] S, with each smooth component assigned a level labeled 1,2 or (λ , j ) where i {1,0, } and 1 j k . Write i i ∈ + ≤ i ≤ i S S1 S2, where S1 consists of the levels labeled either (λ ,0, ) or 1, while S2 = ∪ − · consists of the parts of the domain labeled (λ0, , ) or 2. If k0 0 then S1 S2 + · 6= ∩ 6= . Let (U) be an assignment to each smooth component Sa S of a finite- ; ⊂ energy holomorphic map U a with domain Sa such that If one considers the subset of (U) with domain in S , it is a holomorphic • 1 building in W : (W , J ). 1 = ξ 1 If one considers the subset of (U) with domain in S , it is a holomorphic • 2 building in W : (W , J ). 2 = ξ 2 The maps glue together to give a continuous map (U): S W W . • → 1 ¯ 2 This data will be called a k 1 k0 2 k -holomorphic building. It is said to be −| | | | + stable if both (U) (U) (i 1,2) are stable. i = |Si = Let Uk be a sequence of JRk -holomorphic maps with the same asymptotic orbits and genus; denote the domains by Σk (each a Riemann surface). Let (U) be a building as above with domain S. If Rk is bounded then there is a limit which is a JR -holomorphic building, where R is an accumulation point of ∞ ∞ the sequence. Else, suppose without loss of generality that R . The se- k ↑ ∞ quence Uk converges to the building (U) if there exists a sequence of diffeomorphisms φ : S Σ converging as decorated Riemann surfaces [3] such that k → k C1 For each level (λi , ji )(i { ,0, },1 ji ki ), the maps Uk φk S lie in ∈ − + ≤ ≤ ◦ | (λi ,ji ) the Wi part of (Wξ, JRk ). (λ ,j ) (λ ,j ) C2 There is a sequence of constants c i i such that c i i U φ k k k k S(λi ,ji ) (λ ,j ) ∗ ◦ | converges in C ∞ to U i i (U) S . loc = | (λi ,ji ) C3 The maps G U φ converge in C 0(S,W ) to (U) (where G : W W W ◦ k ◦ k ξ → 1 ¯ 2 is the identification map described above). The asymptotic behavior of finite energy surfaces together with an application of Stokes’ theorem shows - for finite energy solutions of the almost- complex structures that we consider - the energy E(U) is bounded by the action of the positive asymptotic orbits. Then by [3]

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LEMMA 2.1. For J J (Jb , Jb ), or J (λ), or J JR J (J1, J2), the space of J- ∈ − + = ∈ holomorphic ﬁnite energy cylinders positively asymptotic to a closed orbit x of λ is precompact in M J . + 2.1.5. A restricted class of almost-complex structures. Suppose λ are two con- ± tact forms for ξ satisfying λ λ and that L V is a link that is tangent − ≺ + ⊂ to the Reeb vector ﬁelds for both. With the projection π: Wξ V consider 1 → Z π− (L); which is a union of embedded cylinders. We have L = LEMMA 2.2. ZL is an embedded symplectic submanifold of (Wξ,ωξ).

Proof.Z L is embedded, and each component of ZL is Jb(λ)-holomorphic for any λ with kernel ξ and Reeb vector ﬁeld tangent to L, and any J J (ξ). By hypoth- ∈ esis on L there exist such λ. Since each Jb(λ) is ωξ-compatible, we see ZL must be symplectic.

LEMMA 2.3. The subset J (Jb , Jb : ZL) of J (Jb , Jb ) such that T ZL is J invariant − + − + is nonempty and contractible.

The proof of this fact is standard after the observations that (1) ZL is an embedded symplectic manifold, and (2) that regardless of Jb (λ ), ZL is J holo- ± ± morphic in W ±. One considers the space of metrics for which, along ZL, TZL ω and (TZL) (the symplectic complement) are orthogonal and then mimics the usual proof that the space of almost-complex structures is nonempty and contractible (e.g., [33]). We therefore omit the details. Denote the space of smooth paths [0,1] J (Jb , Jb : ZL) by Jτ(Jb , Jb : ZL). Note that if J1 J (Jb , Jb0 : ZL) → − + − + ∈1 − and J2 J (Jb0, Jb : ZL) then for each JR J (J1, J2) the set π− (L) is also JR - ∈ + ∈ holomorphic. 2.2. Review of some facts about Conley–Zehnder Indices. Let x be a closed Reeb orbit, x be the simple Reeb orbit with the same image as x, and Φ be a 1 trivialization Φ: S C x∗ξ. We denote by CZ (x) the Conley–Zehnder index × → Φ of x with respect to the trivialization obtained from Φ by the obvious degree m covering of S1 C (where m is the covering number of x over x), unless it x × x is speciﬁed that Φ is a trivialization of x∗ξ itself (in which case we will compute with respect to Φ). If the trivialization is clear in a given context we drop the subscript Φ from CZΦ. A useful fact about the Conley–Zehnder index is that it grows almost-linearly for Reeb orbits in dimension 3 (see e.g., [32, Theorem 8.3], or [28]) in the following precise sense.

PROPOSITION 2.4. Let γ be a periodic orbit of minimal period T , and Ψ a trivialization of the contact structure over that orbit. The Conley–Zehnder index k CZΨ(γ ) is monotone in k. Moreover, one has the following characterization of the index of its coverings γk : If γ is elliptic, there exists a unique θ such that the complex eigenvalues of • 2πiθ k Φ are e± , and for all covering numbers k, CZ (γ ) 2 k θ 1. T Ψ = b · c + If γ is hyperbolic then there exists an integer n such that CZ (γk ) is k n. • Ψ · JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 423

Another useful fact is a relationship between the Conley–Zehnder index and the winding of eigensections of the asymptotic operator associated to the orbit x. We will briefly describe the asymptotic operator associated to a periodic orbit x of period T - for details we refer the reader to [30, 40]. A choice of 2 J J (ξ) induces a L -inner product for sections of the vector bundle (x )∗ξ ∈ T → R/Z by Z 1 η,ζ (dλ)x(T t)(η(t), Jx(T t) ζ(t))dt. 〈 〉 = 0 · On the space of square-integrable sections with respect this inner product there is an unbounded self-adjoint operator defined by A η J( η T X ) P · = −∇t + ∇η λ where is a torsionless connection on TV , and denotes covariant differen- ∇ ∇t tiation along the curve x (t). In fact, A does not depend on the choice of , T P ∇ and is conjugate to an operator (where S is a smooth loop of symmetric 2 2 × matrices) L i∂ S = − t − which is an unbounded self-adjoint operator in L2(R/Z,R2) with a discrete (real) spectrum consisting of eigenvalues and accumulating only at . A nontriv- ±∞ ial eigenvector v does not vanish, so if v(t) ρ(t)eiθ(t) then the total wind- = ing wind(v) (θ(1) θ(0))/2π is well-defined. It turns out that if v ,v are = − 0 1 eigenvectors for the same eigenvalue then their windings coincide, so we have a well-defined integer wind(ν) associated to every ν σ(L), see [29] for de- ∈ tails. Moreover, for each k Z there are precisely two eigenvalues (multiplicities ∈ counted) with winding equal to k, and ν ν wind(ν ) wind(ν ) whenever 0 ≤ 1 ⇔ 0 ≤ 1 ν ,ν σ(L). 0 1 ∈ DEFINITION 2.5. Let P be a simple, closed Reeb orbit, and k 1 be an integer. ≥ We define αΦ−(P,k) to be the winding number of the eigensection associated to the largest negative eigenvalue of the asymptotic operator of (P,k) (the k-fold covering of P) with respect to Φ, and αΦ+(P,k) to be the winding associated to the least positive eigenvalue (see [39]).

It is proved in [29] that the the eigenvalues can be ordered (with multiplic- ity) in such a way that the winding of the corresponding eigensections is non- decreasing and increases by 1 every second eigenvalue, and that the Conley– Zehnder index is given by

(2.1) CZ (P,k) α−(P,k) α+(P,k) 2α−(P,k) p(P,k), Φ = Φ + Φ = Φ + where p(P,k) is the parity of the Conley–Zehnder index and is either 0 or 1 depending on whether the winding numbers of the ‘extremal’ positive/negative eigenvalues agree or disagree. In particular, the Conley–Zehnder index is odd if and only if the winding of the eigensections associated with the eigenvalues nearest zero differ. This is important because it implies by the asymptotic for- mulas of e.g., [30, 40] that at odd index orbits that the holomorphic curves

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 424 AL MOMIN necessarily wind around the orbit differently (depending on whether they approach positively or negatively).

Using Proposition 2.4 and (2.1) we see easily that αΦ−(P,k) is given by kθ if k n b c P is elliptic, and 2· if it is hyperbolic (even or odd), and αΦ+(P,k) is given by b c k n kθ if P is elliptic, and · if it is hyperbolic. d e d 2 e 2.3. Transversality for cylinders. In this section we assume that λ is a nondegenerate contact form on V . We will consider J-holomorphic cylinders in the symplectization and transversality of the Cauchy–Riemann operator for these cylinders. Here J Jb will be a cylindrical almost-complex structure. The con- = structions of the chain complexes in Section4 requires the Cauchy–Riemann operator to be transverse on enough moduli spaces. In particular we will prove the following transversality Theorem. In order to state it, we recall ﬁrst that a Reeb orbit is said to be SFT-good if it is not an even cover of another orbit with odd Conley–Zehnder index.

THEOREM 2.6. By [13] there is a residual subset Jgen(λ) of J (λ) such that for all somewhere injective J-holomorphic curves the Cauchy–Riemann operator is transverse for J J (λ). Let J J (λ): ∈ gen ∈ gen 1. At a J-holomorphic cylinder U of index Ind(U) 1, the Cauchy–Riemann ≤ operator is transverse at U. In particular the only index 0 cylinders are ≤ trivial cylinders. 2. At a J-holomorphic cylinder U of index Ind(U) 2, and such that both as- = ymptotic orbits are SFT-good, the Cauchy–Riemann operator is transverse at U. Therefore the corresponding moduli spaces (modulo the free smooth R-action) consist of isolated points (index-0 or 1 case) or of isolated intervals or circles (index-2 case).

2.3.1. Some Inequalities. Let π denote the projection of TV to the contact structure ker(λ). Given a finite energy surface U, we use the identification ψ : W λ ξ → R V and let a πR U, u πV U. One often finds the section π Tu of 0,1× = ◦ = ◦ ◦ Ω (T Σ,u∗ξ) to be quite useful. It is proved in [29] that either it vanishes identically or it has isolated zeros. We let windπ(U) denote the oriented count of zeros of this section, which is proved in [29] to be nonnegative. Also, we will use the following topological quantities:

Denote by Γ the set of punctures of the map U and by Γ± the subsets • of positive and negative punctures. Also, #Γeven denotes the number of punctures asymptotic to orbits of even Conley–Zehnder index, and #Γodd denotes the number of punctures asymptotic to orbits of odd Conley– Zehnder index.

• X X (2.2) CZ(U) CZ(z+) CZ(z−) = z Γ − z Γ +∈ + −∈ − JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 425

(this requires a choice of trivialization of U ∗ξ, but is independent of that choice). Ind(U) CZ(U) χ(Σ) #Γ, and 2c (U) Ind(U) χ(Σ) #Γ . • = − + N = − + even We will use the following theorem, proved in [29], to show that certain curves are immersed in order to apply the automatic transversality Theorem 2.8.

THEOREM 2.7 ([29, Theorem 5.8]). Suppose π Tu does not vanish identically. ◦ Then it vanishes only a finite number of times, and 0 wind (U) c (U). ≤ π ≤ N 2.3.2. Review of some results about transversality. The Fredholm theory here was developed in [13]. There the author defines a Banach manifold 1,p;d Bg (x1,...,xk ; y1,..., yl ), 1,p where p 2, of W maps from a surface of genus g with k ‘positive’ punctures > loc and l ‘negative’ punctures into R V which converge exponentially with weight 1,p × d in W to the periodic orbits x1,...,xk at positive punctures and to y1,..., yl at negative punctures. We will really only need the cases g 0,k 1,l 1. = = = Given this structure we have a section ∂ (U) dU J dU j, J = + ◦ ◦ defined over B into a smooth bundle E over B with fibers [ p,δ 0,1 E {U} L (Ω (T Σ˙ C U ∗TW )). = U B × ⊗ ∈ At a zero of this section, U, we define FU to be the projection of D∂J (U) to the vertical part TU E TU B EU . It is then proved in [13] that FU is a Fredholm =∼ ⊕ operator. We say the Cauchy–Riemann operator is transverse or regular at U (or U is a transverse solution, etc.) if FU is a surjective operator, and that a subset of solutions is transverse if each element is a transverse solution. Its index (computed in [13]) gives the dimension of parameterized (j, J)-holomorphic maps near U when U is transverse. The dimension of the moduli space (allowing j to vary and dividing by automorphisms of the domain) is then given ³ dim(R V ) ´¡ ¢ by the topological quantity Ind(U) CZ(U) × 3 χ #Γ which co- = + 2 − Σ − dim(R V) incides with the definition of Ind(U) given earlier since × 2. 2 = We cite a special case of Theorem 0.1 in [43] (valid for dim(R V) 4): × = THEOREM 2.8. [43] Suppose that U is an immersed finite energy surface. Letting g denote the genus of the domain of U, the linearized operator is surjective if Ind(U) 2g #Γ 1. ≥ + even − We also need results for somewhere-injective curves. Following the notation of [13], define the set M M (x ,...,x ; y ..., y ) the set (C, J) of pairs consist- = 1 n 1 n ing of nonparametrized curves C and compatible almost-complex structures J J (where J J (λ) or J (Jb , Jb )) for which C is J-holomorphic (for some ∈ = − + parametrization) positively asymptotic to the Reeb orbits xi and negatively to the y . In case J J (λ) one quotients by the free R-action on solutions as j = JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 426 AL MOMIN well. We have the following from [13, Theorem 1.8 and its corollary] which holds in any dimension:

THEOREM 2.9. The set M carries a structure of a separable Banach manifold. The projection map η: M J , η(C, J) J, is a Fredholm map with Fredholm → = index Ind(C) Ind(U), where U :(S, j)\Γ R V parametrizes C. = 1 → × For regular values J of η, M η− (J) is a smooth, ﬁnite-dimensional man- J = ifold whose dimension agrees with the above index Ind(C). By the Sard–Smale Theorem, the set of regular values is a residual set. Consequently there is a residual set J J of compatible almost-complex structures such that for every gen ⊂ J J if U : S\Γ R V is a somewhere injective ﬁnite energy surface for J, ∈ gen → × then Ind(U) 1 provided π Tu does not vanish identically. ≥ ◦ Theorem 2.9 can also be proved if J J (Jb (λ ), Jb (λ ): Z ) because any some- = − − + + where injective curve for such a J has a point of injectivity in W \(W + W − Z ) ∪ ∪ with the exceptions of (1) components of Z , and (2) holomorphic curves for Jb(λ ) contained entirely in a cylindrical end W ±. We will not be concerned + about transversality of such curves anyway and therefore they can be safely ig- nored. This observation will be needed in Section4.

2.3.3. Proof of Theorem 2.6. Theorem 2.6 is a direct consequence of

PROPOSITION 2.10. There is a residual subset Jgen of dλ-compatible almost- complex structures on ξ such that: 1. At any somewhere injective ﬁnite energy solution, the Cauchy–Riemann operator is transverse, by [13]. 2. For any compatible J (not necessarily in J ), if U M (x; y) is of index gen ∈ J 1, then the linearization of the Cauchy–Riemann operator is surjective at U. 3. Let J be any compatible almost-complex structure, and suppose U is a J- holomorphic ﬁnite-energy cylinder. Then its index is at least 0. 4. For J J , if U M (x; y) is an index-2 cylinder, and both asymptotic ∈ gen ∈ J orbits are SFT-good, then the linearization of the Cauchy–Riemann operator is surjective at U. 5. For J J , all index-zero holomorphic cylinders are trivial cylinders. The ∈ gen linearized Cauchy–Riemann operator is surjective at any trivial cylinder.

Proof. To begin the proof, take Jgen to be as in Theorem 2.9. (1) The first item is an assertion in Theorem 2.9. (2) The map U is an immersion if wind (U) 0. For finite energy cylinders of π = index 1, Theorem 2.7 implies 0 wind (U) 1 2(2) 1 2(1) 0 (since there ≤ π ≤ − + + = must be one odd puncture and one even puncture). Thus, finite energy cylinders of index 1 are always immersed. We have g 0 and #Γ 1 (since the = even = index difference is 1, there is one even and one odd puncture), so by Theorem 2.8 the linearized Cauchy–Riemann operator is surjective. (3) This was already observed in [29]; we include the short proof. π Tu van- ◦ ishes identically on a cylinder if and only if the cylinder is a trivial cylinder. If

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 427 a nontrivial ﬁnite energy cylinder has index less than 0, then by Theorem 2.7 0 2wind (U) 0 4 #Γ 2#Γ 0, which is impossible. ≤ π < − + odd+ even ≤ To prove (4) we will need the following lemma.

LEMMA 2.11. Suppose U is a cylinder of index 2 with asymptotic limits x, y. If x and y have even Conley–Zehnder index and are SFT-good, then U is simply covered.

Proof. Let k be the covering number of U, so U U 0 τ where U is a simple = ◦ k curve and τ is the degree k holomorphic cover of R S1. Choose a trivialization k × Φ0 for U 0∗ξ, which induces trivializations Φ of ξ over U, x and y. We then have CZ (x) CZ (y) 2 by the index formula. Let CZ (x) 2l 2, so CZ (y) 2l. Φ − Φ = Φ = + Φ = Letting x0 and y0 denote the asymptotic limits of U 0„ so x, y are k-covers of x0, y0, by Proposition 2.4 we have k 2l 2, and k 2l, from which it follows that | + | k 2 or 1. Suppose k 2. Then CZ (x0) l 1,CZ (y0) l. Therefore, one = = Φ0 = + Φ0 = of these has odd Conley–Zehnder index, so either x or y is a ‘bad’ orbit. Since both orbits are assumed SFT-good, k 1. = (4) The parity of the Conley–Zehnder indices are either both even or both odd. If they are both even, then by Lemma 2.11 all curves in the index-2 component of the moduli space are simple. Transversality of the index-2 component of the moduli space for J J follows from Theorem 2.9. ∈ gen In the case of an index-2 cylinder with odd asymptotic orbits, we observe

0 wind (U) 2 2(2) 2(1) 2(0) 0, ≤ π ≤ − + + = Ind(U) 2 2g #Γ 1 2(0) 0 1, = ≥ + even − = + − so automatic transversality applies by Theorem 2.7 and Theorem 2.8, in particular for any J chosen for the case of even, SFT-good orbits. (5) Choose any J J . For simply covered index-0 cylinders, the inequality ∈ gen of Corollary 2.9 implies π Tu vanishes identically, which implies the cylinder ◦ is trivial. If a cylinder is multiply covered of index 0, the underlying simple cylinder has index 0 - by the monotone growth of the index of a cylinder under covering. Hence, it is a cover of a trivial cylinder so is trivial itself. Transversality is proved as in [38].

3. INTERSECTIONSANDCOMPACTNESS 3.1. Intersections. We recall the intersection theory described in [39].

DEFINITION 3.1. A stable Hamiltonian structure [39, 16] on a 3-dimensional manifold V is a pair H (λ,ω) such that = 1. λ ω is a volume form for V , ∧ 2. ω is closed, 3. dλ 0. |kerω ≡ It deﬁnes a H -Reeb vector ﬁeld by X λ 1, X ω 0. ¬ ≡ ¬ ≡ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 428 AL MOMIN

For example, if V is a 3-manifold with a contact form λ, the associated stable Hamiltonian structure is H (λ,dλ). In this paper, this is the only case we λ = will consider. Nonetheless, we introduce this notion for two reasons. The ﬁrst is that the reference [39] works at this level of generality, so we can make the discussion compatible with [39] more easily. The second is that stable Hamilton- ian structures can be used under a wider set of circumstances (for instance, one may formulate Hamiltonian Floer homology using stable Hamiltonian structures), so the compactness theory described here for contact 3-manifolds is actually applicable in greater generality and should be useful to study vector ﬁelds associated with more general stable Hamiltonian structures as well. Let W be a 4-manifold. A cylindrical Hamiltonian structure on W is the data (Φ,V,H ) where Φ: W R V is a diffeomorphism and H is a stable Hamil- → × tonian structure on V . A positive (resp. negative) Hamiltonian structured end is the data (W ±,Φ±,V ±,H ±) where W ± is a subset, Φ± : W ± [0, ) V ± is → ± ∞ × a diffeomorphism and H ± is a stable Hamiltonian structure on V ±. We will usually refer only to W ± with the rest of the data implicit. A cobordism is a manifold W such that W \W ± is compact oriented with boundary.

EXAMPLE 3.2. Suppose λ are contact forms with kernel ξ on V , and λ λ ± − ≺ + in W W . We can equip W with the Hamiltonian structured ends W ± = ξ ξ = W ±(λ ),V ± V, H ± (λ ,dλ ) ± = = ± ± µ λ ¶ Φ±(λ) log ,π(λ) . = λ (π(λ)) ± Suppose Wi are two such cobordisms. We say that we can concatenate W1 and W if (V +,H +) (V −,H −). Then we can deﬁne the manifold W W 2 1 1 = 2 2 1 ¯ 2 by compactifying the positive end of W with { } V +, the negative end of W 1 ∞ × 1 2 with { } V − (using the diffeomorphisms Φ± to do so), and identifying them. −∞ × 2 This can be generalized to n-fold concatenations W W for any n 2 as 1 ¯ ··· ¯ n ≥ long as one can concatenate Wi and Wi 1 for each i. + Suppose W1 is a cylindrical Hamiltonian structured manifold i.e., W1 R V , =∼ × and that we can form W W . Then obviously W W is homeomorphic to 1 ¯ 2 1 ¯ 2 W2 and is equipped with the same Hamiltonian ends. This identiﬁcation in convenient and we will use it often in the following without comment.

3.1.1. A homotopy invariant intersection number. Suppose H is a Hamiltonian structure on V , and let W R V . Let (Σ, j,Γ,U,W ) be a C 1 map U : Σ\Γ W . =∼ × → For a closed T -periodic Reeb orbit δ, denote by Zδ the trivial cylinder over δ: Z (s,t) (Ts,δ(T t)). δ =

DEFINITION 3.3. [39] Suppose γ is a simple T -periodic orbit for XH . We say U is asymptotically cylindrical over γm at z Γ if there is a holomorphic embed- ∈ ding φ: [0, ) S1 Σ\{z} with φ(s,t) z (as s ) such that ∞ × → → → ∞ lim ( mT c) U φ(s c,t) 1 Zγm 1 c [0, ) S [0, ) S →∞ − ∗ ◦ + | ∞ × = | ∞ × JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 429

(with convergence in C 1([0, ) S1,W ), and where c U denotes the action by ∞ × ∗ translation of the R co-ordinate by c)). We say U is asymptotically cylindrical if at each puncture z Γ U is asymptotically cylindrical to some H -Reeb orbit ∈ γm. If U is a map into a 4-manifold W with Hamiltonian structured ends H ±, then we say (Σ, j,Γ,U,W ) is asymptotically cylindrical if for each puncture z Γ ∈ there is a neighborhood O of z such that U is contained in one of the ends z |Oz and is asymptotically cylindrical as deﬁned above.

We denote by Cg∞,p ,p (V,H ) the set of smooth asymptotically cylindrical maps of genus g with+ p− positive punctures and p negative punctures, and + − set C ∞(V,H ) to be the union over all g,p ,p 0. Similarly in [39] the author − + ≥ deﬁnes asymptotically cylindrical maps in a 4-manifold W with Hamiltonian structured ends W ± and denotes these by C ∞(W,H +,H −). We will extend this notation to include continuous, piecewise smooth maps which are smooth on neighborhoods of the punctures and satisfy the same asymptotic conditions.

DEFINITION 3.4. Suppose U,V are asymptotically cylindrical maps, and z is m 1 a puncture for V asymptotic to γ . Choose a trivialization Φ: S C γ∗ξ. × → There is a neighborhood Oz of z that is mapped entirely into a cylindrical end and such that

V (φ(s,t)) (ms,exp m h(s,t)). = γ (t) Let β be a cut-off function supported in Oz which is identically 1 on a smaller neighborhood of z. Consider the perturbation

³ £ ¤´ V 0(φ(s,t)) ms,exp m h(s,t) β(φ(s,t)) Φ(mt) ² , = γ (t) + · · which is well-deﬁned for all ² 0 sufﬁciently small. Let V be the map > Φ,β,² that results from making such a perturbation at each puncture of V . Then set ι (U,V ) int(U,V ). It depends only on the homotopy classes of the maps Φ = Φ,β,² and the trivializations Φ, and is symmetric in the arguments U,V (see [39]).

DEFINITION 3.5. Let P be a closed, simple Reeb orbit, and let k1,k2 be two positive integers. ½ ¾ αΦ−(P,k1) αΦ−(P,k2) ΩΦ+(P,k1,k2) k1k2 max , ,(3.1) = k1 k2 ½ ¾ αΦ+(P,k1) αΦ+(P,k2) ΩΦ−(P,k1,k2) k1k2 min , .(3.2) = k1 k2

Let z and z0 be punctures of U and V respectively. If U and V are asymptotic to covers of different Reeb orbits, then we set ΩΦ(z,z0) 0. If U and V are k k = positively asymptotic to covers P 1 ,P 2 of the same Reeb orbit P at z and z0 then we set ΩΦ(z,z0) ΩΦ+(P,k1,k2); if they are negatively asymptotic at z and z0 k k = to covers P 1 ,P 2 of the same Reeb orbit P then we set Ω (z,z0) Ω−(P,k ,k ). Φ = Φ 1 2 JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 430 AL MOMIN

With this notation, set X X Ω (U,V ) Ω (z,z0) Ω (z,z0). Φ = Φ − Φ z Γ+ z Γ− ∈ U ∈ U z0 Γ+ z0 Γ− ∈ V ∈ V It follows from [39, Proposition 4.1 and Lemma 3.4] that ι (U,V ) Ω (U,V ) Φ + Φ depends only on the homotopy classes of U,V .

DEFINITION 3.6. For a pair of punctures z,z0 of U,V , if U,V are asymptotic to covers of different orbits at z,z0 then set ∆(z,z0) 0; else, let P denote the un- = derlying simple orbit, k1(z),k2(z0) the respective covering numbers of the orbit, and set

∆(z,z0) Ω−(P,k (z),k (z0)) Ω+(P,k (z),k (z0)) 0. = Φ 1 2 − Φ 1 2 ≥ It follows from equations (3.1), (3.2) that this difference is indeed trivialization independent. Sum this quantity over all pairs of punctures to get

X X ∆(U,V ) ∆(z,z0) ∆(z,z0) 0. = + ≥ z Γ+ z Γ− ∈ U ∈ U z0 Γ+ z0 Γ− ∈ V ∈ V DEFINITION 3.7. Let [U],[V ] be homotopy classes of asymptotically cylindrical maps with representatives U,V (resp.). Set 1 [U] [V ] ιΦ(U,V ) ΩΦ(U,V ) ∆(U,V ). ∗ = + + 2 By the above comments this only depends on the homotopy classes [U],[V ] of the maps U,V in C ∞(V,H ) (or C ∞(W,H +,H −)) and is therefore well-deﬁned. 1 Asymptotically cylindrical maps Ui in Wi can be concatenated if the asymptotics match in the obvious way so that the maps can be glued together (after choosing asymptotic markers) to form a (piecewise smooth) C 0 map U U in 1 ¯ 2 W W . One can also form n-fold concatenations U U in the obvious 1 ¯ 2 1 ¯ ··· ¯ n way to make a map into W W ; every holomorphic building deﬁnes such 1 ¯···¯ n a concatenation. [39, Proposition 4.3] states (except for the last item which differs slightly; see the remark following the statement):

PROPOSITION 3.8. Let W,W1,...,Wn be 4-manifolds with Hamiltonian structured cylindrical ends. Then:

1. If (Σ, j,Γ,W,U) and (Σ0, j 0,Γ0,W,V ) C ∞(W,H +,H −) then [U] [V ] de- ∈ ∗ pends only on the homotopy classes [U],[V ]. 2.[ U] [V ] [V ] [U]. ∗ = ∗ 1 The concatenation is well-deﬁned only up to a Dehn twist of the domain Σ1 Σ2, since a ◦ gluing of these domains may require a choice of asymptotic markers. This data is determined when the concatenation occurs as a holomorphic building, which is the only case that interests us. Regardless, different choices of asymptotic markers yield the same intersection number (see the proof of [39, Proposition 4.3]).

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3. Using “ ” to denote disjoint union of maps, we have [U U ] [V ] [U ] + 1 + 2 ∗ = 1 ∗ [V ] [U ] [V ]. + 2 ∗ 4. If U ...U and V ...V are concatenations of asymptotically cylindrical 1¯ n 1¯ n maps in W ...W then 1 ¯ n n X [U1 ...Un] [V1 ...Vn] [Ui ] [Vi ]. ¯ ∗ ¯ = i 1 ∗ = REMARK 3.9. The last assertion differs from the one made in [39]; however, it is shown in the proof that the difference

[U U ]˜[V V ] [U ]˜[V ] [U ]˜[V ] 1 ◦ 2 ∗ 1 ◦ 2 − 1 ∗ 1 − 2 ∗ 2 (where ˜ denotes the intersection number minus the term 1 ∆(U,V ), which is ∗ 2 what is used in [39]) is equal to

∆ ∆+((U) ,(V ) ) ∆−((U) ,(V ) ), = 1 1 = 2 2 where we use ∆± to denote the sum of the terms in the deﬁnition of ∆ corresponding to the positive (resp. negative) punctures only (in particular ∆ = ∆+ ∆−). It is easy to see that we get the above identity from our deﬁnition of + “[U] [V ]”. ∗ The statement in [39] is extended to n-fold concatenations above.

Suppose U : Σ\Γ W is in C ∞(W,H +,H −). Considering the oriented blow- → up Σ [3], because U is asymptotically cylindrical it extends over Σ to deﬁne a map U : Σ W . If B is a boundary component of Σ, then U B is a closed → 0 | Reeb orbit in ∂W . Let us consider the subclass C (W ,H +,H −) of those U ∈ C 0(Σ,W ) such that for each boundary component B of Σ, U is a closed Reeb |B orbit in (∂±W ∼ V ±,H ±). By standard arguments the intersection number ex- = 0 tends to the class C (W ,H +,H −) and is homotopy invariant within that class. In particular, the above discussion extends the intersection number to holo- 0 morphic buildings: a building (U) deﬁnes a map (U) C (W ,H +,H −), which ∈ we use to set [(U)] [(V )] ˙ [(U)] [(V )]. ∗ = ∗ Once the intersection number is extended this way, it is clear that

LEMMA 3.10. Suppose U (U) and V (V ) in the SFT-sense. Then lim[U ] k → k → k ∗ [V ] [(U)] [(V )]. k = ∗ Proof. Recalling property C3 of SFT-convergence from Section 2.1.4, G U ◦ k ◦ φk (U) uniformly, and similarly G Vk φk0 (V ) uniformly. Therefore they → ◦ ◦0 → represent the same homotopy classes in C (W ,H +,H −) for k sufﬁciently large and therefore

lim[U ] [V ] lim[U ] [V ] [(U)] [(V )] [(U)] [(V )]. k ∗ k = k ∗ k = ∗ = ∗

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In a concatenation W1 Wn, if both ends of each Wi correspond to con- ¯···¯ 1 tact forms for which L consists of closed orbits, then the cylinder π− (L) Z = L is asymptotically cylindrical for each W , and the concatenated map Z ... Z i L ¯ L identiﬁes with ZL under the homeomorphism W1 ...Wk Wξ. So it makes ¯ =∼ sense to speak about the intersection of a holomorphic building (U) and the cylinder ZL if all the ends of all pieces in the concatenation in which (U) lives are such that L is a closed orbit for the form deﬁning the Hamiltonian structure for that end. Moreover, if we index the levels of (U) by U i i.e., (U) U 1 U k = ¯···¯ then k X i [(U)] [ZL] [U ] [ZL] ∗ = i 1 ∗ = by the levelwise additivity property (Property (4) of Proposition 3.8) of the intersection number.

3.1.2. Positivity and local computations. It is shown in [39] that if U,V are pseudoholomorphic asymptotically cylindrical with no components sharing identical images then [U] [V ] 1 ∆(U,V ), with strict inequality if there is an interior ∗ ≥ 2 intersection. Moreover we see that if U,V are both asymptotic to covers of an orbit γ either of which has odd Conley–Zehnder index then [U] [V ] 0. ∗ > For the following, suppose U,V are asymptotically cylindrical holomorphic maps. The asymptotics formula and the similarity principle implies that if U,V have no components on which they are covers of the same holomorphic curve then the number of intersection points is ﬁnite. Let int(U,V) be the sum of all local intersections, which is nonnegative by e.g., [35,36].

THEOREM 3.11 ([39], Theorem 4.4). Suppose U,V are asymptotically cylindrical pseudoholomorphic maps in an almost-complex cobordism with cylindrical ends, with no common components. Then [U] [V ] int(U,V) δ (U,V) ∗ = + ∞ + 1 ∆(U,V) and δ (U,V ) 0. 2 ∞ ≥ The number δ (U,V ) is described in [39] in terms of local quantities for ∞ holomorphic maps. We refer to [39] for the deﬁnition. For later use we now recall what the quantity δ (U,V ) is in the particular case V ZL where Li is a ∞ = i closed Reeb orbit for both λ in a cobordism of the form W (λ ,λ ) Wξ and 1 ± − + ⊂ Z π− (L ). Li = i In the following formula, let L U(w) mean that U is asymptotic to a cover i ∼ of Li at w, let kw be the covering number of the Li to which U is asymptotic kw 1 at w, let wλ,Φ(U,w) be the winding number of the loop (Φ )− eU , where eU is the eigensection of the asymptotic operator appearing in the asymptotic formula of [40] for U at the puncture w; let αλ±,J,Φ(Li ,k) be the extremal winding k number of the asymptotic operator associated with (Li ,λ, J) and trivialization Φk , and ﬁnally let p (x) {0,1} be the parity of the orbit x with respect to the λ ∈ form λ. Then

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X δ (U, ZLi ) wλ ,Φ(U,w) αλ− ,J ,Φ(Li , kw ) ∞ = − + + + + | | w Γ+ ∈ U Li U(w) (3.3) ∼X wλ ,Φ(U,w) αλ+ ,J ,Φ(Li , kw ). + + − − | | w Γ− − − ∈ U L U(w) i ∼

3.1.3. Moduli spaces. Let (W, J) be a symplectic 4 manifold with Hamiltonian structured ends and compatible J adjusted to the ends. We use MJ (x; y1,..., yk ) to denote the moduli space of ﬁnite energy J-holomorphic spheres with one positive asymptotic orbit x and perhaps several negative asymptotic orbits y1, ...,yk (modulo reparametrizations of the domain). Usually we are interested in moduli space of cylinders with one positive and one negative end, MJ (x; y). If it is clear we may drop the subscript. If J is a cylindrical almost-complex structure J Jb(λ) then we will actually mean the moduli space modulo the further = R-action by translations. We assume that all Hamiltonian structures considered are nondegenerate, so we have the asymptotic formula [30,3] which implies all solutions are asymptotically cylindrical and the intersection theory described above applies. Given a holomorphic curve Z , we denote

M (x; y ,..., y : Z ) {[U] [U] [Z ] 0} M (x; y ,... y ). J 1 k = | ∗ = ⊂ J 1 k For homotopies l J , we write 7→ l [ M{Jl }(x; y1,..., yk ) {l} MJl (x; y1,..., yk )} = l [0,1] × ∈ and [ M{Jl }(x; y1,..., yk : Z ) {l} MJl (x; y1,..., yk : Z )}. = l [0,1] × ∈ 1 In the following, Z will almost always be Z π− (L) (recall π : W V is L = V V ξ → the projection T ∗V V restricted to V ), though we do not specialize yet. → LEMMA 3.12. Suppose Z is a ﬁnite energy surface with all asymptotic orbits el- 1 liptic (e.g., Z π− (L) under condition (E)), and [U] M (x; y ,..., y ) is such L = ∈ J 1 n that no component has image contained in a component of Z . Then [U] ∈ M (x; y, : Z ) if and only if U and Z never intersect and are never asymptotic J ··· to covers of the same asymptotic orbit with the same sign (in the case Z Z this = L is the same as saying x, y D0 and U never intersects Z ). ∉ ± Proof. If U intersects Z , by positivity of intersections U Z 0, and since all · > other terms are nonnegative [U] [Z ] 0. Since the asymptotic orbits of Z are ∗ > elliptic, if U, Z share a common asymptotic orbit we would have ∆(U, Z ) 0. > Therefore in either case [U] [Z ] 0 by Theorem 3.11. If[U] [Z ] 0 both of ∗ > ∗ = these terms must be zero so the converse holds as well.

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3.2. Compactness. Let us call a subset C of M closed under compactification if for every [(U)] ∂C M , each component U i of (U) represents an element i ∈ ⊂ [U ] C. Denoting by MJ (1;k) the union of all MJ (x; y1,..., yk ), then MJ (1; ) S ∈ 2 ∗ = k 0 MJ (1;k) is closed under compactification. Denoting by M ≤ (1;1) the sub- ≥ J set of cylinders of index at most 2, the main obstacle to defining cylindrical 2 contact homology is to show that MJ≤ (1;1) is closed under compactification. Similarly set M (1;k : Z ) to be the union of all MJ (x; y1,..., yk : Z ). Our goal is [a] to show the subsets MJ (1;1 : ZL) of cylinders in MJ (1;k : ZL) connecting asymptotic orbits in the homotopy class of loops [a] are closed under compactification when (λ ,L) (λ ,L) satisfy (E), or when (λ ,L,[a]) satisfy (PLC). + ≥ − ± 3.2.1. Condition (E). We investigate the compactification of finite energy holo- 1 morphic curves in W with intersection number 0 with [Z Z π− (L)] (in ξ = L = V the sense defined above). We assume (λ ,L) (λ ,L) throughout, and later + ≥ − that both satisfy condition (E). We suppose J J (Jb , Jb : Z ) (in the cylindrical ∈ − + case λ λ and J J (λ ); then Z is automatically J-holomorphic). + = − ∈ ± 1 LEMMA 3.13. Suppose (λ ,L) (λ ,L) (with L all elliptic), Z π− (L) and J + ≥ − ± = V ∈ J (Jb , Jb : Z ) (allow the possibility J is cylindrical i.e., λ λ and J Jb Jb − + − = + =1 − = + ∈ J (λ ) J (λ )). Let Li be a connected component of L, and C π− (Li ). Let − = + = U be a possibly branched cover of C with 1 positive puncture and genus 0. Then [U] [C] 1 (1 #Γ ), where #Γ is the number of negative punctures of U. ∗ ≥ 2 − − − Proof. Let S2\Γ(U) be the domain of U, so we have a lift Uˆ : S2\Γ R S1 → × i.e., U C Uˆ . Using the metric g J , since J J (Jb , Jb : Z ) we can identify = ◦ ∈ − + the normal bundle to C with ξ, and with the exponential map and a unitary 1 trivialization Φ: C ∗ξ R S C we can identify a tubular neighborhood of 1 → × × C with (R S ) D², where D² is a disk containing the origin in C. Let us de- × ×ˆ 1 1 note this map f exp Φ− :(R S ) D² W . Consider the pullback bun- = 2 ◦ × × → dle N U ∗ξ over S \Γ(U). Using the trivialization and lift, we have a map = f fˆ (Uˆ Id): S2\Γ(U) D W , with the property that f (z,c) C if and = ◦ × × ² → ∈ only if c 0 (possibly taking D smaller if needed). It is a local-diffeomorphism = ² away from the branch points. Small enough sections of N are identified with functions σ: S2\Γ(U) D → ² by this trivialization, and it defines a map S : S2\Γ(U) W by z f (z,σ(z)). σ → ξ 7→ Choose any section σˆ of N such that near any puncture, using cylindrical coordinates at the puncture ((s,t) • ∈ [0, ) R/Z Dz \{z+} at a positive puncture z+, and (s,t) ( ,0] ∞ × =∼ + ∈ −∞ × R/Z Dz \{z−} at a positive puncture z−) σˆ has the form =∼ − σˆ(s,t) eλse (t) = λ where λ is the extremal eigenvalue of the asymptotic operator of the orbit to which U is asymptotic at that puncture, and eλ(t) represents an eigensection associated to λ, it has only finitely many transverse zeros none of which occur at branch • point,

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it is small enough to be represented by a function σ: U D . • → ² We compute [C] [U] by computing [C] [S ] using the formula of Theorem ∗ ∗ σ 3.11 1 [C] [Sσ] int(C,[Sσ]) δ (C,Sσ) ∆([C],[Sσ]), ∗ = + ∞ + 2 since Sσ is homotopic to U via τ Sτ σ. In view of the representation of σˆ near 7→ · the punctures we have

wλ ,Φ(Sσ,w) αλ− ,Φ(Li ,kw ) 0, if w Γ+(Sσ), (3.4) − + + + = ∈ wλ ,Φ(Sσ,w) αλ+ ,Φ(Li ,kw ) 0, if w Γ−(Sσ). + − − − = ∈

The formula (3.3) for δ applies here, so the equations (3.4) imply δ (C,Sσ) ∞ ∞ = 0. Since all asymptotic limits are elliptic, it follows that each puncture w Γ(S ) ∈ σ contributes ∆( ,w) 1 ( denotes the puncture of C of the same sign as ∞w ≥ ∞w w; this inequality is demonstrated in the proof of Proposition 4.3 item (4) in [39]). Therefore 1 ∆(C,S ) 1 #Γ(U). 2 σ ≥ 2 It remains to compute int(C,S ). Since the map fˆ: R S1 D W is an σ × × ² → orientation-preserving local-diffeomorphism away from branch points, and all zeros occur away from branch points by construction, the computation reduces to a signed count of intersections of this map σ with 0 C (the mapping degree ∈ deg(σ;0)). This degree is computed by the homology class of σ ∂S H1(C\0) 2 | ∈ (where S is S minus the small disk-like neighborhoods Dz± of the punctures Γ(U) on which the asymptotic expression holds, i.e., σ(s,t) eλse (t)). The = λ extremal winding numbers are given by αλ∓ ,Φ(Li ;kw ) where kw is the covering ± number of Li at each puncture w of U, using α− with respect to λ at positive + punctures, and α+ with respect to λ at negative punctures. We recall from − Section 2.2 that there are irrational numbers θi such that ±

i αλ− ,Φ(Li ;kw ) kw θ , + = b +c i i αλ+ ,Φ(Li ;kw ) kw θ kw θ 1. − = d −e = b −c + Therefore, the mapping degree is

i X i k+θ ( kz− θ 1), b +c − z b − −c + − where k+ is the covering number of the positive puncture, and the kz− are the P − covering numbers at the negative punctures z , so k+ z Γ kz− . By the hy- − = −∈ − − pothesis (λ ,L) (λ ,L) we must have that θi θi which implies + ≥ − + ≥ −

i X i k+θ kz− θ 0, b +c − z b − −c ≥ − JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 436 AL MOMIN so the mapping degree is at least #Γ . Summing everything together, − − 1 [C] [U] C Sσ δ (C,Sσ) ∆(C,U) ∗ = · + ∞ + 2 1 #Γ (U) 0 #Γ(U) ≥ − − + + 2 1 (1 #Γ (U)). = 2 − −

REMARK 3.14. If we drop the hypothesis (λ ,L) (λ ,L), then we lose this con- +i ≥P − i trol on this intersection number since k+θ z kz− θ could be negative. b +c − − b − −c LEMMA 3.15. Suppose (λ ,L) (λ ,L) satisfy condition (E) (actually we only + ≥ − need to assume (λ ,L) satisfies (E)) and J J (Jb (λ ), Jb (λ ): Z ). Then for every + ∈ − − + + finite energy J-plane P in W , [P] [Z ] 1/2. If it is not asymptotic to an orbit ξ ∗ ≥ in L then [P] [Z ] 1. ∗ ≥ If (λ ,L) (λ ,L) satisfy condition (PLC) instead, then for every finite energy + ∼ − J-plane P in W [P] [Z ] 1. ξ ∗ ≥ Proof. If P is asymptotic to d L then the first assertion is immediate by the ∈ formula of Theorem 3.11 since the parity term will contribute 1/2 and all terms are nonnegative. Suppose instead that P is asymptotic to an orbit x L . Then ∉ + π P : C V must intersect L by hypothesis, and therefore P must intersect V ◦ → ZL. Since both are J-holomorphic the intersection contributes 1 to the intersection, which bounds below [P] [Z ] 1 by positivity of intersections. ∗ ≥ If (λ ,L) satisfies (PLC) then πV P intersects L by hypothesis, so [P] [Z ] 1 + ◦ ∗ ≥ as explained above.

We will call a puncture of a holomorphic building (U) a free puncture if there is a path from the puncture to an end of the tree such that all nodes along this path represent trivial cylinders.

LEMMA 3.16. Suppose that (λ ,L) (λ ,L), and J J (Jb (λ ), Jb (λ ): Z ). Let + ≥ − ∈ − − + + (U) be any genus 0 holomorphic building with only 1 positive end. Then If no component of (U) is a branched cover of a component of Z , then • [(U)] [Z ] 0. It is strictly greater than zero if either ∗ ≥ – there is an edge in the bubble tree corresponding to an orbit in L, or – the projection of some component intersects L. In particular, if both (λ ,L),(λ ,L) satisfy (E) and some component of (U) + − is a plane then one of these holds necessarily, so [(U)] [Z ] 0. ∗ L > If some components of (U) are branched covers of components of Z , then • 1 [(U)] [Z ] (1 #Γ− ((U),L)), where Γ− ((U),L) denotes the set of free ∗ ≥ 2 − f ree f ree negative punctures belonging to components which are branched covers of components of Z .

Proof. Let Ui be the vertices of (U) which represent curves that are not branched covers of components of Z , and let Vj be the vertices of (U) which represent

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 437 curves that are branched covers of components of Z . The levelwise additivity and disjoint union properties of the intersection number (Proposition 3.8) imply

X X [(U)] [Z ] [ Ui Vj ] [Z ] ∗ = X + X∗ [U ] [Z ] [V ] [Z ]. = i ∗ + j ∗

In the ﬁrst case, the second sum is empty. Since [U ] [Z ] 0, and is strictly i ∗ ≥ positive if it is asymptotic to some x L or if π U i intersects L in its interior ∈ V ◦ (Lemma 3.12), the ﬁrst statement is clear. For the second case, using the estimate from Lemma 3.13 we have X X 1 [Vj ] [Z ] (1 #Γ (Vj )) ∗ ≥ 2 − − Vj Vj 1 X X 1 #Vj ( ). = 2 + −2 Vj Γ (Vj ) −

Consider a negative puncture z Γ (Vj ) which is not a free puncture for (U). ∈ ∪ − There is a ﬁrst vertex on the path below this puncture which does not represent a trivial cylinder, which is either a Ui or a Vj 0 (and which conversely determines

(Vj ,z)). If this vertex is is not a Vj 0 , then it is a Ui asymptotic to an orbit in L and therefore [Ui ] [Z ] 1/2. Else, it is a Vj 0 . Therefore, given a term in the P P 1∗ ≥ sum Vj Γ (Vj ) 2 which does not represent a free negative puncture, we ﬁnd − − a uniquely determined term in the sum P[U ] [Z ] P 1 that combines with i ∗ + Vj 2 it to make the total nonnegative. By this grouping of terms we determine X X [(U)] [Z ] [U ] [Z ] [V ] [Z ] ∗ = i ∗ + j ∗ X 1 X X 1 [Ui ] [Z ] #Vj ( ) = ∗ + 2 + −2 Vj Γ (Vj ) − 1 (1 #Γ− (U,L)). ≥ 2 − f ree

The extra 1 comes from the fact that there is at least one Vj that is not used to P P 1 cancel one of the terms in Vj Γ (Vj ) 2 (there must be one Vj not on a lower − − level than some other Vj ).

Now the following proposition is a consequence of Lemma 3.16 and Lemma 3.12.

PROPOSITION 3.17. Suppose (λ ,L) (λ ,L) satisfy (E), x P (λ ), y P (λ ) + ≥ − ∈ + ∈ − are not in L, and (U) M (x; y) has genus 0. Then [(U)] [Z ] 0. ∈ J ∗ ≥ If [(U)] [Z ] 0, then each level U i of (U) is a cylinder with [U i ] M (a;b : ∗ = ∈ J 0 Z ) (where J 0 {Jb (λ ), J, Jb (λ ) determined by the level i). In particular, this ∈ + + − − JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 438 AL MOMIN holds whenever (U) is a SFT-limit of a sequence Uk in MJ (x; y : ZL), since Lemma 3.10 implies [(U)] [Z ] 0. ∗ = If instead we only have (λ ,L) (λ ,L) and each component of L is elliptic for + ≥ − both λ , then if (U) is a holomorphic building with one positive puncture, no ± asymptotic limits in L (so Γ− ((U),L) 0), and [(U)] [Z ] 0, then no compo- free = ∗ L = nent U i of (U) is asymptotic to L or such that π U i intersects L in the interior. V ◦ In particular, the compactiﬁed projected image π (U) V \L. V ◦ ⊂ 3.2.2. Compactness under splitting. In the following, suppose λ λ0 λ , + Â Â − (λ ,L) (λ0,L) (λ ,L), and L is elliptic for each. Consider almost-complex + ≥ ≥ − structures J1 J (cJ (λ ), Jb0(λ0): ZL), J2 J (cJ (λ ), Jb0(λ0): ZL), and the almost- ∈ − − ∈ − − complex structures J J (J , J ) described in Section 2.1 obtained from J , J . R ∈ 1 2 1 2 PROPOSITION 3.18. Suppose further that (λ ,L) (λ0,L) (λ ,L) all satisfy + ≥ ≥ − condition (E). Then each level U i of (U) is a cylinder [U i ] M (a;b : Z ) where ∈ J J {Jb (λ ), J2, Jb0(λ0), J1, Jb (λ )}. ∈ + + − − Proof. Let (U) be a SFT-limit for the sequence Uk ; by continuity of the intersection number (Lemma 3.10) we must have [(U)] [Z ] 0. Let (U) be the ∗ L = 2 building obtained by looking only at the top 1 k -levels of (U), and (U)1 be | | + the building obtained by looking only at the bottom k 1 k0 levels of (U). Then −| | (U)1 represents the homotopy class [(U)1] of asymptotically cylindrical maps in W (W , J ), (U) represents the homotopy class [(U) ] of asymptotically 1 = ξ 1 2 2 cylindrical maps in W (W , J ), and [(U)] [(U) (U) ]. The map (U) has a 2 = ξ 2 = 2 ¯ 1 2 connected domain, while the map (U)1 may have a disconnected domain: let i us denote by (U1) the connected components of (U)1. Then ³X ´ [(U)] [(U) (U) ] [(U i )] [(U) ]. = 2 ¯ 1 = 1 ¯ 2 The levelwise additivity and disjoint union properties of Proposition 3.8 imply X [(U)] [Z ] [(U) ] [Z ] [(U i )] [Z ]. ∗ L = 2 ∗ L + 1 ∗ L Let k be the number of negative punctures of (U)2. Then there are k compo- i i nents (U1): label them in such a way that for 1 i k 1, the component (U1) is k≤ ≤ − planar (has no negative punctures), while (U1 ) is cylindrical (has one negative puncture asymptotic to y). Suppose the second alternative of Lemma 3.16 holds for (U)2. Then 1 1 [(U)2] [ZL] k. ∗ ≥ 2 − 2 · k If for (U1 ) the positive asymptotic orbit is not in L, the stronger inequality 1 1 [(U)2] [ZL] (k 1) ∗ ≥ 2 − 2 · − holds. By Lemma 3.16, for the k 1 planar components (U i ) − 1 i 1 [(U )] [ZL] . 1 ∗ ≥ 2

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k For (U1 ) (by Lemma 3.16 as well): [(U k )] [Z ] 0. 1 ∗ L ≥ k Moreover, if (U1 ) has a positive asymptotic limit in L, then the last inequality is strengthened to 1/2. This is because if the second alternative to Lemma 3.16 k ≥ k holds for (U1 ), the inequality implies [(U1 )] [Z ] 1/2. If the first holds, then k ∗ ≥ some level of (U1 ) is asymptotic to L: Lemma 3.16 asserts both that level will have strictly positive intersection, and that each level has nonnegative intersection, so by levelwise additivity [(U k )] [Z ] 1/2 again. It follows that whether 1 ∗ ≥ or not (U k ) has positive asymptotic orbit in L, the inequality [(U)] [Z ] 1/2 1 ∗ ≥ holds. Therefore the first alternative to Lemma 3.16 must apply to (U)2. Since the first alternative to Lemma 3.16 holds for (U)2, [(U)2] [Z ] 0. i ∗ k≥ Lemma 3.16 implies that for 1 i k 1, [(U1)] [Z ] 1/2 and that [(U1 )] ≤ ≤ k −1 ∗ ≥ k∗ [Z ] 0. Therefore 0 [(U)] [Z ] − , and therefore k 1 i.e (U) (U ). ≥ = ∗ ≥ 2 = 1 = 1 Lemma 3.16 implies that [(U) ] [Z ] 0. Since [(U) ] [Z ] 0, it must be that 1 ∗ ≥ 2 ∗ ≥ [(U) ] [Z ] [(U) ] [Z ] 0. Lemma 3.16 therefore asserts that (U) has no 2 ∗ = 1 ∗ = 2 planar components and is never asymptotic to an orbit in L - because k 1 = it must be that every level of (U) is a single cylinder. Since [(U) ] [Z ] 0 2 1 ∗ = and #Γ− ((U) ,L) 0 (this is defined in Lemma 3.16), the second alternative free 1 = of Lemma 3.16 cannot hold, so the first alternative to Lemma 3.16 must apply. It implies that (U)1 also has no planar components and is never asymptotic to an orbit in L. Therefore, every level U i for (U) is a cylinder and [U i ] [Z ] 0 ∗ ≥ - it follows by levelwise additivity [U i ] [Z ] 0 for all levels i, in particular ∗ = U i M (a,b : Z ). ∈ J 0 In case (λ ,L) (λ0,L) (λ ,L) do not necessarily satisfy condition (E) (but + ≥ ≥ − are each such that L is elliptic), it will be useful in the proof of Theorem 1.4 to note the following. Let Jn be a sequence of almost-complex structures splitting into J J . 1 ¯ 2 PROPOSITION 3.19. Suppose x, y L and that the sequence U M (x; y : Z ) ∉ n ∈ Jn has the SFT-limit (U) (U) (U) . Then [(U) ] [Z ] 0,[(U) ] [Z ] 0, and = 2 ¯ 1 1 ∗ L = 2 ∗ L = therefore by Lemma 3.16 each component U i of the limit satisfies the following properties: No asymptotic limit of U i is an orbit in L. • π U i has no interior intersections with L. • V ◦ In particular, the compactified image π (U) V \L (in particular it defines a V ◦ i ⊂ homotopy from x to y since its domain is a compactified cylinder). Proof. The proof is similar to the proof of Proposition 3.18. First, by continuity of the intersection number [(U)] [Z ] 0. Let (U i ) be the connected compo- ∗ l = 1 nents of (U)1. Then X 0 [(U)] [Z ] [(U) ] [Z ] [(U i )] [Z ]. = ∗ L = 2 ∗ L + 1 ∗ L Let k be the number of negative punctures of (U) , and l #Γ− ((U) ,L) k 1 2 = free 2 ≤ − (recall from earlier that this is the number of free negative punctures of (U)2

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 440 AL MOMIN asymptotic to a cover of L). Then by Lemma 3.16 1 1 [(U)2] [ZL] l, ∗ ≥ 2 − 2 · and for each puncture of Γfree− ((U)2,L) there is a planar component of (U)1, say (U)1,...,(U)l such that [(U)i ] [Z ] 1 (again by Lemma 3.16). Taken together 1 1 1 ∗ L ≥ 2 this implies [(U)] [ZL] 0, which is impossible so in fact l 0. Thus every ∗ 1 > k =j component (U) ,(U) ,...(U) satisﬁes Γ− ((U) ,L) Γ− ((U) ,L) . Apply- 2 1 1 free 2 = free 1 = ; ing Lemma 3.16 it follows that each term in the above sum is nonnegative, and therefore each term is zero. Reading Lemma 3.16 once again, it now asserts that the components have the properties stated above.

3.2.3. Proper link classes (PLC). If (λ ,L,[a]) satisfy (PLC), we can prove a com- ± pactness theorem for holomorphic cylinders within the class of broken trajec- tories such that all asymptotic limits lie within the class [a]:

PROPOSITION 3.20. Suppose (λ ,L) (λ ,L), and both (λ ,L,[a]) satisfy (PLC). + ∼ − ± Let J J (Jb (λ ), Jb (λ ): Z ) and x be λ -Reeb orbits in the class [a]. If [Uk ] ∈ − − + + ± ± i ∈ MJ (x ;x : Z ) converges to a building [(U)] in MJ , then each level U of (U) + − represents an element in MJ (b;c : Z ) (J 0 {Jb (λ ), J, Jb (λ )}) and b,c [a]. 0 ∈ + + − − ∈ Proof. Condition (PLC) prevents bubbling off as follows. Let Uk be a conver- gent subsequence with limit building (U). Denote the connected components by U i (vertices of the bubble tree) indexed by i I. Let S be the underlying ∈ nodal domain of the building with components Si corresponding to the U i . By properties CHCE2, CHCE3 of SFT-convergence (these are obvious analogs in the setting of the SFT compactness theorem for cylindrical ends of the properties C2, C3 introduced in Section 2.1.4 - the precise statements are found in [3, i i R 1 i Section 8.2]) there are maps φk : S S from S to the domain of Uk such i i → × that U φ i converges to U in C ∞ (up to a translation in cylindrical levels). k ◦ k |S loc Suppose U i is a plane for some i. Then by Lemma 3.15 and condition (PLC) i i i [U ] [Z ] 1. It follows that Uk φk Si Z 0, because Uk φk Si converges to i ∗ ≥ ◦ | · 6= ◦ | 0 i U in Cloc∞ and positivity of intersections implies that all maps C -near U intersect Z as well. Therefore int(U , Z ) 0 and so [U ] [Z ] 0 contrary to the k 6= k ∗ > hypothesis [Uk ] M (x ;x : Z ). This shows that the bubble tree is linear i.e., ∈ + − all components are cylinders. Since x L, it is easy to see as follows that no asymptotic orbit between lev- ∉ els is in L or not in [a]. If not, there would be a ﬁrst level U i with positive asymptotic orbit x0 [a] and negative asymptotic orbit y0 either in L or not i ∈ in [a]. But π U provides a homotopy between x0 and y0, so by condition V ◦ (PLC) there must be an interior intersection with L in either case. But then the Cloc∞ -convergence property of SFT convergence cited in the previous paragraph implies by the same argument used above that for large k, int(U , Z ) 0, k > which violates Theorem 3.11 since U , Z have non-identical images and [U ] k L k ∗ [Z ] 0. After establishing this fact about the asymptotic limits, no U i can have L = identical image with Z and the same Cloc∞ -convergence argument shows that

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 441 int(U i , Z ) 0 for each level. The formula of Theorem 3.11 shows [U i ] [Z ] 0. i = ∗ = So [U ] M (x0; y0 : Z ) and x0, y0 [a]. ∈ J 0 ∈ REMARK 3.21. The same proof goes for the limit of a sequence of such Jk -holomorphic cylinders where the Jk converge in C ∞ to a J J (Jb , Jb : Z ); in partic- ∈ + − ular we get a similar statement when considering M{Jl }(1;1 : Z ). A similar result holds for a sequence of J J (J , J ). In the following, sup- R ∈ 1 2 pose λ λ0 λ , that (λ ,0, ,L,[a]) each satisfy (PLC), and that JR J (J1, J2) + Â Â − + − k ∈ with Rk (where J1 J (Jb , Jb0 : Z ), J2 J (Jb0, Jb : Z )). ↑ ∞ ∈ − ∈ + PROPOSITION 3.22. If (U) is a SFT-limit of a sequence Uk MJ (x; y : Z ) with ∈ Rk R and [x] [y] [a], then each level U i represents a cylinder [U i ] M (b;c : k ↑ ∞ = = ∈ J Z ) where [b] [c] [a] and J {Jb (λ ), J2, Jb0(λ0), J1, Jb (λ )}. = = ∈ + + − − Proof. The proof is similar to the previous proposition. Properties C2, C3 of SFT-convergence and condition (PLC) prohibit the bubbling of planes as in the above proof, so each level U i is connected and cylindrical. Similarly, the arguments in the second paragraph of the proof of Proposition 3.20 prevent any level from intersecting ZL, prevent any level from being asymptotic to any component of L, and force any asymptotic orbit x between levels to be in the homotopy class [a]. So the SFT-limit must have the form stated.

In the proof of Theorem 1.4, we will require a more general fact. Suppose that L satisfies the linking condition of Theorem 1.4, namely L is such that every disk F with boundary ∂F L and [∂F ] 0 ⊂ 6= ∈ H1(L) has an interior intersection with L, and that [a] is a proper link class for L (Definition 1.3). Suppose only that λ ,λ0,λ are each nondegenerate and such that L is closed under the Reeb + − flows. Then

PROPOSITION 3.23. Consider a sequence U of cylinders in MJ (x; y : Z ), with k Rk R and [x] [y] [a]. Then any SFT-limit (U) (U) (U) is such that k ↑ ∞ = = = 1 ¯ 2 [(U) ] [Z ] 0,[(U) ] [Z ] 0, and each component U i of the limit satisﬁes 1 ∗ L = 2 ∗ L = the following properties: No asymptotic limit of U i is an orbit in L. • π U i has no interior intersections with L. • V ◦ In particular, the compactiﬁed images π (U) V \L. V ◦ ⊂ Proof. The assertion that for each component U i that π U i has no interior V ◦ intersections with L is argued using the Cloc∞ convergence properties of SFT- convergence in a familiar way. By properties CHCE2, CHCE3 (analogs of properties C2, C3 in Section 2.1.4—cf. [3, Section 8.2] of SFT-convergence there are i i R 1 i i maps φk : S S from S the domain of U to the domain of Uk such i → × i = that Uk φk Si converges to U in Cloc∞ (up to a translation in cylindrical levels). ◦i | Since U intersects ZL positively, it follows that Uk must intersect ZL for all k

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 442 AL MOMIN large enough, but this contradicts [U ] [Z ] 0. So we conclude U i has no k ∗ L = interior intersections with ZL. The assertion that no asymptotic orbit of any component U i is in L (equivalently, no edge in the bubble tree represents an orbit in L) is argued as follows. Suppose some edge E in the bubble tree represents an orbit in L. Suppose first that the portion of the holomorphic building corresponding to the subtree below this edge is planar. Choose an edge E 0 in this subtree such that no edge lower on the bubble tree lies in L (this is possible because the tree is planar so there are no free negative asymptotic orbits, thus every edge has a vertex below it). Let (P) be the holomorphic building corresponding to the subtree below this edge E 0. Then π (P) has an interior intersection with L, which V ◦ must correspond to an interior intersection of one of its components (since no asymptotic orbit is in L), contradicting the first assertion which we have already proved. Otherwise, the portion below E has one free negative puncture. E is therefore on the unique path from the free positive puncture to the free negative puncture. Without loss of generality, suppose E is the highest edge along this path (i.e., closest to the positive free puncture). Let (X ) be the tree obtained by removing from the original bubble tree the portion below this edge E (by hypothesis E is not the positive free puncture so this is nonempty). By the previous step, any free negative puncture for (X ) bounds a planar building (P) such that each component satisfies the conclusions of this proposition. Hence, its compactified image satisfies π (P) V \L. It will now follow that the interior V ◦ ⊂ of π (X ) V \L, since no edge of (X ) represents an orbit in L (except E) and V ◦ ⊂ we have already shown that no component of (U) has an interior intersection with ZL. However, the negative asymptotic orbit of (X ) is in L (since by hypothesis E represents an orbit in L), and π (X ) realizes a homotopy between this V ◦ orbit in L and the positive asymptotic orbit x [a] through V \L. But this con- ∈ tradicts the hypothesis that [a] is a proper link class, by definition. Therefore no such edge E can exist, which finally proves that no asymptotic orbit of any component U i is in L as claimed.

4. CYLINDRICAL CONTACT HOMOLOGYOF COMPLEMENTS The two crucial requirements for the deﬁnition of contact homology are compactness and regularity of the moduli spaces. Propositions 3.17, 3.20, 3.18, 3.22 supply the required compactness for the spaces MJ (x; y : Z ). In the cylindrical case, Theorem 2.6 provides the required transversality for the moduli spaces (to prove ∂2 0), but in the noncylindrical case these arguments do not ap- = ply and we need to make some hypotheses. When considering noncylindrical almost-complex structures, we will consider only simple homotopy classes of loops in V \L. Then following well-known arguments as in e.g., [17, 13,6, 16], for each ﬁxed choice of cylindrical ends Jb there is a residual subset Jgen ± = Jgen(Jb , Jb : Z ) of admissible almost-complex structures for which the Cauchy– − + Riemann operator will be regular for all J-holomorphic cylinders connecting

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 443 loops in simple classes [a] if J J . Similarly, there are generic paths of ∈ gen almost-complex structures Jτ,gen Jτ,gen(Jb , Jb : Z ). One should specify a Ba- = − + nach space of perturbations of the almost-complex structures; the procedure is now standard so we refer to [17, 13, 16]. The key point is that each holomorphic cylinder with a positive and negative puncture which is not a cover of a cylinder in Z has a point of injectivity outside W W Z where we are free + ∪ − ∪ to inﬁnitesimally perturb the almost-complex structure.

4.1. Boundary Maps; Chain maps and homotopies. With the properties of intersections from Section 3.1, we can describe our cylindrical contact chain complex on V \L in detail and show that it has the required properties. The proofs of Propositions 4.1, 4.2, 4.3 stated below are postponed to the end of the section. Suppose (λ,L) satisfies (E), or that (λ,L,[a]) satisfies (PLC), and choose J 1 ∈ J (λ). Let Z Z be the cylinders over L (Z π− (L)). Let G denote the gen = L = V set of SFT-good Reeb orbits, and G 0 denote G \L. Let I be an ideal in ker(c1(ξ)) where c (ξ): H (V ;Z) Z is the first Chern class of ξ, and let R be the ring 1 2 → R Q[H (V ;Z)/I]. We will denote elements of R as sums Pq e[Ai ], where q = 2 i i ∈ Q,{A } H (V ;Z) is a finite set and [A ] denotes its image in the quotient i ⊂ 2 i H2(V ;Z)/I. Consider the R-module CC (λ rel L) freely generated by elements ∗ q (one for each x G 0). When considering (PLC) we only consider orbits x ∈ in the homotopy class [a] and denote it by CC [a](λ rel L) instead. Grade the module as in [14]. Recall from [14] that this also∗ determines a homology class A H (V ;Z) for each asymptotically cylindrical curve U, and we denote by M A ∈ 2 J the moduli space of solutions representing the class A. Define a degree 1 map ∂L ∂L(J) on the generators of CC (λ rel L) (resp. − = ∗ CC [a](λ rel L)) by summing over index-1 solutions (and extend linearly): ∗

A X ²([U]) n mx x y = · cov([U]) [U] M A (x;y:Z ) ∈ J X (4.1) n n A e[A] R x y = x y ∈ A H2(V ;Z) X∈ ∂L qx nx y qy = y G ∈ 0 Here mx denotes the covering number of the orbit x, and cov(U) denotes the covering number of the map U. The sign ²[U] 1 is determined by compar- = ± ing a choice of coherent orientation [4] with the canonical orientation given by the free R-action (it is well-defined since both asymptotic orbits are SFT-good). Since the moduli spaces we consider are open/closed subsets of the moduli space for (V, J), a coherent orientation chosen in the usual way will restrict to one for our moduli spaces and it does not require further comment (see [4] or [14] for details). Though we suppressed it in the notation, it is an essential in- gredient in the above definition. This sum will always be finite by compactness (Proposition 3.17, or 3.20) and transversality (Theorem 2.6).

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2 PROPOSITION 4.1. Under the above hypotheses ∂L 0. Denote the chain com- [a=] plex (CC (λ rel L),∂L(J)) by CC (λ, J rel L) (or CC (λ, J rel L) for (PLC)). ∗ ∗ Under hypothesis (E), the complex splits according∗ to homotopy classes of loops in V \L: M CC (λ, J rel L) CC [a](λ, J rel L). ∗ = [a] π (ΩV ) ∗ ∈ 0 We remark that this is the usual deﬁnition of the cylindrical contact homology chain complex for the pair (λ , J ) on V \L. |V \L |V \L

4.1.1. Chain maps. Suppose that (λ ,L) (λ ,L) satisfy (E) and [a] is a homo- + ≥ − topy class of simple loops, or that (λ ,L,[a]) (λ ,L,[a]) satisfy (PLC) and [a] + ∼ − is simple. Choosing J Jgen(λ ), we consider the chain complexes ± ∈ ± CC [a](λ , J rel L), CC [a](λ , J rel L), ∗ + + ∗ − − defined in Section 4.1. We will denote their boundary maps ∂L(J ) when it is ± needed to clarify. Also, we use G 0(λ ) to denote the set of SFT-good Reeb orbits ± not in L. Choose and fix a J Jgen(Jb , Jb : Z ). We will define a map ∈ − + Φ[a] (J): CC [a](λ , J rel L) CC [a](λ , J rel L) −+ ∗ + + → ∗ − − For x G 0(λ ), we sum over index-0 components of the moduli space of cylin- ∈ + ders to define as in [14]:

[a] X Φ (J)qx : mx y qy (4.2) −+ = y G 0(λ ) ∈ − A where mx y and mx y are deﬁned as

A X ²(U) m mx , x y = · cov(U) U M A (x;y:Z ) (4.3) ∈ J X A π(A) mx y mx y e R. = A H (V ;Z) ∈ ∈ 2 The sign ²(U) is again determined by the choices of coherent orientations assigned [14,4]. This sum is finite by compactness (Proposition 3.17, or 3.20) and transversality (using Theorem 2.9 and the fact that [a] is simple). It decreases the action by Stokes’ Theorem (see the comment after Lemma 2.1) and preserves the class [a] (else the cylinders would have nontrivial intersection with Z ). Since we only consider homotopy classes of loops with only simple Reeb orbits, the numbers mx and cov(U) are equal to 1, so in fact the first equation in (4.3) simplifies to X m A ²(U). x y = U M A (x;y:Z ) ∈ J

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PROPOSITION 4.2. Under the hypotheses used to deﬁne the operators above (including the regularity hypothesis and that [a] is simple) [a] [a] ∂L(J ) Φ (J) Φ (J) ∂L(J ) 0. − ◦ −+ − −+ ◦ + =

4.1.2. Homotopies between chain maps. Again, let (λ ,L) (λ ,L) and suppose + ∼ − either they both satisfy (E) and (λ ,L) (λ ,L) or that [a] is a proper link class + ≥ − and both satisfy (PLC). Let Jb Jgen(λ ) so the chain complexes ± ∈ ± CC (λ , J rel L) ∗ ± ± are defined as above. Now consider two choices J0, J1 Jgen(Jb , Jb : Z ), which defines chain maps [a] [a] ∈ − + Φ (J0),Φ (J1) as above in the simple homotopy class [a]. Choose a homotopy Jl Jτ,gen(Jb , Jb : Z ) between the two. Define for x G 0(λ ) (summing only ∈ − + ∈ + over the index-0 part of the moduli space M{Jl }(x; y : Z ))

A X ²(µ,U) k mx x y = cov(U) (µ,U) M A (x;y:Z ) ∈ {Jl } X A [A] (4.4) kx y kx y e R = A H (V ;Z) ∈ ∈ 2 [a] X K (Jτ)qx : kx y qy −+ = y G 0(λ ) ∈ − Again, mx ,cov(U) are 1 because [a] is simple, and this sum is ﬁnite by compactness (Proposition 3.17, or 3.20) and transversality (Theorem 2.9). Also, everything again must be oriented appropriately, the details of which we omit. Finally, by “index 0” we refer to the total Fredholm index including the param- eter µ: this means that Ind(µ,U) Ind(U) 1, so we mean that Ind(U) 1. = + = − Then

PROPOSITION 4.3. Under the above hypotheses [a] [a] K (Jτ) ∂L(J ) ∂L(J ) K (Jτ) −+ ◦ + + − ◦ −+ [a] [a] Φ (J1) Φ (J0) = −+ − −+ Proof. The proofs of Proposition 4.1, 4.2, 4.3 are standard in contact homology (see [14] Sections 1.9.1, 1.9.2); the only parts that need to be veriﬁed are that the moduli spaces have the correct boundary structure, which is done as follows: Each proof involves the count of two level holomorphic buildings of cylin- • ders, each in M (1;1 : Z ). For example, in the proof ∂2 0, one counts L = pairs of index-1 cylinders in MJ (1;1 : ZL) which can be concatenated, U U . By levelwise additivity (Proposition 3.8), this conﬁguration also 1 ¯ 2 has intersection number 0 with ZL. After choosing asymptotic markers, the pair can be glued, and by SFT- • continuity of the intersection number (Lemma 3.10), the glued cylinders must be in M (1;1 : ZL).

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By index additivity and transversality, the glued solutions belong to a con- • nected component of the moduli space diffeomorphic to an interval (after quotienting the free R-action in the R-invariant case). By homotopy invariance of the intersection number it is a subset of M (1;1 : ZL). At the other end of the interval, there is an SFT-limit. The compactness • Propositions 3.17 and 3.20 show that the limit is necessarily a broken tra- jectory with each cylinder in M (1;1 : ZL) again; transversality implies there are only two levels and determines the index of both cylinders, so it represents an object corresponding to another term in the algebraic identity. Comparison of the orientations shows that the terms representing oppo- • site ends of the moduli space cancel, proving the required algebraic identity (Proposition 4.1, 4.2, or 4.3). In the cylindrical case the required transversality for nonsimple curves is guaranteed by Theorem 2.6, so ∂2 0 even in nonsimple homotopy classes [a]. L = It is clear that all maps preserve the class [a]; else, some curve in the count (say [U]) is such that π U intersects L, but then [U] [Z ] 0 because int(U,Z) V ◦ ∗ > 6= 0 (see Theorem 3.11), contradicting [U] M (1;1 : Z ). ∈ J 4.2. Consequences of chain maps/homotopies. Most of the following proposition follows already from Propositions 4.1, 4.2, 4.3:

PROPOSITION 4.4. Given (λ ,L) (λ ,L) both satisfying (E), and a simple class + ≥ − [a] of loops in V \L: for each generic J Jgen(λ ,λ : ZL) there is a chain map ∈ − + Φ[a] (J): CC [a](λ , J rel L) CC [a](λ , J rel L). −+ ∗ + + → ∗ − − [a] [a] Moreover, given two such chain maps Φ (J1),Φ (J2) and a generic homotopy −+ −+ Jτ Jτ,gen(λ ,λ : ZL), there is a chain homotopy operator K (Jτ) i.e., ∈ − + [a] [a] Φ (J1) Φ (J2) K (Jτ) ∂ K (Jτ) ∂ . −+ − −+ = ◦ + + ◦ − Therefore, there are natural maps in homology ³ ´ Φ[a] : CCH [a](λ , J rel L) CCH [a](λ , J rel L) −+ ∗ + + → ∗ − − ∗ which furthermore satisfy

[a] [a] [a] (Φ 0) (Φ0 ) (Φ ) . − ∗ ◦ + ∗ = −+ ∗ If (λ ,L) (λ ,L) then this map is an isomorphism. − ≡ + Similarly, if (λ ,L,[a]) (λ ,L,[a]) both satisfy (PLC) and [a] is simple then + ∼ − for each generic J as above there is a chain map Φ[a] (J), and given two such −+ maps and a generic homotopy between them Jτ there is a chain homotopy K (Jτ) satisfying the above identity, so there are natural maps ³ ´ Φ[a] : CCH [a](λ , J rel L) CCH [a](λ , J rel L) −+ ∗ + + → ∗ − − ∗ which obey the above composition law and are also isomorphisms.

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Proof. The proof of Proposition 4.4 is complete once the composition law is verified and we prove that the chain maps give isomorphisms. The composition law is verified by a gluing/compactness argument, and we only need to check that we have compactness under stretching and enough transversality; the required compactness is guaranteed by Propositions 3.18, 3.22, and the required transversality is guaranteed by choosing J , J J and con- 1 2 ∈ gen sidering only simple homotopy classes of loops. Once the composition law is verified, one considers the compositions (Φ (J1) Φ (J2)) (Φ ) , and −+ ◦ +− ∗ =∼ −− ∗ (Φ (J2) Φ (J1)) (Φ ) (under condition (E), both compositions can only +− ◦ −+ ∗ =∼ ++ ∗ be made if (λ ,L) (λ ,L), hence the hypothesis in Proposition 4.4). Using a + ≡ − cylindrical almost-complex structure we find that (Φ ) is the identity map: ±± this is because the index-0 cylinders - whose count defines∗ these chain maps - are precisely the trivial cylinders over closed orbits, so Φ (Jb) is in fact the ±± ∗ identity map at the chain level for cylindrical Jb. This proves the maps (Φ ) ±∓ ∗ are both injective and surjective and therefore isomorphisms.

REMARK 4.5. We would like to point out that the condition “[a] is simple” can be relaxed to the condition “[a] is such that, for one of λ , all closed Reeb or- ± bits in [a] are simple”. First, the chain complexes are deﬁned even for [a] not simple. Then the condition that one of λ has only simple closed orbits in ± [a] guarantees that holomorphic cylinders in a cobordism used to deﬁne the chain maps above are somewhere injective [30, 40] (hence also have an open set of somewhere injective points), and therefore chain maps/homotopies can be constructed for almost-complex structures in Jgen just as above. The conclusion of Proposition 4.4 holds even if the contact form at the opposite end of the cobordism has multiply covered Reeb orbits in [a]. We will use this observation in Section6.

4.3. Action ﬁltration. As in Morse theory, the chain complexes can be ﬁltered [a], N [a] by action. We consider the subspace CC ≤ (λ, J rel L) of CC (λ, J rel L) gen- R ∗ ∗ erated by orbits with action A (x) λ N. By Stokes’ Theorem, the differen- = x ≤ tial decreases the action. For A B (one may take B as well here) there are ≤ = ∞ natural chain maps

[a], A [a], B ιB,A : CC ≤ (λ, J rel L) CC ≤ (λ, J rel L), qx qx ∗ → ∗ 7→ and clearly ιC,B ιB,A ιC,A. The morphisms Φ and homotopies K respect the ◦ = [a] [a] filtration: specifically, given Φ10 : CC (λ0, J0 rel L) CC (λ1, J1 rel L) ∗ → ∗ ¡ [a], A ¢ [a], A Φ10 CC ≤ (λ0, J0 rel L) CC ≤ (λ1, J1 rel L) ∗ ⊂ ∗ again by Stokes’ Theorem, and any two such maps (defined by different choices of J) are chain homotopic. [a], N In fact, in order to construct CC ≤ (λ, J rel L) with N one only re- ∗ < ∞ quires that λ satisfies (E) up to action N. This means that the assertion (E) (resp. (PLC)) about contractible in V \L Reeb orbits holds for orbits with action at most N. Then even though CC [a](λ rel L) may not be defined, for every ∗ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 448 AL MOMIN

A N one can define the chain complexes < [a], A CC ≤ (λ, J rel L) ∗ as above, and they are filtered by the action in the same way. This follows ex- actly as the construction of the above chain complexes. The proofs of compactness are carried out as straightforward extensions of Propositions 3.17, 3.18, 3.20, 3.22, using the fact that any contractible (in V \L) Reeb orbit has action greater than N to rule out bubbling for moduli spaces of holomorphic cylinders. The Fredholm theory remains the same. We will use these filtered chain complexes later.

5. FORMSWITHDEGENERACIES It is possible to obtain existence results that include cases where the form λ may be degenerate or have closed orbits that are contractible in V \L, namely Theorem 1.4. The strategy is to take a sequence of small perturbations of a degenerate form to sufficiently nondegenerate approximating forms and deduce the existence of closed Reeb orbit for the approximating forms by a stretching argument. Then, using the action filtration to bound the action of the orbits found this the way (which provides C 1-bounds), the Arzelà–Ascoli Theorem is used to find the desired Reeb orbit for the original degenerate form. We also briefly justify the use of the Morse–Bott complex (when λ satisfies (PLC) or (E) but is Morse–Bott nondegenerate) to compute CCH [a]([λ] rel L) for L ∗ 6= ; 5.1. Perturbing degenerate contact forms. The proof of the following Lemma closely follows the proof of [9, Lemma 7.1] (pages 36–37).

LEMMA 5.1. Suppose (λ,L) is such that L consists of nondegenerate closed Reeb orbits for λ (including multiple covers). Given any N 0, there is a sequence > λ f λ λ (i.e., f 1 in C ∞(V ;R)) such that (λ ,L) (λ,L) and such that n = n · → n → n ≡ all periodic orbits of the Reeb vector ﬁeld for λn of action at most N are nondegenerate.

Proof. Around each Reeb orbit x of action at most N choose neighborhoods x U V V satisfying ⊂ x ⊂ x ⊂ 1 2 1 2 1. For all such x, Vx S D (2), and in these coordinates Ux S D (1). =∼ × =∼ × 2. The Reeb vector ﬁeld X is transverse to the pages {t} D2(2) of V . λ × x 3. For all t S1, any orbit starting at a point {t} D2(1) (i.e., starting at a ∈ × point in U ) stays inside V for at least time N 1, and thus at least until x x + the ﬁrst return to {t} D2(2) which occurs before N ². × + If x L is an orbit of action at most N, then we choose V in such a way that, in ∉ x addition, V L . Moreover, select the neighborhoods V around the com- x ∩ = ; Li ponents Li of L such that the VLi are mutually disjoint and contain no other orbits of period at most N in VLi (other than iterates of Li ). This is possible because the Li and covers are assumed nondegenerate.

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By the Arzelà–Ascoli Theorem, the set ΓN of periodic orbits of action at most N is compact. By the compactness of the orbit set, it is not difficult to see that the image of these orbits in V (also denoted ΓN ) may be covered by finitely many of the above sets U , i.e., Γ U U . Order them in such a way i N ⊂ 1 ∪ ··· ∪ k that U ,...,U are the neighborhoods of the form U V for components L 1 m Li ⊂ Li i of L among {U1,...,Uk }. We modify λ on V1,...,Vk in such a way that there are no degenerate orbits of period at most N in these Vi by the method of [9, Lemma 7.1]. Choose any of the neighborhood pairs (Ui ,Vi ). Let U be any neighborhood of λ. The proof in [9] shows how one can find a form λ0 U such that ∈ 1. λ0 differs from λ only on a compact subset of Vi , 2. any λ0 orbit that originates in Ui of action at most N remains in Vi for time N 1 ², + − 3. all closed orbits of period at most N contained in Ui are nondegenerate, 4. there are no closed orbits of action at most N originating in the comple- Sk ment V \ j 1 U j (for λn0 sufficiently near λ); therefore all closed orbits are = Sk still contained in j 1 U j . = Proceeding one neighborhood pair at a time, one finds a form λi U such Sk ∈ that j 1 U j still covers all the closed Reeb orbits, but all closed Reeb orbits in Si = j 1 U j are nondegenerate (the above procedure shows how to make all Reeb = Sk orbits in Ui nondegenerate without introducing new ones in V \ j 1 U j , and = by choosing the perturbation small enough all Reeb orbits contained in U j for j i remain nondegenerate). The form λk will be the desired form. ≤ However, we have to take care with the perturbations so they satisfy the conclusions stated above. By hypothesis all periodic orbits originating in U1,. . . ,Um of period at most N are nondegenerate, by the choice of the VLi and the choice of the ordering. In this case, in the above process we do not need to perturb the form for the first m steps, i.e., we can take λm λ. In subsequent steps, the = new perturbations λi ,i m differ from λm only on Si V , which is dis- > j m 1 j joint from L by choice; in particular there is a neighborhood= + N containing L on which λi λ . Therefore in the end (λk ,L) (λ,L). |N = |N ≡ To summarize, for any neighborhood U of λ, there is a λ0 such that

1. λ0 has only nondegenerate periodic orbits of period at most N contained in U1 Uk . ∪ ··· ∪ Sk 2. λ0 has no periodic orbits of period at most N which enter V \ i 1 Ui . = 3. There is a neighborhood N of L such that λ0 λ , thus (λ0,L) (λ,L). |N = |N ≡ The desired sequence is obtained by choosing a countable neighborhood ba- sis U at λ and choosing λ U as above. n n ∈ n Let [(λ,L)] denote the subset of the set of (positive) contact forms µ for ξ (which we will denote Λ(ξ)) such that (µ,L) (λ,L), equipped with the C ∞ ≡ topology as a subspace of Λ(ξ). It is a closed subset of Λ(ξ) so its topology can be obtained from a complete metric space structure on it. Let us denote by

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Λgen(ξ) the set of nondegenerate contact forms - as is well-known this is the intersection of a countable set of open dense sets, and therefore dense in Λgen(ξ). Then in fact

LEMMA 5.2. Suppose λ is such that all orbits in L are nondegenerate. The set [(λ,L)] : ©µ µ [(λ,L)],µ Λ (ξ)ª gen = | ∈ ∈ gen is the intersection of a countable collection of open, dense sets (in [(λ,L)]) and therefore also dense.

Proof. Let GgN denote the subset of Λ(ξ) for which all periodic orbits of action N are nondegenerate, and GN GgN [(λ,L)]. The argument of [9, Lemma 7.1] ≤ = ∩ shows that GgN is open in Λ(ξ), and therefore GN is also open in the subspace [(λ,L)]. In the previous Lemma it was shown that GN is dense in [(λ,L)]. The conclusion follows from the observation that \∞ [(λ,L)]gen GN = N 1 = is a countable intersection of open and dense sets. Finally, we note the following

LEMMA 5.3. If λ is a contact form and L is a closed link for the Reeb vector ﬁeld, then in any C ∞-neighborhood U of λ there is a λ0 such that (λ0,L) (λ,L) and ∼ each component of L (including multiple covers) is nondegenerate for λ0. This can be achieved by an arbitrarily small perturbation to the 2-jet of λ along L (leaving the 1-jet unperturbed so L is still tangent to the Reeb vector ﬁeld). 5.2. The proof of the implied existence Theorem 1.4.

Proof of Theorem 1.4. Let λ0 be any form such that L is closed for the Reeb vector field and elliptic nondegenerate. Suppose (λ,L) (λ0,L) is nondegenerate ≡ and satisfies (E), and by rescaling if necessary that λ0 λ. Choose a constant c ≺ such that also cλ λ0. Choose J J (λ), J 0 J (λ0), and J J (J, J 0 : Z ), ≺ ∈ gen ∈ gen 1 ∈ gen L J J (J 0, J : Z ) (to be specific J is equal to J on W −(cλ)). Then we have 0 ∈ gen L 0 the path J ,R 0 of almost-complex structures in the symplectization splitting R ≥ along λ0 to (W , J ) (W , J ). ξ 0 ¯ ξ 1 Claim: for each R sufficiently large there is a JR -holomorphic cylinder U such that [U] [Z ] 0 with positive asymptotic orbit having λ-action at most N and ∗ L = asymptotic limits in [a]. To see why this is so, we may find a sequence J J n → R with each J J (since J is Baire), and each such defines a morphism n ∈ gen gen [a], N [a], N Φ(J ): CC ≤ (λ, J rel L) CC ≤ (c λ, J rel L) n → · Such a chain map is chain homotopic to the morphism obtained from a cylindrical almost-complex structure Jb, which can be identified with inclusion [a], N [a], N [a], N/c ιN/c,N : CC ≤ (λ, J rel L) CC ≤ (c λ, J rel L) CC ≤ (λ, J rel L) → · =∼ q q x 7→ x

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 451 so Φ(Jn) is nonzero if the homology is nonzero. Let q be a ∂J -closed element of CC [a](λ, J rel L) generated by orbits of action N such that [q] 0 when con- ∗ [a] ≤ 6= sidered as an element of CCH (λ, J rel L). Since Φ(Jn) is chain homotopic to the identity and q is closed but∗ not exact, it is easy to deduce that Φ(J )(q) 0, n 6= so there must be a J ﬁnite energy holomorphic cylinder U with [U ] [Z ] 0 n n n ∗ L = and positive asymptotic orbit of action at most N. Taking a SFT-limit as n , → ∞ by Proposition 3.17 (and using the fact that (λ,L),(cλ,L) satisfy (E)) we obtain a JR -holomorphic cylindrical building, which has a component UR which is a JR - holomorphic cylinder with [U ] [Z ] 0, positive asymptotic orbit of action at R ∗ L = most N, and asymptotic limits in [a] as claimed.

Since R was arbitrary, we make take a sequence of JRn finite energy cylinders, U , with R , each with positive asymptotic orbit of action at most N, n n ↑ ∞ [U ] [Z ] 0, and both asymptotic limits in [a]. To continue, first suppose n ∗ L = that λ0 is nondegenerate up to action N. Then by SFT-compactness we obtain a limit holomorphic building (U) (U) (U) in (W , J ) (W , J ). Proposi- = 0 ¯ 1 ξ 0 ¯ ξ 1 tion 3.19 asserts that the compactified image π (U) V \L. The image is a V ◦ ⊂ piecewise-smooth cylinder with one boundary component equal to x and the other boundary component equal to y. Suppose (U)1 has k free negative punc- 1 k 1 tures. Let (U)0,...,(U)0− be the planar components of (U)0. Consider now the connected subbuilding

³ 1 k 1´ (X ) (U) ...(U) − (U) = 0 + 0 ¯ 1 with one free negative puncture which is a negative asymptotic orbit for (U)1, and therefore a closed Reeb orbit for λ0, say z. The compactified image π (X ) V ◦ is a compact cylinder with boundary components x and z and image V \L, ⊂ and therefore [z] [x] [a]. Thus z is the desired closed λ0-Reeb orbit in [a] = = with action at most N (since action is nonincreasing as one descends levels). To conclude in the case λ0 is degenerate, select λ0 λ0 in C ∞ with (λ0 ,L) n → n ≡ (λ0,L) such that λn0 is nondegenerate up to action N by Lemma 5.2. For each n sufficiently large we obtain a closed λn0 -Reeb orbit xn as above (since eventually λ0 λ). Applying the Arzelà–Ascoli Theorem, we obtain a closed λ0-Reeb orbit n ≺ x as a C ∞ limit of the xn. x cannot be a l-fold cover of a component of L for any l 1, because if it were then one could show that 1 is an eigenvalue ≥ for the corresponding linearized lth-return map of the Reeb flow, contradicting the hypothesis that each cover of L is elliptic nondegenerate. It follows easily 0 that [x] [a], since x x in C ∞, any loop C -close enough to x is in the same = n → homotopy class as x, and every x [a]. The orbit x also has action at most N. n ∈ If instead L satisfies the linking condition of the Theorem and [a] is a proper link class for L, select any (λ0,L,[a]) satisfying (PLC). Assuming first λ is nondegenerate, repeat the above argument using Proposition 3.23 instead. To pass to the degenerate case, use Lemmas 5.3, 5.2 to similarly find approximating nondegenerate Reeb vector fields and a sequence of orbits x [a] with x x with n ∈ n → x a closed λ Reeb orbit. It is clear that x cannot be a cover of a component of

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L since no sequence of loops in [a] can converge to a cover of a component of L: this is because [a] is a proper link class relative to L.

5.3. A brief justification of Morse–Bott computations. Suppose that λ satisfies (E) and is Morse–Bott nondegenerate (a similar argument will work if λ satisfies (PLC) instead). Choose a Morse function on the Reeb orbit sets of λ. The Morse–Bott chain complex CC [a](λ, J rel L) is generated by critical points of the Morse function chosen on the∗ orbit sets, removing the generators corresponding to Reeb orbits with image in L. The differential counts generalized holomorphic cylinders (that is, holomorphic cylinders between orbit sets with cascades along the gradient flow lines of the Morse function, described in detail in [5]) of index 1 which neither intersect nor are asymptotic to L. Using the perturbation of [5, Lemma 2.3] one finds, given any N 0 and > C ∞-neighborhood of λ, a form λN (which in fact only differs from λ on an arbitrarily small tubular neighborhood the orbits sets of action at most N 1) + such that

λ is in the C ∞-neighborhood chosen. • N All closed orbits that have period at most N 1 correspond to critical • + points of a Morse function on the orbit sets of λ and are nondegenerate. There are no contractible (in V \L) Reeb orbits of period at most N (be- • cause a sequence of such orbits for a sequence of forms converging to λ would produce a contractible in V \L Reeb orbit for λ of action at most N by the Arzelà–Ascoli Theorem, contradicting the hypotheses). Such forms are given explicitly in terms of the Morse functions on the orbit set in [5]. Given such a λN , one may construct the cylindrical chain complex (for generic JN and Morse functions) ¡ [a], N ¢ CC ≤ (λN ),∂(λN , JN ) . ∗ For any fixed N, choosing a sequence of such λ λ (with corresponding n → J J), the work of [5] guarantees that for large n the cylindrical contact chain n → complex generated by orbits of action at most N is canonically identified with the Morse–Bott complex for (λ, J) generated by the orbit sets of action at most N, so this portion of the homology can be canonically identified. It is not difficult to see that a sequence U of J -holomorphic cylinders with U Z 0 will n n n ∗ L = converge to a generalized holomorphic cylinder of the type used to construct the differential above. Similarly, if any sequence converges to such a generalized holomorphic cylinder, it is easy to see that limU Z 0. The results of n ∗ L = [5] imply that for large n ¡ [a], N ¢ ¡ [a], N ¢ CC ≤ (λn rel L),∂L(λn, Jn) CC ≤ (λ rel L),∂L(λ, J) ∗ =∼ ∗ as chain complexes. Now given any nondegenerate form λ0 satisfying (E) and (λ0,L) (λ,L), and ≡ J 0 J (λ0), we have chain maps (choosing C 1 so Cλ λ0, c 1 so cλ λ0, ∈ gen > Â < ≺ and n large enough) N/C N N/c CC ≤ (λn, Jn rel L) CC ≤ (λ0, J 0 rel L) CC ≤ (λn, Jn rel L) ∗ → ∗ → ∗ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 453 with the composition chain homotopic to the filtration inclusion ιN/c,N/C . Given [a] [a], N/C any nonzero class in CCH (λ rel L), it is represented in CC ≤ (λn) and [a], N/c ∗ ∗ CC ≤ (λn) by a closed, nonexact element (if N was chosen large enough ∗ and λn is chosen near enough λ). Since the composition above is chain homo- [a], N topic to ι, we get a closed, nonexact element in CC ≤ (λ0 rel L). For any two such choices of representatives, we can find N large∗ enough so the elements [a], N represent the same (nonzero) class in CCH ≤ (λ0 rel L). If this class were [a] ∗ [a], N zero in CCH (λ0, J 0 rel L), then it would be exact in CC ≤ for some N. Thus ∗ [a] ∗ [a] we have an inclusion in homology CCH (λ, J rel L) , CCH (λ0, J 0 rel L). ∗ → ∗ Similar considerations using instead the composition of morphisms

N/C 0 N N/c0 CC ≤ (λ0, J 0 rel L) CC ≤ (λn, Jn rel L) CC ≤ (λ0, J 0 rel L) ∗ → ∗ → ∗ can be used to ﬁnd a surjective map on homology [a] [a] CCH (λ, J rel L) CCH (λ0, J 0 rel L). ∗ ∗

6. EXAMPLESIN S3 We will use two approaches to computing contact homology on some orbit complements in S3. The ﬁrst is to use an integrable model for which the dynamics are known precisely, usually because of some symmetry, and compute using the symmetry to make deductions about holomorphic curves as well. The Morse–Bott technique introduced in [5] is a particularly useful tool when computing using such an approach. The second technique is to use open book decompositions (an approach used in [9] to compute contact homology for a large class of examples). In this section we will give sample computations using both approaches. Since H (S3;Z) 0 we can use Q as the coefﬁcient ring for all chain com- 2 = plexes, and a global trivialization of the contact structure is used to compute all Conley–Zehnder indices (and thus to grade the chain complexes) for the tight contact structure.

6.1. Computations with Morse–Bott contact forms. First, consider the “irrational ellipsoids” (Example 1.2). Since there is a unique orbit in each homotopy class (labeled by linking number `) of loops in S3\P it is trivial to compute

` ` CCH (λ0, J rel P) Q qQ` , CCH (λ0, J rel P,Q) 0. ∗ = · ∗ = In the following examples we will consider Morse–Bott nondegenerate forms, for which we apply the techniques of [5] to compute the contact homology. We extend Example 1.2 as follows. Let θ1,θ2 be any irrational (possibly negative) real numbers. Let γ(t) (x(t), y(t)) for t [0,1] be a smooth embedded = ∈ parameterized curve in the ﬁrst quadrant of R2. Suppose that γ has the following properties:

x(0) 0, y(0) 0, and y0(0) 0; • > = > x(1) 0, y(1) 0, and x0(1) 0; • = > < JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 454 AL MOMIN

The ratio y0(t)/x0(t) is strictly monotone (has nonzero ﬁrst derivative) on • the subdomains of [0,1] for which it is deﬁned; x y0 x0 y 0 for all t [0,1]. Equivalently, γ and γ0 are never co-linear; • ·x0(0)− · > ∈ θ1; •− y0(0) = x0(1) θ2. •− y0(1) = Given such a curve, we can construct a star-shaped hypersurface in C2 R4 = (which is thus of contact type) as follows:

EXAMPLE 6.1. Given such a γ, we define the surface S S as (where r ,θ , i = γ i i = 1,2, are polar coordinates in each R2 factor of R4 R2 R2): = × S S {(r ,θ ,r ,θ ) R4 (r 2,r 2) γ} = γ = 1 1 2 2 ∈ | 1 2 ∈ with the contact form λ0 λ . Alternately, it can be viewed as the contact form = 0|S f 2 λ on S3 {r 2 r 2 1}, where f : S3 (0, ) satisfies f (z) z S . There are · 0 = 1 + 2 = → ∞ · ∈ γ closed orbits H S3 C {0} and H S3 {0} C. 1 = ∩ × 2 = ∩ × The closed orbits H1 and H2 have Conley–Zehnder indices CZ (H k ) 2 k(1 θ ) 1, CZ (H k ) 2 k(1 1/θ ) 1. 1 = b + 1 c + 2 = b + 2 c + It is not difficult to convince oneself that, once given real numbers θ1,θ2, such a curve γ can be constructed. If θ θ 0 are irrational, if one took γ 1 = 2 > to be a straight line then S would be the irrational ellipsoid (though, strictly speaking, it violates the hypothesis that y0/x0 be strictly monotone). First we will show that S is a smooth, star-shaped hypersurface with respect to the origin in R4. Let H(x, y) be any smooth function such that H γ 1 and ◦ ≡ H 0 along γ. Then S is realized as the set H(r 2,r 2) 1 (at least in some ∇ 6= 1 2 = neighborhood of S). We compute ¡r ∂ r ∂ ¢ d(H(r 2,r 2)) 1 r1 + 2 r2 ¬ 1 2 ¡r ∂ r ∂ ¢ 2r (∂ H)(r 2,r 2)dr 2r (∂ H)(r 2,r 2)dr = 1 r1 + 2 r2 ¬ 1 x 1 2 1 + 2 y 1 2 2 2¡r 2(∂ H) r 2(∂ H)¢. = 1 x + 2 y However, since d H γ0 0 we mush have that H(γ(t)) is proportional to j ◦ ≡ ∇ · γ0(t) (y0(t), x0(t)) (since this vector is perpendicular to (x0, y0)) and therefore, = − using x(t) r 2, y(t) r 2, the previous line is proportional to = 1 = 2 2(x(y0) y( x0)) 2(x y0 y x0) 0. + − = · − · > The proportionality constant is nowhere zero, so we see that the original quantity is nowhere vanishing and we can conclude that d H 0 and hence S is a 6= smooth hypersurface in R4. The radial vector field r ∂ r ∂ is transverse 1 r1 + 2 r2 to S and therefore S is star-shaped with respect to the origin. Since the radial vector field is Liouville, the hypersurface is of contact type. The Hamiltonian vector field of H is X 2(∂ H)∂ 2(∂ H)∂ H = x θ1 + y θ2 ¡ ¢ y0(t)∂ x0(t)∂ . ∝ θ1 − θ2

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 455

2 2 2 θ r r2 1 + r2 + 1 = 1 2 1 1+θ2 2 1+θ2 − √ 2 √ 1 2 2 2 1 + 2 + θ1 = 1 r1 θ2 r2 θ2 Slope = θ2 − √1+ 2 √1+ 1 2 r2 Slope (γ′)⊥

Slope (γ′)⊥ Slope = θ2 Slope= θ1

Slope= θ1 r2 2 1 r1

2 r2 Slope = θ2 2 2 2 2 2 + 1 + 2 + θ1 = 1 2 1 2 θ r1 2 r2 2 + 1 = 1 √1+θ2 √1+θ1 r1 θ2 r2 θ2 Slope = θ2 − √1+ 2 − √1+ 1

2 γ r2 Slope ( ′)⊥

Slope (γ′)⊥

Slope= θ1 Slope= θ1 2 2 r1 r1

FIGURE 6.1. Curves γ (given as level sets) with various end- point normal slopes θ1,θ2.

So y0(t)∂ x0(t)∂ positively generates the characteristic foliation. θ1 − θ2 It is clear that r1,r2 are integrals for the flow, so S foliates (on the complement of the Hopf link H H (r 0 r 0)) by tori, parameterized by 1 t 2 = 1 = t 2 = points on the curve (x(t), y(t)) γ(t) on which the flow has slope (y0(t), x0(t)). = − In particular, we find for each t such that

(y0(t), x0(t)) C (p,q), C 0 − = · > is rational a torus foliated by closed Reeb orbits. The signs of p,q will be determined by the direction of the vector γ0(t). The condition that x0(t)/y0(t) is strictly monotone implies that this torus is Morse–Bott, so we have a Morse– Bott ﬂow on S. We shall see that closed orbits in different tori represent different homotopy classes of loops in S3\(H H ) (which is homotopy equivalent 1 t 2 to T 2). Each homotopy class [a] of loops in S3\(H H ) determines a pair of 1 t 2 integers p `(l, H ) and q `(l, H ), and conversely for each pair of integers = 2 = 1 p,q there is a unique homotopy class of loops [a] which determine p,q in this way. In the following we denote the homotopy class determined by the integers p,q by the notation [a] (p,q). = Let us more explicitly describe the set of closed characteristics in S. First, the knots H , H corresponding respectively to the sets r 0, r 0 are closed 1 2 2 = 1 = Reeb orbits on S. Using that the characteristic foliation is positively generated

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 456 AL MOMIN by y0(t)∂ x0(t)∂ , one computes the Conley–Zehnder indices: θ1 − θ2 ¹ º k x0(0) CZ (H1 ) 2 k(1 ) 1 2 k(1 θ1) 1. = − y0(0) + = b + c + ¹ º k y0(1) CZ (H1 ) 2 k(1 ) 1 2 k(1 1/θ2) 1. = − x0(1) + = b + c + The characteristic foliation on the torus at (r 2,r 2) γ(t) is (positively) gen- 1 2 = erated by

X y0(t)∂ x0(t)∂ H = θ1 − θ2 It is convenient to recall the following notation used in the statement of Corol- lary 1.6: given two vectors (x , y ),(x , y ) in the subset S {(x, y) x 0 or y 1 1 2 2 = | ≥ ≥ 0} R2, write (x , y ) (x , y ) if (x , y ) lies counter-clockwise to (x , y ) in S. ⊂ 1 1 < 2 2 2 2 1 1 Then for each vector (p,q) S with integer entries and satisfying ∈ (1,θ ) (p,q) (1/θ ,1) 1 < < 2 we find a point γ(t) on the arc γ with the property that (p,q) is co-linear with the vector (y0(t), x0(t)). Therefore the torus defined by the point γ(t) γ is − ∈ foliated by closed Reeb orbits. It is easily seen that the closed orbits in this torus represent the homotopy class [a] (p,q); since the angle (y0(t), x0(t)) = − varies monotonically we see that for each relatively prime pair (p,q) as such there is a unique Morse–Bott torus of closed orbits representing this homotopy class [a] (p,q). It is also easy to see that besides the components H , H of = 1 2 the link these are all the closed Reeb orbits. In particular, there are no closed Reeb orbits which are contractible in S\H1 H2, and H1, H2 are elliptic, so [a] t CCH (S rel H1 H2) exists. By the Morse–Bott calculation we find that for ∗ t [a] (p,q) satisfying the above conditions and relatively prime (so that [a] = (p,q) 1 = (p,q) is a simple homotopy class), CCH is isomorphic to the H1(S ) up to a grade shift. ∗ Thus, we have computed that the cylindrical contact homology for relatively prime integers (p,q) is  2 Q , if (p,q) S and (1,θ1) (p,q) (1/θ2,1) (p,q)  2 ∈ < < CCH (λ0 rel H1 H2) Q , if (p,q) S and (1/θ2,1) (p,q) (1,θ1) ∗ t = ∈ < <  0, otherwise We remark (without proof since we will not use this fact) that one can show that if 0 θ θ , then the generators have grading 2(p q), 2(p q) 1, and in < 2 < 1 + + − all other cases the generators in the homotopy class (p,q) have grading 2(p + q),2(p q) 1 instead. + + EXAMPLE 6.2. We consider the same form λ0 as above, and (for simplicity) assume that 0 θ θ . Consider instead the two-component link T T T < 1 < 2 = 1 t 2 whose components are T is the one leaf of the foliation of the torus T by the Reeb vector field. • 1 (p,q) T is the one leaf of the foliation of the torus T by the Reeb vector field. • 2 (p0,q0) JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 457 where both pairs (p,q) and (p0,q0) are positive, relatively prime and

q0 q θ1 θ2. < p0 < p <

Choose any relatively prime pair (p00,q00) such that

p p00 p0 . q < q00 < q0 Let [a] [T ] be the homotopy type of a leaf of the foliation of the torus = (p00,q00) T(p00,q00) by the Reeb vector field considered as an element in the space of loops in 3 S \L. Then [T ] is a simple homotopy class and (λ0,T T T ,[T ]) (p00,q00) = 1 t 2 (p00,q00) satisfies (PLC) at least if q00 q,p0 p00. 6= 6= To see this, first compute that if we have K T and K T (l,m) ⊂ (l,m) (l 0,m0) ⊂ (l 0,m0) simple closed Reeb orbits and l/m l 0/m0 then the linking number < `(K ,K ) l 0 m. (l,m) (l 0,m0) = · This shows that the linking numbers

`(T ,[T ]) p00 q `(T ,T ) p0 q, 1 (p00,q00) = · 6= 1 2 = · `(T ,[T ]) p0 q00 `(T ,T ) p0 q, 2 (p00,q00) = · 6= 2 1 = · so [T ] cannot be homotopic to either T or T in the complement of T (p00,q00) 1 2 1 t T since these linking numbers are homotopy invariant in S3\T T . The link- 2 1 t 2 ing numbers of the orbits in T(p00,q00) with T1,T2 show that orbits in different orbit sets do in fact lie in different homotopy classes of loops, so (as in the previous example) the Morse–Bott complex is simply that of a Morse function on the orbit set. Therefore for [T(p00,q00)] ½ [a] [T(p ,q )] Q, 2 (p00 q00) or 2 (p00 q00) 1 CCH = 00 00 (ξ rel T ) ∗ = · + · + + ∗ =∼ 0 otherwise Using the above calculation we derive the following corollary. Note that we are not assuming that the contact form λ is nondegenerate.

COROLLARY 6.3. Let λ be a tight contact form on the 3-sphere. Suppose there is a pair of knots L0 L0 L0 such that there is a contactomorphism taking this = 1 t 2 pair to the pair of torus knots described in Example 6.2: T T T . = 1 t 2 Then for any p00,q00 relatively prime and such that

p p00 p0 , q00 q0, and p p00, q < q00 < q0 6= 6= there is a closed Reeb orbit in the homotopy class [a] [T ] of S3\T. = (p00,q00) Proof. Applying the contactomorphism we can assume without loss of generality that (λ,L0) (λ0,T ) from Example 6.2. Then this follows from Theorem ∼ 1.4 and the computation in Example 6.2, since the homotopy class [a] of K

(which has the homotopy type of the leafs of the foliation of T(p00,q00) by the

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 458 AL MOMIN

Reeb vector ﬁeld in that example) is simple and a proper link class relative to T T . The hypotheses q00 q0,p p00 imply that that the linking numbers 1 t 2 6= 6= `(T ,[a]) `(T ,T ) and `(T ,[a]) `(T ,T ) which ensures that [a] is a proper 1 6= 1 2 2 6= 2 1 link class, though [a] might be a proper knot class even if these inequalities are not satisﬁed.

REMARK 6.4. Similarly, one can consider the cases where at least one of θ1,θ2 is negative and obtain an analogous result for certain cases where p,p0,q,q0 may be negative.

It is also true that (λ0,T,[Hi ]) satisﬁes (PLC) for the knots H1, H2 in Example 6.1 (the components of the Hopf link) if p0 1 and q 1. In this case, it is [Hi ] 6= 6= easiest to compute CCH by choosing θ1,θ2 0 of λ0 appropriately. Select ∗ > them such that ¹ p º 1 p

q < θ2 < q ¹ º q0 q0 θ1 p0 < < p0

This ensures that the only orbits in the homotopy classes [H1],[H2] are the orbits H1, H2 themselves (this can be checked by a simple linking argument which we omit) and therefore

[a] [H1] [a] [H2] CCH = (ξ rel T ) Q q1,CCH = (ξ rel T ) Q q2 ∗ =∼ · ∗ =∼ · where the generators q1,q2 have degrees ¹ º ¹ º q0 p q1 2 1 , q2 2 1 . | | = + p0 | | = + q 6.2. Application to reversible Finsler metrics on S2. The geometric context described in the next three paragraphs is well-known; a clear treatment can be found in [26, Section 4]. Let us recall the double-covering map S3 SU(2) SO(3): =∼ → a2 b2 c2 d 2 2(ad bc) 2( ac bd)  · a ib c id¸ + − − + − + + +  2( ad bc)(a2 b2 c2 d 2) 2(ab cd)  c di a ib 7→ − + − + − + − + − 2(ac bd) 2( ab cd)(a2 b2 c2 d 2) + − + − − + By taking the second and third columns, we obtain a double-covering map G : S3 SU(2) M , where M {(x,v) R3 R3 x 2 v 2 1,x v 0} is the ≡ → 0 0 = ∈ × || | = | | = · = unit tangent bundle of S2 with the round metric a  2(ad bc) 2( ac bd)  b + − +   (a2 b2 c2 d 2) 2(ab cd)  c  7→ − + − + 2( ab cd)(a2 b2 c2 d 2) d − + − − + which gives an explicit formula for the double-covering map S3 S1TS2. → Let

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 459

2 2 ` : TS \0 T ∗S \0 denote the Legendre transform with respect to the • 0 → round Riemannian metric of unit curvature. In general, `F denotes the one obtained from a Finsler metric F . 2 1 F is a Finsler metric on S , while F ∗ is the cometric obtained via F `− . • ◦ F Set M ∗ {F ∗(p) 1} be the unit cosphere bundle with respect to F ∗, and • F = = let M ∗ be the unit cosphere bundle for F the norm for the round 0 = | · | Riemannian metric. 2 g : T ∗S \0 (0, ) is such that F ∗(x,p) g (x,p) p i.e., g (x,p) is the • F → ∞ = F | | F homogeneous degree 0 part of F ∗. Also, let r : M ∗ M ∗ be the map F 0 → F rF (x,p) (x,p/gF (x,p)). = 1 2 G is the map SU(2) S TS M0 considered above. • → =2 λ is the Liouville form on T ∗S , while λ0 is the standard contact form on • 3 S . Note that MF∗ is contact type. 1 LEMMA 6.5. [26] If f `0, then = gF ◦ G∗`∗r ∗(λ M ) 2(f G)λ0 0 F | F∗ = ◦ The Reeb ﬂow of 2(f G)λ is smoothly conjugate to the double cover of the geo- ◦ 0 desic ﬂow on MF .

There is a global frame X1, X2, X3 on M ∗ with the property that X1 is the Reeb vector field, and X , X trivialize ξ kerλ . This can all be pulled back 2 3 MF∗ 3 = | via the double-cover r ` G : S M ∗, so we may use this frame to compute F ◦ 0 ◦ → F Conley–Zehnder indices. The linearized flow along a closed Reeb orbit z(t) can be computed with the following result (see e.g., [26]). Let φt denote the Reeb flow. Let x X (z(0)) y X (z(0)) be in ξ . Let x(t), y(t) solve 0 2 + 0 3 z(0) dφ (z(0)) x(t)X (z(t)) y(t)X (z(t)), x(0) x , y(0) y t = 2 + 3 = 0 = 0 Then x(t), y(t) solves the linear system ½x˙ K (π z(t))y = − ◦ y˙ x = where K : S2 R is determined by the Finsler structure and agrees with the → curvature in the Riemannian case. Poincaré’s rotation number of a geodesic γ can be described as [1] y2n (6.1) 1/ρ lim n = →∞ nL th where yk is the k positive zero of y(t) which solves

y00 K (γ(t)) y 0 + · = and L is the length of the geodesic. This limit can also be described by L #{zeros of y [0,N]} ρ lim ∈ . = N 2 N →∞ Let z be the closed Reeb orbit which double covers γ, such that z has period T . We will now compare the Conley–Zehnder indices CZ (zk ) to the inverse

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 460 AL MOMIN rotation number ρ. The Conley–Zehnder index of zk satisﬁes (see, for example, the Appendix of [32]) CZ (zk ) 2∆ 2. | − | < In the above, 2π∆ is the change in argument of the vector (x(t), y(t)) over the paths domain [0,kT ], where (x, y) satisﬁes the linearized equation associated to z above. However, if (x, y) solves this equation, then y solves the equation (recall G denotes the double cover)

y00 K (G z(t))y 0. + ◦ = Moreover, if there are k zeros of y, then k 1 2∆ k 1. This claim is a con- − < < + sequence of the following observations. First, we notice that all crossings of the x-axis are transverse: this is because if y 0 and y0 0 then y 0 by unique- = = ≡ ness of solutions. Second, at any crossing, by the differential equation we see that if y0 0 then x 0, and if y0 0 then x 0; thus each zero contributes a > > < < transverse counterclockwise crossing of the path (x(t), y(t)) against the x-axis. So 2∆ k 1; hence (for any k): | − | < CZ (z ) #{zeros of y [0,kT ]} 3. | |[0,kT ] − ∈ | < Since z has period T , zk has period kT . Then CZ (zk ) #{zeros of y [0,kT ]} 0 lim | − ∈ | lim3/k 0 ≤ k k ≤ = →∞ so 2L #{zeros of y [0,kT ]} 2ρ lim ∈ = kT 2 kT →∞ T #{zeros of y [0,kT ]} lim ∈ = k 2 kT →∞ 1 #{zeros of y [0,kT ]} lim ∈ = k 2 k →∞ lim CZ (zk )/2k. = k →∞ (The factor of 2 is due to the fact that z is a double cover of the simple geodesic γ, thus the period T of z is equal to 2L, where L is the length of the geodesic γ.) We now use this to derive Corollary 1.8 from Corollary 1.6. We suppose we have a Finsler metric F on S2 and a simple closed curve γ such that the 1 2 curves γ1,γ2 in S TS MF obtained by traversing γ in opposite directions at =∼ unit speed are both F -geodesics, and let ρ1,ρ2 be the inverse Poincaré rotation numbers for γ1,γ2 as defined above. We also suppose that ρ1,ρ2 are irrational. The geodesic flow is a Reeb flow on S1TS2 and we pull this flow back to a tight Reeb flow on S3 using the map G : S3 S1TS2 as in Lemma 6.5. Let us denote → the closed Reeb orbits which map to double covers of γ1,γ2 by z1,z2. It follows from the calculation above k CZ (zi ) 2ρi lim = k 2k →∞ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 461 that the Conley–Zehnder indices of z1,z2 must be CZ (zk ) 2 k(1 (2ρ 1) 1, i 1,2, k 1. i = b + i − c + = ≥ Unless (1,2ρ 1),(2ρ 1,1) 1 − 2 − are co-linear (which is the resonance condition), Corollary 1.6 implies that for each (1,2ρ 1) (p,q) (2ρ 1,1), or (2ρ 1,1) (p,q) (1,2ρ 1) 1 − < < 2 − 2 − < < 1 − there is a closed Reeb orbit in the homotopy class (p,q) in S3\z z . The im- 1 t 2 age of this closed orbit in the unit cotangent bundle MF∗ is a closed geodesic γp,q for F (which may double-cover a simple closed geodesic), and this proves Corollary 1.8. To compare with the geodesics found in [1] in the case F is Riemannian (which are found under weaker regularity hypotheses and without any hypothesis about irrationality of ρ), one can check that given p,q relatively prime and a (p q,2p)-satellite sp q,2p of the geodesic γ that sp q,2p represents - + + 3 + by considering the preimage under the double cover G : S MF of the curve 3 → (sp q,2p ,s˙p q,2p ) MF - the homotopy class (p,q) in S \z1 z2. However, we + + ⊂ t cannot assert using Corollary 1.6 that the geodesics we have found are the (p + q,2p)-satellites found in [1], because there are many different flat knot types lifting to transverse curves in this homotopy class in S3\(z z ). 1 t 2 6.3. Open book decompositions. Contact forms supporting an open book pro- vide a vast class of examples. Let (π,B) be an open book decomposition for (V,ξ). This means that B is a transverse link in V (the binding), π: V \B S1 → is a fibration, and every component B 0 B has a neighborhood U such that 1 ˙2 ⊂ π U\B S D has the form (t,(r,θ)) θ (where (r,θ) are polar coordinates | 0 =∼ × 7→ on the punctured disk). The open book supports a contact structure ξ if there is a contact form λ (called a supporting contact form) with ker(λ) ξ such that = B is closed under the Reeb vector field. • 1 1 The form dλ is symplectic on the pages π− (θ), i.e., dλ 1 0, θ S . • |π− (θ) > ∀ ∈ In particular, the Reeb vector field is transverse to the pages, so there are no contractible orbits in V \B. It turns out that every contact structure is supported by some open book (in fact, infinitely many [21]). We first check an elementary topological fact:

LEMMA 6.6. The binding B satisﬁes the second condition of Theorem 1.4, namely Every disk F with ∂F B and [∂F ] 0 H (B) intersects B in the ⊂ 6= ∈ 1 interior. except when the page is a disk.

2 1 Proof. Let F be any smoothly map D V with ∂F B. Suppose that F − (B) → ⊂ ⊂ ∂D2. Consider the covering map π: R int(S) R int(S)/((t,x) (t 1,h(x)) V \B × → × ∼ − =∼ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 462 AL MOMIN coming from the open book decomposition. For any r 1 we may lift F 2 < |D (r ) to a map G : D2(r ) R int(S), uniquely determined if we choose once and ﬁx 1 → × 2 G(0) π− {F (0)}. Thus we can in fact deﬁne G : int(D ) R int(S). We may ∈ 2 → × homotopy G to G0 : int(D ) {0} int(S) int(S) by scaling the R-coordinate. → × =∼ Let N(B) be the tubular neighborhood of B such that N(B)\B S1 int(N(∂S)), =∼ × where N((∂S) is a collar neighborhood of ∂S chosen small enough that the monodromy h is the identity, so we have an inclusion map |N(∂S) ι: S1 int(N(∂S)) N(B), × → and F 2 ι π G 2 (where r is taken large enough that F 2 B). |∂D (r ) = ◦ ◦ |∂D (r ) |∂D (r ) ⊂ Applying the H1-functor and using the fact that G0 is homotopic to G and

[F 2 ] [F 2 ] 0 H1(N(B)) H1(B) |∂D (r ) = |∂D 6= ∈ =∼ we have

(ι π) [G0 ∂D2(r )] [F ∂D2(r )] 0 H1(N(B)) H1(B) ◦ ∗ | = | 6= ∈ =∼ From this it is easy to conclude that [G0 2 ] k [∂S] H (N(∂S)) for some |∂D (r ) = · ∈ 1 k 0, and therefore is homotopic to a k-fold cover of ∂S with k 0; hence 6= 6= G0 2 is homotopic to k ∂S in S. However, G0 2 is null-homotopic in S, |∂D (r ) · |∂D (r ) while the k-fold cover of ∂S is not if k 0 (unless the page S is a disk). This 6= 1 2 contradiction shows that we cannot have F − (B) ∂D , so F intersects B in ⊂ the interior.

EXAMPLE 6.7. Open Book Decompositions Suppose B supports an open book decomposition (other than the open book decomposition with the disk as the page). Then (λ,B,[a]) satisfies (PLC) for any simple proper link class [a] (relative to B): any disk bounding a contractible Reeb orbit either links nontrivially with the binding (since the Reeb vector field is transverse to the pages), or is asymptotic to a component of the binding (in which case by Lemma 6.6 it has an interior intersection with the binding), and by hypothesis [a] is a proper link class. If the page is not the disk or the annulus, there should be many such [a]. If the binding is elliptic nondegenerate then it satisfies (E) as well.

It follows that CCH [a](ξ rel B) is well-defined for simple proper link classes [a] (relative to B), where∗ ξ is the contact structure supported by the open book. In particular, Theorem 1.4 can be applied to any proper link class [a] for the binding B. In [9], the authors achieve the much more difficult task of computing the cylindrical contact homology of the manifold (V,ξ) (when it is defined) using suitable open book decompositions. For V \B the situation is considerably simpler. First, cylindrical contact homology of the complement will always be defined (at least for simple homotopy classes) and independent of the equivalence classes [λ] of in proper link classes relative to B. Second, we only need ≡ to count a small subset of the holomorphic cylinders and can ignore holomorphic planes (since they will all intersect the binding), so the computation is almost always far easier. Since one can read off computations from [9], we will

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 463 content ourselves to go through a particular example in some detail to illustrate the approach concretely. Recall that the Nielsen fixed point classes of h are the equivalence classes of fixed points under the equivalence relation , where x y if there is a path γ ∼ ∼ from x to y such that γ is homotopic to h(γ) rel end-points. A fixed point for h (or more generally a periodic point) determines a loop in the mapping torus. It is a well-known fact that different Nielsen equivalence classes of fixed points determine different free homotopy classes of loops in the mapping torus. More generally, two loops in the mapping torus arising from two k-periodic orbits of h are freely homotopic if and only if there are points on the corresponding orbits which are hk -equivalent (in this case we will call the orbits Nielsen equivalent as well). Hence, to Nielsen equivalence classes we may assign a unique homotopy class [a] in V \B h.e. Σ(S,h). =∼ 6.4. A concrete example on the figure-eight.

6.4.1. A map on the punctured torus and the torus with one boundary component. Consider the linear map on R2 ·2 1¸ SL (Z) GL (R) 1 1 ∈ 2 ⊂ 2 which fixes the integer lattice. It therefore descends to a map h : T 2 T 2 of → T 2 R2/Z2. Since, moreover, it fixes the integer lattice it also defines a map on = T 2\{0} (R2\Z2)/Z2 which preserves the area form inherited from R2. One may = also wish to think of such a map on a torus with one boundary component, which we model by removing a ²-disk around the each point in the integer lattice and denote T 2\D2(²). The map h can be represented by the time-1 flow t ·x¸ ·2 1¸ ·x¸ y 7→ 1 1 y generated by the symplectic vector field Ã ! ·2 1¸ ·x¸ ·2 1¸ 1 3 p5 ·1 2 ¸ X (x, y) log , log log + sl2(R) = 1 1 · y 1 1 = p 2 2 1 ∈ 5 − and it is straightforward to compute that it is generated by the Hamiltonian flow of a quadratic form Ã ! 1 3 p5 Q(x, y) Ax2 Bx y C y2 log + ¡x2 x y y2¢ = + + = p5 2 − − of signature (1,1). Take φ: R [0, ) to be any smooth function with the prop- → ∞ erties

φ0(x) 0, x R, • ≥ ∀ ∈ φ [0, 1 ] 0, • | 2 ≡ φ [1, ) 1, • | ∞ ≡ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 464 AL MOMIN and denote φ φ( 1 ). Consider the Hamiltonian function ² = ² · 2 2 φ 2 (x y ) Q(x, y) ² + · and denote by h˜ its time-1 map. Since it agrees with h outside of a disk of radius ² we may define h˜ on T 2\D2(²/2) by replacing the map h with h˜ inside the disk ˜ 2 2 of radius ². In this way we get an area-preserving map h² on T \D (²/2) which is the identity on a neighborhood of the boundary. This abstract open book gives rise to S3 with binding a figure-eight knot. The contact structure supported by this open book, which we will denote by ξo, is actually over-twisted. 6.4.2. Realizing the monodromy map by a Reeb flow. Let h : S S be a map that → preserves the area form ω. Let β be a primitive ω dβ. Consider the mapping = torus of h, which is [0,2π] S quotiented by (1,x) (0,h(x)). We wish to find × ∼ 1 a contact form with the Reeb vector field ∂t (we will use t to denote the S “coordinate”), using β to construct it. 1 LEMMA 6.8. (Giroux, see e.g., [11]) Suppose that [h∗β β] 0 in H (S;R) (note − = that h∗β β is always closed because h∗ω ω). Then h can be realized by a Reeb − = flow on the mapping torus of h. 1 LEMMA 6.9. [11] If h∗ I is an isomorphism on H (S;R), we may find a prim- − 1 itive β0 such that h∗β0 β0 is zero in H (S;R). Hence by Lemma 6.8 h can be − realized by a Reeb flow on its mapping torus. ˜ 2 2 Consider specifically the maps h² on T \D (²/2) with the area form inherited from the area form for R2 (thus T 2\D2(²/2) has area 1 π²2/4). We can 2 − assume that the form β has the form ( r 1 )dθ in D2(²). 2 − 2π Let z be the S1 coordinate of the mapping torus, so the contact form dt β + in these coordinates is µr 2 1 ¶ λ dz dθ = + 2 − 2π To construct the open book, this is identified with a neighborhood of the boundary of S1 D2(²/2) with coordinates (s,φ,t) D2(²/2) S1 by × ∈ × s r, φ z, t θ = = = − in which the form is µ 1 s2 ¶ λ dφ dt = + 2π − 2 and the pages are φ const. = We will now extend this form to the interior of D2(²/2) S1 using a well- × known argument [42]. The contact form will be

2 2 1 ¡ 2 2 ¢ f (s )dφ g(s )dt X f 0(s )∂t g 0(s )∂φ + ⇒ λ = f (s2)g(s2) f (s2)g (s2) − 0 − 0 where the functions f ,g have the following properties (and T 0 is a parame- > ter): f (s2) 1 and g(s2) 1 s2/2 for s ²/2; • = = 2π − ≥ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 465

f (s2) s2 and g(s2) 1 Ts2 for s δ for some δ 0 as small as required; • = = − ≤ > f 0g f g 0 0; • − > g 0 0: this ensures that the Reeb vector field is everywhere positively • < transverse to the pages φ const and is the reason we need to assume T = > 0. Moreover, then there is some constant c c 0 such that g 0(s) = ²,T ≥ − ≥ c 0. This bounds the first return time of the Reeb flow to a page φ > = const near the binding B by the constant 2π/c. The explicit construction of such functions is straightforward but slightly te- dious, so we will omit it. We remark as in [42] that it is equivalent to drawing 2 1 a smooth curve in R to the point (1, 2π ) from the point (0,1) with respective velocities at these points (1, T ), and (0, 1), such that the position vector is − − never collinear with the velocity vector (and, by the fourth condition, the velocity vector must point downward at all times as well). We see that the binding has Conley–Zehnder index equal to

CZ (B k ) 2 kT 1 = b c + with respect to the framing given by the angular coordinate φ (i.e., with respect to the vectors ∂ ((r,0),t),∂ ((r,π/2),t) at the point (0,0,t) B). Therefore we r r ∈ will select T irrational so B is elliptic nondegenerate. As constructed, it has an open neighborhood such that no periodic orbit enters the neighborhood. The Reeb flow thus constructed is degenerate, but after a small perturbation can be made nondegenerate without altering the dynamics too much. The pages form a surface of section with bounded return time. Moreover, 2 2 2 there is a page S ∼ T \0 such that the restriction to a subsurface S0 ∼ T \D (²/2) = ˜ = 2 2 of the first-return map is the map h², and to a further subsurface S² T \D (²) =∼ we have h˜ h . ²|S² = |S² Given ² 0 sufficiently small to perform the above construction, let λ be > ² such a contact form.

LEMMA 6.10. For each k 1, there is a ² such that if ² ² then for each non- ≥ k < k trivial Nielsen fixed point class there is a unique nondegenerate closed Reeb orbit 3 2 for λ² in the corresponding homotopy class [a] in S \B Σ(T \0,h). =∼ Proof. The fixed point set of hk for each k is a finite set of points. Therefore, there is a disk D containing the puncture 0 which contains no fixed points of l h for l k. Choose an even smaller disk 0 D0 such that ≤ ∈ Ã k ! [ l 2 h (T \D) D0 . l 1 ∩ = ; = 2 Then if D (²) D0 then the periodic points of period at most k of h˜ and h on ⊂ ² T 2\D are the same. l We may also choose 0 D00 D0 such that h (D00) D0 for 1 l k. Select 2 ∈ ⊂ ⊂ ≤ ≤ ² such that D (² ) D00. Then, if ² ² , all l-periodic orbits of h˜ contained in k k ⊂ < k ² D must enter D00, and are therefore in fact contained in D0. So we only have to 2 examine the cases x˜ D0 and y˜ T \D. ∈ ∈ JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 466 AL MOMIN

Let h : R2 R2 lift h˜ in the sense that π h π h˜ (for π: R2 T 2 the uni- ² → ² ◦ ² = ◦ ² → versal cover), which is uniquely determined by demanding h²(0) 0. Let x˜ be a 2 = l-periodic point in D0 and y˜ a l-periodic point in T \D, and γ˜ a path between these two points, with lifts x, y,γ respectively such that γ(0) x,γ(1) y and x =2 = is contained in the preimage of D0 containing the origin in R . It is easy to see i k that the orbit h²(x) remains in this component for all i and therefore h² (x) x. l l = On the other hand, h²(y) h (y) y. After these observations it is straight- = 6= ˜ l forward to deduce that x˜, y˜ are not Nielsen equivalent for h²: a homotopy rel end-points between γ˜ and h˜l γ˜ would lift to one between γ and hl γ rel end- ² ◦ ² ◦ points, but the end-points of these two paths do not even coincide so there can be no such homotopy in the first place. Thus, y˜,x˜ lie in different equivalence classes. One can show that the loops corresponding to l-periodic points x and y are ˜ i ˜ j homotopic in the mapping torus if and only if there are iterates h²(x), h² (y) ˜ l ˜ l ˜ l which are h²-equivalent. Since x is the only h² fixed point in its h²-equivalence class (which follows by checking directly that the fixed points of the matrix map hl lie in different equivalence classes) this shows that the corresponding orbit is the unique closed Reeb orbit in this homotopy class. It will be hyperbolic and therefore nondegenerate.

Therefore, if [a] is the homotopy class of a fixed point of h, the triple (λ,B,[a]) satisfies (PLC) (and (λ,B) satisfies (E) as well), though λ may not be nondegenerate. Therefore we will need to perturb, and the following Lemma asserts that small nondegenerate perturbations retain important properties:

LEMMA 6.11 (Approximation Lemma). In any C ∞ neighborhood of λ², there is a nondegenerate contact form λ²0 with the following properties: t Let φ denote the Reeb ﬂow of λ0 . There are constants c,C and a page S • ² of the open book decomposition such that for any x S, there is a least ∈ 0 t such that φt (x) S and c t C. In particular, every closed orbit has < ∈ < < nontrivial linking number with B. For each Nielsen ﬁxed point class [a] such that `([a],B) k, if ² ² then • ≤ < k there is a unique nondegenerate closed Reeb orbit for λ²0 in [a].

Proof. By Lemma 6.10 the (degenerate) contact form λ² has the following properties: Let φt denote the Reeb flow. There are constants c,C and a page S of the • open book decomposition such that for any x S, there is a first t 0 t ∈ > such that φ (x) S and c0 t C 0. ∈ < < For each Nielsen fixed point class [a] such that `([a],B) k, if ² ² then • ≤ < k there is a unique nondegenerate closed Reeb orbit for λ²0 in [a].

By Lemma 5.2, in any C ∞-neighborhood of λ², we can find a nondegenerate contact form λ0 such that (λ0 ,B) (λ ,B). In particular, if λ0 is chosen close ² ² ≡ ² ² JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 467 enough, the page S will still be a surface of section for λ²0 because of the Conley– Zehnder index in a neighborhood of the binding, and because λ² has bounded return time on the (compact) complement of this neighborhood in S if λ²0 is chosen close enough. Therefore there are c,C such that if x S then for some ∈ c t C (in fact, the first such t 0) we have φt (x) S. < < > ∈ The second assertion can be deduced from the λ²0 nondegeneracy of the orbits which correspond to fixed points of the unperturbed map h. The following proposition follows immediately:

PROPOSITION 6.12. Let [a] denote a homotopy class in S3\B corresponding to a simple Nielsen fixed point class. Then there is an ² such that for the contact form λ²0 in Lemma 6.11 we have (λ²0 ,B,[a]) is nondegenerate, satisfies (PLC) and for J 0 Jgen ∈ [a] CCH (λ²0 , J 0 rel B) Q. ∗ =∼ It follows that (see the following Remark 6.13)CCH [a] is defined and ∗ CCH [a](ξ rel B) Q. ∗ =∼ REMARK 6.13. While the closed orbit above is assumed simple, it is conceivable that the homotopy class [a] it is in is not (i.e., contain multiply covered loops). However, Remark 4.5 explains why CCH [a]([λ] rel B) is still well-defined. Thus ∗ the chain maps/homotopies with the representative λ²0 make sense if ² is suffi- [a] ciently small and computes CCH (µ, J rel B) for (µ,B) (λ²0 ,B). ∗ ∼ By construction the contact forms λ²,λ²0 give rise to the same over-twisted contact structure ξo which is thus supported by the open book constructed above from the map h. One may apply Theorem 1.4 to deduce, given any contact form for ξo with a closed Reeb orbit transversely isotopic to B, information about the homotopy classes of Reeb orbits in S3\B. However, arguing a little more carefully we can also deduce the following growth rate of periodic orbits in the action (which proves Corollary 1.12):

PROPOSITION 6.14. Suppose µ is a contact form for ξo, and its Reeb vector ﬁeld has a closed periodic orbit transversely isotopic to the binding B. Then the number of geometrically distinct closed Reeb orbits of period at most N • grows at least exponentially in N, and the number of closed orbits x such that `(x,B) k grows exponentially in • ≤ k.

Proof. Choose any λ0 λ0 as in Lemma 6.11, so that (λ0,B,[a]) satisfies (PLC) = ² for each homotopy class [a] corresponding to a nontrivial Nielsen fixed point class for h. Lemma 6.11 asserts that there is a surface of section S such that each point x S has first return time t in bounded by t C for a constant C ∈ < independent of x. In particular, for each closed Reeb orbit we have a bound T (x) C `(x,B). The number of simple nontrivial Nielsen fixed point classes < · of hk is bounded below by Aebk for positive constants A,b. These fixed point classes satisfy 1 `([a],B) k and therefore T C k. Therefore, the number of ≤ ≤ < · JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 468 AL MOMIN closed Reeb orbits of action at most Ck is at least Aebk . By scaling b b/C we = have the number of closed Reeb orbits of action at most k is at least Aebk . 3 Let µ be a contact form on S with kernel ξo with an orbit transversely isotopic to B, and if necessary pullback by a contact isotopy to arrange that this orbit in fact coincides with B. First suppose that µ λ0. Arguing as in the proof ≺ of Theorem 1.4, for each nontrivial simple Nielsen fixed point class [a] such that `([a],B) k we obtain a µ orbit in the same homotopy class with period ≤ bounded by its λ0 period which is itself bounded by C k. Therefore we have an · exponential lower bound on the number of geometrically distinct periodic orbits of period at least N for µ as well. Any µ can be rescaled such that dµ λ0, ≺ and one establishes lower bounds similarly (replacing b with d b). · 6.5. Fibered hyperbolic knots. The arguments in the above example can be re- placed by any open book with one boundary component and pseudo-Anosov monodromy map which is Reeb-realizable. In most examples one must smooth the prong singularities of the pseudo-Anosov map, a procedure we will not discuss (one way to smooth the singularity is discussed in [7, Section 3.2]). Be- cause φ∗ I is invertible (as we observed in the introduction), we can apply − Lemma 6.9 cited above to realize the (smoothed) pseudo-Anosov map by a Reeb vector field, continuing the contact form over the binding as in the last section. The reader should note that the argument that the number of nontrivial Nielsen fixed point classes grows exponentially is generally more com- plicated. Moreover, there may be several nondegenerate fixed points in certain Nielsen equivalence classes, but an appeal to Euler characteristics can be used if needed to show that the homology is nontrivial in nontrivial classes. For details we refer to [9, Section 11.1]. Consider now specifically a tight fibered hyperbolic knot realizing the Ben- nequin bound (1.1) in the tight S3. By [24] (or [15, Theorem 4.1]), tight fibered knots realizing the Bennequin bound are determined by the topological knot type up to transverse isotopy, so we may assume without loss of generality that such a knot is in fact the binding of an open book decomposition supporting the tight S3 with (pseudo-Anosov) monodromy map φ. Since the open book supports the tight contact structure, the kernel of the constructed contact form (which supports this open book) is the tight contact structure on S3 (up to isotopy). Thus Corollary 1.10 is proved in a similar manner as Corollary 1.12/Proposition 6.14 by considering the pseudo-Anosov monodromy map φ in place of the matrix map h. We include some general remarks about tight fibered hyperbolic knots. First, there are approaches to decide whether or not a given knot is fibered ([19, 37]). Second, a knot forms the binding of an open book decomposition for the tight S3 if and only if it has a transverse representative which realizes the Bennequin bound (1.1) in the tight S3 ([24], or [15, Corollary 1.2]). Third, a sufficient (and necessary) condition for the monodromy map to be pseudo-Anosov is that the knot is a hyperbolic knot [41]. Finally, to see that the first property in the definition implies the fourth property as we claimed in the introduction, evaluating

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACTHOMOLOGYOFORBITCOMPLEMENTSANDIMPLIEDEXISTENCE 469

FIGURE 6.2. The 35 tight ﬁbered hyperbolic knots with crossing number at most 12 [12].

JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 470 AL MOMIN the Alexander polynomial ∆(t) det(t I h∗) at t 1 shows that h∗ I is in- = · − = − vertible since ∆(1) 1 for any knot. = ± Matt Hedden pointed out to the author that there is an abundance of knots satisfying all these properties including, for example, the Fintushel–Stern knot (see Figure 6.2, 12n ), also known as the Pretzel knot ( 2,3,7). An easy way 0242 − to see that there are infinitely many examples is to notice that one can positively stabilize many times to make the genus g of the page arbitrarily large, thus increasing the self-linking number of the binding (it will be 2g 1) which − distinguishes the knots. Positive stabilization does not change the manifold or contact structure, so properties (1),(2) continue to hold. It is shown in e.g., [10] that if one positively stabilizes an open book appropriately enough times that the monodromy can be made pseudo-Anosov (thus will satisfy (3)), so this infi- nite family will possess infinitely many tight fibered hyperbolic representatives. Using the search form in [12] and the observations above, the 35 tight fibered hyperbolic knots with crossing number at most 12 were found and are listed in Figure 6.2 (by knot diagram and Alexander–Briggs notation).

Acknowledgments. This work is a continuation of the author’s Ph.D. thesis; I thank my adviser Helmut Hofer for his invaluable guidance and support. I thank Alberto Abbondandolo, Umberto Hryniewicz, Richard Siefring, Robert Vandervorst, and Chris Wendl for inﬂuential discussions, and Matt Hedden for answering some questions about ﬁbered knots in S3. Finally, I thank Peter Al- bers and Pedro Salomão for many helpful suggestions, and the anonymous ref- eree for a careful reading and for generously offering numerous detailed im- provements to the readability.

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AL MOMIN : Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47906, USA

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