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Contact Homology of Orbit Complements and Implied Existence

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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 ¸ and associated Reeb vector field X¸. 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 ques- tions 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 ¸ imposing as few conditions on the or- bit set as we can manage. We will show in detail one approach to such a the- ory 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 com- plement, holomorphic curves, intersection theory. INTHEPUBLICDOMAINAFTER 2039 409 ©2011 AIMSCIENCES 410 AL MOMIN 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 ¸ on V is called a contact form if ¸ d¸ is everywhere nonzero. Such a one-form ^ uniquely determines its Reeb vector field X by the equations ¸(X ) 1 and ¸ ¸ ´ d¸(X , ) 0. It also determines a distribution » ker¸, which is a contact ¸ ¢ ´ Æ structure, and two forms ¸ induce the same contact structure if and only if § ¸ f ¸ for some nowhere vanishing function. We note that different choices Å Æ ¢ ¡ 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 be- tween 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 hy- potheses below, which are not mutually exclusive. They simplify the construc- tion 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 (¸,L) in which ¸ is a nondegenerate contact form on V and L is a link composed of closed orbits of the Reeb vector field of ¸. 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 subman- ifold 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 ¢ ! Æ (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 ») are nonreal. DEFINITION 1.1. We will say (¸,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- ² degenerate elliptic, and each contractible Reeb orbit y not in L “links” with L. To be precise, we ² 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- 2¼iθ genvalues of the linearized return map are e§ for some irrational real num- ber θ. We shall see the hypotheses of Definition 1.1 force the compactness of JOURNALOF MODERN DYNAMICS VOLUME 5, NO. 3 (2011), 409–472 CONTACT HOMOLOGY OF ORBIT COMPLEMENTS AND IMPLIED EXISTENCE 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 ) Æ £ i i (i 1,2) on each R2 factor of R4 and let E be the ellipsoid determined by the Æ equation r 2 r 2 1 2 1 a Å b Æ where a,b are positive constants. The one-form 2 X 1 2 ¸0 ri dθi Æ i 1 2 Æ restricted to E defines a contact form which we denote ¸. If the ratio a/b is ir- rational, 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 Æ \ £ Æ \ £ 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. Æ Å b Å Æ Å a Å Since the linking numbers `(P k ,Q) k,`(P,Qk ) k, the pairs (¸,P), (¸,Q), (¸,P Æ Æ 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 ¸, and let [a] be the homotopy class of a loop a in the complement. DEFINITION 1.3. We say that (¸,L,[a]) satisfies the “proper link class” condition (PLC) if for any connected component x L, no representative γ [a] can be ho- ² ½ 2 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 ¢ Æ ¢ Æ £ ½ class [a] will be called a proper link class for L. for every disk F with boundary @F y a closed (nonconstant, but possibly ² Æ 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 Æ 0 Æ 1 each s 1 the loop a lies in the complement V \L. 6Æ 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 restric- tive. 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.
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