Homological invariants of codimension 2 contact submanifolds
aa r X i v : . [ m a t h . S G ] S e p HOMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACTSUBMANIFOLDS
LAURENT C ˆOT´E AND FRANC¸ OIS-SIMON FAUTEUX-CHAPLEAU
Abstract.
Codimension 2 contact submanifolds are the natural generalization of transverseknots to contact manifolds of arbitrary dimension. In this paper, we construct new invariantsof codimension 2 contact submanifolds. Our main invariant can be viewed as a deformationof the contact homology algebra of the ambient manifold. We describe various applicationsof these invariants to contact topology. In particular, we exhibit examples of codimension 2contact embeddings into overtwisted and tight contact manifolds which are formally isotopicbut fail to be isotopic through contact embeddings. We also give new obstructions to certainrelative symplectic and Lagrangian cobordisms.
Contents
1. Introduction 21.1. Overview 21.2. Energy and positivity of intersection 31.3. Legendrian invariants and the surgery formula 51.4. Applications to contact and Legendrian embeddings 51.5. Applications to symplectic and Lagrangian cobordisms 71.6. Context and related invariants 91.7. Notation and conventions 91.8. Acknowledgements 102. Geometric preliminaries 102.1. Symplectic cobordisms 102.2. Homotopy classes of asymptotically cylindrical maps 163. Intersection theory for punctured holomorphic curves 173.1. Normal dynamics and adapted contact forms 173.2. Definition of the intersection number 203.3. Positivity of intersection 213.4. The intersection number for buildings 233.5. Open book decompositions 283.6. The intersection number for cycles 294. Energy and twisting maps 314.1. Standard setups 314.2. Twisting maps 324.3. The energy of a symplectic cobordism 365. Enriched setups and twisted moduli counts 425.1. Enriched setups 425.2. Twisting maps associated to enriched setups 436. Construction of the main invariants 51
Date : September 16, 2020. G -manifolds 94A.2. The group of almost G -structures on the sphere 98References 991. Introduction
Overview.
The purpose of this paper is to introduce new invariants of codimension 2contact submanifolds. Given a closed, co-oriented contact manifold (
Y, ξ ) and a codimension2 contact submanifold (
V, ξ | V ) with trivial contact normal bundle, our main constructionproduces a unital, Z / Q [ U ]-algebra(1.1) CH • ( Y, ξ, V ; r ) . This invariant can be viewed as a deformation of the contact homology algebra CH • ( Y, ξ ); seeRemark 1.1 below. In particular, there is a natural map(1.2) ev U =1 : CH • ( Y, ξ, V ; r ) → CH • ( Y, ξ )obtained by setting U = 1.The algebra CH • ( Y, ξ, V ; r ) is generated by (good) Reeb orbits for an auxiliary contact form λ on ( Y, ξ ). The form λ is required to be adapted to r ∈ R ( Y, ξ, V ), which means in particularthat V is preserved by the Reeb flow of λ . Here R ( Y, ξ, V ) is a non-empty set defined laterwhose elements encode possible Reeb dynamics near V of a contact form on Y .The differential is defined as in ordinary contact homology by counting pseudo-holomorphiccurves in the symplectization ˆ Y , where the additional U variable keeps track of the intersectionnumber of curves with the symplectization ˆ V ⊂ ˆ Y . More precisely, we fix an almost-complexstructure J : ξ → ξ which is compatible with the symplectic form dλ and preserves ξ | V ⊂ T Y | V . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 3
We then consider ˆ J -holomorphic curves in ˆ Y , where ˆ J = − ∂ t ⊗ λ + R λ ⊗ dt + J . The differentialis defined on generators by (roughly) the following formula:(1.3) d ( γ ) = X β ∈ π ( ˆ Y ,γ ⊔ Ω − ) M ( β ) U ˆ V ∗ β +Γ − ( β,V ) γ . . . γ l , where Ω − = γ ⊔ · · · ⊔ γ l and Γ − ( β, V ) = { γ i ⊂ V } is the number of negative orbits of β which are contained in V .The moduli counts appearing in (1.3) are defined as in Pardon’s construction of contacthomology, via his theory of virtual fundamental cycles [Par16, Par19]. The pairing ( − ∗ − ) isa count of intersections between ˆ V and β which was introduced by Siefring [Sie11, MS19].We also construct a closely related invariant(1.4) g CH • ( Y, ξ, V ; r )which we call reduced . This is a unital, Z / Q -algebra which is generated by Reeborbits in the complement of V . The differential counts pseudo-holomorphic curves which donot intersect ˆ V . For appropriately chosen pairs r , r ′ ∈ R ( Y, ξ, V ), we have a morphism of Q -algebras CH • ( Y, ξ, V ; r ′ ) → g CH • ( Y, ξ, V ; r ) . The reduced invariant g CH • ( − ; − ) carries less information, but is easier to compute. Remark . The Q [ U ]-algebra CH • ( Y, ξ, V ; r ) can be viewed as a deformation of the contacthomology Q -algebra CH ( Y, ξ ) in the following way. First, recall that for a ring R and adifferential graded R -algebra ( A, d ), a formal deformation of (
A, d ) is the data of a differential d t := d + td + t d + . . . on the R [[ t ]]-algebra A [[ t ]] satisfying the graded Leibnitz rule, whereeach d i is an endomorphism of A (see [GW99]). Now, let us set U = e t in (1.3) and expand in t . We then get d ( γ ) = X β ∞ X k =0 t k M ( β ) ( ˆ V ∗ β + Γ − ( β, V )) k k ! γ . . . γ l . Thus CH • ( Y, ξ, V ; r ) is indeed a deformation of ordinary contact homology, which can berecovered by sending t → − ( β, V ) = 0, the coefficient M ( β ) ( ˆ V ∗ β +Γ − ( β,V )) k k ! = M ( β ) ( ˆ V ∗ β ) k k ! could naturally be interpreted as a count of pseudo-holomorphic curves which send k markedpoints in the source to the pseudo-holomorphic divisor ˆ V . Of course, it is far from clear howto make such a count rigorous, since curves may develop asymptotic intersections “at infinity”(which might correspond to marked points moving off to infinity).1.2. Energy and positivity of intersection.
In order to ensure that (1.3) defines a differ-ential over Q [ U ], we need to ensure that ˆ V ∗ β + Γ − ( β, V ) ≥ M ( β ) = 0. If M ( β )is nonempty and at least one of the asymptotic orbits of β is disjoint from V , then this is aconsequence of the familiar phenomenon of positivity of intersection. Indeed, in this case β admits a ˆ J -holomorphic representative u which is not contained in ˆ V . Positivity of intersectionthen implies that ˆ V ∗ β = ˆ V ∗ u ≥ Since [Par19] uses virtual techniques to define contact homology without making any transversality as-sumptions, it is possible for the compactification M ( β ) to be nonempty even if M ( β ) is empty. Positivity ofintersection still holds when this happens, but the proof requires a bit more work. Details can be found insections 3.4 and 5.2. LAURENT C ˆOT´E AND FRANC¸ OIS-SIMON FAUTEUX-CHAPLEAU
The situation is more complicated when all of the asymptotic orbits of β are contained in V . Indeed, in this case, the ˆ J -holomorphic representatives of β may be contained in ˆ V andpositivity of intersection fails in general. However, one can show that there is a universal lowerbound on the intersection number(1.5) ˆ V ∗ β ≥ − Γ − ( β, V ) . This explains the appearance of the correction term Γ − ( β, V ) in (1.3).In order to construct CH • ( Y, ξ, V ; r ), it is not enough to define a differential: one also needsto define continuation maps, composition homotopies, etc. These maps are defined by countingcurves in more complicated setups. For example, the continuation map is obtained by countingcurves in a suitably marked exact relative symplectic cobordism ( ˆ X, ˆ λ, H ). More precisely, oneobtains an algebra map similar to (1.3) by counting ˆ J -holomorphic curves in ( ˆ X, ˆ λ ) weightedby their intersection number with H , for a compatible almost-complex structure ˆ J which agreeswith ˆ J ± near the ends.Unfortunately, for an arbitrary relative symplectic cobordism, a lower bound of the type(1.5) fails to hold. A key step in constructing the invariants (1.1) is to identify a sufficientlylarge class of relative symplectic cobordisms for which such a lower bound does hold. Thisleads us to introduce notions of energy for exact symplectic cobordisms and almost-complexstructures on exact relative symplectic cobordisms. These energy notions are developed inSection 4 and are of central importance in this paper.We prove that a lower bound of the type as in (1.5) holds under a certain condition whichrelates the behavior of λ ± near V ± to the energy of ˆ J . We also prove analogous statementsfor other related setups. This allows us to prove that CH • ( Y, ξ, V ; r ) is well-defined. We alsoprove that an exact relative symplectic cobordism ( ˆ X, ˆ λ, H ) induces a map(1.6) CH • ( Y + , ξ + , V + ; r + ) → CH • ( Y − , ξ − , V − ; r − )provided that a certain inequality is satisfied, where the inequality involves r ± and the energyof the (sub)cobordism H ⊂ ( ˆ X, ˆ λ ).Energy considerations play a similarly central role in our construction of the reduced in-variant g CH ( − ; − ). Although the continuation map for the reduced invariant does not countcurves contained in H , one needs to ensure that sequences of curves disjoint from H do notdegenerate into H . This requires hypotheses on the energy of the relevant cobordism. In gen-eral, the arguments involved in constructing CH • ( − ; − ) and g CH • ( − ; − ) turn out to be verysimilar.Energy is not in general well-behaved under gluing symplectic cobordisms, unless one ofthem happens to be a symplectization. As a result, cobordism maps cannot be composedarbitrarily. This lack of functoriality of the invariants (1.1) and (1.4) can be remedied byconsidering variants of these invariants which are obtained by taking certain (co)limits over r ∈ R ( Y, ξ, V ); see Section 6.3. These variants are fully functorial but also seem harder tocompute.
Remark . The apparent failure of positivity of intersection in the absence of energy boundsis not a deficiency of our method: one can construct examples involving 4-dimensional ellipsoidswhich show that certain cobordisms which violate our energy bounds cannot induce maps onthe invariants (1.1) and (1.4). An exact relative symplectic cobordism is the data of an exact symplectic cobordism ( ˆ X, ˆ λ ) which lookslike ( ˆ Y ± , ˆ λ ± ) near the ends, and a codimension 2 symplectic submanifold H ⊂ ˆ X which looks like ˆ V ± near theends; see Definition 2.18 for the details. OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 5
Legendrian invariants and the surgery formula.
Contact homology is one of manyinvariants which can be constructed using the framework of Symplectic Field Theory (SFT).SFT was first introduced by Eliashberg–Givental–Hofer [EGH00] and provides (among otherthings) a mechanism for constructing invariants in symplectic and contact topology by countingpunctured pseudo-holomorphic curves in symplectic manifolds with cylindrical ends.In some of the later sections of this paper, we discuss how the invariants (1.1) and (1.4)are related to other SFT-type invariants. For computational purposes, it is particularly usefulto explore the behavior of the invariants (1.1) and (1.4) under Weinstein handle attachment,following the work of Bourgeois–Ekholm–Eliashberg [BEE12].To this end, we introduce analogs of (1.1) and (1.4) for Legendrian submanifolds. With(
Y, ξ, V ) as above, suppose that Λ ⊂ ( Y − V, ξ ) is a Legendrian submanifold. We then define(under mild topological assumptions) invariants(1.7) L ( Y, ξ, V, Λ; r ) and e L ( Y, ξ, V, Λ; r ) . The first invariant can be thought of as a deformation of the Chekanov-Eliashberg dg algebraof Λ ⊂ ( Y, ξ ), while the second invariant is a reduced version.We describe a surgery exact sequence which relates (linearized versions of) the invariants(1.1), (1.4) and (1.7) under Weinstein handle attachments. This surgery exact sequence is ananalog of the surgery exact sequence for linearized contact homology of Bourgeois–Eklholm–Eliashberg [BEE12, Thm. 5.2].Whereas contact homology has been rigorously defined by Pardon [Par19], most of theremaining aspects of SFT unfortunately have yet to be rigorously defined in full generality.Our paper mirrors the general state of the theory. Indeed, the invariants (1.1) and (1.4) areconstructed fully rigorously, using Pardon’s work. However, our discussion of the surgeryformula (and of the Legendrian invariants therein) is not fully rigorous, and is justified onsimilar grounds as the original work of Bourgeois–Ekholm–Eliashberg [BEE12].To aid the reader in identifying the parts of this paper which are not fully rigorous, allstatements in this paper which depend on unproved assumptions are labeled by a star.
Theproofs of starred statements depend only on a limited set of assumptions which are clearlyidentified. We believe that these assumptions should be entirely believable to experts in thefield. Indeed, they are essentially analogs of the assumptions in [BEE12] and can be justifiedon the same grounds. We expect that if [BEE12] can be made rigorous (say in the frameworkof polyfolds, or using Pardon’s techniques [Par19]), then extending this to our context shouldpose no substantial additional difficulties.1.4.
Applications to contact and Legendrian embeddings.
Transverse knots are im-portant objects of study in three dimensional contact topology. The notion of a codimension2 contact embedding generalizes transverse knots to contact manifolds of arbitrary dimen-sion. However, until recently, it was not understood whether the high-dimensional theory ofcodimension 2 contact embeddings is interesting from the perspective of contact topology, orwhether it reduces entirely to differential topology.
Definition 1.3.
Given a pair of contact manifolds ( V m − , ζ ) and ( Y n − , ξ ), a contact em-bedding is a smooth embedding i : ( V, ζ ) → ( Y, ξ )such that i ∗ ( ξ | i ( V ) ) = ζ . Such a map is also referred to as an isocontact embedding in theliterature (see e.g. [CE20, EM02, PP18]), but we will not use this terminology. A contactsubmanifold ( V, ξ | V ) ⊂ ( Y, ξ ) is a submanifold with the property that ξ | V is a contact structure. LAURENT C ˆOT´E AND FRANC¸ OIS-SIMON FAUTEUX-CHAPLEAU
Observe that if 2 n − m − Example 1.4.
Let π : Y − B → S be an open book decomposition which supports the contactstructure ξ on Y (see Definition 3.25). Then the binding ( B, ξ | B ) ⊂ ( Y, ξ ) is a codimension 2contact submanifold.
Example 1.5 (see Definition 8.1) . Let (
Y, ξ ) be a contact manifold and let Λ ֒ → Y be aLegendrian embedding. Then the Weinstein neighborhood theorem furnishes an embedding τ (Λ) : ( ∂ ( D ∗ Λ) , ξ std ) ֒ → ( Y, ξ )which is canonical up to isotopy through codimension 2 contact embeddings. We refer to τ (Λ)as the contact pushoff of Λ ֒ → Y . By abuse of notation, we will routinely identify τ (Λ) withits image.As is customary in contact and symplectic topology, there is a notion of a formal contactembedding. This notion encodes certain necessary bundle-theoretic conditions which must besatisfied by any (genuine) contact embedding. It is then natural to seek to understand to whatextent the space of genuine contact embeddings of ( V, ζ ) into (
Y, ξ ) differs from the space offormal contact embeddings.In case V is a closed manifold of codimension at least 4 with respect to Y , or open and ofcodimension at least 2, then an h-principle due to Gromov (see [EM02, Thm. 12.3.1 and Rmk.12.3]) implies that the space of contact embeddings is essentially equivalent to the space offormal contact embeddings. Thus, in these settings, the theory of contact embeddings reducesto differential topology.In contrast, a recent breakthrough result due to Casals and Etnyre [CE20] shows thatthis h-principle fails in general for codimension 2 contact embeddings of closed manifolds.More precisely, Casals and Etnyre [CE20, Thm. 1] exhibited a pair of contact embeddingsof ( D ∗ S n − , ξ ) = ∂ ∞ ( T ∗ S n − , λ can ) into the standard contact sphere ( S n − , ξ std ) which areformally isotopic but are not isotopic through contact embeddings. They also proved [CE20,Thm. 4] that there are in fact infinitely many pairs of codimension 2 contact embeddings intothe standard contact sphere which are formally isotopic but fail to be isotopic through contactembeddings.There has also been recent work to establish existence results for codimension 2 contactembeddings under certain conditions [PP18, Laz20]. This culminates in a full existence h-principle for codimension 2 contact embeddings due to Casals–Pancholi–Presas [CPP19], whichstates that any formal codimension 2 contact embedding is formally isotopic to a genuinecontact embedding.The invariants constructed in this paper can be used to distinguish pairs of formally isotopiccontact embeddings which are not isotopic through contact embeddings. We illustrate twotypes of applications, applying respectively to contact embeddings into overtwisted contactmanifolds and into the standard contact sphere.Let us begin with the overtwisted case. In Construction 10.6, we describe a procedure forconstructing pairs of formally isotopic contact embeddings into overtwisted contact manifoldswhich are not isotopic through contact embedings. Theorem 1.6.
Let i and j be the (formally isotopic) contact embeddings into the overtwistedmanifold ( Y, ξ ) constructed according to Construction 10.6. Then i and j are not isotopicthrough contact embeddings. In fact, i is not isotopic to any reparametrization of j in the OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 7 source, meaning that i ( V ) and j ( V ) are not isotopic as codimension contact submanifolds of ( Y, ξ ) . Theorem 1.6 can be proved using either of the invariants (1.1) or (1.4). To the best of ourknowledge, it cannot be proved in general using invariants already in the literature. However,for some examples, Theorem 1.6 essentially reduces to the statement that the binding of an openbook decomposition is tight, a fact which was shown by Etnyre and Vela-Vick [EVV10, Thm.1.2] in dimension 3 and Klukas [Klu18, Cor. 3] in general.In some cases (see Corollary 10.10), the embeddings i and j in fact coincide with the contactpushoffs of Legendrian embeddings. It is not hard to show that an isotopy of Legendrianembeddings induces an isotopy of their contact pushoffs. Thus the invariants (1.1) and (1.4)also distinguish Legendrian embeddings in overtwisted contact manifolds. We note it is ingeneral difficult to distinguish Legendrians in overtwisted contact manifolds since Legendriancontact homology (assuming that it can be defined) vanishes for formal reasons.Our second application concerns codimension 2 contact embeddings into the standard con-tact spheres ( S n − , ξ std ). More precisely, we use the reduced invariant (1.4) to distinguishformally isotopic contact embeddings of ( S ∗ S n − , ξ ) = ∂ ∞ ( T ∗ S n − , λ can ) into ( S n − , ξ std ),thus reproving the main result of Casals and Etnyre [CE20, Thm. 1] in dimensions 4 n − n > Theorem* 1.7 (see Theorem* 10.18) . Let ( V, ξ ) be the ideal boundary of ( T ∗ S n − , λ can ) .Then for n > , there exists a pair of formally isotopic contact embeddings i , i : ( V, ξ ) → ( S n − , ξ std ) which are not isotopic through contact embeddings. The embeddings we exhibit in fact coincide exactly with those exhibited by Casals andEtnyre in proving [CE20, Thm. 1.1] (see Remark 10.21). However, the proofs that theseembeddings are not contact isotopic are entirely different. Casals and Etnyre consider a doublebranched cover along the contact submanifolds i ( V ) and i ( V ). Using symplectic homology,they prove that the two branched covers do not admit the same fillings. This implies that i ( V ) and i ( V ) cannot be isotopic, since otherwise they would have contactomorphic branchedcovers.In contrast, our proof of Theorem* 1.7 uses the invariant g CH ( − ; − ) introduced in thispaper. Roughly speaking, we prove Theorem* 1.7 by partially computing (linearizations of) g CH ( − ; − ) associated to the two embeddings under consideration, and observing that they donot match. Our computations rely crucially on our version of the surgery formula discussedin Section 1.3 as well as the well-definedness of the invariants therein. This explains why thistheorem statement is starred, following the convention stated in Section 1.3. We also remarkthat although Theorem* 1.7 only applies to spheres of dimension 4 n −
1, we expect that thesame invariant also distinguishes embeddings into spheres of dimension 4 n −
3. However,proving this would likely require more involved computations than those carried out in thispaper.1.5.
Applications to symplectic and Lagrangian cobordisms.
Consider a pair of contactmanifolds ( Y ± , ξ ± ) and codimension 2 contact submanifolds ( V ± , ξ ± | V ± ) ⊂ ( Y ± , ξ ± ).An exact relative symplectic cobordism from ( Y + , ξ + , V + ) to ( Y − , ξ − , V − ) is a triple ( ˆ X, ˆ λ, H ),where ( ˆ X, ˆ λ ) is an exact symplectic cobordism from ( Y + , ξ + ) to ( Y − , ξ − ) and H ⊂ ˆ X is acodimension 2 symplectic submanifold which coincides near the ends with the symplectization LAURENT C ˆOT´E AND FRANC¸ OIS-SIMON FAUTEUX-CHAPLEAU of V ± ; see Definition 2.18. In the special case where ˆ X is the symplectization of Y ± and H isdiffeomorphic to R × V ± , we speak of a symplectic concordance from V + to V − .These notions were first considered by Bowden in his PhD thesis. Using gauge theory, heexhibited certain restrictions on symplectic cobordisms between transverse links in contact3-manifolds [Bow10, Sec. 7].The following theorem provides a constraint on exact symplectic cobordisms between certainpairs of codimension 2 contact submanifolds of an ambient overtwisted manifold. To the best ofour knowledge, this is the first negative result in the literature on relative symplectic cobordismsin dimensions greater than three. Theorem 1.8.
Let V = i ( B ) , V ′ = j ( B ) be the codimension contact submanifolds of theovertwisted contact manifold ( Y, ξ ) as described in Construction 10.6. Then there does not existan exact relative symplectic cobordism ( ˆ X, ˆ λ, H ) from ( Y, ξ, V ′ ) to ( Y, ξ, V ) with H ( H ; Z ) = H ( H ; Z ) = 0 . In particular, there is no symplectic concordance from V ′ to V . One can similarly consider Lagrangian cobordisms and concordances between Legendriansubmanifolds. An exact Lagrangian cobordism from ( Y + , ξ + , Λ + ) to ( Y − , ξ − , V − ) is a triple( ˆ X, ˆ λ, L ) where ( ˆ X, ˆ λ ) is an exact symplectic cobordism from ( Y + , ξ + ) to ( Y − , ξ − ) and L ⊂ ˆ X is a Lagrangian submanifold which coincides near the ends with the Lagrangian lifts of Λ ± ; seeDefinition 2.21. If ˆ X is the symplectization of Y − and L = R × Λ − , one speaks of a Lagrangianconcordance from Λ + to Λ − .The theory of Lagrangian cobordisms has been extensively developed in the literature fromvarious perspectives (see e.g. [CDRGG20, Ekh12, Pan17, ST13]). While a great deal is knownin ( R n +1 , ξ std ) and certain other tight contact manifolds, we are not aware of any resultsconstraining cobordisms and concordances in overtwisted contact manifolds; see Remark 1.10.The following theorem provides a first result in this direction. Theorem 1.9.
Let Λ , Λ ′ be the Legendrian submanifolds of the overwisted contact manifold ( Y, ξ ) as constructed in Construction 10.9. Then Λ ′ is not concordant to Λ . In contrast, a result of Eliashberg and Murphy [EM13, Thm. 2.2] implies that Λ is concordantto Λ ′ . Remark . It is a basic fact that exact Lagrangian cobordisms induce morphisms on Legen-drian contact homology which behave well under composition of cobordisms [EN19, Sec. 5.1].This leads to a myriad of interesting obstructions to the existence of Lagrangian cobordismsand concordances. One can also obtain many interesting obstructions using finite-dimensionalinvariants (which are closely related to Legendrian contact homology) coming from generatingfunctions or sheaf-theory; see e.g. [ST13, Pan17].One drawback of these approaches is that they are necessary blind on overwisted contactmanifolds. Indeed, even if Legendrian contact homology could be rigorously defined in fullgenerality following the framework of [EGH00, Sec. 2.8], it would provide no informationfor Legendrians in overtwisted contact manifolds: being a module over the contact homologyalgebra, it would vanish. In contrast, the invariants developed in this paper do give informationabout Legendrians even in the overwisted case.Our final result states that certain Lagrangian concordances cannot be displaced from acodimension 2 symplectic submanifold. More precisely, let (
Y, ξ ) = obd( T ∗ S n − , id) and let V ⊂ Y be the binding of the open book. Let Λ ⊂ Y the zero section of a page and let Λ ′ beobtained by stabilizing Λ in the complement of V . It can be shown [CM19, Prop. 2.9] thatΛ ⊂ ( Y, ξ ) is a loose Legendrian; hence Λ , Λ ′ are Legendrian isotopic in ( Y, ξ ). OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 9
Theorem* 1.11.
Any Lagrangian concordance from Λ ′ to Λ must intersect the symplectizationof V . In contrast, work of Eliashberg and Murphy [EM13, Thm. 2.2] implies that there exists aLagrangian concordance from Λ to Λ ′ which is disjoint from the symplectization of V .Our proof of Theorem* 1.11 uses the deformed versions of the Chekanov-Eliashberg dg alge-bra in (1.7). Hence the statement is starred according to the convention stated in Section 1.3.1.6. Context and related invariants.
The invariants constructed in this paper, when spe-cialized to contact 3-manifolds, are related to other invariants in the literature. The mostclosely related invariant is due to Momin [Mom11]. Given a contact 3-manifold ( Y , ξ ), Mominconsiders the set of pairs ( λ, L ) where λ is a contact form and L ⊂ Y is a link of Reeb orbits of λ . Two such pairs ( λ, L ) , ( λ ′ , L ′ ) are said to be equivalent if L = L ′ and each component orbit(and all its multiple covers) has the same Conley-Zehnder index. Under certain assumptionson ( Y, λ, L ), Momin defines an invariant which we denote by CH mo • ( Y, [( λ, L )]). This is a Z -graded Q -vector space which depends only on Y and the equivalence class of ( λ, L ).The invariant constructed by Momin is in general distinct from the invariants described inthis paper. In particular, he considers cylindrical contact homology, whereas we work withordinary contact homology. However, in the special case where ( Y , ξ ) is the standard contactsphere (or more generally a subcritical Stein manifold with c ( ξ ) = 0) and L ⊂ ( Y, ξ ) is acollection of Reeb orbits which bound a symplectic submanifold H ⊂ B , then we expect that(1.8) CH mo • ( Y, [ λ, L ]) = g CH ˜ ǫ • ( Y, ξ, L ; r ) , for suitable r which depends on the equivalence class of ( λ, L ). Here the right hand side is acertain linearized version of CH • ( Y, ξ, L ; r ) which depends on the relative filling ( B , λ std , H );see Section 7.3.Momin’s work has led to beautiful applications to Reeb dynamics on contact 3-manifolds(see e.g. [AP20, HMS15]). It would be interesting to explore whether the invariants developedin this paper can be used in studying Reeb dynamics in higher dimensions.Another related invariant is Hutchings’ “knot-filtered embedded contact homology” [Hut16].The setting for this invariant is a contact 3-manifold ( Y, ξ ) with H ( Y ; Z ) = 0. Given atransverse knot L ⊂ ( Y, ξ ) and an irrational parameter θ ∈ R − Q , Hutchings defines a filtrationon embedded contact homology with values in Z + Z θ which is an invariant of ( L, θ ). The basicidea is to choose a contact form ξ = ker λ so that L is a Reeb orbit, and to filter the generatorsof embedded contact homology by their linking number with L . Positivity of intersectionconsiderations imply that the differential decreases the linking number for orbits which aredisjoint from L . However, the situation is more complex when the differential involves L ,which explains why the filtration is only valued in Z + Z θ .One could presumably carry over Hutchings’ construction to the context of (cylindrical)contact homology in dimension 3. We expect that the resulting invariant would carry re-lated information to the one defined by Momin or to the invariants constructed in this paper.However, we do not have a precise formulation of what this relationship should be.We remark that the invariants introduced by Momin and Hutchings are built using tech-niques from 4-dimensional symplectic topology which cannot be generalized to higher dimen-sions. In contrast, the invariants introduced in this paper are constructed by a different ap-proach which ultimately relies on Pardon’s robust virtual fundamental cycles package [Par16].1.7. Notation and conventions.
All manifolds in this paper are assumed to be smooth. If M is a manifold, a ball B ⊂ M is an open subset diffeomorphic to the open unit disk andwhose closure is embedded and diffeomorphic to the closed unit disk. If ( M, ω ) is symplectic, a Darboux ball B ⊂ M is a ball which is symplectomorphic to the open unit disk equippedwith (some constant rescaling of) the standard symplectic form.Let ( Y n − , ξ = ker λ ) be a closed co-oriented contact manifold. The Reeb vector fieldassociated to the contact form λ will be denoted by R λ .Let γ be a Reeb orbit of period T >
0, parametrized so that λ ( γ ′ ) = T . Given a choiceof dλ -compatible almost complex structure J on ξ , we can define the asymptotic operator A γ : Γ( γ ∗ ξ ) → Γ( γ ∗ ξ ) by A γ = − J ( ∇ t − T ∇ R λ ), where ∇ is some symmetric connection on Y .The Conley-Zehnder index of a Reeb orbit γ relative to a trivialization τ of γ ∗ ξ will bedenoted by CZ τ ( γ ).1.8. Acknowledgements.
We are grateful to Yasha Eliashberg for suggesting this project,and for many helpful discussions. We have also benefited from discussions and correspon-dence with C´edric De Groote, Georgios Dimitroglou Rizell, Sheel Ganatra, Oleg Lazarev, JoshSabloff and Kyler Siegel. The first author was supported by a Stanford University BenchmarkGraduate Fellowship for part of the period during which this work was carried out.2.
Geometric preliminaries
Symplectic cobordisms.
Let (
Y, ξ ) be a closed co-oriented contact manifold. The sym-plectization of (
Y, ξ ) is the exact symplectic manifold (
SY, λ Y ) where SY ⊂ T ∗ Y is the totalspace of the bundle of positive contact forms on Y (i.e. a point ( p, α ) ∈ T ∗ Y is in SY ifand only if α : T p Y → R vanishes on ξ p and the induced map T p Y /ξ p → R is an orientation-preserving isomorphism) and λ Y is the restriction of the tautological Liouville form on T ∗ Y .Given a choice of positive contact form α for ( Y, ξ ), there is a canonical identification(2.1) σ α : ( R × Y, e s α ) → ( SY, λ Y )given by σ α ( s, p ) = ( p, e s α p ). We will refer to ( ˆ Y , ˆ α ) := ( R × Y, e s α ) as the symplectization of( Y, α ).A subset U ⊂ SY will be called a neighborhood of + ∞ (resp. of −∞ ) if it contains σ α ([ N, ∞ ) × Y ) (resp. σ α (( −∞ , − N ] × R ) for N > α ). Definition 2.1.
Given a contactomorphism f : ( Y, ξ ) → ( Y, ξ ), we define its symplectic lift ˜ f : ( SY, λ Y ) → ( SY, λ Y )˜ f ( p, α ) = ( f ( p ) , α ◦ ( df p ) − ) . One can verify that ˜ f ∗ λ Y = λ Y , so ˜ f is in particular a symplectomorphism. Moreover, if f t : ( Y, ξ ) → ( Y, ξ ) is a family of contactomorphisms, then ˜ f t is Hamiltonian; see [Cha10, Prop.2.2]. Definition 2.2.
Let ( Y + , ξ + ) and ( Y − , ξ − ) be closed co-oriented contact manifolds. An exact symplectic cobordism from ( Y + , ξ + ) to ( Y − , ξ − ) is an exact symplectic manifold ( ˆ X, ˆ λ )equipped with embeddings e + : SY + → ˆ X (2.2) e − : SY − → ˆ X (2.3)satisfying the following properties: • ( e ± ) ∗ ˆ λ = λ Y ± ; OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 11 • there exists a neighborhood U + ⊂ SY + of + ∞ and a neighborhood U − ⊂ SY − of −∞ such that the restriction of e ± to U ± is proper, the images e + ( U + ) and e − ( U − ) aredisjoint and the complement ˆ X \ ( e + ( U + ) ∪ e − ( U − )) is compact. Definition 2.3 (cf. [Par19, Sec. 1.3]) . Let ( Y + , λ + ) and ( Y − , λ − ) be closed manifoldsequipped with contact forms. A (strict) exact symplectic cobordism from ( Y + , λ + ) to ( Y − , λ − )is an exact symplectic manifold ( ˆ X, ˆ λ ) equipped with embeddings e + : R × Y + → ˆ X (2.4) e − : R × Y − → ˆ X (2.5)satisfying the following properties: • ( e ± ) ∗ ˆ λ = ˆ λ ± ; • there exists an N ∈ R such that the restrictions of e + to [ N, ∞ ) × Y + and of e − to( −∞ , − N ] × Y − are proper and that the images e + ([ N, ∞ ) × Y + ) and e − (( −∞ , − N ] × Y − ) are disjoint and together cover a neighborhood of infinity (i.e. the complement oftheir union is compact). Notation 2.4.
Let ( ˆ X, ˆ λ ) be an exact symplectic cobordism from ( Y + , ξ + ) to ( Y − , ξ − ) in thesense of Definition 2.2. Given any choice of contact forms λ ± on ( Y ± , ξ ± ), one can obtain fromˆ X a cobordism from ( Y + , λ + ) to ( Y − , λ − ) in the sense of Definition 2.3 by pre-composing theembeddings (2.2)–(2.3) with the canonical identifications R × Y ± → SY ± induced by λ ± . Wewill denote this cobordism by ( ˆ X, ˆ λ ) λ + λ − or simply by ˆ X λ + λ − when this creates no ambiguity.Similarly, any cobordism ( ˆ X, ˆ λ ) in the sense of Definition 2.3 can be viewed as a cobordismin the sense of Definition 2.2 as well. Remark . In light of the above discussion, Definition 2.2 and Definition 2.3 are essentiallyequivalent. However, it will be convenient for us to be able to discuss symplectic cobordismswithout fixing a particular choice of contact forms on the ends, so we adopt Definition 2.2 asour main definition moving forward.
Example 2.6 (Symplectizations) . The symplectization (
SY, λ Y ) of a contact manifold ( Y, ξ )is canonically endowed with the structure of an exact symplectic cobordism in the sense ofDefinition 2.2 by letting e + = e − = id. The additional data of a pair of contact forms λ + , λ − for ( Y, ξ ), endows (
SY, λ Y ) with the structure of a strict exact symplectic cobordism in thesense of Definition 2.3 and we write ( SY, λ Y ) λ + λ − .We note that a different choice of e + , e − would endow ( SY, λ Y ) with a priori non-equivalentsymplectic cobordism structure. In the sequel, we always assume unless otherwise specifiedthat symplectizations are endowed with the canonical symplectic cobordism structure.
Definition 2.7.
Let ( ˆ X , ˆ λ ) and ( ˆ X , ˆ λ ) be exact symplectic cobordisms from ( Y , ξ )to ( Y , ξ ) and from ( Y , ξ ) to ( Y , ξ ) respectively. Fix a real number t ≥ µ t : SY → SY denote multiplication by e t . The t -gluing of ˆ X and ˆ X , denoted by ˆ X t ˆ X ,is the smooth manifold obtained by gluing ˆ X and ˆ X along the maps SY ˆ X SY ˆ X µ t (2.2) Since µ ∗ t λ Y = e t λ Y , there is, for any s ∈ R , a Liouville form on ˆ X t ˆ X which agreeswith e t + s ˆ λ on ˆ X and with e s ˆ λ on ˆ X . We will denote it by ˆ λ t,s ˆ λ . Note that ( ˆ X t ˆ X , ˆ λ t,s ˆ λ ) is canonically equipped with the structure of an exact symplecticcobordism from ( Y , ξ ) to ( Y , ξ ) via the embeddings SY SY ˆ X ˆ X t ˆ X SY SY ˆ X ˆ X t ˆ X µ − t − s (2.2) µ − s (2.3) The precise choice of s doesn’t really matter since the forms ˆ λ t,s ˆ λ , s ∈ R , are all constantmultiples of each other. When t = 0, it is natural to choose s = 0, and we will denote the result-ing cobordism simply by ( ˆ X X , ˆ λ λ ). There is no obvious choice for t >
0, but for thesake of definiteness we set ˆ λ t ˆ λ := ˆ λ t, − t/ ˆ λ and will refer to ( ˆ X t ˆ X , ˆ λ t ˆ λ )as “the” t -gluing of ( ˆ X , ˆ λ ) and ( ˆ X , ˆ λ ). Remark . When t = s = 0, it follows directly from the definition that the gluing operation isassociative: (cid:0) ( ˆ X X ) X , (ˆ λ λ ) λ (cid:1) and (cid:0) ˆ X X X ) , ˆ λ λ λ ) (cid:1) arecanonically isomorphic. Remark . Multiplication by e t on SY corresponds to translation by t in the R coordinateunder the identification SY ∼ = R × Y induced by a choice of contact form on Y . Definition 2.7is therefore consistent with the notion of “ t -gluing” in [Par19, Sec. 1.5]. Definition 2.10.
Let ( ˆ X , ˆ λ ) and ( ˆ X , ˆ λ ) be cobordisms from ( Y + , ξ + ) to ( Y − , ξ − ). An isomorphism of exact symplectic cobordisms φ : ( ˆ X , ˆ λ ) → ( ˆ X , ˆ λ ) consists of a diffeomor-phism φ : ˆ X → ˆ X such that φ ∗ ˆ λ = ˆ λ and which is compatible with the ends in the sensethat the following diagram commutes: ˆ X SY + SY − ˆ X φ (2.2)(2.2) (2.3)(2.3) Example 2.11.
Let ( ˆ X, ˆ λ ) be an exact symplectic cobordism from ( Y + , ξ + ) to ( Y − , ξ − ). Thenfor any t ≥ s ∈ R , the glued cobordisms ( SY + t ˆ X, λ Y + t,s ˆ λ ) and ( ˆ X t SY − , ˆ λ t,s λ Y − )are canonically isomorphic to ( ˆ X, ˆ λ ). Definition 2.12. A one-parameter family of exact symplectic cobordisms from ( Y + , ξ + ) to( Y − , ξ − ) is a manifold ˆ X equipped with a family of Liouville forms { ˆ λ t } t ∈ I (where I ⊂ R isan interval), together with embeddings e + t : SY + → ˆ X (2.6) e − t : SY − → ˆ X (2.7)as in Definition 2.2. We will always assume that the family is fixed at infinity, meaning thatfor every compact subinterval [ a, b ] ⊂ I , • { ˆ λ t } t ∈ [ a,b ] is constant outside of a compact subset of ˆ X ; • { e + t } t ∈ [ a,b ] (resp. { e − t } t ∈ [ a,b ] ) is independent of t on some neighborhood of + ∞ in SY + (resp. of −∞ in SY − ). OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 13
Two cobordisms ( ˆ X , ˆ λ ) and ( ˆ X , ˆ λ ) are said to be deformation equivalent if there exists aone-parameter family ( ˆ W , ˆ µ t ) t ∈ [0 , such that ( ˆ X , ˆ λ ) is isomorphic to ( ˆ W , ˆ µ ) and ( ˆ X , ˆ λ ) isisomorphic to ( ˆ W , ˆ µ ). The deformation class of a cobordism ( ˆ X, ˆ λ ) will be denoted by [ ˆ X, ˆ λ ]. Example 2.13.
Given ( ˆ X , ˆ λ ) and ( ˆ X , ˆ λ ) as in Definition 2.7, the glued cobordisms( ˆ X t ˆ X , ˆ λ t ˆ λ ) t ∈ [0 , ∞ ) form a one-parameter family. Similarly, for any fixed t ≥ X t ˆ X , ˆ λ t,s ˆ λ ) s ∈ R is a one-parameter family. As a corollary, we have that the defor-mation class [ ˆ X t ˆ X , ˆ λ t,s ˆ λ ] is independent of both t and s . Proof.
We will construct a two-parameter family φ t,s : ˆ X X → ˆ X t ˆ X of diffeomor-phisms, with φ , = id, such that the forms φ ∗ t,s (ˆ λ t,s ˆ λ ) agree with ˆ λ λ outside of acompact set (depending on t, s ) and form a smooth family.In order to simplify the notation, we fix contact forms λ i on ( Y i , ξ i ), i = 0 , ,
2, so that wecan view the symplectization of Y i as a product R × Y i . For C > X and ˆ X asˆ X = ( −∞ , × Y ∪ ¯ X ∪ [ C, ∞ ) × Y ˆ X = ( −∞ , − C ] × Y ∪ ¯ X ∪ [ − , ∞ ) × Y where ¯ X ⊂ ˆ X is a compact submanifold with boundary { } × Y ⊔ { C } × Y , and similarlyfor ¯ X . This induces a decomposition of ˆ X t ˆ X of the formˆ X t ˆ X = ( −∞ , − C ] × Y ∪ ¯ X ∪ [ − , t + 1] × Y ∪ ¯ X ∪ [ C, ∞ ) × Y for any t ≥
0. Hence, in order to define φ t,s , it suffices to make a choice of: • a smooth family of diffeomorphisms f t : [ − , → [ − , t + 1] which coincide with theidentity near − t near 1; • a smooth family of diffeomorphisms g t,s : [ C, ∞ ) → [ C, ∞ ) which coincide with theidentity near C and with translation by − t − s at infinity; • a smooth family of diffeomorphisms h t,s : ( −∞ , − C ] → ( −∞ , − C ] which coincide withthe identity near − C and with translation by − s at infinity.We of course also require that f , g , and h , be the identity on their respective domains. (cid:3) Proposition 2.14.
The deformation class of ( ˆ X X , ˆ λ λ ) only depends on the defor-mation classes of ( ˆ X , ˆ λ ) and ( ˆ X , ˆ λ ) .Proof. Let ( ˆ W , ˆ µ ,s ) s ∈ [0 , and ( ˆ W , ˆ µ ,s ) s ∈ [0 , be one-parameter families of exact sym-plectic cobordisms from ( Y , ξ ) to ( Y , ξ ) and from ( Y , ξ ) to ( Y , ξ ) respectively. Thenegative end (2.7) of ( ˆ W , ˆ µ ,s ) will be denoted by e − s : SY → ˆ W and the positive end(2.6) of ( ˆ W , ˆ µ ,s ) will be denoted by e + s : SY → ˆ W . By definition, we can find a neigh-borhood U + ⊂ SY of + ∞ and a neighborhood U − ⊂ SY such that the restriction of e ± s to U ± is independent of s . This common restriction will be denoted by e ± .Fix a large t > V := µ − t ( U + ) ∩ U − is nonempty. Let ˆ W V ˆ W be the space obtained by gluing ˆ W \ e − ( U − \ V ) and ˆ W \ e + ( U + \ µ t ( V )) along the maps Strictly speaking, the underlying manifold of ˆ X t ˆ X depends on t , so in order to obtain a family in thesense of Definition 2.12 one needs to choose suitable diffeomorphisms ˆ X t ˆ X ∼ = ˆ X X . V ˆ W \ e − ( U − \ V ) U + ˆ W \ e + ( U + \ µ t ( V )) e − µ t e + As a smooth manifold, ˆ W V ˆ W is canonically identified with ˆ W t ˆ W . Thus we canview ˆ µ ,s t ˆ µ ,s as a Liouville form on ˆ W V ˆ W for each s , and this makes ( ˆ W V ˆ W , ˆ µ ,s t ˆ µ ,s ) s ∈ [0 , into a one-parameter family of cobordisms. In particular, it follows that ( ˆ W t ˆ W , ˆ µ , t ˆ µ , )and ( ˆ W t ˆ W , ˆ µ , t ˆ µ , ) are deformation equivalent. (cid:3) Corollary 2.15.
There is a well-defined gluing operation on deformation classes of exactsymplectic cobordisms given by (2.8) [ ˆ X , ˆ λ ] X , ˆ λ ] = [ ˆ X t ˆ X , ˆ λ t,s ˆ λ ] for any t ≥ and s ∈ R . Proposition 2.16.
The gluing operation (2.8) is associative.Proof.
This follows from Remark 2.8. (cid:3)
Let V ⊂ ( Y, ξ ) be a contact submanifold. There is a canonical exact symplectic embedding(
SV, λ V ) → ( SY, λ Y ); it corresponds to the obvious inclusion R × V → R × Y under theidentifications SV ∼ = R × V and SY ∼ = R × Y induced by a choice of contact form on Y . Convention 2.17.
In this paper, all contact submanifolds are compact and without boundary.
Definition 2.18.
Let V + ⊂ ( Y + , ξ + ) and V − ⊂ ( Y − , ξ − ) be contact submanifolds of the samecodimension, and let ( ˆ X, ˆ λ ) be an exact symplectic cobordism from ( Y + , ξ + ) to ( Y − , ξ − ). Wesay that a smooth submanifold H ⊂ ˆ X is cylindrical with ends V ± if it is closed (as a subset)and there exist neighborhoods U ± ⊂ SY ± of ±∞ such that( e ± ) − ( H ) ∩ U ± = SV ± ∩ U ± , where e ± : SY ± → ˆ X are the ends (2.2)–(2.3) of ( ˆ X, ˆ λ ).If H is a symplectic cylindrical submanifold of ( ˆ X, ˆ λ ), then we say that ( ˆ X, ˆ λ, H ) is an exact relative symplectic cobordism from ( Y + , ξ + , V + ) to ( Y − , ξ − , V − ). Note that in this case,the restrictions of e ± to SV ± ∩ U ± endow ( H, ˆ λ | H ) with the structure of an exact symplecticcobordism from ( V + , ξ + | V + ) to ( V − , ξ − | V − ). Example 2.19. If V is a contact submanifold of ( Y, ξ ), then, as noted above, SV can beviewed as a symplectic submanifold of ( SY, λ Y ), and ( SY, λ Y , SV ) is canonically endowedwith the structure of an exact relative symplectic cobordism in the sense of Definition 2.18 byletting e + = e − = id. Notation 2.20.
Let ˆ X, ˆ λ, H be as in Definition 2.18. As explained in Notation 2.4, a choiceof contact forms ker λ ± = ξ ± endows ( ˆ X, ˆ λ ) with the structure of a strict relative symplecticcobordism. We analogously speak of a strict relative exact symplectic cobordism and write( ˆ X, ˆ λ, H ) λ + λ − when we wish to emphasize that we are fixing contact forms λ ± on the ends.Let Λ ⊂ ( Y, ξ ) be a Legendrian submanifold. The
Lagrangian lift of Λ is the Lagrangiansubmanifold L = { ( p, α ) ∈ SY ⊂ T ∗ Y | p ∈ Λ } ⊂ ( SY, λ Y ) . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 15
Definition 2.21.
Let Λ + ⊂ ( Y + , ξ + ) and Λ − ⊂ ( Y − , ξ − ) be Legendrian submanifolds and let( ˆ X, ˆ λ ) be an exact symplectic cobordism from ( Y + , ξ + ) to ( Y − , ξ − ). We say that a Lagrangiansubmanifold L ⊂ ( ˆ X, ˆ λ ) is cylindrical with ends Λ ± if it is closed (as a subset) and there existneighborhoods U ± ⊂ SY ± of ±∞ such that( e ± ) − ( L ) ∩ U ± = L ± ∩ U ± , where e ± : SY ± → ˆ X are the ends (2.2)–(2.3) of ( ˆ X, ˆ λ ) and L ± are the Lagrangian lifts of Λ ± .The data of a triple ( ˆ X, ˆ λ, L ) is called an (exact) Lagrangian cobordism from ( Y + , ξ + , Λ + )to ( Y − , ξ − , Λ − ). Definition 2.22.
The set of equivalence classes of cylindrical codimension 2 submanifolds ofˆ X with ends V ± , where two submanifolds are equivalent if they are isotopic via a compactlysupported isotopy, will be denoted by Ω n − ( ˆ X, V + ⊔ V − ). Definition 2.23.
A contact submanifold V ⊂ ( Y, λ ) is said to be a strong contact submanifold if it is (set-wise) invariant under the Reeb flow of λ on Y . We will also say that ( ˆ X, ˆ λ, H ) λ + λ − is a strong relative cobordism if both V + ⊂ ( Y + , λ + ) and V − ⊂ ( Y − , λ − ) are strong contactsubmanifolds. Definition 2.24.
Let ( ˆ X , ˆ λ , H ) and ( ˆ X , ˆ λ , H ) be exact relative symplectic cobor-disms from ( Y , ξ , V ) to ( Y , ξ , V ) and from ( Y , ξ , V ) to ( Y , ξ , V ) respectively. Forany sufficiently large real number t ≥ H t H sits naturally inside ( ˆ X t ˆ X , ˆ λ t ˆ λ )as a symplectic submanifold, and ( ˆ X t ˆ X , ˆ λ t ˆ λ , H t H ) is a relative cobordismfrom ( Y , ξ , V ) to ( Y , ξ , V ). We will refer to it as the t -gluing of ( ˆ X , ˆ λ , H ) and( ˆ X , ˆ λ , H ). Definition 2.25.
Let ( ˆ X , ˆ λ , H ) and ( ˆ X , ˆ λ , H ) be relative cobordisms from ( Y + , ξ + , V + )to ( Y − , ξ − , V − ). An isomorphism of exact relative symplectic cobordisms φ : ( ˆ X , ˆ λ , H ) → ( ˆ X , ˆ λ , H ) is an isomorphism φ : ( ˆ X , ˆ λ ) → ( ˆ X , ˆ λ ) in the sense of Definition 2.10 whichmaps H diffeomorphically onto H . Example 2.26.
Let ( ˆ X, ˆ λ, H ) be an exact symplectic relative cobordism from ( Y + , ξ + , V + )to ( Y − , ξ − , V − ). Then for any t ≥
0, the glued cobordisms ( SY + , SV + ) t ( ˆ X, H ) and( ˆ
X, H ) t ( SY − , SV − ) are defined and canonically isomorphic to ( ˆ X, ˆ λ, H ). Definition 2.27. A one-parameter family of exact relative symplectic cobordisms from ( Y + , ξ + , V + )to ( Y − , ξ − , V − ) is a manifold ˆ X equipped with a family of Liouville forms { ˆ λ t } t ∈ I , a family ofsymplectic submanifolds H t ⊂ ( ˆ X, ˆ λ t ), and embeddings e + t : SY + → ˆ X (2.9) e − t : SY − → ˆ X (2.10)as in Definition 2.18. We will always assume that the family is fixed at infinity, meaning thatfor every compact subinterval [ a, b ] ⊂ I , • { ˆ λ t } t ∈ [ a,b ] and { H t } t ∈ [ a,b ] are constant outside of a compact subset of ˆ X ; • { e + t } t ∈ [ a,b ] (resp. { e − t } t ∈ [ a,b ] ) is independent of t on some neighborhood of + ∞ in SY + (resp. of −∞ in SY − ).Two relative cobordisms ( ˆ X , ˆ λ , H ) and ( ˆ X , ˆ λ , H ) are said to be deformation equivalent if there exists a one-parameter family ( ˆ W , ˆ µ t , K t ) t ∈ [0 , such that ( ˆ X , ˆ λ , H ) is isomorphic to ( ˆ W , ˆ µ , K ) and ( ˆ X , ˆ λ , H ) is isomorphic to ( ˆ W , ˆ µ , K ). The deformation class of acobordism ( ˆ X, ˆ λ, H ) will be denoted by [ ˆ X, ˆ λ, H ]. Example 2.28.
Given ( ˆ X , ˆ λ , H ) and ( ˆ X , ˆ λ , H ) as in Definition 2.24, the gluedcobordisms ( ˆ X t ˆ X , ˆ λ t ˆ λ , H t H ) t ∈ [ N, ∞ ) form a one-parameter family for N > t ≫
0, ( ˆ X t ˆ X , ˆ λ t,s ˆ λ , H t H ) s ∈ R is aone-parameter family. As in Example 2.13, it follows that the deformation class[ ˆ X t ˆ X , ˆ λ t,s ˆ λ , H t H ]is independent of t ≫ s ∈ R . Proposition 2.29.
The deformation class of ( ˆ X t ˆ X , ˆ λ t ˆ λ , H t H ) only dependson the deformation classes of ( ˆ X , ˆ λ , H ) and ( ˆ X , ˆ λ , H ) .Proof. The proof of Proposition 2.14 also works in the relative case as long as t > (cid:3)
Corollary 2.30.
There is a well-defined gluing operation on deformation classes of exactrelative symplectic cobordisms given by (2.11) [ ˆ X , ˆ λ , H ] X , ˆ λ , H ] = [ ˆ X t ˆ X , ˆ λ t,s ˆ λ , H t H ] for any t ≫ and s ∈ R . Proposition 2.31.
The gluing operation (2.11) is associative.Proof.
Let ( ˆ X i,i +1 , ˆ λ i,i +1 , H i,i +1 ) be a relative cobordism from ( Y i , ξ i , V i ) to ( Y i +1 , ξ i +1 , V i +1 ), i ∈ { , , } , and fix t , t ≫
0. Note that (cid:0) ( ˆ X t ˆ X ) t ˆ X , ( H t H ) t H (cid:1) and (cid:0) ˆ X t ( ˆ X t ˆ X ) , H t ( H t H ) (cid:1) can be canonically identified as pairs of smoothmanifolds. Hence, it suffices to show that there exist s , s ∈ R such that(ˆ λ t ,s ˆ λ ) t ,s ˆ λ = ˆ λ t ,s (ˆ λ t ,s ˆ λ ) . One can easily see from Definition 2.7 that taking s = 0 and s = − t works. (cid:3) Homotopy classes of asymptotically cylindrical maps.Definition 2.32.
Suppose that ( ˆ X, ˆ λ ) is an exact symplectic cobordism from ( Y + , λ + ) to( Y − , λ − ). Given a closed surface Σ and finite subsets p + , p − ⊂ Σ, a smooth map u : Σ − ( p + ⊔ p − ) → ˆ X is said to be asymptotically cylindrical if it converges exponentially near eachpuncture z ∈ p + ⊔ p − to a trivial cylinder over a Reeb orbit.More precisely, given any choice of translation invariant metric on R × Y ± , we require thatthere exists a choice of cylindrical coordinates near each z ∈ p ± such that u takes the form u ( s, t ) = exp ( T s,γ z ( t )) h ( s, t )for | s | large, where γ z is a Reeb orbit of period T and h ( s, t ) is a vector field which decays tozero with all its derivatives as | s | → ∞ . Remark . There is also a notion of an asymptotically cylindrical submanifold which willnot be needed in this paper.
Definition 2.34 (cf. [Par19, Sec. 1.2(I)]) . Let ( ˆ X, ˆ λ ) be an exact symplectic cobordism from( Y + , λ + ) to ( Y − , λ − ) and let Γ ± be a finite set of Reeb orbits in ( Y ± , λ ± ). OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 17
By truncating the ends of ˆ X , we obtain a compact submanifold X ⊂ ˆ X with boundary ∂X = Y + ⊔ Y − . We define the set of homotopy classes π ( ˆ X, Γ + ⊔ Γ − ) by(2.12) π ( ˆ X, Γ + ⊔ Γ − ) := [( S, ∂S ) , ( X , Γ + ⊔ Γ − )] / Diff (
S, ∂S ) , where S is a compact connected oriented surface of genus 0 equipped with a homeomorphism ∂S → Γ + ⊔ Γ − , and Diff( S, ∂S ) is the group of diffeomorphisms of S which fix ∂S pointwise. Remark . The right-hand side of (2.12) is independent of the choice of truncation X up tocanonical bijection. In the case where ( ˆ X, ˆ λ ) = ( R × Y, e s λ ) is the symplectization of a contactmanifold ( Y, λ ), we can take X = { } × Y and (2.12) becomes identical to [Par19, (1.2)].For any choice of truncation X ⊂ ˆ X , there is a canonical retraction π : ˆ X → X inducedby the Liouville flow. If u : Σ − ( p + ⊔ p − ) → ˆ X is an asymptotically cylindrical map, then thecomposition π ◦ u can be extended to a map(2.13) ¯ u : (Σ , ∂ Σ) → ( X , Γ + ⊔ Γ − ) , where Σ is a compactification of Σ − ( p + ⊔ p − ) obtained by adding one boundary circle foreach puncture. The homotopy class [ u ] ∈ π ( ˆ X, Γ + ⊔ Γ − ) of u is defined to be the equivalenceclass of (2.13). Definition 2.36.
Let ( ˆ X, ˆ λ, L ) be an exact Lagrangian cobordism from ( Y + , λ + , Λ + ) to( Y − , λ − , Λ − ) (see Definition 2.21).Given a surface with boundary Σ and finite subsets p + , p − ⊂ int(Σ) and c + , c − ∈ ∂ Σ, asmooth map u : Σ − ( p ± ∪ c ± ) is said to be cylindrical if it converges asymptotically near eachinterior puncture to a trivial cylinder over a Reeb orbit, and it converges exponentially neareach boundary puncture to a trivial strip over a Reeb chord.Let Γ ± be a finite set of Reeb orbits in ( Y ± , λ ± ) and let Γ Λ ± be a finite ordered set of Reebchords of Λ ± ⊂ ( Y ± , λ ± ). We let p ± be a finite set equipped with bijections γ ± : p ± → Γ ± and we let c = c + ⊔ c − be a finite ordered set equipped with order-preserving bijections a ± : c ± → Γ Λ ± . Then we let π ( ˆ X ; Γ Λ + , Γ Λ − , Γ + , Γ − )be the set of equivalence classes of maps from Σ − ( p ± ∪ c ± ) to ˆ X which are asymptotic to γ ± p at p ∈ p ± (resp. a ± c at c ∈ c ± ), where two such maps u, v are equivalent if there exists acompactly supported diffeomorphism φ of Σ − ( p ± ∪ c ± ) such that u and v ◦ φ are homotopic(through cylindrical maps).3. Intersection theory for punctured holomorphic curves
Normal dynamics and adapted contact forms.
We begin with some definitionswhich are used throughout this paper.
Definition 3.1. A contact pair is a datum ( Y, ξ, V ) consisting of a closed co-oriented contactmanifold (
Y, ξ ) and a (possibly empty) codimension 2 contact submanifold (
V, ξ | V ) ⊂ ( Y, ξ ).We require in addition that the normal contact distribution ξ | V /T V is trivial. A choice of(homotopy class of) trivialization τ is called a framing .It will be convenient to allow the case where Y = V = ∅ and ξ ∅ is understood as the uniquecontact structure on the empty set. Definition 3.2.
Given a contact pair (
Y, ξ, V ) with V non-empty, let R ( Y, ξ, V ) be the set oftriples r = ( α V , τ, r ) for r ≥
0, where: • α V ∈ Ω ( V ) is a non-degenerate contact form for ξ | V ; • τ is a trivialization of ξ | ⊥ V ; • if r >
0, we have (1 /r ) Z ∩ S ( α V ) = ∅ , where S ( α V ) is the action spectrum of α V .We let R + ( Y, ξ, V ) ⊂ R ( Y, ξ, V ) be the subset of those triples ( α V , τ, r ) with r > V = ∅ (with Y possibly also empty), we define R ( Y, ξ, ∅ ) = { ( α ∅ , τ ∅ , } , where α ∅ , τ ∅ are understood as a contact form and normal trivialization on the empty set. We also define R + ( Y, ξ, ∅ ) = R ( Y, ξ, ∅ ). Definition 3.3.
Given a contact pair (
Y, ξ, V ) and r = ( α V , τ, r ) ∈ R ( Y, ξ, V ), we say that acontact form ker λ = ξ is adapted to r if: • λ is non-degenerate; • λ | V = α V ; • V is a strong contact submanifold of ( Y, λ ); • If r = 0, then CZ τN ( γ ) = 0 for all Reeb orbits γ ⊂ V . If r >
0, then CZ τN ( γ ) = 1+2 ⌊ rT γ ⌋ for all Reeb orbits γ ⊂ V , where T γ is the period of γ .In case V = ∅ , any contact form λ is considered to be adapted to the unique element( α ∅ , τ ∅ , ∈ R ( ∅ , ξ ∅ , ∅ ).Given a contactomorphism f : ( Y, ξ, V ) → ( Y ′ , ξ ′ , V ′ ), we write f ∗ r = ( f ∗ α V , f ∗ τ, r ) ∈ R ( Y ′ , ξ ′ , V ′ ). If φ t : V → Y is an isotopy of contact embeddings where φ is the tautologicalembedding V id −→ V ⊂ Y and φ ( V ) = V ′ , then φ t extends to a family of contactomorphisms f t . We then write ( φ ) ∗ r := ( f ) ∗ r (this is independent of the choice of extension).We say that λ is hyperbolic near V if it is adapted to some datum ( α V , τ, λ is positive elliptic near V if it is adapted to some datum ( α V , τ, r ) for r >
0. It will beconvenient to refer to r ≥ rotation parameter . Remark . Our insistence on allowing the case where Y = ∅ in the above definitions isexplained by the need to treat Liouville manifolds as special cases of Liouville cobordisms inthe arguments of Section 5.We will prove in Proposition 3.8 that adapted contact forms always exist, i.e. for any contactpair ( Y, ξ, V ) and r = ( α V , τ, r ) ∈ R ( Y, ξ, V ) there exists a contact form adapted to r . Thefirst step is to construct a suitable local model. Construction 3.5.
Let (
V, α V ) be a contact manifold and let φ : D → R > be a smoothpositive function which has a nondegenerate critical point at 0 and satisfies φ (0) = 1. Wedefine α φV = φ − ( α V + λ D )where λ D = ( x dy − y dx ) is the usual Liouville form on D . This is a contact form on V × D whose restriction to V = V × { } coincides with α V . Its Reeb vector field is given by R φ = ( φ − Z D φ ) R V + X φ where Z D = ( x∂ x + y∂ y ) is the Liouville vector field of λ D and X φ = − ( ∂ y φ ) ∂ x + ( ∂ x φ ) ∂ y is the Hamiltonian vector field of φ with respect to the symplectic form ω D = dλ D . Ourassumptions on φ imply that R φ = R V on V × { } , so that ( V, α V ) is a strong contactsubmanifold of ( V × D , α φV ). We will let S φ = (cid:18) ∂ xx φ (0) ∂ yx φ (0) ∂ xy φ (0) ∂ yy φ (0) (cid:19) ∈ R × denote the Hessian of φ at the origin. Since S φ is symmetric and nondegenerate, its eigenvaluesare real and nonzero, so its signature Sign( S φ ) is one of 0 , ±
2. We will say that φ is hyperbolic if Sign( S φ ) = 0, positive elliptic if Sign( S φ ) = 2 and negative elliptic if Sign( S φ ) = −
2. In the
OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 19 elliptic case, we define c φ = p det( S φ ) / (2 π ); this is a positive real number since det( S φ ) > ξ φ ) | V = ( ξ φ ) | ⊤ V ⊕ ( ξ φ ) | ⊥ V mentionned in section 3.2 is given by( ξ φ ) | ⊤ V = ξ V and ( ξ φ ) | ⊥ V = T D . We will let τ φ denote the trivialization of ( ξ φ ) | ⊥ V by { ∂ x , ∂ y } .Say that α φV is non-degenerate on V if every Reeb orbit of α V is non-degenerate when viewedas a Reeb orbit of α φV . Proposition 3.6.
Suppose that α V is non-degenerate. (1) If φ is hyperbolic, then α φV is non-degenerate on V . (2) If φ is elliptic, then α φV is non-degenerate on V if and only if (1 /c φ ) Z ∩ S ( α V ) = ∅ , where S ( α V ) denotes the action spectrum of α V .Proof. Let γ be a Reeb orbit of period T contained in V . Recall that γ is non-degenerate ifand only if its asymptotic operator is non-degenerate (i.e. has trivial kernel; see [Wen16, Sec.3.3, Ex. 3.23]). Choose a trivialization τ and an almost complex structure J on ( ξ φ ) | γ whichpreserve the splitting ( ξ φ ) | γ = ( ξ V ) | γ ⊕ T D and coincide with τ φ and J respectively on T D ,where J denotes the standard almost complex structure on R = T D . The asymptoticoperator A γ is compatible with this splitting and can therefore be written as A γ = A ⊤ γ ⊕ A ⊥ γ .The tangential part A ⊤ γ is non-degenerate since it coincides with the asymptotic operator of γ as a Reeb orbit in V . The normal part A ⊥ γ is given explicitly by A ⊥ γ = − J ∂ t − T S φ (this follows from a short computation using the formula for the Reeb vector field R φ given inConstruction 3.5). Define a path Ψ of symplectic matrices by Ψ( t ) = exp( tT J S φ ). Then A ⊥ γ is non-degenerate if and only if Ψ(1) doesn’t have 1 as an eigenvalue.If φ is hyperbolic, then the eigenvalues of Ψ(1) are exp( ± T p | det( S φ ) | ). Since det( S φ ) = 0,this proves that γ is non-degenerate.If φ is elliptic, then the eigenvalues of Ψ(1) are exp( ± iT p det( S φ )). Hence, γ is non-degenerate if and only if T p det( S φ ) is not an integer multiple of 2 π , i.e. T / ∈ (1 /c φ ) Z . Itfollows that λ φ is non-degenerate if and only if (1 /c φ ) Z ∩ S ( α V ) = ∅ , as claimed. (cid:3) The important feature of Construction 3.5 is that the normal Conley-Zehnder indices of theReeb orbits in V can be computed explicitly. Proposition 3.7.
Assume α φV is non-degenerate on V . If φ is hyperbolic, then CZ τ φ N ( γ ) = 0 for every Reeb orbit γ contained in V . If φ is elliptic, then CZ τ φ N ( γ ) = ± (1 + 2 ⌊ c φ T γ ⌋ ) for every Reeb orbit γ contained in V , where T γ > denotes the period of γ and the sign is + or − depending on whether φ is positive elliptic or negative elliptic.Proof. We have CZ τ φ N ( γ ) = CZ(Ψ), where Ψ( t ) = exp( tT J S φ ) is the path of symplecticmatrices defined in the proof of Proposition 3.6 (see [Wen16, Sec. 3.4]). Proposition 41 of[Gut12] implies that CZ(Ψ) = 0 if Sign( S φ ) = 0 and thatCZ(Ψ) = ± (1 + 2 ⌊ c φ T ⌋ )if Sign( S φ ) = ± (cid:3) Proposition 3.8.
Fix a contact pair ( Y, ξ, V ) and an element r = ( α V , τ, r ) ∈ R ( Y, ξ, V ) . Let ( V, τ ) be a framed codimension contact submanifold of Y , and let α V be a non-degeneratecontact form on ( V, ξ | V ) . Then there exists a contact form λ on ( Y, ξ ) which is adapted to r .Proof. Let φ be as in Construction 3.5. The standard neighborhood theorem for contactsubmanifolds (see [Gei08, Thm. 2.5.15]) implies that the inclusion map V → Y extends to acontact embedding ι : ( V × D ǫ , ker( α φV )) → ( Y, ξ ) such that ι ∗ τ is homotopic to τ φ . Hencethere exists a contact form λ for ξ such that ι ∗ λ = α φV near V . It remains to show that, if φ ischosen appropriately, then we can modify λ away from V so that it becomes non-degenerate.By [ABW10, Thm. 13], it suffices to choose a φ satisfying the following two conditions: • α φV is non-degenerate on V ; • all the Reeb orbits of α φV in V × D ǫ are contained in V .In case r = 0, we can take φ = 1 + x − y . Indeed, Proposition 3.6 implies that α φV isnon-degenerate on V . Moreover, if γ : S → V × D ǫ is a Reeb orbit of α φV , then its projectionto D ǫ is an orbit of X φ = − y∂ x + 2 x∂ y since R φ = ( φ − Z D φ ) R V + X φ . The only such orbitis the constant one at the origin, so γ is contained in V .In case r >
0, we can take φ = 1 + πr ( x + y ). Proposition 3.6 implies that α φV is non-degenerate on V since c φ = p (2 πr ) / (2 π ) = r . Since R φ = R V + X φ , every Reeb orbit γ of α φV is of the form γ = ( γ V , γ φ ) where γ V is an orbit of R V and γ φ is an orbit of X φ with thesame period T >
0. From the formula X φ = − πry∂ x + 2 πrx∂ y , we see that if γ φ were notconstant, we would have T ∈ (1 /r ) Z , contradicting our assumption on r (see Definition 3.2).Thus γ is contained in V . (cid:3) Definition of the intersection number.
We will make use in this paper of an intersec-tion theory for asymptotically cylindrical maps and submanifolds. The four-dimensional theorywas constructed by Siefring [Sie11] and assigns an integer to a pair of asymptotically cylindricalmaps in a 4-dimensional symplectic cobordism (see also the survey by Wendl [Wen17]). Thehigher-dimensional theory, also due to Siefring, assigns an integer to the pairing of a codimen-sion 2 (asymptotically) cylindrical hypersurface with a (asymptotically) cylindrical map. Adetailed overview can be found in [MS19].Consider a contact manifold ( Y n − , ξ = ker λ ) and a strong contact submanifold ( V n − , λ | V ).Observe that the contact distribution splits naturally along V as ξ | V = ξ | ⊤ V ⊕ ξ | ⊥ V , where ξ | ⊤ V = ξ | V ∩ T V and ξ | ⊥ V is the symplectic orthogonal complement of ξ | ⊤ V ⊂ ξ | V withrespect to dλ .Let γ : S → V be a Reeb orbit and let J be a dλ -compatible almost complex structure on ξ . If J respects the above splitting, then so does the associated asymptotic operator, whichwe can therefore write as A γ = A ⊤ γ ⊕ A ⊥ γ . If we choose a trivialization τ of ξ | γ which is alsocompatible with the splitting, we can define CZ τT ( γ ) := CZ τ ( A ⊤ γ ) and CZ τN ( γ ) := CZ τ ( A ⊥ γ ).We call these respectively the tangential and normal Conley-Zehnder indices of γ with respectto τ .We define the integers α τ ; − N ( γ ) := ⌊ CZ τN ( γ ) / ⌋ , α τ ;+ N ( γ ) := ⌈ CZ τN ( γ ) / ⌉ . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 21
Let p N ( γ ) = α τ ;+ N ( γ ) − α τ ; − N ( γ ) ∈ { , } be the (normal) parity of γ and observe that it isindependent of the choice of trivialization. We have(3.1) CZ τN ( γ ) = 2 α τ ; − N ( γ ) + p N ( γ ) = 2 α τ ;+ N ( γ ) − p N ( γ )from which it also follows that p N ( γ ) ≡ CZ τN ( γ ) mod 2.Let us now consider an exact symplectic cobordism ( ˆ X, ˆ λ ) from ( Y + , λ + ) to ( Y − , λ − ). Let( V ± , λ ± | V ) ⊂ ( Y ± , λ ± ) be strong contact submanifolds and let H ⊂ ˆ X be a codimension 2submanifold with cylindrical ends V + ⊔ V − .We let τ denote a choice of trivialization of ( ξ ± ) ⊥ along every Reeb orbit in V ± . Werequire that the trivialization along a multiply covered orbit be pulled back from the chosentrivialization along the underlying simple orbit. Let u : Σ − ( p + u ⊔ p − u ) → ˆ X be a map whichis positively/negatively asymptotic at z ∈ p ± u to the Reeb orbit γ z . Now set u • τ H := u τ · H, where u τ is a perturbation of u which is transverse to H and constant with respect to τ atinfinity, and ( − · − ) is the usual algebraic intersection number for transversely intersectingsmooth maps. Definition 3.9.
The generalized intersection number u ∗ H ∈ Z of u and H is defined by(3.2) u ∗ H = u • τ H + X z ∈ p + u α τ ; − N ( γ z ) − X z ∈ p − u α τ ;+ N ( γ z ) Proposition 3.10.
The intersection number u ∗ H only depends on the equivalence classes of u in π ( ˆ X, Γ + ⊔ Γ − ) and H in Ω n − ( ˆ X, V + ⊔ V − ) .Proof. The intersection number u τ · H is clearly invariant under compactly supported isotopiesof H .Given a truncation X ⊂ ˆ X , we can proceed as in Section 2.2 to associate to u τ a map¯ u τ : Σ → X . Let H = H ∩ X . If we choose X sufficiently large (so that H is cylindrical inits complement), then H will be a submanifold with boundary ∂H = H ∩ ∂X = V + ⊔ V − .Note that ¯ u τ · H only depends on [ u ] ∈ π ( ˆ X, Γ + ⊔ Γ − ). Moreover, we have ¯ u τ · H = u τ · H ;indeed, if X is sufficiently large, then the intersections of ¯ u τ with H are exactly the same asthose of u τ with H . (cid:3) Positivity of intersection.
We now discuss positivity of intersection for the Siefringintersection number. Given a contact manifold (
Y, ξ = ker λ ) and an almost-complex structure J on ξ , we adopt the usual convention of letting ˆ J denote the induced almost complex structureon the symplectization. An almost complex structure on a cobordism ( ˆ X, ˆ λ ) between twocontact manifolds ( Y ± , λ ± ) is called cylindrical if it agrees at infinity with ˆ J ± for some choiceof dλ ± -compatible almost complex structures J ± on ker( λ ± ). A cylindrical almost complexstructure which is compatible with d ˆ λ is called adapted . Proposition 3.11 (see Cor. 2.3 and Thm. 2.5 in [MS19]) . Let ( ˆ X, ˆ λ ) be an exact symplecticcobordism from ( Y + , λ + ) to ( Y − , λ − ) . Let u and H denote an asymptotically cylindrical mapand a cylindrical submanifold of codimension in ˆ X respectively.Suppose that u and H are ˆ J -holomorphic for some adapted almost complex structure ˆ J on ˆ X . If the image of u is not contained in H , then Im( u ) ∩ H is a finite set and u ∗ H ≥ u · H. (Note that by ordinary positivity of intersection for two pseudo-holomorphic submanifolds, thisimplies that u ∗ H ≥ and that Im( u ) and H are disjoint if u ∗ H = 0 .) When the image of u is contained in H , positivity of intersection does not hold. Thefollowing computation, which will be useful to us later, is one example of this. The notation b γ refers to the trivial cylinder R × S → ˆ Y over the Reeb orbit γ ; similarly, ˆ V = R × V ⊂ ˆ Y is the cylinder over the strong contact submanifold V . Corollary 3.12.
Let γ be a Reeb orbit in Y . If γ is contained in V , then b γ ∗ ˆ V = − p N ( γ ) . Proof.
By definition, b γ ∗ ˆ V = b γ τ · ˆ V + α τ ; − N ( γ ) − α τ ;+ N ( γ ) . We can choose the perturbation b γ τ so that its image is disjoint from ˆ V . The result followssince α τ ;+ N ( γ ) − α τ ; − N ( γ ) = p N ( γ ) by definition. (cid:3) Remark . If γ is disjoint from V , then b γ ∗ ˆ V = 0.Corollary 3.12 shows that positivity of intersection fails for curves contained in ˆ V . However,we still have a lower bound on the intersection number u ∗ ˆ V when u = b γ is a trivial cylinder(namely, b γ ∗ ˆ V ≥ − V ⊂ ( Y, ξ ) is acodimension 2 contact submanifold with trivial normal bundle, then it is always possible tochoose a contact form λ for ξ so that V is a strong contact submanifold of ( Y, λ ) and thatthe intersection number u ∗ ˆ V is bounded below for all asymptotically cylindrical curves u contained in ˆ V . More precisely: • if the Reeb vector field R λ has “hyperbolic normal dynamics” near V , then u ∗ ˆ V = 0; • if R λ has “elliptic normal dynamics” near V , then u ∗ ˆ V ≥ − p − u , where p − u denotes thenumber of negative punctures of u .We will also give an analoguous result for cylindrical submanifolds of symplectic cobordisms H ⊂ ( ˆ X, ω ) with trivial normal bundle.
Proposition 3.14.
Fix a contact pair ( Y, ξ, V ) and a datum r ∈ R ( Y, ξ, V ) . Consider acontact form λ on ( Y, ξ ) which is adapted to r , and an almost-complex structure J on ξ whichis compatible with dλ and which preserves ξ | V . Suppose that u is a ˆ J -holomorphic curve whoseimage is entirely contained in ˆ V . (1) If λ is hyperbolic near V , then u ∗ ˆ V = 0 . (2) If λ is positive elliptic near V , then u ∗ ˆ V ≥ − p u , where p u denotes the number ofpunctures (positive and negative) of u .Proof. In the hyperbolic case, this follows from Proposition 3.15.In the positive elliptic case, we have by definition that α τ ; − N ( γ z ) = ⌊ CZ τN ( γ z ) / ⌋ = ⌊ rT z ⌋ for z ∈ p + u and α τ ;+ N ( γ z ) = ⌈ CZ τN ( γ z ) / ⌉ = 1 + ⌊ rT z ⌋ for z ∈ p − u . Using the trivial bounds x − < ⌊ x ⌋ ≤ x and the fact that u τ · ˆ V = 0, we obtain u ∗ ˆ V > X z ∈ p + u ( rT z − − X z ∈ p − u (1 + rT z ) ≥ − p u + r X z ∈ p + u T z − X z ∈ p − u T z . The fact that u is ˆ J -holomorphic implies that P z ∈ p + u T z − P z ∈ p − u T z is nonnegative (see [Wen16,p. 60]). Thus u ∗ ˆ V ≥ − p u as desired. (cid:3) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 23
We will need an analogue of Proposition 3.14 for cobordisms. Note that if V ⊂ Y is acodimension 2 contact submanifold, then the normal bundle of ˆ V = R × V ⊂ R × Y = ˆ Y canbe identified with the pullback of ξ | ⊥ V under the projection ˆ V → V . Hence, any trivialization τ of ξ | ⊥ V induces a trivialization of the normal bundle of ˆ V , which we will denote by ˆ τ . Proposition 3.15.
Fix contact pairs ( Y ± , ξ ± , V ± ) and elements r ± = ( α ± V , τ ± , r ± ) ∈ R ( Y ± , ξ ± , V ± ) .Let λ ± be contact forms on ( Y ± , ξ ± ) which are adapted to r ± , and let ( ˆ X, ˆ λ, H ) λ + λ − be a strongrelative symplectic cobordism from ( Y + , ξ + , V + ) to ( Y − , ξ − , V − ) . We assume that there existsa global trivialization τ of the normal bundle of H which coincides with ˆ τ ± near ±∞ .Let ˆ J be an adapted almost complex structure on ˆ X such that H is ˆ J -holomorphic. Let u bean asymptotically cylindrical map in ˆ X which is ˆ J -holomorphic and whose image is entirelycontained in H . Following the notation of Proposition 3.14, we have: (1) If λ ± is hyperbolic near V ± , then u ∗ H = 0 . (2) If λ + is positive elliptic near V + , then u ∗ H > − p u + r + P z ∈ p + u T z − r − P z ∈ p − u T z . Inparticular, if u has no negative puncture, then u ∗ H ≥ .Proof. We have u ∗ H = u τ · H + X z ∈ p + u α τ ; − N ( γ z ) − X z ∈ p − u α τ ;+ N ( γ z ) . We can choose the perturbation u τ so that it is disjoint from H , so u τ · H = 0.In the hyperbolic case, we have CZ τN ( γ z ) = 0 for all z ∈ p + u ⊔ p − u , which implies that u ∗ H = 0.In the positive elliptic case, we argue as in Proposition 3.14 to find that u ∗ H > X z ∈ p + u ( r + T z − − X z ∈ p − u (1 + r − T z ) ≥ − p u + r + X z ∈ p + u T z − r − X z ∈ p − u T z . (cid:3) The intersection number for buildings.
The definition of contact homology involvesnot just pseudo-holomorphic curves but pseudo-holomorphic buildings . In [Par19], Pardondefines four categories of labelled trees S ∗ ( ∗ = I , II , III , IV) whose objects represent the com-binatorial types of the pseudo-holomorphic buildings needed to define contact homology. Themorphisms in these categories correspond to gluing a subset of the curves appearing in a givenbuilding.In this section, we define an intersection number for buildings of asymptotically cylindricalmaps and buildings of asymptotically cylindrical codimension 2 submanifolds. Since the dif-ferences between S I , S II , S III and S IV don’t matter for this purpose, we start by defining acategory b S of labeled trees which only keeps track of the information needed for intersectiontheory (in particular, there are obvious “forgetful” functors S ∗ → b S ).The category b S = b S ( { ˆ X ij } ij ) depends on the following data:(i) An integer m ≥ m + 1 co-oriented contact manifolds ( Y i , ξ i ), eachequipped with a choice of contact form λ i (0 ≤ i ≤ m ).(ii) For each pair of integers 0 ≤ i ≤ j ≤ m , an exact symplectic cobordism ( ˆ X ij , ˆ λ ij )with positive end ( Y i , ξ i ) and negative end ( Y j , ξ j ). We require that ˆ X ii = SY i bethe symplectization of Y i and that ˆ X ik = ˆ X ij X jk for i ≤ j ≤ k (this makes sense inlight of Remark 2.8). An object T ∈ b S is a finite directed forest (i.e. a finite collection of finite directed trees).We require that every vertex has a unique incoming edge. Edges which are adjacent to onlyone vertex are allowed; we will refer to them as input or output edges depending on whetherthey are missing a source or a sink. The other edges will be called interior edges. We also havethe following decorations: • For each edge e ∈ E ( T ), a symbol ∗ ( e ) ∈ { , . . . , m } such that ∗ ( e ) = 0 for input edgesand ∗ ( e ) = m for output edges, together with a Reeb orbit γ e in ( Y ∗ ( e ) , λ ∗ ( e ) ). • For each vertex v ∈ V ( T ), a pair ∗ ( v ) = ( ∗ + ( v ) , ∗ − ( v )) ∈ { , . . . , m } such that ∗ + ( v ) ≤ ∗ − ( v ) and a homotopy class β v ∈ π ( ˆ X ∗ ( v ) , γ e + ( v ) ⊔ { γ e − } e − ∈ E − ( v ) ), where e + ( v ) denotes the unique incoming edge of v and E − ( v ) denotes the set of its outgoingedges. We require that ∗ ( e + ( v )) = ∗ + ( v ) and ∗ ( e − ) = ∗ − ( v ) for every e − ∈ E − ( v ).We will let Γ + T and Γ − T denote the collections of Reeb orbits associated to the input and outputedges of an object T ∈ b S . In the case where T is a tree, the unique element of Γ + T will bedenoted by γ + T .A morphism π : T → T ′ consists of a contraction of the underlying forests (meaning that T ′ is identified with the forest obtained by contracting a certain subset of the interior edges of T ) subject to the following conditions: • For every non-contracted edge e ∈ E ( T ), we require that ∗ ( π ( e )) = ∗ ( e ) and γ π ( e ) = γ e . • For every vertex v ∈ V ( T ), we have ∗ + ( π ( v )) ≤ ∗ + ( v ) and ∗ − ( π ( v )) ≥ ∗ − ( v ). • For every vertex v ′ ∈ V ( T ′ ), we require that β v ′ = π ( v )= v ′ β v .Note that for any morphism T → T ′ , we have Γ + T = Γ + T ′ and Γ − T = Γ − T ′ . Remark . For every T ∈ b S , we get a morphism T → T max by contacting all of the interioredges of T . Each component of T max is a tree with a unique vertex. In the case where T is connected, we will write β T = v β v ∈ π ( ˆ X m , γ + T ⊔ Γ − T ) for the homotopy class labelingthe unique vertex of T max . Note that for every morphism T → T ′ , we have T max = T ′ max . Inparticular, if T and T ′ are trees, then β T = β T ′ . Definition 3.17.
Let T ∈ b S and let { T i } i denote its connected components. The intersectionnumber T ∗ H of T with a codimension 2 cylindrical submanifold H ⊂ ˆ X m is defined to be T ∗ H = X i β T i ∗ H. By Proposition 3.10, this intersection number only depends on the class of H in Ω n − ( ˆ X m , V ⊔ V m ). By Remark 3.16, it is “invariant under gluing”: Proposition 3.18.
Let
T, T ′ ∈ b S . If there exists a morphism T → T ′ , then T ∗ H = T ′ ∗ H . Suppose now that m = 0, so that objects T ∈ b S represent buildings of curves in thesymplectization ˆ Y of a single contact manifold ( Y, λ ) := ( Y , λ ), and that H = ˆ V := R × V is the trivial cylinder over some strong contact submanifold V ⊂ Y of codimension 2. In thatcase, the intersection number T ∗ H can be expressed more explicitly as follows. Proposition 3.19.
For any T ∈ b S , we have (3.3) T ∗ ˆ V = X v ∈ V ( T ) β v ∗ ˆ V − X e ∈ E int ( T ) b γ e ∗ ˆ V .
Proof.
The proof will be by induction on the number of interior edges. If this number is zero,then (3.3) is true by definition. Otherwise, pick an edge e ∈ E int ( T ) and contract it to obtain OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 25 a morphism π : T → T ′ where T ′ has one less interior edge than T . We can assume inductivelythat T ′ satisfies (3.3). Since T ∗ ˆ V = T ′ ∗ ˆ V , it suffices to show that β v + ∗ ˆ V + β v − ∗ ˆ V − b γ e ∗ ˆ V = β v ′ ∗ ˆ V , where v + and v − are the source and sink of e respectively and v ′ = π ( v + ) = π ( v − ).To do this, start by picking curves u ± : Σ ± → ˆ Y representing the classes β v ± . Fix a choiceof cylindrical coordinates near the positive puncture of u − and near the negative puncture of u + corresponding to e . We can assume that u ± is cylindrical at infinity, so that there exists aconstant C > u ± ( s, t ) = ( T e s, γ e ( t ))for ∓ s ≥ C . Now let Σ + = Σ + \ (( −∞ , − C ) × S ) , Σ − = Σ − \ ((3 C, ∞ ) × S ) , and let Σ = Σ + − be obtained by indentifying [ − C, − C ] × S ⊂ Σ + with [ C, C ] × S ⊂ Σ − via translation by 4 C . The curve u + u − : Σ → ˆ Y which is given by τ CT e ◦ u + on Σ + and τ − CT e ◦ u − on Σ − (where τ s : ˆ Y → ˆ Y denotes translation by s ) then represents the homotopyclass β v + β v − = β v ′ .Choose a trivialization τ of ξ | ⊥ V along the relevant Reeb orbits and use it to produce per-turbations u τ ± , ( u + u − ) τ as in section 3.2. We can do this in such a way that ( u + u − ) τ isobtained by gluing u τ + and u τ − . Then( u + u − ) τ · ˆ V = u τ + · ˆ V + u τ − · ˆ V , so ( u + u − ) ∗ ˆ V = u τ + · ˆ V + u τ − · ˆ V + α τ ; − N ( γ + ) − X z ∈ p − u + u − α τ ;+ N ( γ z )= u τ + · ˆ V + u τ − · ˆ V + α τ ; − N ( γ + ) + α τ ;+ N ( γ e ) − X z ∈ p − u + α τ ;+ N ( γ z ) − X z ∈ p − u − α τ ;+ N ( γ z )= u + ∗ ˆ V + u − ∗ ˆ V + α τ ;+ N ( γ e ) − α τ ; − N ( γ e )= u + ∗ ˆ V + u − ∗ ˆ V + p ( γ e ) . We have p ( γ e ) = − b γ e ∗ ˆ V by Corollary 3.12, so this implies that β v ′ ∗ ˆ V = ( u + u − ) ∗ ˆ V = u + ∗ ˆ V + u − ∗ ˆ V − b γ e ∗ ˆ V = β v + ∗ ˆ V + β v − ∗ ˆ V − b γ e ∗ ˆ V , as desired. (cid:3)
Definition 3.20.
Given T ∈ b S , we say that T is representable by a holomorphic building if there exists a dλ -compatible almost complex structure J on ξ such that, for every vertex v ∈ V ( T ), β v ∈ π ( ˆ Y , γ e + ( v ) ⊔ { γ e − } e − ∈ E − ( v ) ) admits a ˆ J -holomorphic representative. We saythat T is representable by a ˆ J -holomorphic building if we wish to specify ˆ J . Corollary 3.21.
Let T ∈ b S . Suppose that there exists a morphism T ′ → T and a dλ -compatible almost complex structure J on ξ such that T ′ is representable by a ˆ J -holomorphicbuilding. Suppose also that ˆ V is ˆ J -holomorphic. (1) If λ is hyperbolic near V , then T ∗ ˆ V ≥ . (2) If λ is positive elliptic near V , then T ∗ ˆ V ≥ − Γ − ( T, V ) , where Γ − ( T, V ) denotes thenumber of output edges e of T such that γ e is contained in V .Proof. By Proposition 3.18, T ∗ ˆ V = T ′ ∗ ˆ V . By Proposition 3.19, T ′ ∗ ˆ V = X v ∈ V ( T ′ ) β v ∗ ˆ V − X e ∈ E int ( T ′ ) b γ e ∗ ˆ V .
In the hyperbolic case, we have β v ∗ ˆ V ≥ b γ e ∗ ˆ V = 0. Thus T ′ ∗ η ≥ β v ∗ ˆ V ≥
0, unless the holomorphic representative of β v isentirely contained in ˆ V , in which case Proposition 3.14 only tells us that β v ∗ ˆ V ≥ − E − ( v ).However, by Corollary 3.12, we have − E − ( v ) = X e ∈ E − ( v ) b γ e ∗ ˆ V .
Given v ∈ V ( T ′ ), let us denote by Γ − ( v, V ) the number of output edges e ∈ E − ( v ) suchthat γ e ⊂ V . Appealing again to Proposition 3.19, we have: T ′∗ ˆ V = X v ∈ V ( T ′ ) β v ∗ ˆ V − X e ∈ E int ( T ′ ) b γ e ∗ ˆ V = X v ∈ V ( T ′ ) β v ∗ ˆ V − X e ∈ E − ( v ) ˆ γ e ∗ ˆ V + X e ∈ E − ( T ) b γ e ∗ ˆ V = X v ∈ V ( T ′ ) ( β v ∗ ˆ V + Γ − ( v, V )) − Γ − ( T ′ , V ) ≥ − Γ − ( T, V ) , where we have used the fact that T, T ′ have the same exterior edges in the last line. Thiscompletes the proof. (cid:3) More generally, suppose we are given the following data, where m is now allowed to be anynonnegative integer: • For each 0 ≤ i ≤ m , a strong contact submanifold V i ⊂ Y i of codimension 2. • For each 0 ≤ i ≤ j ≤ m , a homotopy class η ij ∈ Ω n − ( ˆ X ij , V i ⊔ V j ). We requirethat η ii := [ ˆ V i ] be the homotopy class of ˆ V i = R × V i and that η ik = η ij η jk for any i ≤ j ≤ k .Let η := η m ∈ Ω n − ( ˆ X m , V ⊔ V m ). Proposition 3.22.
Let T ∈ b S . Then (3.4) T ∗ η = X v ∈ V ( T ) β v ∗ η ∗ ( v ) − X e ∈ E int ( T ) b γ e ∗ ˆ V ∗ ( e ) . Proof.
We will say a vertex v ∈ V ( T ) is a symplectization vertex if ∗ ( v ) = ii for some i and a cobordism vertex otherwise. This induces a partition E int ( T ) = E ss ( T ) ⊔ E sc ( T ) ⊔ E cc ( T ) ofthe set of interior edges according to the types of the vertices they are adjacent to. Similarly,the set of exterior edges admits a partition E ext ( T ) = E s ( T ) ⊔ E c ( T ).We can (and will) assume without loss of generality that E ss ( T ) is empty. Indeed, let T → T ′ be the morphism obtained by contracting all the edges in E ss ( T ). Replacing T with T ′ doesn’t OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 27 change the left-hand side of (3.4) by Proposition 3.18 and doesn’t change the right-hand sideby Proposition 3.19.Let I = V ( T ) ⊔ E c ( T ) ⊔ E cc ( T ) and choose a familty of curves { u i } i ∈ I with the followingproperties: • For each v ∈ V ( T ), u v is a curve in the homotopy class β v which is cylindrical atinfinity. • For each e ∈ E c ( T ) ⊔ E cc ( T ), u e = b γ e is the trivial cylinder over the Reeb orbit γ e .For t > u i ’s to obtain a curve u in X t X t X t · · · t X mm ∼ = X m representing T . We can also choose representatives H ij of η ij so that H ii = ˆ V i and H := H m coincides with H t H t · · · t H mm .As in the proof of Proposition 3.19, we can choose perturbation u τ , { u τi } so that u τ · H = X i ∈ I u τi · H i , where H i = H ∗ ( v ) for i = v ∈ V ( T ) and H i = ˆ V ∗ ( e ) for i = e ∈ E c ( T ) ⊔ E cc ( T ). The difference P i u i ∗ H i − u ∗ H is therefore equal to X e ∈ E sc ( T ) ⊔ E c ( T ) α τ ; − N ( γ e ) − α τ ;+ N ( γ e ) + 2 X e ∈ E cc ( T ) α τ ; − N ( γ e ) − α τ ;+ N ( γ e )= X e ∈ E sc ( T ) ⊔ E c ( T ) b γ e ∗ ˆ V ∗ ( e ) + 2 X e ∈ E cc ( T ) b γ e ∗ ˆ V ∗ ( e ) = X e ∈ E int ( T ) b γ e ∗ ˆ V ∗ ( e ) + X e ∈ E c ( T ) ⊔ E cc ( T ) b γ e ∗ ˆ V ∗ ( e ) Since u i ∗ H i = b γ e ∗ ˆ V ∗ ( e ) for i = e ∈ E c ( T ) ⊔ E cc ( T ), we conclude that u ∗ H = X v ∈ V ( T ) u v ∗ H ∗ ( v ) − X e ∈ E int ( T ) b γ e ∗ ˆ V ∗ ( e ) , which implies (3.4). (cid:3) Definition 3.23.
Given T ∈ b S , we say that T is representable by a holomorphic building iffor every vertex v ∈ V ( T ), there exists an adapted almost complex structure ˆ J v on ˆ X ∗ ( v ) suchthat β v ∈ π ( ˆ X ∗ ( v ) , γ e + ( v ) ⊔ { γ e − } e − ∈ E − ( v ) ) and η ∗ ( v ) ∈ Ω n − ( ˆ X ∗ ( v ) , V ∗ + ( v ) ⊔ V ∗ − ( v ) ) admitˆ J v -holomorphic representatives.For r ≥
0, let us set ˜ δ ( r ) = ( r = 0 , Proposition 3.24.
Let T ∈ b S . Suppose that there exists a morphism T ′ → T where T ′ isrepresentable by a holomorphic building.Suppose that λ i is either hyperbolic or positive elliptic near V i for all ≤ i ≤ m , and let r i ≥ be the rotation parameter (see Definition 3.3).If β v ∗ η ∗ ( v ) ≥ − ˜ δ ( r ∗ − ( v ) ) { E − ( v ) } , then T ∗ η ≥ − δ ( r m )Γ − ( T, V m ) .Proof. By Proposition 3.18, we have T ∗ η = T ′ ∗ η . Given v ∈ V ( T ′ ), let us denote byΓ − ( v, V ∗ − ( v ) ) the number of output edges e ∈ E − ( v ) such that γ e ⊂ V ∗ − ( v ) . Arguing as in the proof of Corollary 3.21, we obtain from Proposition 3.22 that T ′ ∗ ˆ V = X v ∈ V ( T ′ ) β v ∗ η ∗ ( v ) − X e ∈ E int ( T ′ ) b γ e ∗ ˆ V ∗ ( e ) = X v ∈ V ( T ′ ) β v ∗ η ∗ ( v ) − X e ∈ E − ( v ) b γ e ∗ ˆ V ∗ ( e ) + ˜ δ ( r m ) X e ∈ E − ( T ′ ) b γ e ∗ ˆ V ∗ ( e ) = X v ∈ V ( T ′ ) ( β v ∗ ˆ V + ˜ δ ( r ∗ − ( v ) ) − ( v, V ∗ − ( v ) )) − ˜ δ ( r m )Γ − ( T ′ , V ) ≥ − ˜ δ ( r m )Γ − ( T, V ) , where we have used the fact that T, T ′ have the same exterior edges in the last line. Thiscompletes the proof. (cid:3) Open book decompositions.
In this section, we consider normal Reeb dynamics forbindings of open book decompositions. Let us begin by recalling the definition of an openbook decomposition.
Definition 3.25 ([Gir02]) . An open book decomposition ( Y, B, π ) of a closed, oriented n -manifold Y consists of the following data:(i) An oriented, closed, codimension-2 submanifold B ⊂ Y with trivial normal bundle.(ii) A fibration π : Y − B → S which coincides with the angular coordinate in someneighborhood B × { } ⊂ B × D = B × { ( x, y ) | x + y < } .The submanifold B ⊂ Y is called the binding and the fibers of π are called pages .Observe that the data of an open book decomposition induces a natural trivialization of thenormal bundle to the binding.We also recall what it means for an open book decomposition to support a contact structure. Definition 3.26 ([Gir02]) . Given an odd-dimensional manifold Y n − , an open book decom-position ( Y, B, π ) is said to support a contact structure ξ if there exists a contact form ξ = ker α such that the following properties hold:(i) The restriction of α to B is a contact form.(ii) The restriction of dα to any page π − ( θ ) is a symplectic form.(iii) The orientation of B induced by α coincides with the orientation of B as the boundaryof the symplectic manifold ( P θ , dα ), where P θ = π − ( θ ) is any page.Such a contact form is called a Giroux form (and is also said in the literature to be adapted to the open book decomposition).
Remark . Condition (ii) in the above definition is equivalent to the Reeb vector field of α being transverse to the pages.For future convenience, we state the following definition. Definition 3.28.
Let G be the set of contact pairs ( Y, ξ, V ) having the property that ξ issupported by an open book decomposition π : Y − V → S with binding V . Lemma 3.29.
Let ( Y, B, π ) be an open book decomposition supporting the contact structure ξ and let α B be a contact form for ( B, ξ | B ) . Then there exists a Giroux form α with the propertythat α | B = α B . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 29
Proof.
According to the proof of [DGZ14, Prop. 2], there exists a Giroux form α and a tubularneighborhood B × D ǫ of the binding on which π = θ and α = g ( α | B + λ D ), where g : B × D ǫ → R is a positive smooth function and λ D = ( x dy − y dx ).Note that g ( α | B + λ D ) is a Giroux form on ( B × D ǫ , B × { } , π = θ ) if and only if ∂g/∂r < r >
0. In particular, this means that for any positive smooth function ˜ g : B × D ǫ → R suchthat ∂ ˜ g/∂r < r > g = g near B × ∂D ǫ , there exists a unique Giroux form ˜ α on( Y, B, π ) which coincides with α outside of B × D ǫ and satisfies ˜ α = ˜ g ( α | B + λ D ) on B × D ǫ .Since α | B and α B define the same contact structure, we can write α B = (1 + h )( α | B ) forsome smooth function h : B → R . Since any (nonzero) constant multiple of α is also a Girouxform, we can assume without loss of generality that h > σ : [0 , ǫ ] → R be a nonincreasing smooth function such that σ ( r ) = 1 for r near 0 and σ ( r ) = 0 for r near ǫ . Then ˜ g = (1 + σ ( r ) h ) g satisfies the conditions stated in the previous paragraph, and thecorresponding Giroux form ˜ α restricts to α B on the binding. (cid:3) Lemma 3.30.
Let ( Y, B, π ) be an open book decomposition supporting the contact structure ξ ,let α be a Giroux form, and let f : [0 , → R be a smooth positive function such that f (0) = 1 and f ′ ( r ) < for r > . There exists a Giroux form ˜ α which coincides with α away fromthe binding and an embedding φ : B × D ǫ → Y (for some small ǫ > ) with the followingproperties: (1) ˜ α | B = α | B . (2) The projection π ◦ φ is given by ( r, θ ) θ on B × D ǫ − B × { } . (3) φ ∗ ˜ α = f ( r )( ˜ α | B + λ D ) .Proof. According to the proof of [DGZ14, Prop. 2], there exists an embedding φ : B × D ǫ → Y satisfying (ii) and such that φ ∗ α = g ( α | B + λ D ), where g : B × D ǫ → R is a positive smoothfunction with g = 1 on B × { } . The Giroux condition implies that ∂g/∂r < r > h : B × D ǫ → R be a positive smooth function such that h = f near B × { } , h = g near B × ∂D ǫ , and ∂h/∂r < r >
0. Let ˜ α be the unique contact form on Y which coincideswith α outside the image of φ and satisfies φ ∗ ˜ α = h ( α | B + λ D ). Then ˜ α is a Giroux form andsatisfies conditions (1) – (3). (cid:3) Corollary 3.31.
Consider an open book decomposition ( Y, B, π ) which supports a contactstructure ξ and let τ denote the induced trivialization of the normal bundle of B ⊂ Y . Choosean element r = ( α B , τ, r ) ∈ R ( Y, ξ, B ) with r > . Then there exists a Giroux form α which isadapted to r (see Definition 3.3).Proof. Let us fix an auxiliary Giroux form α such that α | B = α B (such a form exists byLemma 3.29). Let κ = πr and define f ( s ) = (1 + κs ) − for s ∈ [0 , f (0) = 1and f ′ ( s ) <
0, it follows from Lemma 3.30 that there exists a Giroux form ˜ α satisfying theconditions in stated in this lemma. One can easily compute, just as in Proposition 3.7, that ˜ α is adapted to r .As we observed in the proof of Proposition 3.8, there exists a neighborhood U of B with thefollowing properties: • ˜ α is nondegenerate on U ; • all the Reeb orbits in U are contained in B .According to [ABW10, Thm. 13], we can obtain a nondegenerate contact form by multiplying˜ α by a smooth function g : Y → R + with g ≡ B (so g ˜ a is still adapted to r ). Moreover,we can assume that g − C -small and hence that g ˜ α is still a Giroux form. (cid:3) The intersection number for cycles.
For future reference, we collect some basic factsabout intersection numbers for cycles in oriented manifolds.
Definition 3.32.
Let M be an oriented, compact manifold of dimension n , possibly withboundary. Let S , S ⊂ M be disjoint submanifolds. Then we can define a pairing − ∗ − : H k ( M, S ; Z ) × H n − k ( M, S ; Z ) → Z , ( A, B ) A ∗ B, where A ∗ B is a signed count of intersections between cycles representing A and B (thesecycles can be assumed to intersect transversally after an arbitrarily small homotopy). It is afolklore result which is beyond the scope of this paper that this count is graded-symmetric andwell-defined.If A, B are (the pushforward of the fundamental class of) oriented manifolds, then A ∗ B coincides with the usual intersection number for submanifolds. By abuse of notation, we willview the intersection pairing as being defined on both cycles and oriented submanifolds. Remark . Let n ( S i ) be a tubular neighborhood of S i . One alternative way to define theintersection number is to use the isomorphisms H k ( M, S ; Z ) = H k ( M − n ( S ) , ∂n ( S ); Z ) = H n − k ( M − n ( S ) , ∂n ( S ) ∪ ∂M ; Z )furnished by excision and Lefschetz duality to evaluate A against B . We prefer to work withthe more geometric definition above, although both perspectives can be shown to be equivalent. Definition 3.34.
Fix a closed manifold Y of dimension m ≥ V ⊂ Y . Suppose that H ( Y ; Z ) = H ( Y ; Z ) = 0.Let γ : S → Y − V be a loop. The linking number of γ with respect to V is denotedlink V ( γ ) and defined as follows: link V ( γ ) := V ∗ C γ , where C γ is a cycle bounding γ . This is well-defined due to our assumption that H ( Y ; Z ) = 0.Suppose now that Λ ⊂ Y − V is a submanifold with π (Λ) = π (Λ) = 0. Let c : [0 , → Y − V be a path with the property that c (0) , c (1) ∈ Λ. Let c : S → Y − V be a loop obtained byconnecting c (1) to c (0) by a path in Λ. The (path) linking number of c with respect to V isdenoted link V ( c ; Λ) and is defined as follows:link V ( c ; Λ) := V ∗ C c , where C c is a cycle bounding c . This is independent of c since π (Λ) = 0 and independent of C c since H ( Y ; Z ) = 0. Remark . Fix an open book decomposition (
Y, B, π ) and let γ : S → Y − B be a loop.Then it is not hard to show that we havelink B ( γ ) := deg( π ◦ γ ) . Similarly, suppose that Λ ⊂ Y is a submanifold which is contained in a page of ( Y, B, π ).Let c : [0 , → Y − B be a path with the property that c (0) , c (1) ∈ Λ. Then the composition π ◦ c : [0 , → S induces a map c : [0 , / { , } → S . We then have:link B ( c ; Λ) := deg c. Lemma 3.36.
Let Y ± be oriented manifolds with Y + = ∅ and let B ± ⊂ Y ± be orientedsubmanifolds. Let W be an oriented, smooth cobordism from Y + to Y − and let H ⊂ W bean oriented sub-cobordism from B + to B − (i.e. H is an embedded submanifold which admitsa collar neighborhood near the boundary of W .) Suppose that H ( Y ± ; Z ) = H ( Y ± ; Z ) = H ( W, Y + ; Z ) = 0 . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 31
Let Σ be a Riemann surface with k +1 boundary components labelled γ + , γ − , . . . , γ − k . Supposethat u : (Σ , ∂ Σ) → ( W, ∂W ) is a smooth map sending γ + into Y + − B + and γ − i into Y − − B − .Then (3.5) link B + ( γ + ) − k X i =1 link B − ( γ − i ) = H ∗ u (Σ) , where we have identified the boundary components of Σ with the restriction of u to thesecomponents.Proof. Choose a 2-chain B − ∈ C ( Y ± ; Z ) with dB − = γ − ∪ · · · ∪ γ − k . Glue B − to u (Σ) alongthe γ − i and call the resulting chain C ∈ C ( W ; Z ). We now have H ∗ C = H ∗ u (Σ) + k X i =1 link B − ( γ − i ) . By the long exact sequence of the triple (
W, Y + , γ + ) and our assumption that H ( W, Y + ; Z ) =0, the natural map H ( Y, γ + ; Z ) → H ( W, γ + ; Z ) is surjective. Let ˜ C ∈ H ( Y + , γ + ; Z ) be alift of C ∈ H ( W, γ + ; Z ). Then H ∗ C = H ∗ ˜ C = B + ∗ ˜ C = link B + ( γ + ) . (cid:3) Lemma 3.37.
We carry over the setup and notation from Lemma 3.36. In addition to thedata considered there, let Λ ± ⊂ Y ± − B ± be an oriented, smooth submanifold and let Λ ⊂ W be an oriented sub-cobordism from Λ + to Λ − which is disjoint from H . We suppose in additionthat π (Λ ± ) = π (Λ ± ) = 0 .Let Σ be a closed, oriented surface of genus zero with s + 1 boundary components labeled γ ∗ , γ , . . . , γ n . For σ ∈ N + , we place σ disjoint marked points on γ ∗ , this partitioning γ ∗ into σ sub-intervals. Let us label these subintervals by the symbols c + , b , c − , b , c − , . . . , b ( σ − σ , c − σ , b σ ,in the order induced by the orientation.Suppose now that u : (Σ , ∂ Σ) → ( W, ∂W ∪ Λ) is a smooth map sending ( c + , ∂c + ) into ( Y + − B + , Λ + ) , sending ( c − i , ∂c − i ) into ( Y − − B − , Λ − ) , sending b i ( i +1) into Λ , and sending the γ − i into Y − − B − .Then (3.6) link B + ( c + ; Λ + ) − σ X i =1 link B − ( c − i ; Λ − ) − s X i =1 link B − ( γ − i ) = u ∗ [Σ] ∗ H, where we have again identified the boundary components of Σ with the restriction of u to thesecomponents. (cid:3) Energy and twisting maps
Standard setups.
Contact homology is defined in [Par19] by counting pseudo-holomorphiccurves in four setups which we now recall.
Setup I.
A datum D for Setup I consists of a triple ( Y, λ, J ), where Y is a closed manifold, λ is a non-degenerate contact form on Y and J is a dλ -compatible almost complex structure on ξ = ker λ . Setup II.
A datum D = ( D + , D − , ˆ X, ˆ λ, ˆ J ) for this Setup II consists of • data D ± = ( Y ± , λ ± , J ± ) as in setup I; • an exact symplectic cobordism ( ˆ X, ˆ λ ) with positive end ( Y + , λ + ) and negative end( Y − , λ − ); • a d ˆ λ -tame almost complex structure ˆ J on ˆ X which agrees with ˆ J ± at infinity. Setup III.
A datum D = ( D + , D − , ( ˆ X , ˆ λ t , ˆ J t ) t ∈ [0 , ) for this setting consists of • data D ± = ( Y ± , λ ± , J ± ) as in setup I; • a family of exact symplectic cobordisms ( ˆ X, ˆ λ t ) t ∈ [0 , with positive end ( Y + , λ + ) andnegative end ( Y − , λ − ); • a d ˆ λ t -tame almost complex structure ˆ J t on ˆ X which agrees with ˆ J ± at infinity.Note that for every t ∈ [0 , D t = t = ( D + , D − , ˆ X, ˆ λ t , ˆ J t ) as in Setup II. Setup IV.
A datum D = ( D , D , ( ˆ X ,t , ˆ λ ,t , ˆ J ,t ) t ∈ [0 , ∞ ) ) for this setting consists of • data D = ( D , D , ˆ X , ˆ λ , ˆ J ) D = ( D , D , ˆ X , ˆ λ , ˆ J )as in setup II, where D i = ( Y i , λ i , J i ), i = 0 , , • a family of exact symplectic cobordisms ( ˆ X ,t , ˆ λ ,t ) t ∈ [0 , ∞ ) with positive end ( Y , λ )and negative end ( Y , λ ), which for t large coincides with the t -gluing of ( ˆ X , ˆ λ )and ( ˆ X , ˆ λ ); • a d ˆ λ ,t -tame almost complex structure ˆ J ,t on ˆ X ,t which agrees with ˆ J , ˆ J atinfinity and is induced by ˆ J , ˆ J for t large.4.2. Twisting maps.
Contact homology is defined by counting pseudo-holomorphic curves inSetups I-IV. In each setting, Pardon defines a category S = S ( D ) depending on some datum D .Each object T ∈ S is a labelled tree representing a certain class of pseudo-holomorphic curves.The moduli space of curves in that class will be denoted by M ( T ); it has a well-defined virtualdimension vdim( T ). The morphisms of S are obtained by contracting edges. This correspondsto gluing pseudo-holomorphic curves. The compactified moduli space M ( T ) is defined by M ( T ) := G T ′ → T M ( T ′ ) / Aut( T ′ /T ) . Theorem 1.1 in [Par19] gives virtual moduli counts M ( T ) vir ∈ Q (which are zero forvdim( T ) = 0) satisfying 0 = X codim( T ′ /T )=1 | Aut( T ′ /T ) | M ( T ′ ) vir (4.1) M ( i T i ) vir = 1 | Aut( { T i } i / T i ) | Y i M ( T i ) vir (4.2)This can be used to define the various maps involved in the definition of contact homology(e.g. the differential d ) and show that they satisfy the expected relations (e.g. d = 0).In order to define our invariants, we will proceed as follows. First, we will use Siefring’sintersection theory to define maps ψ : S → R . Here R could be any Q -algebra, though wewill only use R = Q [ U ] and R = Q . We will then use these maps to define “twisted” modulicounts ψ M ( T ) := M ( T ) · ψ ( T ) ∈ R . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 33
The maps ψ will have the property that ψ ( T ′ ) = ψ ( T ) for every morphism T ′ → T (4.3) ψ ( i T i ) = Y i ψ ( T i )(4.4)which implies that equations 4.1 and 4.2 still hold if M is replaced by ψ M .The properties which must be satisfied by the maps ψ in order to obtain twisted countswhich are suitable for defining our invariants can be conveniently axiomatized in the notion ofa twisting map . We now define precisely this notion in each of the four setups. Setup I.
Fix a datum D for Setup I. Let S = ∅ I denote the full subcategory of S I spanned byobjects T for which the moduli space M ( T ) is nonempty. Definition 4.1.
Let R be a Q -algebra. The set Ψ I ( D ; R ) of R -valued twisting maps consistsof all maps ψ : S = ∅ I ( D ) → R satisfying the following two properties: • for any morphism T ′ → T , ψ ( T ′ ) = ψ ( T ); • for any concatenation { T i } i , ψ ( i T i ) = Q i ψ ( T i ).Fix a twisting map ψ ∈ Ψ I ( D ; R ). Let CC • ( Y, ξ, ψ ) λ := M n ≥ Sym n R (cid:16) M γ ∈P good o γ (cid:17) be the free supercommutative Z / R -algebra generated by the good Reeb orbits.The grading of a Reeb orbit is given by its parity , which is defined as(4.5) | γ | = sign det( I − A γ ) ∈ {± } = Z / , where A γ is the linearized Poincar´e return map of ξ along γ (see [Par19, Sec. 2.13]).Theorem 1.1 of [Par19] provides a set of perturbation data Θ I ( D ) and associated virtualmoduli counts M I ( T ) vir θ ∈ Q satisfying (4.1) and (4.2). We define the twisted moduli counts ψ M I ( T ) vir θ := M I ( T ) vir θ · ψ ( T ) ∈ R . It follows easily from Definition 4.1 that the twisted moduli counts also satisfy (4.1) and(4.2). We may therefore endow CC • ( Y, ξ, ψ ) λ with a differential d ψ,J,θ , which is given by(4.6) d ψ,J,θ ( o γ + ) = X Γ − →P good µ ( γ + , Γ − ; β )=1 | Aut | · ψ M I ( γ + , Γ − ; β ) vir J,θ o Γ − . The homology of ( CC • ( Y, ξ, ψ ) λ , d ψ,J,θ ) is a supercommutative Z / R -algebrawhich is denoted CH • ( Y, ξ, ψ ) λ,J,θ . Setup II.
Fix a datum D for Setup II. Suppose now we are given a map of Q -algebras m : R + → R − and twisting maps ψ ± ∈ Ψ I ( D ± ; R ± ). Definition 4.2.
The set Ψ II ( D ; ψ + , ψ − ) consists of all maps ψ : S = ∅ II ( D ) → R − satisfying thefollowing two properties: • for any morphism T ′ → T , ψ ( T ′ ) = ψ ( T ); • for any concatenation { T i } i , ψ ( i T i ) = Y T i ∈S +I m ( ψ + ( T i )) Y T i ∈S II ψ ( T i ) Y T i ∈S − I ψ − ( T i ) . Fix a twisting map ψ ∈ Ψ II ( D ; ψ + , ψ − ). Theorem 1.1 of [Par19] provides a set of pertur-bation data Θ II ( D ) together with a forgetful map Θ II ( D ) → Θ I ( D + ) × Θ I ( D − ) and associatedvirtual moduli counts M II ( T ) vir θ ∈ Q . We define the twisted moduli counts ψ M II ( T ) vir θ := M II ( T ) vir θ · ψ ( T ) ∈ R − . For any θ ∈ Θ II ( D ) mapping to ( θ + , θ − ) ∈ Θ I ( D + ) × Θ I ( D − ), we obtain a unital R + -algebramap Φ( ˆ X, ˆ λ, ψ ) ˆ J,θ : CC • ( Y + , ξ + , ψ + ) λ + ,J + ,θ + → CC • ( Y − , ξ − , ψ − ) λ − ,J − ,θ − which maps o γ + to X Γ − →P good ( Y − ) µ ( γ + , Γ − ; β )=0 | Aut | · ψ M II ( γ + , Γ − ; β ) virˆ J,θ o Γ − . This is a chain map since it follows from Definition 4.2 that the twisted moduli counts satisfy(4.1) and (4.2).
Setup III.
Fix a datum D for Setup III. There are three types of concatenations { T i } i in S III = S III ( D ):(1) { T i } ⊂ S +I ⊔ S t =0II ⊔ S − I , in which case s ( i T i ) = { } ;(2) { T i } ⊂ S +I ⊔ S t =1II ⊔ S − I , in which case s ( i T i ) = { } ;(3) { T i } ⊂ S +I ⊔ S III ⊔ S − I and T i ∈ S III for a unique i = i , in which case s ( i T i ) = s ( T i ).Suppose now we are given a map of Q -algebras m : R + → R − and twisting maps ψ ± ∈ Ψ I ( D ± ; R ± ), ψ ∈ Ψ II ( D t =0 ; ψ + , ψ − ) and ψ ∈ Ψ II ( D t =1 ; ψ + , ψ − ). Definition 4.3.
The set Ψ
III ( D ; ψ , ψ ) consists of all maps ψ : S = ∅ III ( D ) → R − satisfying thefollowing properties: • for any morphism T ′ → T , ψ ( T ′ ) = ψ ( T ); • for any concatenation { T i } i of the first type, ψ ( i T i ) = Y T i ∈S +I m ( ψ + ( T i )) Y T i ∈S t =0II ψ ( T i ) Y T i ∈S − I ψ − ( T i ) • for any concatenation { T i } i of the second type, ψ ( i T i ) = Y T i ∈S +I m ( ψ + ( T i )) Y T i ∈S t =1II ψ ( T i ) Y T i ∈S − I ψ − ( T i ) • for any concatenation { T i } i of the third type, ψ ( i T i ) = Y T i ∈S +I m ( ψ + ( T i )) ψ ( T i ) Y T i ∈S − I ψ − ( T i ) . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 35
Fix a twisting map ψ ∈ Ψ III ( D ; ψ , ψ ). Theorem 1.1 of [Par19] provides a set of perturbationdata Θ III ( D ) together with a forgetful map Θ III ( D ) → Θ II ( D ) × Θ I ( D + ) × Θ I ( D − ) Θ II ( D ) andassociated virtual moduli counts M III ( T ) vir θ ∈ Q . We define the twisted moduli counts ψ M III ( T ) vir θ := M III ( T ) vir θ · ψ ( T ) ∈ R − . If ( ˆ X, ˆ λ t ) is a family of exact cobordisms, then for any θ ∈ Θ III ( D ), we obtain an R + -linearmap K ( ˆ X, { λ t } t , ψ ) ˆ J t ,θ : CC • ( Y + , ξ + , ψ + ) λ + ,J + ,θ + → CC • +1 ( Y − , ξ − , ψ − ) λ − ,J − ,θ − which sends the monomial Q i ∈ I o γ + i to X { Γ − i →P good ( Y − ) } i ∈ I vdim( { γ + i , Γ − i ; β i } i ∈ I )=0 | Aut | · ψ M III ( { γ + i , Γ − i ; β i } i ∈ I ) virˆ J t ,θ Y i ∈ I o Γ − i . Equations (4.1) and (4.2) applied to the twisted moduli counts imply that this is a chainhomotopy between Φ( ˆ X, ˆ λ , ψ ) ˆ J ,θ and Φ( ˆ X, ˆ λ , ψ ) ˆ J ,θ and hence that the induced mapson homology CH • ( Y + , ξ + , ψ + ) λ + ,J + ,θ + CH • ( Y − , ξ − , ψ − ) λ − ,J − ,θ − Φ( ˆ X, ˆ λ ,ψ ) ˆ J ,θ Φ( ˆ X, ˆ λ ,ψ ) ˆ J ,θ are equal. Setup IV.
Fix a datum D for Setup IV. There are three types of concatenations { T i } i in S IV = S IV ( D ):(1) { T i } ⊂ S ⊔ S ⊔ S , in which case s ( i T i ) = { } ;(2) { T i } ⊂ S ⊔ S ⊔ S ⊔ S ⊔ S , in which case s ( i T i ) = {∞} ;(3) { T i } ⊂ S ⊔ S IV ⊔ S and T i ∈ S IV for a unique i = i , in which case s ( i T i ) = s ( T i ).Suppose now we are given maps of Q -algebras m : R → R , m : R → R and twistingmaps ψ i ∈ Ψ I ( D i ; R i ) ( i = 0 , , ψ ij ∈ Ψ II ( D ij ; ψ i , ψ j ) ( ij = 01 , , m = m ◦ m : R → R . Definition 4.4.
The set Ψ IV ( D ; { ψ ij } ) consists of all maps ψ : S = ∅ IV ( D ) → R satisfying thefollowing properties: • for any morphism T ′ → T , ψ ( T ′ ) = ψ ( T ); • for any concatenation { T i } i of the first type, ψ ( i T i ) = Y T i ∈S m ( ψ ( T i )) Y T i ∈S ψ ( T i ) Y T i ∈S ψ ( T i ) • for any concatenation { T i } i of the second type, ψ ( i T i ) = Y T i ∈S m ( ψ ( T i )) Y T i ∈S m ( ψ ( T i )) Y T i ∈S m ( ψ ( T i )) Y T i ∈S ψ ( T i ) Y T i ∈S ψ ( T i ) • for any concatenation { T i } i of the third type, ψ ( i T i ) = Y T i ∈S m ( ψ ( T i )) ψ ( T i ) Y T i ∈S ψ ( T i ) . Fix a twisting map ψ ∈ Ψ IV ( D ; { ψ ij } ). Theorem 1.1 of [Par19] provides a set of per-turbation data Θ IV ( D ) together with a forgetful map Θ IV ( D ) → Θ II ( D ) × Θ I ( D ) × Θ I ( D ) (Θ II ( D ) × Θ I ( D ) Θ II ( D )) and associated virtual moduli counts M IV ( T ) vir θ ∈ Q . We definethe twisted moduli counts ψ M IV ( T ) vir θ := M IV ( T ) vir θ · ψ ( T ) ∈ R . As in the previous section, we obtain an R -linear map CC • ( Y , ξ , ψ ) λ ,J ,θ → CC • +1 ( Y , ξ , ψ ) λ ,J ,θ which is a chain homotopy between the maps Φ( ˆ X , ˆ λ , ψ ) ˆ J ,θ and Φ( ˆ X , ˆ λ , ψ ) ˆ J ,θ ◦ Φ( ˆ X , ˆ λ , ψ ) ˆ J ,θ , so that the diagram CH • ( Y , ξ , ψ ) λ ,J ,θ CH • ( Y , ξ , ψ ) λ ,J ,θ CH • ( Y , ξ , ψ ) λ ,J ,θ Φ( ˆ X , ˆ λ ,ψ ) ˆ J ,θ Φ( ˆ X , ˆ λ ,ψ ) ˆ J ,θ Φ( ˆ X , ˆ λ ,ψ ) ˆ J ,θ commutes.4.3. The energy of a symplectic cobordism.
In this section, we introduce a notion ofenergy for (families of strict) exact symplectic cobordisms, and for certain classes of almost-complex structures.
Notation 4.5.
Recall that a strict exact symplectic cobordism from ( Y + , λ + ) to ( Y − , λ − ) isthe data of an exact symplectic cobordism ( ˆ X, ˆ λ ) and embeddings(4.7) e ± : ( R × Y ± , ˆ λ ± ) → ( ˆ X, λ )which preserve the Liouville forms and satisfy certain additional properties stated in Defini-tion 2.3.When we consider strict exact symplectic cobordisms in this section, we will routinely abusenotation by identifying subsets of R × Y ± with their image under e ± . We hope that this abusewill make the section easier to read without introducing any substantial ambiguities.We begin with the following definition. Definition 4.6.
Let ( ˆ X, ˆ λ ) be a strict exact symplectic cobordism (see Definition 2.3) from( Y + , λ + ) to ( Y − , λ − ). A Type A cobordism decomposition is the data of a pair of hypersurfaces H − = {− C − } × Y − H + = { C + } × Y + such that(4.8) (( −∞ , − C − ) × Y − ) ∩ ( C + , ∞ ) × Y + ) = ∅ . (If Y − = ∅ , we set H − = ∅ , C − = 0 and we consider that (4.8) is tautologically satisfied.) Welet Σ( ˆ X, ˆ λ ) = Σ( ˆ X, ˆ λ ; λ + , λ − ) be the set of all such cobordism decompositions. OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 37
Definition 4.7.
Let ( ˆ X, ˆ λ ) be as in Definition 4.6 and let σ ∈ Σ( ˆ X, ˆ λ ) be a Type A cobordismdecomposition. We let E ( σ ) := C − + C + be the energy of the decomposition σ . We define E ( ˆ X, ˆ λ ) = E ( ˆ X, ˆ λ ; λ + , λ − ) := inf σ ∈ Σ( ˆ X, ˆ λ ) E ( σ ) ∈ R ∪ {−∞} to be the energy of ( ˆ X, ˆ λ ) (this is well-defined since a cobordism decomposition clearly alwaysexists). We note that this energy may in general be negative. Given C ∈ R , let Σ( ˆ X, ˆ λ ) 0. Hence E ( SY, λ Y ) ≡ Proof of Lemma 4.8. Identifying ( SY, λ Y ) with ( ˆ Y , ˆ λ − ) = ( R × Y, e s λ − ), we can embed ( Y + , λ + )as the graph of f over { }× Y . This gives us a Type A decomposition of SY with C − = − min f and C + = 0. (cid:3) Lemma 4.10. We have E ( ˆ X, ˆ λ ) = −∞ if and only if Y − = ∅ .Proof. Suppose that Y − = ∅ . For volume reasons, the Liouville flow of any slice {− C − } × Y − must intersect any slice { C }× Y + in finite time, and vice versa. Using this, it is straightforwardto give an absolute lower bound (depending only on ( ˆ X, ˆ λ ; λ + , λ − )) on the energy of anycobordism decomposition.Suppose now that Y − = ∅ . The backwards Liouville flow of any slice { C + } × Y + is definedfor all time, which implies that we can find a cobordism decomposition of arbitrarily negativeenergy. (cid:3) Lemma 4.11. Fix a (strict) exact symplectic cobordism ( ˆ X, ˆ λ ) from ( Y + , λ + ) to ( Y − , λ − ) .Then Σ( ˆ X, ˆ λ ) Let ( ˆ X, ˆ λ t ) t ∈ [0 , be a one-parameter family of (strict) exact symplecticcobordisms (cf. Definition 2.12). A one-parameter family of Type A cobordism decompositionsis just the data of a family of hypersurfaces H − ( t ) = {− C − ( t ) } × Y − H + ( t ) = { C + ( t ) } × Y + such that(4.9) (( −∞ , − C − ( t )) × Y − ) ∩ (( C + ( t ) , ∞ ) × Y + ) = ∅ . (If Y − = ∅ , we again set H − ( t ) = ∅ , C − ( t ) = 0 and we consider that (4.9) is tautologicallysatisfied). We let Σ( ˆ X, ˆ λ t ) t ∈ [0 , be the set of all such families of cobordism decompositions. Definition 4.13. With the notation as above, with define the energy of a family of Type Acobordism decompositions σ ∈ Σ( ˆ X, ˆ λ t ) t ∈ [0 , to be E ( σ ) := sup t ( C − ( t ) + C + ( t )).Let ( ˆ X , ˆ λ ) (resp. ( ˆ X , ˆ λ )) be a strict exact symplectic cobordism from ( Y , λ ) to( Y , λ ) (resp. from ( Y , λ ) to ( Y , λ )). Let ( ˆ X, ˆ λ t ) t ∈ [0 , ∞ ) be a one-parameter family ofstrict exact symplectic cobordisms which agrees for t ≥ a large enough with the t -gluing( ˆ X t ˆ X , ˆ λ t ˆ λ ) t ∈ [ a, ∞ ) ; see Definition 2.7. For t ≥ a , note that there are canonicalLiouville embeddings ι ,t : ( ˆ X , ˆ λ ) ( ˆ X , e t/ ˆ λ ) ( ˆ X t ˆ X , ˆ λ t ˆ λ ) ι ,t : ( ˆ X , ˆ λ ) ( ˆ X , e − t/ ˆ λ ) ( ˆ X t ˆ X , ˆ λ t ˆ λ ) . µ t/ µ − t/ Definition 4.14. A Type B cobordism decomposition of ( ˆ X, ˆ λ t ) t ∈ [0 , ∞ ) is the data of a familyof hypersurfaces H ( t ) = {− C ( t ) } × Y H ( t ) = { C ( t ) } × Y and a Liouville embedding ([ − C ( t ) , C ′ ( t )] × Y , e s λ ) ֒ → ( ˆ X, λ t )) such that(4.10) ( −∞ , − C ( t )) × Y , ( − C ( t ) , C ′ ( t )) × Y , ( C ( t ) × ∞ ) × Y are pairwise disjoint. (In case Y − = ∅ , we set H ( t ) = ∅ , C ( t ) = 0 and replace (4.10) by thecondition that ( − C ( t ) , C ′ ( t )) × Y , ( C ( t ) × ∞ ) × Y and pairwise disjoint.)We let H ( t ) = {− C ( t ) } × Y H ′ ( t ) = { C ′ ( t ) } × Y . This data is required to satisfy the following hypotheses:(1) C ′ (0) = − C (0),(2) for t large enough, H , H ′ ( t ) (resp. H ( t ) , H ( t )) are in the image of the canonicalembedding ι ,t (resp. ι ,t ). Moreover, their restriction defines a Type A decompositionon ( ˆ X , ˆ λ ) (resp. on ( ˆ X , ˆ λ )) which is independent of t . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 39 We let Σ B (( ˆ X , ˆ λ t ) t ∈ [0 , ∞ ) ) denote the set of all such cobordism decompositions. We willwrite Σ( − ) instead of Σ B ( − ) when the subscript is understood from the context. Definition 4.15. It follows from property (1) of Definition 4.14 that a Type B cobordismdecomposition σ ∈ Σ B (( ˆ X, ˆ λ t ) t ∈ [0 , ∞ ) ) induces a Type A cobordism decomposition σ ∈ Σ A ( ˆ X, ˆ λ ) by taking H − = H (0) and H + = H (0). We say that σ is induced at zero by σ .Similarly, property (2) of Definition 4.14 states that a Type B cobordism decomposition σ ∈ Σ B (( ˆ X, ˆ λ t ) t ∈ [0 , ∞ ) ) induces a pair of Type A decompositions σ ∈ Σ A ( ˆ X , ˆ λ ) and σ ∈ Σ A ( ˆ X , ˆ λ ). We say that the pair ( σ , σ ) is induced at infinity by σ . Definition 4.16. With the notation as above, we define the energy of a Type B cobordismdecomposition σ ∈ Σ B (( ˆ X, ˆ λ t )) to be E ( σ ) := sup t ( C ( t ) + C ( t ) − C ( t ) − C ′ ( t )). We let E (( ˆ X, ˆ λ t )) := inf σ ∈ Σ B (( ˆ X, ˆ λ t ) E ( σ ) ∈ R ∪ {−∞} . Given C ∈ R , let Σ( ˆ X, ˆ λ t ) Let σ be a Type B cobordism decomposition. Suppose that σ is induced at zeroby σ and that ( σ , σ ) is induced at infinity. Then E ( σ ) ≤ E ( σ ) and E ( σ ) + E ( σ ) ≤E ( σ ) .Proof. The first claim follows from (1) in Definition 4.14 and the definition of energy for TypeA and Type B cobordism decomposition. The second claim follows similarly from (2) inDefinition 4.14. (cid:3) Corollary 4.18. We have E (( ˆ X , ˆ λ t )) = −∞ if and only if Y = ∅ .Proof. One direction follows from Lemma 4.10 and Lemma 4.17. The other direction can bechecked by inspection, using the backwards Liouville flow as in the proof of the correspondingstatement in Lemma 4.10. (cid:3) Definition 4.19. Let ( ˆ X, ˆ λ ) and ( ˆ X ′ , ˆ λ ′ ) be exact symplectic cobordisms. For C ∈ R , letΣ A (( ˆ X, ˆ λ ) , ( ˆ X ′ , ˆ λ ′ )) Given C ∈ R such that Σ( X ,t , ˆ λ ,t ) Suppose that ( ˆ X ,t , ˆ λ ,t ) is the ( t + T ) -gluing of two exact symplectic cobor-disms ( ˆ X , ˆ λ ) and ( ˆ X , ˆ λ ) , for T > fixed and t ∈ [0 , ∞ ) (cf. Example 2.13). Supposethat either ( ˆ X , ˆ λ ) or ( ˆ X , ˆ λ ) is a symplectization (see Example 2.6). Then E ( ˆ X ,t , ˆ λ ,t ) = E ( ˆ X , ˆ λ ) + E ( ˆ X , ˆ λ ) .Proof. By Lemma 4.10 and Corollary 4.18, we may assume that ˆ X has a non-empty negativeend.We only treat the case where ( ˆ X , ˆ λ ) is a symplectization and T = 0 since the other casesare analogous.Choose σ so that E ( σ ) ≤ E ( ˆ X , ˆ λ )+ ǫ and choose σ so that E ( σ ) ≤ E ( ˆ X , ˆ λ )+ ǫ .Let ˜ X ⊂ ˆ X and ˜ X ⊂ ˆ X be the Liouville subdomains which determine the Type Adecompositions σ and σ respectively.Note that ˆ X ,t comes equipped with tautological embeddings ι ,t : ˆ X → ˆ X ,t and ι ,t :ˆ X → ˆ X ,t (see Definition 2.7). For T ′ large enough and t ≥ T ′ , note that ι ,t ( H − ) is inthe image of ι ,t ( H ) under the Liouville flow. These hypersurfaces therefore bound Liouvilledomains ([ − C ( t ) , C ′ ( t )] × Y , e s λ ).Let f : [0 , ∞ ) → R be a function which equals − ( C ( T ′ )+ C ′ ( T ′ )) on [0 , T ′ ], is non-decreasingon [ T ′ , T ′ + 1] and is zero on [ T ′ + 1 , ∞ ). Let τ f : ˆ X × [0 , ∞ ) → ˆ X be defined by τ f ( x, t ) = φ f ( t )+ T ′ +1 − t ( x ), where φ − ) is Liouville flow on ˆ X . Now define a map ι ,t : ˆ X → ˆ X ,t byletting ι ,t ( x ) = ι ,t ◦ τ f ( x, t ).We now define the data of a Type B cobordism decomposition by letting H ( t ) = ι ,t ( H − ) , H ( t ) = ι ,t ( H ) , H ′ = ι ,t ( H − ( t )) , H = ι ,t ( H ( t )) . One can check that this data indeed defines a Type B cobordism decomposition, which hasenergy precisely equal to E ( σ )+ E ( σ ) ≤ E ( ˆ X , ˆ λ )+ E ( ˆ X , ˆ λ )+2 ǫ . Since ǫ was arbitrary,we conclude that E ( ˆ X ,t , ˆ λ ,t ) ≤ E ( ˆ X , ˆ λ ) + E ( ˆ X , ˆ λ ). (cid:3) We now discuss almost-complex structures for Setups II-IV. Setup II. Fix a datum D = ( D + , D − , ˆ X, ˆ λ, ˆ J ) for Setup II, D ± = ( Y ± , λ ± , J ± ). Let( V ± , ξ ± | V ± , τ ± ) ⊂ ( Y ± , ξ ± ) be framed codimension 2 contact submanifolds and let α ± := λ ± | V ± . Let J ± be dλ ± -compatible almost-complex structures on ξ ± ⊂ T Y ± which preserve ξ ± ∩ T V ± .Let H ⊂ ˆ X be a codimension 2 symplectic submanifold such that ( ˆ X, ˆ λ, H ) is an exactrelative symplectic cobordism from ( Y + , ξ + , V + ) to ( Y − , ξ − , V − ). We will also consider (seeNotation 2.4) the strict symplectic cobordisms ( ˆ X, ˆ λ ) λ + λ − and ( H, ˆ λ | H ) α + α − . Definition 4.22. Fix a Type A cobordism decomposition σ ∈ Σ( H, ˆ λ | H ), which is specifiedby a pair of hypersurfaces H − = {− C − } × V − and H + = { C + } × V + . We say that analmost-complex structure ˆ J on ˆ X is adapted to σ if the following properties hold: • ˆ J is compatible with d ˆ λ , • ˆ J coincides with ˆ J ± near the ends (where ˆ J ± is the canonical cylindrical almost-complex structure induced on ( ˆ Y , ˆ λ ± ) by J ± ), • H ⊂ ˆ X is a ˆ J -complex hypersurface, • ˆ J preserves ker α + ⊂ T V + on [ C , ∞ ) × V + (resp. preserves ker α − ⊂ T V − on( −∞ , − C ] × V − ) and the induced almost-complex structure is dα + -compatible (resp. dα − -compatible).In case V − = ∅ , the conditions involving V − are considered to be vacuously satisfied. OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 41 Definition 4.23. Given an almost-complex structure ˆ J on ( ˆ X, ˆ λ ) we define its energy E ( ˆ J ) := inf {E ( σ ) | σ ∈ Σ( H, ˆ λ | H ) , ˆ J is adapted to σ } ∈ R ∪ {±∞} . We define E ( ˆ J ) = ∞ if ˆ J is not adapted to any cobordism decomposition.Let J ( ˆ X, ˆ λ, H ) The set J ( ˆ X, ˆ λ, H ) Fix a datum D for Setup IV. We write D = ( D , D , ( ˆ X ,t , ˆ λ ,t , ˆ J ,t ) t ∈ [0 , ∞ ) ),where D = ( D , D , ˆ X , ˆ λ , ˆ J ) D = ( D , D , ˆ X , ˆ λ , ˆ J ) D i = ( Y i , λ i , J i ) ( i = 0 , , V i , ξ i | V i , τ i ) ⊂ ( Y i , ξ ) be framed codimension 2 contact submanifolds and set α V i = λ i | V i . Let H ⊂ ˆ X , H ⊂ ˆ X , and ( H ,t ⊂ ˆ X ,t ) t ∈ [0 , ∞ ) be cylindrical symplecticsubmanifolds such that ( ˆ X ,t , ˆ λ ,t , H ,t ) t ∈ [0 , ∞ ) is a family of relative symplectic cobordismsthat agrees for t large with the t -gluing of the relative symplectic cobordisms ( ˆ X , ˆ λ , H )and ( ˆ X , ˆ λ , H ). Note that { H ,t } forms a family of Liouville manifolds with respect to(the restriction of) ˆ λ ,t . Definition 4.25. Fix a Type B cobordism decomposition σ ∈ Σ B ( ˆ H ,t , ˆ λ ,tH ,t ). Recallthat σ consists in the data of hypersurfaces H ( t ) = {− C ( t ) } × V , H ( t ) = {− C ( t ) } × V , H ′ ( t ) = { C ′ ( t ) } × V , H ( t ) = { C ( t ) } × V . We say that an almost-complex structureˆ J ,t is adapted to σ if the following properties hold: • ˆ J ,t is compatible with d ˆ λ ,t , • ˆ J ,t coincides with ˆ J (resp. ˆ J ) near the positive (resp. negative) end, • H ,t is a ˆ J ,t -complex hypersurface, and ˆ J ,t is compatible with the restriction of d ˆ λ ,t to H ,t , • ˆ J ,t preserves ker α on [ C ( t ) , ∞ ) × V (resp. ker α on [ − C ( t ) , C ′ ( t )] × V , resp.ker α on ( −∞ , − C ( t )] × V ). Moreover, the induced almost-complex structure is dα -compatible (resp. dα -compatible, resp. dα -compatible).In case V = ∅ , all conditions involving V are considered to be vacuously satisfied. Definition 4.26. Given a family of almost-complex structures ˆ J t , we define its energy E ( ˆ J t ) := inf {E ( σ ) | σ ∈ Σ( ˆ X ,t , ˆ λ ,t ) , ˆ J t is adapted to σ } ∈ R ∪ {±∞} . If ˆ J t is not adapted to any cobordism decomposition, we set E ( ˆ J t ) = ∞ .Let J ( ˆ X ,t , ˆ λ ,t , H ,t ) be the set of almost-complex structures adapted to some Type Bdecomposition σ ∈ Σ B ( H ,t , ˆ λ ,t | H ,t ). For C ∈ R , let J ( ˆ X ,t , ˆ λ ,t , H ,t ) The set J ( ˆ X ,t , ˆ λ ,t , H ,t ) Lemma 4.28. Suppose that J ( ˆ X ,t , ˆ λ ,t , H ,t ) Enriched setups. The construction of our invariants follows the same general schemeas Pardon’s construction of contact homology. However, we work with a class of “enriched”setups I*-IV*, which contain more information than the standard setups I-IV considered byPardon.We will show in Section 5.2 that the data associated to our enriched setups give rise totwisting maps. These twisting maps are constructed using Siefring’s intersection theory, andwill be used to define “twisted” moduli counts, following the approach outlined in Section 4.2.Given a datum D for any of Setups I*-IV*, there is a “forgetful functor” which allows oneto view D as a datum of Setup I-IV. However, it is not the case that every datum of SetupI-IV admits an enrichment. We will show in Section 6.1 that the class of enriched data is largeenough for the purpose of defining invariants in the spirit of contact homology. Setup I*. A datum D = (( Y, ξ, V ) , r , λ, J ) for Setup I* consists of: • A contact pair ( Y, ξ, V ), • an element r = ( α V , τ, r ) ∈ R ( Y, ξ, V ), • a contact form ker λ = ξ which is adapted to r , • an almost-complex structure J which is compatible with dλ and preserves ξ | V . Setup II*. A datum D = ( D + , D − , ˆ X, ˆ λ, H, ˆ J ) for Setup II* consists of: OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 43 • data D ± = (( Y ± , ξ ± , V ± ) , r ± , λ ± , J ± ) as in Setup I*; • an exact relative symplectic cobordism ( ˆ X, ˆ λ, H ) with positive end ( Y + , λ + , V + ) andnegative end ( Y − , λ − , V ± ); • an d ˆ λ -tame almost complex structure ˆ J on ˆ X which agrees with ˆ J ± at infinity.This datum is moreover subject to the following conditions: • there exists a trivialization of the normal bundle of H which restricts to τ + (resp. τ − )the positive (resp. negative) end; • r + ≥ e E ( ˆ J ) r − . Setup III*. A datum D = ( D + , D − , ˆ X, ˆ λ t , H t , ˆ J t ) t ∈ [0 , for Setup III* consists of: • data D ± = (( Y ± , ξ ± , V ± ) , r ± , λ ± , J ± ) as in Setup I*; • a family of exact relative symplectic cobordisms ( ˆ X, ˆ λ t , ˆ H t ) for t ∈ [0 , Y + , λ + , V + ) and negative end ( Y − , λ − , V ± ); • a family d ˆ λ t -tame almost complex structures ˆ J t on ˆ X , which agree with ˆ J ± at infinity.This datum is moreover subject to the following conditions: • there exists a trivialization of the normal bundle of H which restricts to τ + (resp. τ − )the positive (resp. negative) end; • r + ≥ e E ( ˆ J t ) r − . Setup IV*. A datum D = ( D , D , ( ˆ X ,t , ˆ λ ,t , H ,t , ˆ J ,t ) t ∈ [0 , ∞ ) ) for Setup IV* consistsof: • a datum D = ( D , D , ˆ X , ˆ λ , H , ˆ J ) for Setup II*; • a datum D = ( D , D , ˆ X , ˆ λ , H , ˆ J ) for Setup II*; • data D i = (( Y i , ξ i , V i ); r i , λ i , J i ) for Setup I*, for i = 0 , , • a family of cylindrical symplectic submanifolds H ,t ⊂ ˆ X ,t , for t ∈ [0 , ∞ ), such that( ˆ X ,t , ˆ λ ,t , H ,t ) t ∈ [0 , ∞ ) is a family of exact relative symplectic cobordisms that agreesfor t large with the t -gluing of the relative symplectic cobordisms ( ˆ X , ˆ λ , H ) and( ˆ X , ˆ λ , H ).This datum is moreover subject to the following conditions: • there exists a trivialization of the normal bundle of H ,t which restricts to τ (resp. τ ) at the positive (resp. negative) end; • there exists a trivialization of the normal bundle of H which restricts to τ (resp. τ )the positive (resp. negative) end; • there exists a trivialization of the normal bundle of H which restricts to τ (resp. τ )the positive (resp. negative) end; • r ≥ e E ( ˆ J ,t ) r ; • r ≥ e E ( ˆ J ) r ; • r ≥ e E ( ˆ J ) r .5.2. Twisting maps associated to enriched setups. In this section, we construct twistingmaps on the contact homology algebra. These maps depend on geometric data involvingcodimension-2 contact submanifolds and relative symplectic cobordisms. Throughout thissection, it will be convenient to set ˜ δ ( r ) = ( r = 0 , Setup I*. Let D = (( Y, ξ, V ) , r , λ, J ) be a datum for Setup I*, where r = ( α V , τ, r ). There isan obvious functor from S I ( D ) to the category b S ( ˆ Y ) defined in section 3.4. We therefore havea well-defined intersection number T ∗ ˆ V for T ∈ S I ( D ).We now introduce twisting maps associated to the above setup. Definition 5.1. We define a map ψ V ( T ) : S = ∅ I ( D ) → Q [ U ] by ψ V ( T ) = U T ∗ ˆ V +˜ δ ( r )Γ − ( T,V ) , where Γ − ( T, V ) denotes the number of output edges e of T such that the corresponding Reeborbit γ e is contained in V . Corollary 3.21 ensures that the exponents appearing in thesedefinitions are nonnegative. Remark . Corollary 3.21 only applies to trees T such that M ( T ) = ∅ . This is why thedefinition of twisting maps only requires them to be defined on S = ∅ and not on the wholecategory S . Definition 5.3. Suppose that r > 0. We define a map e ψ V : S = ∅ I ( D ) → Q by e ψ V ( T ) = ( T ∗ ˆ V = 0 and | γ e | ∩ V = ∅ for every e ∈ E ( T )0 otherwiseWe must now check that the maps in Definitions 5.1–5.3 satisfy the axioms of Definition 4.1. Proposition 5.4. The map ψ V introduced in Definition 5.1 is a twisting map.Proof. It follows from Proposition 3.18 that ψ ell ,V ( T ) = ψ ell ,V ( T ′ ) for any morphism T → T ′ .Let { T i } i be a concatenation in S = ∅ I . We need to show that ψ ell ,V ( i T i ) = Q i ψ ell ,V ( T i ), i.e.(5.1) ( i T i ) ∗ ˆ V + ˜ δ ( r )Γ − ( i T i , V ) = X i ( T i ∗ ˆ V + ˜ δ ( r )Γ − ( T i , V )) . Suppose first r = 0. Since the contact form λ is hyperbolic near V , we have p N ( γ ) = 0for every Reeb orbit γ contained in V by Proposition 3.7. Remark 3.13 and Corollary 3.12therefore imply that ˆ γ ∗ ˆ V = 0 for all Reeb orbit γ in Y . By Proposition 3.19, this means that T ∗ ˆ V = P v ∈ V ( T ) β v ∗ ˆ V for all T ∈ S I . In particular,( i T i ) ∗ ˆ V = X v ∈ V ( i T i ) β v ∗ ˆ V = X i X v ∈ V ( T i ) β v ∗ ˆ V = X i T i ∗ ˆ V . Suppose now r > 0. Since the contact form λ is elliptic near V , we have p N ( γ ) = 1 for everyReeb orbit γ contained in V by Proposition 3.7. Remark 3.13 and Corollary 3.12 thereforeimply that ˆ γ ∗ ˆ V is equal to − γ is contained in V and 0 otherwise. By Proposition 3.19,this means that(5.2) T ∗ ˆ V = X v ∈ V ( T ) β v ∗ ˆ V + Γ int ( T, V )for all T ∈ S I , where Γ int ( T, V ) denotes the number of edges e ∈ E int ( T ) such that γ e iscontained in V . Equation (5.1) is therefore equivalent toΓ int ( i T i , V ) + Γ − ( i T i , V ) = X i (Γ int ( T i , V ) + Γ − ( T i , V )) . The result now follows from the observation that there is a (canonical) label preserving bijectionbetween E int ( i T i ) ∪ E − ( i T i ) and ∪ i ( E int ( T i ) ∪ E − ( T i )) (this is an immediate consequence OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 45 of the definition: every interior edge of T j corresponds to an interior edge of i T i , and everyoutput edge of T j corresponds either to an interior or an output edge of i T i depending onwhether it is identified with another edge in the concatenation or not). (cid:3) It will be convenient to introduce the following definition. Definition 5.5. Given a tree T ∈ S I , a vertex v ∈ V ( T ) is bad if it is an interior vertex and | γ e | ⊂ V for all e ∈ e + ( v ) ⊔ E − ( v ). All other vertices are said to be good . These sets aredenoted V b ( T ) ⊂ V ( T ) and V g ( T ) ⊂ V ( T ) respectively. Proposition 5.6. The map e ψ V introduced in Definition 5.3 is a twisting map.Proof. Fix a tree T ∈ S = ∅ I . We first show that e ψ V ( T ′ ) = e ψ V ( T ) for any tree T ′ ∈ S = ∅ I admitting a morphism T ′ → T . Observe that we may assume without loss of generality that T ′ is representable by a ˆ J -holomorphic building (see Definition 3.20). Indeed, since T, T ′ ∈ S = ∅ I ,there exists T ′′ → T ′ → T such that T ′′ is representable by a ˆ J -holomorphic building. So wemay as well prove that e ψ V ( T ′′ ) = e ψ V ( T ′ ) and e ψ V ( T ′′ ) = e ψ V ( T ).Let us therefore fix T ′ ∈ S = ∅ I such that T ′ is representable by a ˆ J -holomorphic building, anda morphism T ′ → T . It follows from Proposition 3.18 that T ′ ∗ ˆ V = T ∗ ˆ V . Note that T ′ , T havethe same exterior edges. If one of these edges is contained in V , then e ψ V ( T ′ ) = e ψ V ( T ) = 0.So we can assume that the exterior edges of T ′ , T are not contained in V .Suppose now that T ′ has an interior edge contained in V . For i = 0 , , 2, let X i ≥ e ∈ E ( T ′ ) such that | γ e | ⊂ V and e is adjacent to exactly i bad vertices. Byassumption, we have X + X + X ≥ T ′ ∗ ˆ V = X v ∈ V g ( T ′ ) β v ∗ ˆ V + X v ∈ V b ( T ′ ) β v ∗ ˆ V + X + X + X . According to Proposition 3.11, we also have that P v ∈ V g ( T ′ ) β v ∗ ˆ V ≥ T ′ is representable by a ˆ J -holomorphic building).If there are no bad vertices, then we have that P v ∈ V b ( T ′ ) β v ∗ ˆ V = 0, X = X = 0 and X ≥ 1. So T ′ ∗ ˆ V > X ≤ V b ( T ′ ) − 1. Moreover,given v ∈ V b ( T ′ ), Proposition 3.14 together with the fact that T ′ is representable by a ˆ J -holomorphic building imply that β v ∗ ˆ V ≥ − p v , where p v is the number of edges adjacent to v .It follows that P v ∈ V b ( T ′ ) β v ∗ ˆ V + X + X ≥ ( V b ( T ′ ) − X − X )+ X + X = V b ( T ′ ) − X ≥ T ′ ∗ ˆ V > e ψ H ( T ′ ) = e ψ H ( T ) = 0 if T ′ has an interior edge contained in V .We are left with the case where T ′ and hence T have no edges contained in V . It’s thenimmediate that e ψ V ( T ′ ) = e ψ V ( T ).If { T i } i is a concatenation, then the argument is the same as in the proof of Proposition 5.4since every edge in i T i appears in at least one of the T i . (cid:3) Setup II*. Fix a datum D = ( D + , D − , ˆ X, H, ˆ λ, ˆ J ) for Setup II*, where we write D ± =(( Y ± , ξ ± , V ± ) , r ± , λ ± , J ± ) and r ± = ( α ± , τ ± , r ± ).We now introduce the following twisting maps. Definition 5.7. We define a map ψ H : S = ∅ II ( D ) → Q [ U ] by ψ H ( T ) = U T ∗ H +˜ δ ( r − )Γ − ( T,V − ) . Definition 5.8. Suppose that r − > 0. We define a map e ψ H : S = ∅ II ( D ) → Q by e ψ H ( T ) = ( T ∗ H = 0 and | γ e | ∩ V ± = ∅ for every e ∈ E ( T )0 otherwiseWe need to verify that the above definitions satisfy the axioms of twisting maps. Thefirst step is to prove that the ψ H ( T ) are non-negative powers of U . This is the content ofCorollary 5.11, whose proof requires some preparatory lemmas. Lemma 5.9. For n ≥ , suppose that β ∈ π ( ˆ X, γ + ⊔ ( ∪ ni =1 γ − i )) is represented by a ˆ J -holomorphic curve u : ˙Σ → ˆ X which is contained in H . Then T + − e −E ( ˆ J ) ( P ni =1 T − i ) ≥ ,where T + (resp. T − i ) is the period of γ + ⊂ ( Y + , λ + ) (resp. the period of γ − i ⊂ ( Y − , λ − ) ).Proof. The claim is trivial if E ( ˆ J ) = ∞ , so let us assume that ˆ J ∈ J ( ˆ X, ˆ λ, H ). We maytherefore fix a Type A decomposition σ of ( H, λ H ), which is specified by a pair of hypersurfaces H − = {− C } × V − and H + = { C + } × V + .It will be convenient to define the regions R − := ( −∞ , − C ] × V − , R + := [ C × ∞ ) × V + and ˜ H = H − int( R − ) − int( R + ). Let us first assume that u is transverse to the boundary of˜ H . Consider now the sum Z u − ( R − ) e − C u ∗ dα − + Z u − ( ˜ H ) u ∗ d ˆ λ + e C Z u − ( R + ) u ∗ dα + . Each summand is non-negative due to the fact that u is ˆ J -holomorphic and that ˆ J is adaptedto σ . By Stokes’ theorem, the sum of the integrals is e C T + − e − C ( P ni =1 T − i ) ≥ 0. Thisimplies that T + ≥ e −E ( σ ) ( P ni =1 T − i ) and the claim follows by the definition of E ( ˆ J ).If u is not transverse to the boundary of ˜ H , observe by Sard’s theorem that transversalitycan be achieved for a sequence of domains ˜ H n := ˜ H ∪ [ − C n , − C ] ∪ [ C , C n ], where { C ni } ∞ n =0 ismonotonically decreasing and C ni → C i . It’s easy to verify that ˆ J is still adapted to the TypeA decompositions induced by the boundary of ˜ H n , so the above argument goes through. Thedesired equality now follows by passing to the limit. (cid:3) Lemma 5.10. For n ≥ , suppose that β ∈ π ( ˆ X, γ + ⊔ ( ∪ ni =1 γ − i )) is represented by a ˆ J -holomorphic curve u : ˙Σ → ˆ X . (Note that unlike in Lemma 5.9, we allow n = 0 in which casethe union is interpreted as being empty.)Then β ∗ H ≥ ˜ δ ( r − ) n u where n u is the total number of negative punctures of u contained in V ± ⊂ Y ± .Proof. According to Proposition 3.11, we only need to consider the case where the image of u is contained in H . By definition of a datum for Setup II*, the trivializations τ ± extend toa global trivialization τ of the normal bundle of H , which implies that u τ · H = 0. Using thefact that u τ · H = 0, we have (see Definition 3.9 and the proof of Proposition 3.14) u ∗ H = α τ ; − N ( γ + ) − n X i =1 α τ ;+ N ( γ − i )= ⌊ CZ τN ( γ + ) / ⌋ − n X i =1 ⌈ CZ τN ( γ − i ) / ⌉ = ⌊ r + T + ⌋ − n X i =1 ( δ ( r − ) + ⌊ rT i ⌋ ) , OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 47 where the sum is interpreted as zero if u has no negative punctures.If u has no negative punctures or if r − = 0, then the lemma is automatically verified. Hencewe only need to consider the case where n ≥ r − > 0. Let p u = n u + 1 be the totalnumber of punctures (positive and negative) of u contained in V ± ⊂ Y ± .Using the trivial bounds x − < ⌊ x ⌋ ≤ x , we obtain u ∗ H > ( r + T + − − n X i =1 (1 + r − T − i ) = − p u + r + T + − r − n X i =1 T − i . Using now Lemma 5.9 and the fact that r + ≥ e E ( ˆ J ) r − , we have − p u + r + T + − r − n X i =1 T − i ≥ − p u + r + ǫ −E ( ˆ J ) n X i =1 T − i − r − n X i =1 T − i ≥ − p u . The claim follows. (cid:3) Corollary 5.11. We have T ∗ H ≥ − ˜ δ ( r − )Γ − ( T, V − ) for any T ∈ S = ∅ II ( D ) . Hence ψ H ( T ) ∈ Q [ U ] .Proof. This follows immediately by combining Lemma 5.10 and Proposition 3.24. (cid:3) Proposition 5.12. Let { T i } i be a concatenation in S II . Then we have: ( i T i ) ∗ H + ˜ δ ( r − )Γ − ( i T i , V − ) = X T i ∈S +I ( T i ∗ ˆ V + + ˜ δ ( r + )Γ − ( T i , V + ))+ X T i ∈S II ( T i ∗ H + ˜ δ ( r − )Γ − ( T i , V − )) + X T i ∈S − I ( T i ∗ ˆ V − + ˜ δ ( r − )Γ − ( T i , V − )) . Proof. As in the proof of Proposition 5.4, our assumptions imply that ˆ γ ∗ V ± = − ˜ δ ( r ± ) if γ iscontained in V ± and 0 otherwise. By Proposition 3.22, we have(5.3) T ∗ H = X v ∈ V ( T ) ∗ ( v )=00 β v ∗ ˆ V + + X v ∈ V ( T ) ∗ ( v )=01 β v ∗ H + X v ∈ V ( T ) ∗ ( v )=11 β v ∗ ˆ V − + ˜ δ ( r + )Γ int ( T, V + ) + ˜ δ ( r − )Γ int ( T, V − )for all T ∈ S II . By applying this formula to T = i T i (and also using (5.2)), we see that itsuffices to prove that˜ δ ( r + )Γ int ( i T i , V + ) + ˜ δ ( r − )Γ int ( i T i , V − ) + ˜ δ ( r − )Γ − ( i T i , V − )= X T i ∈S +I ˜ δ ( r + )Γ int ( T i , V + ) + ˜ δ ( r − )Γ − ( T i , V + )+ X T i ∈S II ˜ δ ( r + )Γ int ( T i , V + ) + ˜ δ ( r − )Γ int ( T i , V − ) + ˜ δ ( r − )Γ − ( T i , V − )+ X T i ∈S − I ˜ δ ( r − )Γ int ( T i , V − ) + ˜ δ ( r − )Γ − ( T i , V − ) . As in the proof of Proposition 5.4, this is just a matter of understanding how the edges of i T i are obtained from the edges of the T i ’s. (cid:3) Corollary 5.13. Under the assumptions of Proposition 5.12, ψ H ∈ Ψ II ( D ; ψ V + , ψ V − ) . Proof. Proposition 3.18 implies that ψ H ( T ) = ψ H ( T ′ ) for any morphism T → T ′ . Proposi-tion 5.12 implies that ψ H acts correctly on concatenations. (cid:3) It remains to treat the twisting map e ψ H . We will need the following definition. Definition 5.14. Given a tree T ∈ S II , a vertex v ∈ V ( T ) is bad if it is an interior vertexand | γ e | ⊂ V ± for all e ∈ e + ( v ) ⊔ E − ( v ). All other vertices are said to be good . These sets aredenoted V b ( T ) ⊂ V ( T ) and V g ( T ) ⊂ V ( T ) respectively. Proposition 5.15. Under the assumptions of Definition 5.8, e ψ H ∈ Ψ II ( D ; e ψ V + , e ψ V − ) .Proof. Consider a tree T ′ ∈ S = ∅ II with a morphism T ′ → T . We wish to show that e ψ H ( T ′ ) = e ψ H ( T ). As in the proof of Proposition 5.6, we may assume that T ′ is representable by abuilding (see Definition 3.23).It follows from Proposition 3.18 that T ′ ∗ H = T ∗ H . Note that T ′ , T have the same exterioredges. If one of these edges is contained in V ± , then e ψ H ( T ′ ) = e ψ H ( T ) = 0. So we can assumethat the exterior edges of T ′ , T are not contained in V ± .Suppose now that T ′ has an interior edge contained in V ± . Arguing as in the proof ofProposition 5.6, let X i ≥ i = 0 , , 2) denote the number of edges e ∈ E ( T ′ ) such that | γ e | ⊂ V ± and e is adjacent to exactly i bad vertices. By assumption X + X + X ≥ T ′ ∗ ˆ V = X v ∈ V g ( T ′ ) β v ∗ ˆ V + X v ∈ V b ( T ′ ) β v ∗ ˆ V + X + X + X . According to Proposition 3.14 and the fact that T ′ is representable by a building, we have that P v ∈ V g ( T ′ ) β v ∗ ˆ V ≥ P v ∈ V b ( T ′ ) β v ∗ ˆ V = X = X = 0 and X ≥ 1. Hence T ′ ∗ ˆ V ≥ X ≤ V b ( T ′ ) − 1. According toLemma 5.10 and the fact that T ′ is representable by a building, we have that P v ∈ V b ( T ) β v ∗ ˆ V + X + X ≥ V b ( T ′ ) − X − X + X + X = V b ( T ′ ) − X ≥ 1. It thus follows againthat T ′ ∗ ˆ V ≥ e ψ V ( T ′ ) = e ψ V ( T ) = 0 if T ′ has an interior edge contained in V ± .We are left with the case where T ′ and hence T have no edges contained in V ± . It’s thenimmediate that e ψ H ( T ′ ) = e ψ H ( T ).If { T i } i is a concatenation, then the argument is the same as in the proof of Proposition 5.12since every edge in i T i appears in at least one of the T i . (cid:3) Setup III*. Fix a datum D = ( D + , D − , ˆ X, ˆ λ t , H t , ˆ J t ) t ∈ [0 , for Setup III*, where D ± =(( Y ± , ξ ± , V ± ) , r ± , λ ± , J ± ).We now introduce the following twisting maps. Definition 5.16. We define a map ψ H t : S = ∅ III ( D ) → Q [ U ] by ψ H t ( T ) = U T ∗ H t +˜ δ ( r − )Γ − ( T,V − ) . Definition 5.17. Suppose that r − > 0. We define a map e ψ H t : S = ∅ III ( D ) → Q by e ψ H t ( T ) = ( T ∗ H t = 0 and | γ e | ∩ V ± = ∅ for every e ∈ E ( T )0 otherwise OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 49 There is no difference between S III and S II from the point of view of the intersection theorydefined in section 3.4. It can therefore be shown by essentially the same arguments as in theprevious section that the above definitions do indeed satisfy the axioms for twisting maps. Corollary 5.18. We have ψ H t ∈ Ψ III ( D ; ψ H , ψ H ) and e ψ H t ∈ Ψ III ( D ; e ψ H , e ψ H ) . Setup IV*. Fix datum D = ( D , D , ( ˆ X ,t , ˆ λ ,t , H ,t , ˆ J ,t ) t ∈ [0 , ∞ ) ) for Setup IV*. Here,we have that: • D = ( D , D , ˆ X , ˆ λ , H , ˆ J ) is a datum for Setup II*; • D = ( D , D , ˆ X , ˆ λ , H , ˆ J ) is a datum for Setup II*; • D i = (( Y i , ξ i , V i ); r i , λ i , J i ) is a datum for Setup I*, for i = 0 , , Definition 5.19. We define ψ H ,t : S = ∅ IV ( D ) → Q [ U ] by ψ H ,t ( T ) = U T ∗ η +˜ δ ( r )Γ − ( T,V ) . Definition 5.20. Suppose r , r > 0. Then we may define e ψ H ,t : S = ∅ IV ( D ) → Q by e ψ H ,t ( T ) = ( T ∗ η = 0 and | γ e | ∩ V i = ∅ for all e ∈ E ( T ) , i ∈ { , , } U appearing in Definition 5.19 are non-negative. Thiswill be the content of Corollary 5.23, which requires some preparatory lemmas. Lemma 5.21. For n ≥ , suppose that u : ˙Σ → ˆ X ,t is ˆ J ,t -holomorphic with positive orbit γ + and negative orbits ∪ ni =1 γ − i . Then we have T + − e −E ( ˆ J ,t ) P i T − i ≥ , where T + (resp. T − i ) is the period of γ + ⊂ V (resp. γ − i ⊂ V ).Proof. The proof is analogous to that of Lemma 5.9. If E ( ˆ J ,t ) = ∞ , the result is trivial. Hencewe may assume that ˆ J ,t ∈ J ( ˆ X ,t , ˆ λ ,t , H ,t ) and fix a type B cobordism decomposition σ ,t of ( H ,t , ˆ λ ,t | H ,t ) to which ˆ J ,t is adapted. The decomposition σ ,t is specified by afamily of hypersurfaces H ( t ) = {− C ( t ) } × V , H ( t ) = {− C ( t ) } × V , H ′ ( t ) = { C ′ ( t ) } × V , H ( t ) = { C ( t ) } × V .It will be convenient to define the regions R ( t ) = ( −∞ , − C ( t )] × V , R ( t ) = [ C ( t ) , ∞ ) × V and R ( t ) = [ − C ( t ) , C ′ ( t )] × V . We let ˜ H ,t ⊔ ˜ H ,t be the connected components of X ,t − int( R ( t ) ∪ R ( t ) ∪ R ( t )).Let us first assume that the image of u intersects the boundaries of ˜ H ,t and ˜ H ,t transver-sally. We then have the following computations: • R u − ( R ( t )) u ∗ α = R u − ( H ( t )) u ∗ α − P ni =1 T − i ≥ • R u − ( H ,t ) u ∗ d ( e s α ) = e − N R u − ( H ( t )) u ∗ α − e − C R u − ( H ( t )) u ∗ α ≥ • R u − ( R ( t )) u ∗ α = R u − ( H ′ ( t )) u ∗ α − R u − ( H ( t )) u ∗ α ≥ • R u − ( ˜ H ,t ) u ∗ d ( e s α ) = e C R u − ( H ( t )) u ∗ α − e N ′ R u − ( H ′ ( t )) u ∗ α ≥ • R u − ( R ( t ) u ∗ α = T + i − R u − ( H ( t )) u ∗ α ≥ T + − e −E ( σ ,t ) P i T − i ≥ u does not intersect the boundaries of ˜ H ,t and ˜ H ,t transversally, wecan still prove that T + − e −E ( σ ,t ) P i T − i ≥ The lemma now follows from the definition of E ( ˆ J ,t ). (cid:3) Lemma 5.22. For n ≥ , suppose that u : ˙Σ → ˆ X ,t is a ˆ J ,t -holomorphic curve in the class β ∈ π ( ˆ X ,t , γ + ⊔ ( ∪ ni =1 γ − i )) for t < ∞ . Then β ∗ [ H ,t ] ≥ − ˜ δ ( r ) n u , where n u is the totalnumber of negative punctures. (Note that unlike in Lemma 5.21, we allow n = 0 here in whichcase the union is interpreted as being empty.)Proof. We argue as in the proof of Lemma 5.10. It is enough to consider the case where theimage of u is contained in H ,t . The trivialization τ extends to a global trivialization along H ,t , implying that β · τ H ,t = 0.We thus have u ∗ H ,t = α τ ; − N ( γ + ) − n X i =1 α τ ;+ N ( γ − i )= ⌊ CZ τN ( γ + ) / ⌋ − n X i =1 ⌈ CZ τN ( γ − i ) / ⌉ = ⌊ r T + ⌋ − n X i =1 ( δ ( r ) + ⌊ rT i ⌋ ) , where the sum is interpreted as zero if u has no negative punctures. Thus the lemma is verifiedif n = 0 or r = 0. It remains only to consider the case where n ≥ r > x − < ⌊ x ⌋ ≤ x , we obtain u ∗ ˆ V > X z ∈ p + u ( r T z − − X z ∈ p − u (1 + r T z ) ≥ − p u + e E ( σ ) r ( X z ∈ p + u T z ) − r X z ∈ p − u T z . It follows from Lemma 5.21 that e ˆ J ,t r ( P z ∈ p + u T z ) − r P z ∈ p − u T z ≥ 0. The claim follows. (cid:3) Corollary 5.23. We have that T ∗ η + ˜ δ ( r )Γ − ( T, V ) ≥ .Proof. We need to consider two cases. If s ( T ) ∈ [0 , ∞ ), then the claim follows by combiningProposition 3.24 and Lemma 5.22. If s ( T ) = {∞} , then the argument is the same as in theproof of Corollary 5.11. (cid:3) Proposition 5.24. Let { T i } i be a concatenation in S IV of type 2 (see page 35). Then we have ( i T i ) ∗ η + ˜ δ ( r )Γ − ( T, V )= X T i ∈S ( T i ∗ ˆ V + ˜ δ ( r )Γ − ( T i , V )) + X T i ∈S ( T i ∗ H + ˜ δ ( r )Γ − ( T i , V )) + X T i ∈S ( T i ∗ ˆ V + ˜ δ ( r )Γ − ( T i , V ))+ X T i ∈S ( T i ∗ H + ˜ δ ( r )Γ − ( T i , V )) + X T i ∈S ( T i ∗ ˆ V + ˜ δ ( r )Γ − ( T i , V )) . Proof. As in the proof of Proposition 5.4, our assumptions imply that ˆ γ ∗ V j = − γ iscontained in V j and 0 otherwise. Proposition 3.22 implies that( i T i ) ∗ η = X v ∈ V ( i T i ) ∗ ( v )=00 β v ∗ ˆ V + X v ∈ V ( i T i ) ∗ ( v )=01 β v ∗ H + X v ∈ V ( i T i ) ∗ ( v )=11 β v ∗ ˆ V + X v ∈ V ( i T i ) ∗ ( v )=12 β v ∗ H + X v ∈ V ( i T i ) ∗ ( v )=22 β v ∗ ˆ V + ˜ δ ( r )Γ int ( i T i , V ) + ˜ δ ( r )Γ int ( i T i , V ) + ˜ δ ( r )Γ int ( i T i , V ) . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 51 As in the proof of Proposition 5.12, it follows that the result is equivalent to˜ δ ( r )Γ int ( i T i , V ) + ˜ δ ( r )Γ int ( i T i , V ) + ˜ δ ( r )Γ int ( i T i , V ) + ˜ δ ( r )Γ − ( T, V )= X T i ∈S ˜ δ ( r )Γ int ( T i , V ) + ˜ δ ( r )Γ − ( T i , V )+ X T i ∈S ˜ δ ( r )Γ int ( T i , V ) + ˜ δ ( r )Γ int ( T i , V ) + ˜ δ ( r )Γ − ( T i , V )+ X T i ∈S ˜ δ ( r )Γ int ( T i , V ) + ˜ δ ( r )Γ − ( T i , V )+ X T i ∈S ˜ δ ( r )Γ int ( T i , V ) + ˜ δ ( r )Γ int ( T i , V ) + ˜ δ ( r )Γ − ( T i , V )+ X T i ∈S ˜ δ ( r )Γ int ( T i , V ) + ˜ δ ( r )Γ − ( T i , V ) , which is a consequence of the way the edges of i T i are obtained from the edges of the T i ’s. (cid:3) Corollary 5.25. We have ψ H ,t ∈ Ψ IV ( D ; ψ H , ψ H , ψ H , ) and we have that e ψ H ,t ∈ Ψ IV ( D ; e ψ H , e ψ H , e ψ H , ) .Proof. Proposition 5.24 shows that ψ H ,t and e ψ H ,t act correctly on concatenations of type 2;the proof that they behave well with respect to the other two types of concatenation is virtuallyidentical. Proposition 3.18 implies that ψ H ,t ( T ) = ψ H ,t ( T ′ ) for any morphism T → T ′ . Theargument that e ψ H ,t ( T ) = e ψ H ,t ( T ′ ) is essentially a combination of Proposition 5.15 andProposition 5.24 and is left to the reader. (cid:3) The results from the previous sections can be conveniently packaged into the followingtheorem. Theorem 5.26 (cf. Thm. 1.1 in [Par19]) . Let D be a datum for any one of Setups I*-IV*. Thenthere exists a set of perturbation data θ ( D ) and twisted moduli counts ψ M ∈ Q [ U ] , ˜ ψ M ∈ Q satisfying the obvious analogs of (i)–(v) in Thm. 1.1 in [Par19] .Proof. There is a forgetful functor from the data from the enriched setups I*-IV* to the datafor the ordinary setups I-IV considered in [Par19]. So the set of perturbation data is furnishedby Thm. 1.1 in [Par19]. We showed in Section 5.2 a datum for setups I*-IV* gives rise totwisting maps, from which we may define our twisted moduli counts as in Section 4.2. Theproperties (i)-(iv) are tautological and (v) is essentially equivalent to the axioms of the axiomsof twisting maps, as explained in Section 4.2. (cid:3) Construction of the main invariants In this section, we construct the invariants which are the central objects of this paper. Tothe data of a contact pair ( Y, ξ, V ) and an element r ∈ R ( Y, ξ, V ), we associate a unital, Z / Q [ U ]-algebra CH • ( Y, ξ, V ; r ) . There is a natural map to ordinary contact homology CH • ( Y, ξ, V ; r ) → CH • ( Y, ξ ) whichcomes from setting U = 1.A contactomorphism f : ( Y, ξ, V ) → ( Y ′ , ξ ′ , V ′ ) induces an identification CH • ( Y, ξ, V ; r ) = CH • ( Y ′ , ξ ′ , V ′ ; f ∗ r ). An exact relative symplectic cobordism ( ˆ X, ˆ λ, H ) from ( Y + , ξ + , V + ) to ( Y − , ξ − , V − ) satis-fying an energy condition induces a map CH • ( Y + , ξ + , V + ; r + ) → CH • ( Y − , ξ − , V − ; r − ). Un-fortunately, our notions of energy are not well behaved under compositions of arbitrary relativesymplectic cobordisms, so the composition of maps is not always defined.We also define various related invariants, including a reduced version which only countsReeb orbits in the complement of a codimension 2 submanifold, and “asymptotic invariants”which have good functoriality properties.6.1. Construction and basic properties of the invariants. As usual, we let ( Y n − , ξ )be a co-oriented contact manifold. Setup I*. Fix a contact pair ( Y, ξ, V ) and an element r ∈ R ( Y, ξ, V ). According to Proposi-tion 3.8, we may choose a contact form ξ = ker λ which is adapted to r . Let J : ξ → ξ be a dλ -compatible almost complex structure which preserves ξ | V . We therefore obtain a datum D for Setup I*. Theorem 5.26 applied to D gives rise to a Z / 2- graded, unital Q [ U ]-algebra CH • ( Y, ξ, V ; r ) λ,J,θ for any choice of perturbation datum θ ∈ Θ I ( D ). Setup II*. Fix pairs D ± = (( Y ± , ξ ± , V ± ) , r ± , λ ± , J ± ) of data for Setup I*, where we write r ± = ( α ± , τ ± , r ± ). Let ( ˆ X, ˆ λ, H ) be an exact relative symplectic cobordism with positive end( Y + , λ + , V + ) and negative end ( Y − , λ − , V − ), and suppose that there exists a trivialization ofthe normal bundle of H which restricts to τ ± on the positive/negative end. Proposition 6.1. Suppose that r + > e E ( H, ˆ λ | H ) r − . Then there is an induced map on homology (6.1) Φ( ˆ X, ˆ λ, H ) ˆ J,θ : CH • ( Y + , ξ + , V + ; r + ) λ + ,J + ,θ + → CH • ( Y − , ξ − , V − ; r − ) λ − ,J − ,θ − . If α + = α − and ( H, ˆ λ | H ) is a symplectization, then the same conclusion holds provided that r + ≥ r − .Proof. According to Lemma 4.24, we can choose an almost-complex structure ˆ J on ˆ X whichis d ˆ λ -compatible and agrees with ˆ J ± at infinity, and such that r + ≥ e E ( ˆ J ) r − . We thus obtaina datum D = ( D + , D − , ˆ X, H, ˆ λ, ˆ J ) for Setup II*.Given ( θ + , θ − ) ∈ Θ I ( D + ) × Θ I ( D − ), Theorem 5.26 thus provides a perturbation datum θ ∈ Θ II ( D ) with θ ( θ + , θ − ), and twisted moduli counts which give rise to the map (6.1). (cid:3) Setup III*. We have the following proposition. Proposition 6.2. Under the assumptions of Proposition 6.1, the map (6.1) is independent ofthe pair ( ˆ J , θ ) .Proof. Let ( ˆ J , θ ) and ( ˆ J , θ ) be two possible choices of such pairs. Let us first treat thecaes where ( H, ˆ λ | H ) is not a symplectization. For any ǫ > 0, Lemma 4.24 provides an interpo-lating family of almost-complex structure { ˆ J t } t ∈ [0 , such that E ( ˆ J t ) ≤ max( E ( ˆ J ) , E ( ˆ J )) + ǫ .Choosing ǫ small enough so that r + > e E ( ˆ J t ) r − , we thus get a datum D for Setup III*.Theorem 5.26 now provides perturbation data θ ∈ Θ III ( D ) mapping to ( θ , θ ), and a chainhomotopy between the maps Φ( ˆ X, ˆ λ, H ) ˆ J ,θ and Φ( ˆ X, ˆ λ, H ) ˆ J ,θ .If α + = α − and ( H, ˆ λ | H ) is a symplectization, then Lemma 4.24 implies that we may repeatthe above argument for a family of almost-complex structures ˆ J t which have vanishing energy.Tracing through the proof, it is straightforward to check that desired conclusion goes throughprovided that r + ≥ r − . (cid:3) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 53 Setup IV*. Let us consider data ˜ D = ( ˜ D , ˜ D , ( ˆ X ,t , ˆ λ ,t ) t ∈ [0 , ∞ ) ), where˜ D = ( D , D , ˆ X , ˆ λ , H )˜ D = ( D , D , ˆ X , ˆ λ , H ) D i = (( Y i , ξ i , V i ) , r i , λ i , J i ) ( i = 0 , , D , ˜ D , ˜ D are “partial data” for Setups II* and IV*, since they do not contain anyinformation about almost-complex structures. These “partial data” are assumed to obey allthe axioms stated in Section 5.1 which do not involve complex structures.The D i are (ordinary) data for Setup I*. Proposition 6.3. Suppose that the following conditions hold: • r > e E ( H ,t , ˆ λ ,t | H ,t ) r , • r > e E ( H , ˆ λ | H ) r , • r > e E ( H , ˆ λ | H ) r .Then the following diagram commutes: CH • ( Y , ξ , V ; r ) λ ,J ,θ CH • ( Y , ξ , V ; r ) λ ,J ,θ CH • ( Y , ξ , V ; r ) λ ,J ,θ Φ( ˆ X , ˆ λ ,H )Φ( ˆ X , ˆ λ ,H )Φ( ˆ X , ˆ λ ,H ) If α = α = α and if ( H , ˆ λ | H ) , ( H , ˆ λ | H ) , ( H ,t , ˆ λ ,t | H ,t ) are symplectiza-tions, then the conclusion still holds if we only assume that r i ≥ r j for i ≥ j .Proof. According to Lemma 4.27, one can choose a family of almost complex structures ˆ J ,t sothat r > e E ( ˆ J ,t ) r . Moreover, by Lemma 4.28, one may also assume that r > e E ( ˆ J ) r and r > e E ( ˆ J ) r , where J and J are the almost-complex structures induced at infinity by J ,t .We therefore obtain a datum for Setup IV* by considering D = ( ˜ D , ˆ J ) , D = ( ˜ D , ˆ J )and D = ( ˜ D , ˆ J ,t ).Theorem 5.26 applied to D now implies the commutativity of the above diagram.Under the additional hypotheses that α = α = α and that the relevant cobordisms aresymplectizations, Lemma 4.27 and Lemma 4.28 allow us to work with (families of) almost-complex structures with vanishing energy. Retracing through the above argument, we findthat the desired conclusion follows if r i ≥ r j for i ≥ j . (cid:3) Proposition 6.4. Let ( ˆ X, ˆ λ, H ) be a relative symplectic cobordism from ( Y + , ξ + , V + ) to ( Y − , ξ − , V − ) . Let ( ˆ V ± , ˆ λ ± ˆ V ) be the Liouville structure induced on ˆ V ± from the canonical Li-ouville structure of the symplectization ( ˆ Y ± , λ Y ± ) . For i ∈ { , } , consider elements r ± i ∈ R ( Y ± , ξ ± , V ± ) and let λ ± i be a contact form on Y ± which is adapted to r ± i . Suppose finallythat we have: (1) r + i > e E ( H, ˆ λ | H ) r − i ; (2) r +1 > e E ( ˆ V , ˆ λ +ˆ V ) r +2 and r − > e E ( ˆ V , ˆ λ − ˆ V ) r − .Then the following diagram commutes: CH • ( Y + , ξ + , V + ; r +1 ) λ +1 ,J + ,θ + CH • ( Y + , ξ + , V + ; r +2 ) λ +2 ,J + ,θ + CH • ( Y − , ξ − , V − ; r − ) λ − ,J − ,θ − CH • ( Y − , ξ − , V − ; r − ) λ − ,J − ,θ − Φ( ˆ X, ˆ λ,H ) Φ( ˆ Y + , ˆ V + ) Φ( ˆ X, ˆ λ,H )Φ( ˆ Y − , ˆ V − ) As usual, if α +1 = α − = α +2 = α − and ( H, ˆ λ | H ) is a symplectization, then it is enough toassume that r + i ≥ r − i , r +1 ≥ r +2 and r − ≥ r − .Proof. Observe first that the conditions (1-2) along with Proposition 6.1 ensure that the mapsappearing in the commutative diagram are well-defined. Let us now consider the strict exactsymplectic cobordisms ( ˆ X, ˆ λ, H ) λ +1 λ − and ( ˆ Y − , λ − , ˆ V − ) λ − λ − . For t ∈ [0 , ∞ ) and T > t + T )-gluing ( ˆ X t , ˆ λ t , H t ); cf. Definition 2.24. According toLemma 4.21, we have that (cf. Notation 2.20) E (( H t , ˆ λ t | H t ) α +1 α − ) = E (( H, ˆ λ | H ) α +1 α − ) + E (( ˆ V − , ˆ λ − ˆ V ) α − α − ) . It then follows from (1-2) that r +1 > e E (( H t , ˆ λ t | Ht ) α +1 α − ) r − .We can now appeal to Proposition 6.3, which implies that the composition Φ( ˆ Y , ˆ V − ) ◦ Φ( ˆ X, ˆ λ, H ) agrees with the map induced by ( ˆ X , ˆ λ , H ) = ( ˆ X, ˆ λ, H ) λ +1 λ − ; see Example 2.11.The same argument shows that composition along the upper right hand side of the diagramagrees with the map induced by ( ˆ X, ˆ λ, H ) λ +1 λ − . This proves the claim. (cid:3) We obtain the following corollary by putting together the results of the previous section. Corollary 6.5. Consider a contact pair ( Y, ξ, V ) and fix an element r ∈ R ( Y, ξ, V ) . Let D ± = ( Y, ξ, V ) , r , λ ± , J ± ) be a pair of data for Setup I* and fix θ ± ∈ Θ I ( Y ± , λ ± , J ± ) .The map (6.2) Φ( ˆ Y , ˆ λ, ˆ V ) : CH • ( Y, ξ, V ; r ) λ + ,J + ,θ + → CH • ( Y, ξ, V ; r ) λ − ,J − ,θ − defined in Proposition 6.1 is an isomorphism.Proof. In light of Proposition 6.4 and Lemma 4.8, it’s enough to consider the case λ + = λ − = λ and J + = J − = J . Let θ ∈ Θ II ( ˆ Y , ˆ λ, ˆ J ) be a lift of ( θ + , θ − ) under the forgetful mapΘ II ( ˆ Y , ˆ λ, ˆ J ) → Θ I ( Y, λ, J ) × Θ I ( Y, λ, J ). The proof of [Par19, Lem. 1.2] can be adapted toshow that the mapΦ( ˆ Y , ˆ λ, ˆ V ) ˆ J,θ : CC • ( Y, ξ, V ; r ) λ,J,θ + → CC • ( Y, ξ, V ; r ) λ,J,θ − is an isomorphism of chain complexes: one simply needs to observe that the twisted counts oftrivial cylinders coincide with the usual counts. (cid:3) We now arrive at the definition of our main invariants. Definition 6.6 (Full invariant) . Consider a contact pair ( Y, ξ, V ) and choose an element r ∈ R ( Y, ξ, V ). Let(6.3) CH • ( Y, ξ, V ; r ) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 55 be the limit (or equivalently the colimit) of { CH • ( Y, ξ, V ; r ) λ,J,θ } λ,J,θ along the maps (6.2).Proposition 6.4 and Corollary 6.5 imply that CH • ( Y, ξ, V ; r ) is canonically isomorphic to CH • ( Y, ξ, V ; r ) λ,J,θ for any admissible choice of ( λ, J, θ ).Given s ∈ Q , define CH U = s • ( Y, ξ, V ; r ) := CH • ( Y, ξ, V ; r ) ⊗ Q [ U ] Q , where the map Q [ U ] → Q sends U s . There is a natural evaluation morphism of Q [ U ]-algebras(6.4) ev U = s : CH • ( Y, ξ, V ; r ) → CH U = s • ( Y, ξ, V ; r ) . It follows tautologically from the construction that CH U =1 • ( Y, ξ, V ; r ) = CH • ( Y, ξ ). Theinvariant CH • ( Y, ξ, V ; r ) therefore admits a Q [ U ] algebra morphism to ordinary contact ho-mology (which is viewed as a Q [ U ]-algebra by letting U act by the identity).We can also define a “reduced” variant of the invariants (6.3) which are based on the twistingmap ˜ ψ . These invariants are naturally Q -algebras (as opposed to Q [ U ]-algebras) and only takeinto account Reeb orbits in the complement of the codimension 2 submanifold.More precisely, given a datum (( Y, ξ, V ) , r , λ, J ) for Setup I*, we may proceed as in Sec-tion 6.1(I*) and let ( g CC • ( Y, ξ, V ; r ) λ , d e ψ,J,θ )be the complex generated by the (good) Reeb orbits not contained in V ⊂ Y , for some per-turbation datum θ ∈ Θ I ( D ).By repeating the above arguments with the twisting maps e ψ − in place of the twisting maps ψ − , one can establish the obvious analog of Proposition 6.4 and Corollary 6.5. In particular,given choices of data ( λ + , J + , θ + ) , ( λ − , J − , θ − ) as in Corollary 6.5, there is an isomorphism(6.5) Φ( ˆ Y , ˆ λ, ˆ V ) : g CH • ( Y, ξ, V ; r ) λ + ,J + ,θ + → g CH • ( Y, ξ, V ; r ) λ + ,J + ,θ + . In particular, the following definition makes sense. Definition 6.7 (Reduced invariant) . Consider a contact pair ( Y, ξ, V ) and fix an element r ∈ R + ( Y, ξ, V ). Let(6.6) g CH • ( Y, ξ, V ; r )be the limit (or equivalently the colimit) of the algebras { g CH • ( Y, ξ, V ; r ) λ,J,θ } λ,J,θ along themaps (6.5).For future reference, we record the following corollary of the above discussion. Corollary 6.8. Let ( Y ± , ξ ± , V ± ) be contact pairs and choose elements r ± = ( α ± , τ ± , r ± ) ∈ R ( Y ± , ξ ± , V ± ) . Consider an exact relative symplectic cobordism ( ˆ X, ˆ λ, H ) with positive end ( Y + , ξ + , V + ) and negative end ( Y − , ξ − , V − ) , and suppose that τ + , τ − extend to a global triv-ialization of the normal bundle of H . If r + ≥ e E (( H, ˆ λ | H ) α + α − ) r − , then there is an induced map Φ( ˆ X, ˆ λ, H ) : CH • ( Y + , ξ + , V + ; r + ) → CH • ( Y − , ξ − , V − ; r − ) . Similarly, suppose that ( ˆ X, ˆ λ t , H t ) t ∈ [0 , is a family of exact relative symplectic cobordismswith ends ( V ± , ξ ± , V ± ) and such that τ ± extends to a global trivialization of the normal bundleof H t . If r + ≥ e E ( H t , (ˆ λ t ) | Ht ) r − , then Φ( ˆ X, ˆ λ , H ) = Φ( ˆ X, ˆ λ , H ) . The analogous statement holds for the reduced invariants g CH • ( − ) . (Bi)gradings. The Z / CH • ( − ) , g CH • ( − )shall be referred to as the homological grading . As in the case of (ordinary) contact homology,the homological grading can be lifted to a Z -grading under certain topological assumptions.We will also to refer to this Z -grading as the homological grading when it exists. Definition 6.9 (see Sec. 1.8 in [Par19]) . Let ( Y n − , ξ, V ) be a contact pair and choose r ∈ R ( Y, ξ, V ). Suppose that H ( Y ; Z ) = 0 and c ( ξ ) = 0. Then the homological Z / Z -grading defined on generators by(6.7) | γ | = CZ τ ( γ ) + n − , where τ is any trivialization of the contact distribution along γ (this is independent of τ dueto our assumption that c ( ξ ) = 0). Remark . In Definition 6.9, our assumption that c ( ξ ) = 0 is equivalent to the statementthat the canonical bundle Λ n − C ξ is trivial. The grading in general depends on a trivializationof the canonical bundle; however, our assumption that H ( Y ; Z ) = 0 along with the universalcoefficients theorem implies that H ( Y ; Z ) = 0. Hence the canonical bundle admits a uniquetrivialization. Lemma 6.11 (see (2.50) in [Par19]) . With the notation of Corollary 6.8, let us suppose that H ( Y ± ; Z ) = 0 and that c ( ξ ± ) = c ( T X ) = 0 . Then the cobordism maps described in Corol-lary 6.8 preserve the homological Z -grading. (cid:3) Under certain topological assumptions, the reduced invariant g CH • ( − ; − ) admits an addi-tional Z -grading which we will refer to as the linking number grading . Definition 6.12. Let ( Y, ξ, V ) be a contact pair and choose r ∈ R + ( Y, ξ, V ). Suppose that H ( Y ; Z ) = H ( Y ; Z ) = 0. Then the linking number grading | · | link on g CH • ( Y, ξ, V ; r ) is givenon generators by (see Definition 3.34)(6.8) | γ | link = link V ( γ ) . The grading of a word of generators is then defined to be the sum of the grading of each letter.One can verify using Lemma 3.36 that this grading is well-defined.We let(6.9) g CH • , • ( Y, ξ, V ; r )be the (super)-commutative bigraded Q -algebra, where • the first bullet refers to the homological Z -grading (which exists in view of our topo-logical assumption and the universal coefficients theorem, see Definition 6.9); • the second bullet refers to the linking number Z -grading.We sometimes drop the second grading in our notation, so the reader should keep in mindthat the notation g CH • ( − ; − ) always refers to the homological grading.We have the following lemma as a consequence of Lemma 3.36 and Lemma 6.11. Lemma 6.13. With the notation of Corollary 6.8, suppose that H ( Y ± ; Z ) = H ( X, Y + ; Z ) =0 . Then the cobordism maps described in Corollary 6.8 preserve the linking number Z -grading.In case we also have that c ( ξ ± ) = c ( T X ) = 0 , then the cobordism maps preserve the ( Z × Z ) -bigrading (6.9) . (cid:3) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 57 Asymptotic invariants. Given a contact pair ( Y, ξ, V ) and a trivialization τ of thecontact normal bundle ξ | V /T V , let R τ ( Y, ξ, V ) = { r = ( α, τ ′ , r ′ ) ∈ R ( Y, ξ, V ) | τ ′ = τ } ⊂ R ( Y, ξ, V ) . We equip R τ ( Y, ξ, V ) with a partial order (cid:22) defined by setting ( α − V , τ, r − ) (cid:22) ( α + V , τ, r + ) if r + ≥ e − min f r − , where α + V = e f α − V . It’s easy to check that this is indeed a partial order. Welet (cid:22) op denote the opposite order.We now define a functor F ( Y, ξ, V ) from the poset ( R τ ( Y, ξ, V ) , (cid:22) op ) to the category of Q [ U ]-algebras. On objects, the functor takes r to CH • ( Y, ξ, V ; r ). It remains to define thefunctor on morphisms.Given elements r ± = ( α ± V , τ, r ± ) ∈ R τ ( Y, ξ, V ), let λ ± be a contact form on Y which isadapted to r ± . Consider the symplectization ( ˆ Y , ˆ λ, ˆ V ) λ + λ − . If r − (cid:22) r + , then Lemma 4.8,Proposition 6.1 and Proposition 6.4 imply that there is a mapΦ( ˆ Y , ˆ λ, ˆ V ) : CH • ( Y, ξ, V ; r + ) → CH • ( Y, ξ, V ; r − ) . This defines F ( Y, ξ, V ) on morphisms. One can check using Proposition 6.4 that F ( Y, ξ, V ) isindeed a functor.We can similarly define a functor F + ( Y, ξ, V ) from ( R τ + ( Y, ξ, V ) , (cid:22) op ) to the category of Q -algebras using g CH • ( − ). Definition 6.14 (Asymptotic invariants) . Noting that the category of Q [ U ]-algebras is com-plete and co-complete, we let CH ←−− • ( Y, ξ, V ; τ )denote the limit of the Q [ U ]-algebras { CH • ( Y, ξ, V ; r ) } over the poset ( R τ ( Y, ξ, V ) , (cid:22) op ).We let CH −−→ • ( Y, ξ, V ; τ )denote the colimit of the Q [ U ]-algebras { CH • ( Y, ξ, V ; r ) } over the poset ( R τ ( Y, ξ, V ) , (cid:22) op ).We let g CH ←−− • ( Y, ξ, V ; τ ) and g CH −−→ • ( Y, ξ, V ; τ ) be defined similarly over the category of Q -algebras.It’s easy to check that ( R τ ( Y, ξ, V ) , (cid:22) op ) is a filtered poset. In particular, (co)limits can becomputed by restricting to (co)final subsets. In particular, given any contact form α V , the set { ( α V , τ, r ) | r ∈ R + − S ( α V ) } is (co)final and we can therefore compute: CH −−→ • ( Y, ξ, V ; τ ) = lim −→ r ∈ R + −S ( α V ) CH • ( Y, ξ, V ; ( α V , τ, r )) ,CH ←−− • ( Y, ξ, V ; τ ) = lim ←− r ∈ R + −S ( α V ) CH • ( Y, ξ, V ; ( α V , τ, r )) . and similarly for g CH ←−− • ( Y, ξ, V ; τ ) and g CH −−→ • ( Y, ξ, V ; τ ).In contrast to the invariants defined in Section 6.1, the asymptotic invariants introducedin Definition 6.14 are functorial under compositions of arbitrary relative symplectic cobor-disms which respect normal trivializations. In particular, there are no energy conditions (cf.Corollary 6.8). The following definition makes this precise. Definition 6.15. Let RelContact n be the category whose objects are pairs (( Y n − , ξ, V ) , τ ),where ( Y n − , ξ, V ) is a contact pair and τ is a trivialization of ξ | V /T V . A morphism from(( Y + , ξ + , V + ) , τ + ) to (( Y − , ξ − , V − ) , τ − ) is a (deformation class of) relative exact symplectic cobordism ( ˆ X, ˆ λ, H ) from ( Y + , ξ + , V + ) to ( Y − , ξ − , V − ) such that H admits a normal trivial-ization which restricts to τ ± at the ends. The composition of morphisms is defined by gluingrelative symplectic cobordisms as in Definition 2.7.Let Ring Z / R be the category of Z / R -algebras. Then CH ←−− • ( − ) , CH −−→ • ( − ) definefunctors RelContact n → Ring Z / Q [ U ] and g CH ←−− • ( − ) and g CH −−→ • ( − ) define functors RelContact n → Ring Z / Q . A verification of this is tedious and essentially consists of repeating the arguments of Sec-tion 6.1. Remark . The invariant CH −−→ • ( Y, ξ, V ; τ ) can be constructed directly without taking a limit.Indeed, given any element r = ( α, τ, ∈ R τ ( Y, ξ, V ), observe that r is maximal with respectto the partial order (cid:22) op . It follows that we have a canonical isomorphism CH −−→ • ( Y, ξ, V ; τ ) = CH • ( Y, ξ, V ; r ).6.4. Mixed morphisms. Consider a contact pair ( Y, ξ, V ) and elements r ± = ( α ± , τ ± , r ± ) ∈ R + ( Y, ξ, V ). In this section, we exhibit a Q -algebra map CH U =0 • ( Y, ξ, V ; r + ) → g CH • ( Y, ξ, V ; r − )under certain assumptions on r + , r − . Precomposing with (6.4) gives a Q -algebra map CH • ( Y, ξ, V ; r + ) → g CH • ( Y, ξ, V ; r − ) . Let us begin by considering a datum D = ( D + , D − , ˆ X, H, ˆ λ, ˆ J ) for Setup II*, where we let D ± = (( Y, ξ, V ) , r ± , λ ± , J ± ). Definition 6.17. Suppose that the following assumptions hold:(i) r + ≥ e E ( ˆ J ) r − ,(ii) r + > /R min α , where R min α denotes the smallest action of all Reeb orbits of α .Then we may define a twisting map ψ mix : S = ∅ II ( D ) → Q by ψ mix ( T ) = ( T ∗ ˆ V = 0 and | γ e | ∩ V − = ∅ for every e ∈ E ( T );0 otherwise. Proposition 6.18. ψ mix ( − ) is a twisting map. We begin with a crucial lemma. Lemma 6.19. Suppose that β ∈ π ( ˆ Y , γ + ⊔ ( ∪ ni =1 γ − i )) is represented by a ˆ J -holomorphic curve u : ˙Σ → ˆ X which is contained in ˆ V . Suppose moreover that r + , r − satisfy the assumptions (i)and (ii) in Definition 6.17. Then β ∗ ˆ V ≥ − p u , where p u is the total number of punctures(positive and negative) of u contained in V ± .Proof. By Lemma 5.9, we have that r + T + − r − P ni =1 T − i ≥ r + T + − r − T + e E ( ˆ J ) ≥ T + ( r + − r − e E ( ˆ J ) ) ≥ R min α (( r + / − r − e E ( ˆ J ) ) + r + / ≥ R min α r + / ≥ 1. The lemma now follows fromProposition 3.15. (cid:3) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 59 Definition 6.20. Given a tree T ∈ S = ∅ II , we say that a vertex v ∈ V ( T ) is bad if all adjacentedges are contained in V ± . Otherwise, we say that v ∈ V ( T ) is good . We denote by V b ( T )(resp. V g ( T )) the set of bad (resp. good) vertices of T . Proof of Proposition 6.18. Choose a tree T ′ ∈ S = ∅ II .Let T ′ → T be a morphism. It follows from Proposition 3.18 that T ′ ∗ ˆ V = T ∗ ˆ V . If T ′ hasno edges contained in V − , then neither does T ′ and we see that ψ mix ( T ′ ) = ψ mix ( T ).Let us now suppose that T ′ has an edge contained in V − . Note that T ′ , T have the sameexterior edges. If one of these edges is contained in V − , then ψ mix ( T ′ ) = ψ mix ( T ) = 0. Let ustherefore assume that the exterior edges of T ′ , T are not contained in V − .We are left with the case where T ′ has at least one interior edge contained in V − . If T ′ hadno bad vertices, then it would follow from Proposition 3.22 that there are no interior edgescontained in V ± , which is a contradiction. It follows that T ′ has at least one bad vertex.Let E int b ( T ′ ) ⊂ E int ( T ′ ) be the set of interior edges which occur as an outgoing edge of somebad vertex. According to Proposition 3.22, we have T ′ ∗ ˆ V = P v ∈ V g ( T ′ ) β v ∗ ˆ V + P v ∈ V b ( T ′ ) β v ∗ ˆ V + | E int ( T ′ ) − E int b ( T ′ ) | + | E int b ( T ′ ) | ≥ P v ∈ V b ( T ′ ) β v ∗ ˆ V + | E int b ( T ′ ) | = P v ∈ V b ( T ′ ) ( β v ∗ ˆ V + p − v ),where p − v denotes the number of outgoing edges of v . (Here, we have used the fact that theoutgoing edges of a bad vertex are all interior edges, which follows from our assumption thatthe exterior edges of T ′ , T are not contained in V − .)It now follows from Lemma 6.19 and the fact that T ′ has at least one bad vertex that P v ∈ V b ( T ′ ) ( β v ∗ ˆ V + p v ) ≥ P v ∈ V b ( T ′ ) (2 − p v + p − v ) ≥ P v ∈ V b ( T ′ ) ≥ 1. Hence ψ mix ( T ′ ) = ψ mix ( T ) = 0. This completes the proof that ψ mix ( T ′ ) = ψ mix ( T ).If { T i } i is a concatenation, then the argument is the same as in the proof of Proposition 5.12since every edge in i T i appears in at least one of the T i . (cid:3) We now state the main result of this section. Proposition 6.21. Consider a contact pair ( Y, ξ, V ) and elements r ± = ( α ± , τ ± , r ± ) ∈ R + ( Y, ξ, V ) . Suppose that r ′ > e E ( ˆ V , ˆ λ | ˆ V ) r − and that r + > /R min α .Then there is a map of Q -algebras (6.10) CH U =0 • ( Y, ξ, V ; r + ) → g CH • ( Y, ξ, V ; r − ) . Proof. The argument is essentially the same as the proof of Proposition 6.1. Choose data ofType I* D ± = (( Y, ξ, V ) , r ± , λ ± , J ± ). Now consider the symplectization ( ˆ Y , ˆ λ, ˆ V ). Lemma 4.24furnishes an almost-complex structure ˆ J on ˆ Y which is d ˆ λ -compatible and agrees with ˆ J ± atinfinity, and such that r + ≥ e E ( ˆ J ) r − . It now follows as in Section 4.2(II) that we have a Q -algebra chain map(6.11) Φ( ˆ Y , ˆ λ, ψ mix ) ˆ J, Θ : CC U =0 • ( Y, ξ, V ; r ′ ) J + ,θ + → g CC • ( Y, ξ, V ; r ) J − ,θ − , for perturbation data Θ ( θ + , θ − ). (cid:3) Remark . The proof of Proposition 6.21 does not show that (6.10) is independent ofauxiliary choices (i.e. ˆ J , J ± , Θ , θ ± ). To show this, one needs to extend the definition of thetwisting map ψ mix to Setups III* and IV*. One can then prove analogs of Theorem 5.26 andCorollary 6.8 for ψ mix . All of the ingredients for doing this are already in place, but we omitthe details since Proposition 6.21 is sufficient for our purposes. Corollary 6.23. Suppose that g CH • ( Y, ξ, V ; r ) = 0 for some r = ( α V , τ, r ) ∈ R + ( Y, ξ, V ) .Letting r ′ = ( α ′ V , τ ′ , r ′ ) , we have CH • ( Y, ξ, V ; r ′ ) = 0 provided that r ′ is large enough. Inparticular, CH ←−− • ( Y, ξ, V ; τ ) = 0 . (cid:3) Augmentations and linearized invariants Differential graded algebras. Let R be a commutative ring of characteristic zero.Throughout this section, all R -modules and R -algebras are assumed to be flat over R . Un-less otherwise specified, all dg algebras are assumed to be unital, Z -graded, not necessarilycommutative, and equipped with a differential of degree − Definition 7.1. An augmentation of a dg algebra A over a ring R is a morphism of dg algebras ǫ : A → R , where R is viewed as a dg algebra concentrated in degree 0. A dg algebra equippedwith an augmentation is said to be augmented. Definition 7.2. Given an augmentation ǫ : ( A, d ) → R , we consider the graded R -module A ǫ := ker ǫ/ (ker ǫ ) . The differential d descends to a differential d ǫ on A ǫ . The resultingdifferential graded module ( A ǫ , d ǫ ) is called the linearization of ( A, d ) at the augmentation ǫ (it is sometimes also called the “indecomposable quotient” in the literature).Let us say that a dg algebra is action-filtered if the underlying graded algebra is the freealgebra on a free graded module U having the following property: U admits a basis { x α | α ∈A} for some well-ordered set A such that dx α is a sum of words in the letters x β for β < α .Note that the dg algebras which arise in Symplectic Field Theory are automatically actionfiltered. Definition 7.3. Given a dg algebra A , a cylinder object for A is a dg algebra C along withmorphisms A ⊕ A C A ( i ,i ) p such that p induces an isomorphism in homology and p ◦ i = p ◦ i = id.Two morphisms f, g : A → B are said to be (left) homotopic if there exists a cylinder object C and a diagram A C AB i f i g It can be shown [DG] that (left) homotopy is an equivalence relation on action filtered dgalgebras. We may therefore introduce the following definition. Definition 7.4. Let hdga be the category whose objects are (unital, Z -graded) action-filtereddg R -algebras and whose morphisms are homotopy classes of morphisms of dg R -algebras. Wewrite R − hdga when we wish to emphasize the underlying ring.We similarly let hcdga be the category whose objects are commutative action-filtered dg R -algebras, and whose morphisms are homotopy classes of commutative dg R -algebra morphisms.A dg algebra (resp. commutative dg algebra) which is equipped with a homotopy class ofaugmentations is said to be augmented as an object of hdga (resp. hcdga ).The following lemma is folklore; a proof in the case of non-negatively graded dg algebrasover Z / Lemma 7.5. A morphism of augmented, action-filtered dg algebras induces an isomorphismof linearizations if it induces an isomorphism on homology. (cid:3) Applying Lemma 7.5 twice to the diagram which defines a homotopy of dg algebra mor-phisms, one concludes that the linearization of an action-filtered augmented dg algebra only OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 61 depends on the homotopy class of the augmentation. In particular, any action-filtered aug-mented dg algebra, viewed as an object of hdga or hcdga , admits a linearization which iswell-defined up to isomorphism. Remark . Let ( A, d ) be a dg algebra, where A is the free R -algebra generated by the set { x α | α ∈ A} . Let ǫ : A → R be the unique R -algebra map sending the generators x α to zero.Then ǫ is an augmentation if and only if dx α is contained in the ideal ( x β | β ∈ A ) for all α ∈ A (or equivalently, iff the differential has no constant term). If ǫ is an augmentation, it iscalled the zero augmentation .Suppose now that ( A, d ) is the (possibly deformed) contact algebra of some contact manifold,i.e. ( A, d ) is the commutative R -algebra generated by good Reeb orbits (for R = Q , Q [ U ]) andthe differential is defined as in Section 4.2. Suppose that ǫ : A → R is the zero augmentation.Then ker ǫ is the free R -module space generated by (good) Reeb orbits and d ǫ counts curvewith one input and one output (i.e. d ǫ is defined as in (4.6), where the sum is restricted tocurves with | Γ − | = 1). It follows that the homology of the complex ( A ǫ , d ǫ ) can be interpretedas the (possibly deformed) cylindrical contact homology.7.2. Cyclic homology.Definition 7.7. Let S be a countable, well-ordered set equipped with a map | · | : S → Z . Let A = R h S i be the free Z -graded R -algebra generated by S , where the Z -grading is induced byextending | · | multiplicatively.Let d : A → A be a differential of degree − 1. Let A := A/R , and consider the cyclic permuta-tion map τ : A → A which is defined on monomials by τ ( γ . . . γ l ) = ( − | γ | ( | γ | + ··· + | γ l | ) γ . . . γ l γ and extended R -linearly.We let A τ := A/ (1 − τ ) be the Z -graded R -module of coinvariants. Observe that d passesto the quotient. We denote the induced differential by d τ .We now define HC • ( A ) := H • ( A τ , d τ )and refer to this invariant as the reduced cyclic homology of the dg algebra ( A, d ). Remark . Definition 7.7 agrees with other definitions of reduced cyclic homology of dgalgebras (such as [Lod98, Sec. 5.3]) which may be more familiar to the reader, when both aredefined. We adopt the present definition for consistency with [BEE12].In the special case where A is the Chekanov-Eliashberg dg algebra of a Legendrian knot ina contact manifold satisfying the assumptions of [BEE12, Sec. 4.1], the algebraic invariantsconsidered in [BEE12, Sec. 4] can be translated as follows: HC • ( A ) = L H cyc ( A ), L H Ho + ( A ) = HH • ( A ) and L H Ho ( A ) = HH • ( A ). Here HH • ( − ) and HH • ( − ) denote respectively Hochschildhomology and reduced Hochschild homology.We record the following computation which will be useful to us later on. Lemma 7.9. Under the assumptions of Definition 7.7, if ( A, d ) is acyclic, then (7.1) HC k ( A ) = ( R if k is odd and positive, otherwise.Proof. Let us first prove the lemma under the assumption that S is a finite set. Note firstthat the Hochschild homology of an acyclic finitely-generated dg algebra vanishes identically.Moreover, we have an exact triangle(7.2) R [0] → HH • ( A ) → HH • ( A ) [ − −−→ , which implies that HH • ( A ) is just a copy of R concentrated in degree 1.We now consider the following Gysin-type exact triangle (see [BEE12, Prop. 4.9]):(7.3) HC • ( A ) [ − −−→ HC • ( A ) [+1] −−→ HH • ( A ) [0] −→ The desired result now follows immediately by induction, using (7.3) and the fact that HC • ( A ) vanishes in sufficiently large positive and negative degrees due to our finiteness hy-potheses. We remark that (7.3) is constructed by from a spectral sequence, whose convergencecan only be verified under finiteness assumptions.Let us now drop our assumption that S is a finite set. We instead consider an exhaustionof S by finite subsets S (1) ⊂ S (2) ⊂ . . . ⊂ S . Let ( A ( k ) , d ) ⊂ ( A, d ) be the dg sub-algebragenerated by S ( k ) . One can readily verify thatlim −→ HC • ( A ( k ) ) = HC • (lim −→ A ( k ) ) = HC • ( A ) . Observe that ( A ( k ) , d ) is acyclic for k large enough and satisfies the assumption of Defini-tion 7.7. Since we have already proved the lemma under the assumption that S is finite, itis enough to prove that the natural maps HC • ( A ( k ) ) → HC • ( A ( k +1) ) are isomorphisms for k large enough.To this end, note that the exact triangles (7.2) and (7.3) can be shown to be functorialunder morphisms of bounded dg algebras. Since quasi-isomorphisms induce isomorphisms onHochschild homology, it follows from (7.2) that the natural map HH • ( A ( k ) ) → HH • ( A ( k +1) )is an isomorphism. Since HH • ( A ( k ) ) = HH • ( A ( k +1) ) is concentrated in degree 1, and since HC i ( A ( k ) ) and HC i ( A ( k +1) ) vanish for | i | sufficiently large, the desired claim can be checkedby inductively applying the five-lemma (cf. [Lod98, Sec. 2.2.3]). (cid:3) Augmentations from relative fillings.Assumption 7.10. The constructions of Section 6 can be lifted to the category hcdga . Remark . In the context of ordinary contact homology, the analog of Assumption 7.10 iswidely believed by experts and is work in progress of De Groote [DG]. In fact, the originalwork of [EGH00] claims something stronger, namely that the contact homology algebra iswell-defined up to isomorphism in the category of dg algebras (i.e. without quotienting byhomotopy equivalence). The ongoing development of polyfolds should eventually enable oneto establish this. Definition* 7.12. Fix a contact pair ( Y, ξ, V ) and an element r ∈ R ( Y, ξ, V ). Let A ( Y, ξ, V ; r ) ∈ Q [ U ] − hcdga be the limit (or equivalently the colimit) of the dg algebras { ( CC • ( Y, ξ, V ; r ) λ , d ψ,J,θ ) } λ,J,θ under the lifts of the maps (6.2) which are furnished by Assumption 7.10.Given r ∈ R + ( Y, ξ, V ), we define e A ( Y, ξ, V ; r ) ∈ Q − hcdga analogously. Remark . With the notation of Definition* 7.12, suppose that H ( Y ; Z ) = H ( Y ; Z ) = 0. Combining Assumption 7.10 with the discussion of Section 6.2, it then followsthat e A ( Y, ξ, V ; r ) is a ( Z × Z )-bigraded differential algebra, where the differential has bidegree( − , OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 63 Definition* 7.14. Given an augmentation ǫ : A ( Y, ξ, V ; r ) → Q [ U ], we let A ǫ ( Y, ξ, V ; r ) be thelinearized chain complex (in the sense of Definition 7.2) with respect to ǫ and let CH ǫ • ( Y, ξ, V ; r )be the resulting homology.We have analogous invariants in the reduced case, which are denoted by e A ˜ ǫ ( Y, ξ, V ; r ) and g CH ˜ ǫ • ( Y, ξ, V ; r ) for an augmentation ˜ ǫ : e A ( Y, ξ, V ; r ) → Q ]. Definition 7.15. Given a contact manifold ( Y, ξ ) and a codimension 2 contact submanifold( V, ξ | V ), a relative filling ( ˆ X, ˆ λ, H ) is a relative symplectic cobordism from ( Y, ξ, V ) to theempty set.Let ( ˆ X, ˆ λ, H ) be a relative filling of ( Y, ξ, V ) and fix r ∈ R ( Y, ξ, V ). Suppose that τ extends to a normal trivialization of H . Then Lemma 6.11 and Assumption 7.10 furnishan augmentation ǫ ( ˆ X, ˆ λ, H ) : A ( Y, ξ, V ; r ) → Q [ U ]. Similarly, we have an augmentation˜ ǫ ( ˆ X, ˆ λ, H ) : e A ( Y, ξ, V ; r ) → Q .If we suppose that H ( Y ; Z ) = H ( Y ; Z ) = H ( X, Y ; Z ) = 0 and c ( T X ) = 0, thenLemma 6.13 and Assumption 7.10 imply that ˜ ǫ ( ˆ X, ˆ λ, H ) preserves the ( Z × Z )-bigrading definedin Remark 7.13. It follows that the linearized complex(7.4) e A ˜ ǫ ( Y, ξ, V ; r )inherits a ( Z × Z )-bigrading with differential of bidegree ( − , g CH ˜ ǫ • ( Y, ξ, V ; r )is a ( Z × Z )-bigraded Q -vector space.We end this section by collecting some useful lemmas which will be used later. The readeris referred to Section 3.5 for a review of open book decompositions. Lemma* 7.16. Suppose that ( ˆ W , ˆ λ, H ) is a relative filling of ( Y, ξ, V ) . Suppose that V isthe binding of an open book decomposition ( Y, V, π ) which supports ξ . Fix an element r =( α V , τ, r ) ∈ R + ( Y, ξ, V ) where τ is the canonical trivialization induced by the open book.Suppose that H admits a normal trivialization which restricts to τ . Suppose also that H ( Y ; Z ) = H ( Y ; Z ) = H ( W, Y ; Z ) = 0 and that c ( T W ) = 0 . Then the augmentation ˜ ǫ : e A ( Y, ξ, V ; r ) → Q is the zero augmentation. In particular, it depends only on ( ˆ W , ˆ λ ) and not on H .Proof. It is shown in Corollary 3.31 that there exists a non-degenerate contact form α for ( Y, ξ )which is adapted to r and has the property that all Reeb orbits are transverse to the pagesof the open book decomposition. Given auxiliary choices of almost-complex structures andperturbation data, the augmentation ˜ ǫ counts (possibly broken) holomorphic planes u whichare asymptotic to a Reeb orbit γ disjoint from H , and such that [ u ] ∗ H = 0. However, ourtopological assumptions and Lemma 3.36 implies that [ u ] ∗ H is precisely the linking numberof γ with the binding V , which is strictly positive by assumption (see Remark 3.35). (cid:3) Lemma* 7.17. Let ( ˆ X, ˆ λ, H ) and ( ˆ X ′ , ˆ λ ′ , H ′ ) be relative symplectic fillings of ( Y, ξ, V ) and ( Y ′ , ξ ′ , V ′ ) . Let f : ( Y, ξ, V ) → ( Y ′ , ξ ′ , V ′ ) be a contactomorphism. Suppose that there exist asymplectomorphism φ : ( ˆ X, ˆ λ ) → ( ˆ X ′ , ˆ λ ′ ) which coincides near infinity with the induced map ˜ f : SY → SY ′ .Given any r ∈ R ( Y, ξ, V ) , we have: (7.5) CH • ( Y, ξ, V ; r ) Q [ U ] CH • ( Y ′ , ξ ′ , V ′ ; f ∗ r ) Q [ U ] CH • ( Y ′ , ξ ′ , V ′ ; f ∗ r ) Q [ U ] Φ( ˆ X, ˆ λ,H )= Φ( ˆ X ′ ,φ ∗ ˆ λ,φ ( H ))= Φ( ˆ X ′ , ˆ λ ′ ,φ ( H )) The analogous statement holds for g CH • ( − ) (with Q in place of Q [ U ] ). In addition, if H ( Y ; Z ) = H ( Y ; Z ) = 0 and c ( T X ) = 0 , then all arrows can be assumed to preserve thebigrading in Definition 6.12.Proof. The commutativity of the top square is essentially tautological; more precisely, it followsfrom the functoriality of the moduli counts in Theorem 5.26. The commutativity of the bottomsquare follows from the observation that φ preserves the Liouville form outside a compact set.Hence ( ˆ X ′ , φ ∗ ˆ λ ) and ( ˆ X ′ , ˆ λ ′ ) are deformation equivalent. It follows by Corollary 6.8 that theyinduce the same morphism on homology. The fact that the maps preserve the bigradings on g CH ( − ; − ) (under the above topological assumptions) is a consequence on Lemma 6.13. (cid:3) Corollary* 7.18. Let ( ˆ X, ˆ λ, H ) (resp. ( ˆ X, ˆ λ, H ′ ) be relative fillings for ( Y, ξ, V ) (resp. ( Y, ξ, V ′ ) ).Suppose that V is the binding of an open book decomposition of ( Y, ξ ) and fix r = ( α, τ, r ) ∈ R ( Y, ξ, V ) , where τ is induced by the open book. Let f : ( Y, ξ, V ) → ( Y, ξ, V ′ ) be a contacto-morphism.Suppose that H ( Y ; Z ) = H ( Y ; Z ) = H ( X, Y ; Z ) = 0 and that c ( T X ) = 0 . Then g CH ˜ ǫ • , • ( Y, ξ, V ; r ) = g CH ˜ ǫ ′ • , • ( Y, ξ, V ′ ; f ∗ r ) , where both augmentations are induced by the relative fillings and the ( Z × Z ) -bigrading is definedin Definition 6.12.Proof. Indeed, since the lift of a contactomorphism to the symplectization is a Hamiltoniansymplectomorphism, it is easy to construct a symplectic automorphism of ( X, d ˆ λ ) satisfyingthe conditions of Lemma* 7.17 (see e.g. [Cha10, Sec. 3.2]). The claim now follows fromLemma* 7.16. (cid:3) Invariants of Legendrian submanifolds Invariants of contact pushoffs.Definition 8.1 (see Def. 3.1 in [CE20]) . Let ( Y, ξ ) be a contact manifold and let Λ ֒ → Y be a Legendrian embedding. By the Weinstein neighborhood theorem, the map extends toan embedding Op(Λ) ⊂ ( J Λ , ξ std ) → ( Y, ξ ), where Op(Λ) ⊂ ( J Λ , ξ std ) denotes an openneighborhood of the zero section.Let τ (Λ) be the induced codimension 2 contact embedding τ (Λ) : ∂ ( D ∗ ǫ,g Λ) = ∂ ( D ∗ ǫ,g Λ) × ⊂ T ∗ Λ × R = J Λ ֒ → ( Y, ξ ) . Here D ∗ ǫ,g Λ is the sphere bundle of covectors of length ǫ with respect to some metric g , which isa contact manifold with respect to the restriction of the canonical 1-form on T ∗ Λ. We refer to τ (Λ) as the contact pushoff of Λ ֒ → Y . Standard arguments establish that the contact pushoffis canonical up to isotopy through codimension 2 contact embeddings. By abuse of notation,we will routinely identify τ (Λ) with its image. OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 65 It follows that CH • ( Y, ξ, τ (Λ); r ) , g CH • ( Y, ξ, τ (Λ); r )can be viewed as invariants of Λ.8.2. Deformations of the Chekanov-Eliashberg dg algebra. In the spirit of the previoussections, we now consider deformations of the Chekanov-Eliashberg dg algebra of a Legendrianinduced by a codimension 2 contact submanifold.We begin with some preliminary definitions. Definition 8.2. Let Λ ⊂ ( Y, ξ ) be a Legendrian submanifold. Given a contact form ker α = ξ ,consider a Reeb chord c : [0 , R ] → Y . The linearized Reeb flow defines a path of sym-plectomorphisms P r : ξ | c (0) → ξ | c ( r ) . We say that the Reeb chord c is non-degenerate if P R ( T c (0) Λ) ∩ T c ( R ) Λ = { } . Definition 8.3 (cf. Sec. 2.1 in [BEE12]) . With the notation of Definition 8.2, let V n − C ( ξ, dα )be the canonical bundle of ξ and suppose that it admits a trivialization σ . Let Λ , . . . , Λ k bean enumeration of the components of Λ. Suppose that each Λ i has vanishing Maslov class.Suppose first of all that k = 1 (i.e. Λ is connected). Given a non-degenerate Reeb chord c , pick a path c − in Λ connecting c ( R ) to c (0). Observe that V n − T c − Λ i ⊂ V n − C ( ξ, dα ) is apath of Lagrangian subspaces along c − . We call this path L c − . The parallel transport map P r also defines a path of Lagrangian subspaces V n − P r ( T c (0) Λ i ) ⊂ V n − C ( ξ, dα ) along c . We callthis path L c .Let ˜ c = c − ∗ c be obtained by concatenating c − and c (the concatenation is from left toright). Now consider the path of Lagrangian subspaces L ˜ c = L c − ∗ L c ∗ P + , where P + is apositive rotation from P R ( T c (0) Λ) to T c ( R ) Λ (this is well-defined by our assumption that c isnon-degenerate).The Conley-Zehnder index for chords of c with respect to σ is denoted CZ + ,σ ( c ) and definedby(8.1) CZ + ,σ ( c ) = µ σ ( L ˜ c ) , where µ σ ( − ) is the Maslov index with respect to σ [MS17, Thm. 2.3.7]. This definition isindependent of the choice of c − due to our assumption that Λ has vanishing Maslov class.Note also that the resulting index depends on σ , but its parity does not.In case k > 1, the definition of the Conley-Zehnder index for chords is more complicated, anddepends on additional choices. We refer the reader to [BEE12, Sec. 2.1] (we warn the readerthat there is a typo in the formula stated there: the correct formula for the Conley-Zehnderindex for chords should read CZ + ,σ ( c ) = | c | − φ − − φ Λ ( x )) /π + ( n − / Remark . It may happen that a Reeb orbit can also be viewed as a Reeb chord with samestarting and end point. In this case, we have in general that CZ + ( c ) = CZ( c ).Let us now consider a contact pair ( Y, ξ, V ) and a Legendrian submanifold Λ ⊂ ( Y − V, ξ ).We let Λ , . . . , Λ k be an enumeration of the connected components of Λ. It will be convenientto assume that H (Λ k ; Z ) = 0. Definition/Assumption* 8.5. Fix r ∈ R ( Y, ξ, V ). Let us also choose the following additionaldata: • a contact form λ ∈ Ω ( Y ) which adapted to r and has the property that all Reeb orbitsand Λ-Reeb chords are non-degenerate • a dλ -compatible almost-complex structure J on ξ which preserves T V . Given a class β ∈ π ( ˆ Y ; c + , Γ − Λ , Γ), we let M (Γ − , Γ +Λ , Γ − Λ ; β ) J be the moduli space of connected ˆ J -holomorphic curves, modulo R -translation representing theclass β . (Here we follow the notation of Section 2.2, where c + is a Reeb chord of Λ ⊂ ( Y, λ ),Γ − Λ = { c − , . . . , c − σ } is an ordered collections of (not necessarily distinct) Reeb chords, andΓ − is a collection of Reeb orbits.) Since V ⊂ ( Y − Λ , λ ) is a strong contact submanifold,a straightforward extension of Siefring’s intersection theory defines an intersection numberˆ V ∗ β ∈ Z .Let us now consider the semi-simple ring R = ⊕ ki =1 Q [ U ] , and let e , . . . , e k be the idempotents corresponding to the unit in each summand.Let CL • ( Y, ξ, V, Λ; r ) λ be the free R algebra generated by (good) Reeb orbits of ( Y, α ) andReeb chords of Λ, subject to the following relations: • γ γ = ( − | γ || γ | γ γ for Reeb orbits a, b , • If c ij is a Reeb chord from Λ i to Λ j , then e k c ij e l = δ jk c ij δ il .We assume that there exists a suitable virtual perturbation framework so that one can definea differential d J on generators as follows: • for a Reeb chord c , we let d J ( c ) = X | Aut | M ( c + , Γ − Λ , Γ − ; β ) J U ˆ V ∗ β c − . . . c − σ γ . . . γ s , where the sum is over choices of β ∈ π ( ˆ Y ; c + , Γ − Λ , Γ), for all possible choices of Γ − Λ , Γ − ; • for a Reeb orbit γ , we let d J ( γ ) be the usual deformed contact homology differential,as constructed (using Pardon’s perturbation framework) in Section 6.1.This induces a differential d J on CL • ( Y, ξ, V, Λ; r ) λ by the graded Leibnitz rule.We assume that ( CL • ( Y, ξ, V, Λ; r ) λ , d J ) is independent of λ, J up to canonical isomorphismin Q [ U ]- hdga . We denote the resulting object by(8.2) L ( Y, ξ, V, Λ; r )and we let CH • ( Y, ξ, V, Λ; r ) be its homology. We assume that L ( Y, ξ, V, Λ; r ) satisfies thelimited functoriality described in Proposition* 8.7. Definition/Assumption* 8.6. Carrying over the hypotheses and notation from Defini-tion/Assumption* 8.5, let us consider the semi-simple ring˜ R = ⊕ ki =1 Q , where we again let e , . . . , e k be the idempotents corresponding to the unit in each summand.We let g CL • ( Y, ξ, V, Λ; r ) λ be the free ˜ R algebra generated by (good) Reeb orbits of ( Y, α ) which are not contained in V and Λ Reeb chords, subject to the following relations: • γ γ = ( − | γ || γ | γ γ for Reeb orbits a, b , • If c ij is a Reeb chord from Λ i to Λ j , then e k c ij e l = δ jk c ij δ il .This algebra is again Z / Z -graded when the canonical bundle istrivialized. We assume that a suitable virtual perturbation framework has been chosen so thatone can define a differential ˜ d L on generators as follows: OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 67 • for a Reeb chord c , we let d J ( c ) = X | Aut | M ( c + , Γ − Λ , Γ − ; β ) J δ ( ˆ V ∗ β ) c − . . . c − σ γ . . . γ s , where δ : R → { , } satisfies δ (0) = 1 and δ ( s ) = 0 for s = 0 and the sum is over allpossible choices of homotopy classes as in Definition/Assumption* 8.5. • for a Reeb orbit γ , we let ˜ d J ( γ ) be the reduced contact homology differential associatedto the twisted moduli counts ˜ ψ M , which only counts curves disjoint from ˆ V This induces a differential ˜ d J on CL • ( Y, ξ, V, Λ; r ) λ by the graded Leibnitz rule.We assume that ( g CL • ( Y, ξ, V, Λ; r ) λ , ˜ d J ) is independent of λ, J up to canonical isomorphismin Q - hdga . We denote the resulting object by(8.3) e L ( Y, ξ, V, Λ; r )and we let CH • ( Y, ξ, V, Λ; r ) be its homology. We also assume that e L ( Y, ξ, V, Λ; r ) satisfies thelimited functoriality described in Proposition* 8.7. Proposition* 8.7 (cf. Corollary 6.8) . Let ( Y ± , ξ ± , V ± ) be contact pairs and choose elements r ± = ( α ± , τ ± , r ± ) ∈ R ( Y ± , ξ ± , V ± ) . Consider an exact relative symplectic cobordism ( ˆ X, ˆ λ, H ) with positive end ( Y + , ξ + , V + ) and negative end ( Y − , ξ − , V − ) , and suppose that τ + , τ − extendto a global trivialization of the normal bundle of H .Suppose that L ⊂ ( ˆ X, ˆ λ, H ) is a cylindrical Lagrangian submanifold which is disjoint from H , with ends Λ ± ⊂ ( Y ± − V ± , ξ ± ) .If r + ≥ e E (( H, ˆ λ H ) α + α − ) r − , then there is an induced map Φ( ˆ X, ˆ λ, H, L ) : L ( Y + , ξ + , V + ; r + ) → L ( Y − , ξ − , V − ; r − ) . The analogous statement holds for the reduced invariants e L ( − ) . Definition* 8.8. Let ǫ : A ( Y, ξ, V ; r ) → Q [ U ] be an augmentation. Then we let L ǫ ( Y, ξ, V, Λ; r ) := L ( Y, ξ, V, Λ; r ) ⊗ A ( Y,ξ,V ; r ) Q [ U ] , with differential d L ⊗ 1. This is naturally also a differential graded Q [ U ] algebra.We similarly define e L ǫ ( Y, ξ, V, Λ; r ) := e L ( Y, ξ, V, Λ; r ) ⊗ e A ( Y,ξ,V ; r ) Q , which is naturally a differential graded Q -algebra. Remark . The algebra L ǫ ( Y, ξ, V, Λ; r ) is the twisted analog of the Legendrian homology dgalgebra (or Chekanov-Eliashberg dg algebra) described in [BEE12, Sec. 4.1].We now discuss gradings on the above Legendrian invariants. Definition* 8.10. Let ( Y, ξ, V ) be a contact pair and choose r ∈ R ( Y, ξ, V ). Let Λ ⊂ ( Y − V, ξ )be a Legendrian submanifold. Suppose that H ( Y ; Z ) = 0 and that c ( ξ ) = 0. Then the Leg-endrian homological Z / L ( Y, ξ, V, Λ; r ) (resp. e L ( Y, ξ, V, Λ; r ) for r ∈ R + ( Y, ξ, V ; r ))lifts to a canonical Z -grading given on orbits by (6.7) and given on chords by(8.4) | c | = CZ + ,τ ( c ) − , which is well-defined due to our topological assumptions.The invariants L ǫ ( Y, ξ, V, Λ; r ) , HC ( L ǫ ( Y, ξ, V, Λ; r )) and (for r ∈ R + ( Y, ξ, V ; r )) e L ǫ ( Y, ξ, V, Λ; r ) , HC ( e L ǫ ( Y, ξ, V, Λ; r ))inherit a Z -grading which we also refer to as the homological grading. Lemma* 8.11. With the notation of Proposition* 8.7, suppose that H ( Y ± ; Z ) = 0 and that w ( L ) = c ( ξ ± ) = c ( T X ) = 0 . Then the cobordism maps described in Corollary 6.8 preservethe Legendrian homological Z -grading. (cid:3) As in Section 6.2, there is also a linking number Z -grading on the reduced Legendrianinvariants under certain topological assumptions. Definition* 8.12. Let ( Y, ξ, V ) be a contact pair and choose r ∈ R + ( Y, ξ, V ) Let Λ ⊂ ( Y, ξ )be a Legendrian submanifold. Suppose that H ( Y ; Z ) = H ( Y ; Z ) = π (Λ) = π (Λ). Then the linking number grading | · | link on e L ( Y, ξ, V, Λ; r ) is given on Reeb chords by(8.5) | c | link = link V ( c ; Λ) . It is given on Reeb orbits by (6.8). The grading is extended to arbitrary words by defining thegrading of word to be the sum of the gradings of its letters. One can verify using Lemma 3.37that this grading is well-defined.We let(8.6) e L • , • ( Y, ξ, V, Λ; r )be the bigraded differential Q -algebra of bidegree ( − , • the first bullet refers to the (Legendrian) homological Z -grading (which is well-definedin view of our topological assumptions and the universal coefficients theorem, see Def-inition* 8.10); • the second bullet refers to the (Legendrian) linking number grading.We also have the following lemma which follows from Lemma* 8.11 and Lemma 3.37. Lemma* 8.13. With the notation of Proposition* 8.7, suppose that H ( Y ± ; Z ) = H ( Y ± ; Z ) = H ( X, Y + ; Z ) = 0 and that π (Λ ± ) = π (Λ ± ) = 0 . Then the cobordism maps described inProposition* 8.7 preserve the linking number Z -grading. In case we also have that w (Λ ± ) = c ( ξ ± ) = c ( T X ) = 0 , then the cobordism maps preserve the ( Z × Z ) -bigrading (8.6) . (cid:3) Corollary* 8.14. Consider a contact pair ( Y, ξ, V ) and an element r ∈ R + ( Y, ξ, V ) . Let Λ ⊂ ( Y, ξ ) be a Legendrian submanifold. Let ( W, λ, H ) be a relative filling for ( Y, ξ, V ) andlet ˜ ǫ : ˜ A ( Y, ξ, V ; r ) → Q be the induced augmentation. Suppose that H ( Y ; Z ) = H ( Y ; Z ) = H ( W, Y ; Z ) = 0 , that π (Λ) = π (Λ) = 0 and that w (Λ) = c ( ξ ) = c ( T W ) = 0 .Then e L ˜ ǫ • , • ( Y, ξ, V ; r ) inherits the structure of a ( Z × Z ) -bigraded Q -algebra with differential of bidegree ( − , .Moreover, HC • , • ( e L ˜ ǫ ( Y, ξ, V ; r )) inherits the structures of a ( Z × Z ) -bigraded Q -vector space.Proof. According to Lemma 3.36 and our topological hypotheses, the augmentation ˜ ǫ preservesthe linking number. The first claim follows. For the second claim, note that both the homo-logical grading and linking number grading are preserved by the cyclic permutation operator τ , and hence pass to reduced cyclic homology (see Definition 7.7). (cid:3) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 69 The effect of Legendrian surgery. The familiar procedure of attaching a handle indifferential topology can be performed in the symplectic category. There are various essentiallyequivalent approaches to doing this in the literature. For concreteness, we exclusively followin this paper the construction described in [vK17, Sec. 3.1] which we now summarize. Construction 8.15 (Attaching a handle) . Let ( X n , λ ) be a Liouville cobordism with positiveboundary ( Y n − , ξ = ker( λ )). Let Λ ⊂ ( Y − V, ξ ) be an isotropic sphere with trivializedconformal symplectic normal bundle (the latter condition is vacuous if Λ is a Legendrian).Choose an arbitrary open neighborhood U of Λ which we refer to as the attaching region .We may now glue a model handle H along Y inside U , following the detailed constructiongiven in [vK17, Sec. 3.1]. The gluing is carried out by identifying the Liouville flow near Λwith the flow on H . We note that this gluing procedure involves some auxiliary choices whichwe do not state here.The outcome of the procedure (for any of the above auxiliary choices) is a Liouville cobordism( X, λ ) with positive boundary ( Y, ξ = ker( λ | Y )). We say that this domain is obtained from( X , λ ) by attaching a handle along Λ, or Legendrian surgery on Λ. As it well-known fromdifferential topology, Y differs from Y by surgery along Λ.In [BEE12], Bourgeois, Ekholm and Eliashberg studied the effect of handle attachment onvarious flavors of symplectic and contact homology. In particular, they describe exact se-quences which should govern the change in these invariants and describe the moduli spaces ofholomorphic curves which underly the existence of these exact sequences. While there is wide-spread agreement in the community about the validity of these exact sequences, their existence(and the invariance of the terms appearing in them) has not been rigorously established in theliterature.We expect that the surgery exact sequence for linearized contact homology described in[BEE12, Thm. 5.1] directly generalizes to the setting of our deformed invariants. In particu-lar, the arguments sketched in [BEE12, Sec. 6] also apply directly our setting, up to routinemodifications.Let us now state precisely the expected surgery formulas for our deformed invariants.Let ( Y n − , ξ ) be a contact manifold and let ( V, ξ | V ) ⊂ ( Y , ξ ) be a codimension 2 contactsubmanifold with trivial contact normal bundle. Let ( X , λ ) be a Liouville domain withpositive boundary ( Y , ξ = ker λ ) and let H ⊂ ( X, λ ) be a symplectic submanifold which ispreserved set-wise by the Liouville flow near ∂X = Y and such that ∂H = V .Let Λ ⊂ ( Y − V, ξ ) be an isotropic sphere with trivialized conformal symplectic normalbundle (the latter condition is vacuous if Λ is a Legendrian). Let ( X, λ ) be the Liouville domainobtained by attaching a Weinstein handle along Λ (see [Gei08, Sec. 6.2]) and let ( Y, ξ = ker λ )be the positive boundary.We may assume that the attaching region is disjoint from V ⊂ Y . By abuse of notation,we therefore view V as a codimension 2 contact submanifold of ( Y , ξ ) and ( Y, ξ ) and view H as a submanifold of X and X . We also identify R ( Y , ξ , V ) = R ( Y, ξ, V ).We let ( ˆ X , ˆ λ , H ) be the completion of ( X , λ , H ) and let ( ˆ X, ˆ λ, H ) be the completion of( X, λ, H ) There are a natural (strict) markings e : R × Y → ˆ X (8.7) ( t, y ) ψ t ( y ) , (8.8) e : R × Y → ˆ X (8.9) ( t, y ) ψ t ( y ) , (8.10) where ψ (resp. ψ ) is the Liouville flow in ˆ X (resp. in ˆ X ).We let ǫ (( ˆ X , ˆ λ , H ) , e ) : A ( Y, ξ, V ) → Q [ U ] and ǫ (( ˆ X, ˆ λ, H ) , e ) : A ( Y, ξ, V ) → Q [ U ] bethe induced augmentations. Finally, in order to have well-defined homological Z -gradings, weassume that H ( Y ; Z ) = H ( Y ; Z ) = 0 and that c ( T X ) = c ( T X ) = 0. Theorem* 8.16 (cf. Thm. 5.1 in [BEE12]) . If Λ is a Legendrian sphere, we have the followingexact triangle, where the top horizontal arrow is the natural map induced by an exact relativesymplectic cobordism. (8.11) CH ǫ •− ( n − ( Y, ξ, V ; r ) CH ǫ •− ( n − ( Y , ξ , V ; r ) HC • ( L ǫ ( Y , ξ , V, Λ; r )) [ − If dim Λ = k ≤ n − , then we have that (8.12) H ∗ (Cone( CH ǫ •− ( n − ( Y, ξ, V ; r ) → CH ǫ •− ( n − ( Y , ξ , V ; r ))) = ( Q [ U ] if ∗ = n − k + 2 N otherwise. Theorem* 8.17 (cf. Thm. 5.1 in [BEE12]) . With the above setup and r ∈ R + ( Y, ξ, V ) , con-sider the augmentation ˜ ǫ (( ˆ X , ˆ λ , H ) , e ) : ˜ A ( Y, ξ, V ) → Q and its pullback ˜ ǫ (( ˆ X, ˆ λ, H ) , e ) :˜ A ( Y, ξ, V ) → Q .If Λ is a Legendrian sphere, we have the following exact triangle, where the top horizontalarrow is the natural map induced by an exact relative symplectic cobordism. (8.13) g CH ˜ ǫ •− ( n − ( Y, ξ, V ; r ) g CH ˜ ǫ •− ( n − ( Y , ξ , V ; r ) HC • ( e L ˜ ǫ ( Y , ξ , V, Λ; r )) [ − If dim Λ = k ≤ n − , then we have that (8.14) H ∗ (Cone( g CH ˜ ǫ •− ( n − ( Y, ξ, V ; r ) → g CH ˜ ǫ •− ( n − ( Y , ξ , V ; r )) = ( Q if ∗ = n − k + 2 N otherwise.Remark . With the setup of Theorem* 8.17, let us in addition assume that H ( Y ; Z ) = H ( X , Y ) = H ( Y ; Z ) = H ( X, Y ; Z ) = 0. Then Lemma 6.13 and Corollary* 8.14 provide anadditional linking number Z -grading on the invariants appearing in the surgery exact sequences.The resulting ( Z × Z )-bigrading is preserved by the maps in the surgery exact sequence.Indeed, Lemma 6.13 ensures the top horizontal map preserves the linking number Z -grading.The bottom right map counts holomorphic disks with one positive interior puncture, k -negativeboundary punctures, and with boundary mapping to S Λ (the relevant moduli space is describedin [BEE12, Sec. 2.6]). Hence one can readily verify (cf. Lemma 3.37) that this map alsopreserves the linking number grading. Finally, the bottom left map is defined algebraically asthe connecting map in the long exact sequence. Since the internal differentials of the relevantchain complexes preserve the linking number grading, this connecting map does too. OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 71 Some computations Vanishing results. Recall that a contact manifold ( Y n − , ξ ) is said to be overtwisted if it contains an overtwisted disk; see [BEM15, Sec. 1]. In general, if ( Y, ξ ) is overtwisted and C ⊂ Y is a closed subset, then ( Y − C, ξ ) may not be overtwisted. Definition 9.1 (see [Etn13]) . Given an overtwisted contact manifold ( Y, ξ ), a contact sub-manifold ( V, ξ | V ) is said to be loose if ( Y − V, ξ ) is overtwisted. A Legendrian submanifold ℓ ⊂ ( Y, ξ ) is said to be loose if ( Y − ℓ, ξ ) is overtwitsed. This is consistent with Murphy’sdefinition of a loose Legendrian (recall that a Legendrian in an arbitrary contact manifold issaid to be loose if it admits a loose chart ; see [Mur19]).In this section, we prove that the invariants constructed in Section 6.1 vanish on loose contactsubmanifolds. It follows that the corresponding Legendrian knot invariants also vanish on looseLegendrians. Theorem 9.2. Suppose that ( Y, ξ, V ) is a contact pair such that ( Y, ξ ) is overtwisted and ( V, ξ | V ) is loose. Given any element r ∈ R ( Y, ξ, V ) , we have CH • ( Y, ξ, V ; r ) = g CH • ( Y, ξ, V ; r ) = 0 . We collect some definitions which will be useful in proving Theorem 9.2. Recall the defi-nition of an almost-contact structure stated in Definition 10.1. We let alm U( n − ( S n − ) bethe set of (homotopy classes of) almost-contact structures on S n − . It follows by the maintheorem of [BEM15] that alm U( n − ( S n − ) is in canonical correspondence with the set of over-twisted contact structures on the sphere, a fact which will be used implicitly in the proof ofTheorem 9.2.In Appendix A, we study connected sums of almost-contact manifolds (more generally ofalmost G -manifolds). In particular, we prove (see Corollary A.13) that for any fixed element β ∈ alm U( n − ( S n − ), the operation of connected sum endows alm U( n − ( S n − ) with a groupstructure with identity element β . The isomorphism class of the resulting group is independentof β (see Remark A.14). This is a precise version of a folklore result in contact topology; seee.g. [CMP19, Sec. 6].For the remainder of this section, we fix β ∈ alm U( n − ( S n − ) to be the almost-contactstructure induced by the standard contact structure on the sphere.Given a pair of contact manifolds ( M , α ) , ( M , α ), one can also consider their connectedsum ( M M , α α ), which is obtained by gluing-in a neck along Darboux balls in M , M .This operation is discussed in Remark A.10. As noted there, the two a priori different notionsof a connected sum of (almost-)contact manifolds commute with the forgetful map from contactmanifolds to almost-contact manifolds. Proof of Theorem 9.2. It is enough to prove that the invariants vanish for a particular choiceof non-degenerate contact form ˜ α on Y which is adapted to r . To construct such a form, wefollow arguments of Bourgeois and Van Koert in [BvK10, Sec. 6.2].Using Construction 3.5, we define an auxiliary contact form α in a small neighborhood N of V with the property that V is a strong contact submanifold and that α is adapted to r .After possibly shrinking N , we can assume that ( Y − N , ξ ) is overtwisted. We now extend α arbitrarily to a globally-defined, non-degenerate contact form on ( Y, ξ ). (Since α V is non-degenerate, Construction 3.5 produces a non-degenerate contact form on N , so it extendsunproblematically to a global non-degenerate contact form).Choose a Darboux ball B ⊂ Y whose closure is disjoint from N . Let B ′ ⊂ B be a smallerDarboux ball and let A = B − B ′ . Let β denote the almost-contact structure on B obtained by restricting ξ . As in Appendix A.2 let alm U( n − ( B, A ; β ) be the set of almost-contactstructures on B which agree with β near A .In Appendix A.2, we describe a group action of alm U( n − ( S n − ) on alm U( n − ( B, A ; β ),which is obtained by connect-summing with an almost-contact sphere along a disk whoseclosure is disjoint from A (see (A.1)).Bourgeois and van Koert construct a special overtwisted contact form ( S n − , α L ) on thesphere (see [BvK10, Sec. 2.2]). We now form the connected sum of ( S n − , α L ) with ( Y, α ),where we assume that the gluing happens entirely inside of B ′ . It now follows by the aboveremarks that we can further connect sum with another overtwisted contact sphere ( S n − , α ′ L )so that ( B, α α L α ′ L | B ) is formally contact isotopic to ( B, α | B ), through a contact isotopyfixed near A . Unwinding the definitions, (see Proposition A.9 and Remark A.10), this meansprecisely that there exists a diffeomorphism ψ : B → B S n − S n − fixed near the boundaryand a formal contact isotopy from ker ψ ∗ ( α α L α ′ L ) | B to ξ | B = ker α | B , which is fixed near A .On the one hand, the arguments of [BvK10, Sec. 6.2] produce a Reeb orbit γ contained inthe region of the connected sum( Y S n − S n − , α α L α ′ L )corresponding to ( S n − , α L ) such that γ bounds a single, transversally cut-out J -holomorphicplane, for some suitable J on the symplectization. By suitably adjusting the necks along whichone forms the connected sum, they show that this plane can be assumed to stay entirely inthe region corresponding to ( S n − , α L ) , i.e. it cannot cross the necks.On the other hand, if we extend ψ to a diffeomorphism Y → Y S n − S n − by letting it bethe identity outside of B , we observe that ker ψ ∗ ( α α L α ′ L ) is formally isotopic to ξ = ker α .Moreover, these contact structures agree on Y − B ⊃ N . Since ( Y − N , ξ ) is overtwisted,it follows from the relative h-principle for overtwisted contact structures (see [BEM15, Thm.1.2]) that there is a smooth isotopy φ t fixed on N so that ˜ α := φ ∗ ψ ∗ ( α α L α ′ L ) is a contactform for ( Y, ξ ). By construction, ˜ α = α on N , so ˜ α is adapted to r . However, it follows fromthe previous paragraph that CH • ( Y, ξ, V ; r ) vanishes when we compute it using the form ˜ α .An analogous argument shows that g CH • ( Y, ξ, V ; r ) vanishes as well. (cid:3) We also state a vanishing result for the deformed Chekanov-Eliashberg dg algebra of certainloose Legendrians. To set the notation, let us now assume that ( Y n − , ξ, V ) is an arbitrarycontact pair. Fix r ∈ R ( Y, ξ, V ). Proposition* 9.3. Suppose that Λ ⊂ ( Y − V, ξ ) is a loose Legendrian submanifold. Then L ( Y, ξ, V, Λ; r ) and e L ( Y, ξ, V, Λ; r ) are acyclic. Given augmentations ǫ : A ( Y, ξ, V ; r ) → Q [ U ] and ˜ ǫ : ˜ A ( Y, ξ, V ; r ) → Q , the invariants L ǫ ( Y, ξ, V, Λ; r ) and e L ǫ ( Y, ξ, V, Λ; r ) are also acyclic.Proof. The argument is the same as that which shows that the (undeformed) Chekanov-Eliashberg dg algebra of a loose Legendrian is acyclic (see e.g. [Mur19, Sec. 5]): up to Legen-drian isotopy in Y − V , we can find a chord c of arbitrarily small action which bounds a singlehalf-disk. This disk can be assumed to stay in a small ball disjoint from V for action reasons.Hence we have d ( c ) = 1. (cid:3) Nonvanishing results: bindings of open books. The following theorem is the mainresult of this section. Theorem 9.4. Consider a contact pair ( Y, ξ, V ) . Suppose that Y admits an open book decom-position ( Y, B, π ) which supports the contact structure ξ and realizes V = B as its binding. OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 73 Viewing ( B, τ ) as a framed contact submanifold, where τ denote the trivialization of B ⊂ Y induced by the open book decomposition, we have g CH • ( Y, ξ, B ; r ) = 0 for any r = ( α B , τ, r ) ∈ R + ( Y, ξ, B ) . By combining Theorem 9.4 with Corollary 6.23, we obtain the following result. Corollary 9.5. Under the hypotheses of Theorem 9.4, if r ′ is large enough and we write r ′ = ( α B , τ, r ′ ) , then CH • ( Y, ξ, B ; r ′ ) = 0 . Proof of Theorem 9.4. According to Corollary 3.31, the open book decomposition ( Y, B, π )supports a non-degenerate Giroux form α which is adapted to r , for any r = ( α B , τ, r ) ∈ R ( Y, ξ, B ) with r > g CC • ( Y, ξ, B ; r ) generated by (good) Reeb orbits of α not containedin B . After fixing an almost complex structure J : ξ → ξ which is compatible with dα andpreserves ξ | B , and a choice of perturbation data θ ∈ Θ I (( Y, ξ, B ) , α, J ), we get a differential d J = d ( ˜ ψ B , J, θ ) and the homology of the resulting chain complex is (canonically isomorphicto) g CH • ( Y, ξ, B ; r ).Let us suppose for contradiction that g CH • ( Y, ξ, B ; r ) = 0. This means that 1 is in the imageof the differential. By the Leibnitz rule, this implies that there exists some good Reeb orbit γ : S → Y and a relative homotopy class β ∈ π ( Y, γ ) such that the twisted moduli countof planes positively asymptotic to γ in the homotopy class β is non-zero. To state this moreformally in the language of Section 3.4, let T ∈ S I (( Y, ξ, B ) , α, J ) be the tree with a single inputedge e and a single vertex v , where e is decorated with the Reeb orbit γ and v is decoratedwith the β ∈ π ( Y, γ ). Then we have that e ψ B ( T ) = 0.In particular, this implies that M ( T ) = ∅ . Hence there exists T ′ → T such that T ′ is representable by a J -holomorphic building. The proof of Proposition 5.6 shows that wemay assume that T ′ does not have any edges contained in B (since otherwise we would have e ψ B ( T ′ ) = e ψ B ( T ) = 0).It follows by Proposition 3.19 that T ′ ∗ ˆ B = P v ∈ V ( T ′ ) β v ∗ ˆ B , and Corollary 3.21 implies thatall the terms on the right-hand side are non-negative. Since e ψ B ( T ′ ) = e ψ B ( T ) = 0, it followsby definition of the reduced twisting maps that T ′ ∗ ˆ V = 0. Hence β v ∗ ˆ B = 0 for all v ∈ V ( T ′ ).For topological reasons, there exists ˜ v ∈ V ( T ′ ) such that ˜ v has a single incoming edge andno outgoing edges. Hence ˜ v is represented by a J -holomorphic plane u which is asymptotic tosome Reeb orbit ˜ γ .By positivity of intersection (see Proposition 3.11) and the fact that β ˜ v ∗ ˆ B = 0, the imageof u is contained in R × ( Y \ B ). Thus ˜ γ is contractible in Y \ B , which implies that thecomposition π ◦ ˜ γ : S → Y \ B → S has degree 0. This is a contradiction: since α is a Giroux form, π ◦ ˜ γ must be an immersion(by Remark 3.27) and hence have nonzero degree. (cid:3) Given a Legendrian embedding Λ ֒ → ( Y, ξ ), its contact pushoff τ (Λ) ֒ → ( Y, ξ ) is a codimen-sion 2 contact embedding. As mentioned in Section 8.1, once can view CH • ( Y, ξ, τ (Λ); r ) , g CH • ( Y, ξ, τ (Λ); r )as invariants of Λ. It is natural to attempt to compute these invariants on loose legendrians . If ( Y, ξ ) is an overtwisted contact manifold and Λ ⊂ ( Y, ξ ) is a Loose Legendrian, then( Y − τ (Λ) , ξ ) is a overtwisted. Hence Theorem 9.2 implies that CH • ( Y, ξ, τ (Λ); r ) = g CH • ( Y, ξ, τ (Λ); r ) = 0 . In contrast, suppose that ( Y, ξ ) = obd( T ∗ S n − , id) for n ≥ B be the bind-ing. The zero section of any page is a loose Legendrian by [CM19, Prop. 2.9]. Moreover,it follows easily from the definition of the contact pushoff (see Definition 8.1) that we have B = τ (Λ). Corollary 9.5 thus implies that g CH • ( Y, ξ, Λ; r ) = 0 for all r ∈ R + ( Y, ξ, B ) and that CH • ( Y, ξ, Λ; r ′ ) = 0 for suitable r ′ ∈ R ( Y, ξ, V ).Since CH • ( − ) is a deformation of contact homology, it is natural to expect that this defor-mation should be trivial on a the pushoff of a loose legendrian. Conjecture 9.6. Let Λ ⊂ ( Y, ξ ) be a loose Legendrian and let τ ( V ) be its contact pushoff.Then given any r ∈ R ( Y, ξ, τ (Λ)) , we have CH • ( Y, ξ, τ (Λ); r ) = CH • ( Y, ξ ) ⊗ Q Q [ U ] . That is, the pushoff of a loose Legendrian induces the trivial deformation of contact homology. This conjecture is consistent with the computations carried out in Section 9.3, but we haveno further evidence to support it. We do not have a conjectural characterization of g CH • ( − )on the pushoff of a loose Legendrian.We end this section with an analog of Theorem 9.4 for deformed (reduced) Chekanov-Eliashberg dg algebra introduced in Section 8.2. Theorem* 9.7. Let ( Y, ξ, V ) be a contact pair and let Λ ⊂ ( Y − V, ξ ) be a Legendrian sub-manifold. Suppose that ξ supports an open book decomposition π with binding B = V , suchthat Λ is contained in a single page. Let τ be the trivialization of ξ | V /T V induced by the openbook. Then we have e L ( Y, ξ, V, Λ; r ) = 0 . Proof. The proof is identical to that of Theorem 9.4; namely, one argues that the image ofany Reeb orbit or chord under the differential cannot contain a term of degree zero, whichimmediately implies the claim. (cid:3) We note that is was proved by Honda and Huang [HH19, Cor. 1.3.3] that any Legendrian Λ ina contact manifold ( Y, ξ ) is contained in the page of some compatible open book decomposition.Hence it follows from Theorem* 9.7 that every Legendrian is tight in the complement of somecodimension 2 contact submanifold.9.3. Explicit computations in open books. We now perform certain explicit computationsin open book decompositions which will be used in applications in the next sections. We assumethroughout this section that n ≥ S n − with a Riemannian metric g having the property that all geodesics arenon-degenerate. Such metrics, which are typically referred to as “bumpy” in the literature,are generic in the space of Riemannian metrics (see [Abr70] or [Kli77, 3.3.9]). It can be shown[Abr70] that any manifold endowed with a bumpy metric admits a closed geodesic of minimallength. We let ρ > S n − , g ).To set the stage for this section, it will be useful to recall some general facts about coordinatesystems. Given a system of local coordinates ( q , . . . , q m ) on some manifold M , the dual OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 75 coordinates ( p , . . . , p m ) in the fibers of the T ∗ M are characterized by the property that( q , . . . , q m , p , . . . , p m ) = m X i =1 p i dq i . Unless otherwise indicated, a pair ( q , p ) refers to a system of local coordinates in the cotangentbundle of a manifold, where p is dual to q .It will sometimes also be useful to work with Riemannian normal coordinates. Recall that ona Riemannian manifold ( M, h ), a system of normal coordinates ( x , . . . , x m ) has the propertythat for any vector a ∈ T x M , the path θ a t is a geodesic. If ( q , . . . , q n ) is a system ofRiemannian normal coordinates, then the path t ( γ ( t ) , ˙ γ ♭ ) can be written in coordinates( q , p ) as(9.1) t ( a t, a ) ∈ T ∗ M. Let us now consider the Liouville manifold(9.2) ( ˆ W , ˆ λ a ) = ( D × T ∗ S n − , ˆ λ a := 1 a s dθ + λ std )for a > 0, where we have chosen local coordinates ( s, θ, q , p ).Let φ : ˆ W → R be the function φ ( s, θ, q , p ) = s + k p k . We consider the Liouville domain(9.3) ( W , λ a ) = ( { φ ≤ } , λ a := ˆ λ a | W ) , and its contact-type boundary(9.4) ( Y , ξ ) = ( { φ = 1 } , ξ = ker λ ) , where λ = (ˆ λ a ) | Y is the induced contact form.Consider also the codimension 2 contact submanifold V = { φ = 1 , s = 0 } ⊂ ( Y , ξ )and the Legendrian(9.5) Λ := { φ = 1 , θ = constant , s = 1 , k p k = 0 } . We define α := ( λ ) | V and let τ be the (unique) trivialization of ξ V /T V . We set r = ( α, τ, a ) ∈ R ( Y , ( ξ ) | V , V ) . Finally, we let H = { } × T ∗ S n − ⊂ ˆ W . Lemma 9.8. The manifolds W , Y have vanishing first and second homology and cohomologywith Z coefficients. In addition, we have H ( V ; Z ) = 0 and w (Λ) = 0 .Proof. The first claim is proved in Corollary 10.17. To compute H ( V ; Z ), note that V is thesphere bundle associated to T ∗ S n − . Hence, we have a fibration S n − ֒ → V → S n − givingrise to a Gysin sequence(9.6) · · · → H k ( S n − ; Z ) → H k ( V ; Z ) → H k − ( n − ( S n − ; Z ) → . . . Taking k = 1 immediately gives the desired result for n ≥ 4. For n = 3, one can just usethat V = RP , which has vanishing first cohomology.Finally, note that Λ = S n − , which has vanishing homology (with any coefficients) in degrees1 ≤ i ≤ n − 2. Hence w (Λ) = 0 for n ≥ 4. If n = 3, then one can simply note that w is themod 2 reduction of the first Chern class, and therefore vanishes. (cid:3) Observe that there is a natural marking e : R × Y → ( ˆ W , ˆ λ a , H )( t, y ) ψ t ( y ) , where ψ ( − ) is the Liouville flow associated to ˆ λ .This endows ( ˆ W , ˆ λ , H ) with the structure of a (strict) relative exact symplectic cobordism.We thus obtain an augmentation ˜ ǫ : ˜ A ( Y , ξ , V ; r ) → Q . It follows from Lemma 9.8 and the discussion following Definition* 7.14 that ˜ A ( Y , ξ , V ; r )and ˜ A ˜ ǫ ( Y , ξ , V ; r ) admit a ( Z × Z )-bigrading.We now analyze the structure of ( Y , λ ) in more detail. First, observe that ( Y − V, λ | Y − V )is strictly contactomorphic to( S × D ∗ S n − , α V := 1 a (1 − k p k ) dθ + λ std )via the map S × D ∗ S n − → Y − V ( θ, q , p ) ( p − k p k , θ, q , p )where D ∗ S n − = { ( q , p ) ∈ T ∗ S n − | k p k < } . We let N ⊂ Y denote the image of S × S n − under this map; equivalently, N = {k p k = 0 } . The complement ( Y − N , λ | Y −N ) is strictlycontactomorphic to ( B × U, α N := a ( x dy − y dx ) + p − x − y α U ), where B ⊂ R denotesthe open unit disk and ( U, α U ) denotes the unit cotangent bundle of ( S n − , g ), equippedwith the contact form α U := λ std induced by the canonical Liouville form on T ∗ S n − . Acontactomorphism is given by B × U → Y − N ( x, y, q , p ) ( x, y, q , p − x − y p )Our first task is to study the Reeb orbits of λ which are in the complement of N . Inparticular, we wish to show that they are nondegenerate for a generic choice of a , and moreoverthat their Conley-Zehnder indices depend linearly on a . This is the content of Proposition 9.9and Corollary 9.10 below. Proposition 9.9. Let γ U : R / Z → U be a Reeb orbit of α U of period T U . Then (1) the map γ : R / Z → B × Ut (0 , , γ U ( t )) is a Reeb orbit of α N of period T = T U ; (2) given any r ∈ (0 , and integers m, n > such that aT U π p − r = mn , the map γ : R / Z → B × Ut ( r cos(2 πmt ) , r sin(2 πmt ) , γ U ( nt )) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 77 is a Reeb orbit of α N of period T = (2 − r ) 2 πma = (2 − r ) nT U p − r . Every Reeb orbit of α N is of the form (1) or (2) for some choice of γ U , r , m , n .If α U is nondegenerate and a satisfies (9.7) a − / ∈ [ q ∈ Q > π √ q S ( α U ) , where S ( α U ) ⊂ R is the action spectrum of α U , then α N is nondegenerate. Moreover, givenany trivialization τ of γ ∗ U ker( α U ) , there exist trivializations τ i of γ ∗ i ker( α N ) ( i = 1 , ) suchthat CZ τ ( γ ) = 1 + 2 (cid:22) T a π (cid:23) + CZ τ ( γ U ) and CZ τ ( γ ) = 1 + 2 (cid:22) T aπ (2 − r ) (cid:23) + CZ τ ( γ nU ) . If τ extends to a disk spanning γ U , then τ i extends to a disk spanning γ i .Proof. The Reeb vector field of α N is given by R α N = 12 − x − y (cid:16) a ( x∂ y − y∂ x ) + 2 p − x − y R U (cid:17) where R U denotes the Reeb vector field of α U . A simple computation shows that γ and γ are Reeb orbits with periods as claimed, and that there are no other orbits.Note that the contact structure ξ = ker( α N ) splits as ξ = h e , e i ⊕ ker( α U )where e and e are the vector fields on B × U defined by e = ∂ x + ya p − x − y R U e = ∂ y − xa p − x − y R U In particular, given a trivialization τ of γ ∗ U ker( α U ), we get trivializations τ = h γ ∗ e , γ ∗ e i⊕ τ and τ = h γ ∗ e , γ ∗ e i ⊕ τ n of γ ∗ ξ and γ ∗ ξ , where τ n denotes the trivialization of ( γ nU ) ∗ ker( α U )induced by τ .We have L e R α N = − ∂ x (cid:18) ay − x − y (cid:19) e + ∂ x (cid:18) ax − x − y (cid:19) e = a (2 − x − y ) (cid:0) − xye + (2 + x − y ) e (cid:1) L e R α N = ∂ y (cid:18) ay − x − y (cid:19) e − ∂ y (cid:18) ax − x − y (cid:19) e = a (2 − x − y ) (cid:0) − (2 − x + y ) e + 2 xye (cid:1) Moreover, for any vector field X on U such that X ∈ ker( α U ), we have L X R α N = 2 p − x − y − x − y L X R U Hence, if Ψ i ( t ) : ξ γ i (0) → ξ γ i ( t ) denotes the linearized Reeb flow along γ i (viewed as a matrixvia the trivialization τ i ), i = 1 , 2, then Ψ ′ i ( t ) = S i ( t )Ψ i ( t ) with S ( t ) = aT U (cid:18) − 11 0 (cid:19) ⊕ T U S U ( t ) S ( t ) = 2 πm − r (cid:18) − r sin(4 πmt ) − r cos(4 πmt )2 + r cos(4 πmt ) r sin(4 πmt ) (cid:19) ⊕ nT U S U ( nt )where S U ( t ) is the matrix such that the linearized Reeb flow Ψ U : ( ξ U ) γ U (0) → ( ξ U ) γ U ( t ) of R U along γ U satisfies Ψ ′ U ( t ) = T U S U ( t )Ψ U ( t ).It follows that CZ τ ( γ ) = CZ( ψ ) + CZ τ ( γ U ) and CZ τ ( γ ) = CZ( ψ ) + CZ τ ( γ nU ), where ψ and ψ are paths of 2 × ψ i ( t ) = exp( P i ( t )) with P ( t ) = t aT U (cid:18) − 11 0 (cid:19) P ( t ) = Z t πm − r (cid:18) − r sin(4 πms ) − r cos(4 πms )2 + r cos(4 πms ) r sin(4 πms ) (cid:19) ds = 2 πm − r r πm (cos(4 πmt ) − − t + r πm sin(4 πmt )2 t + r πm sin(4 πmt ) − r πm (cos(4 πmt ) − ! Note that P ( t ) and P ( t ) are diagonalizable with eigenvalues ± πiλ ( t ) and ± πiλ ( t ) re-spectively, where λ ( t ) = t aT U πλ ( t ) = 12 − r r m t − r π (1 − cos(4 πmt ))It follows that ker( ψ i ( t ) − Id) is either R or 0 depending on whether λ i ( t ) is an integer or not.Assumption (9.7) implies that λ i (1) is not an integer and hence that ψ i (1) doesn’t have 1 asan eigenvalue, i.e. ψ i is nondegenerate. Since − J P ′ i ( t ) is positive-definite for all t , it followsfrom [Gut12, Prop. 52] thatCZ( ψ i ) = 1 + 2 { t ∈ (0 , | λ i ( t ) ∈ Z } . Since λ i is strictly increasing with λ i (0) = 0 and λ i (1) / ∈ Z , the right-hand side is equal to1 + 2 ⌊ λ i (1) ⌋ . Thus CZ( ψ ) = 1 + 2 (cid:22) aT U π (cid:23) = 1 + 2 (cid:22) aT π (cid:23) CZ( ψ ) = 1 + 2 (cid:22) m − r (cid:23) = 1 + 2 (cid:22) aT π (2 − r ) (cid:23) as desired. (cid:3) Corollary 9.10. Suppose that γ is a closed Reeb orbits of ( Y , ξ = ker λ ) which is containedin the complement of N ⊂ Y . Then CZ τ ( γ ) > j aρπ k , where τ is a trivialization which extends to a spanning disk and ρ > is as on page 74. OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 79 Proof. It is well-known that the Reeb orbits on U correspond bijectively to geodesics on( S n − , g ); with our normalization, the action of a closed Reeb orbit equals twice the lengthof the corresponding unit speed geodesic (see e.g. [Gei08, Sec. 1.5]). Moreover, according to[EGH00, Prop. 1.7.3], given a Reeb orbit ˜ γ which corresponds to a geodesic γ , we have µ M (˜ γ ) = CZ τ ( γ ) , where µ M is the Morse index of the geodesic and τ extends to a spanning disk (see Re-mark 9.11). Since the Morse index of a geodesic is non-negative by definition, the corollaryfollows from Proposition 9.9. (cid:3) Remark . The trivialization considered in [EGH00, Prop. 1.7.3] is in fact constructed asfollows. Choose a spanning disk ˜ v : D → U ⊂ T ∗ S n − for ˜ γ and let v := π ◦ ˜ v , where π : T ∗ S n − → S n − is the projection. Let { σ , . . . , σ n − } be a trivialization of v ∗ T S n − . Forpoints π : ˜ x x , let Q x ;˜ x : π − ( x ) → T ˜ x ( π − ( x )) be the canonical identification. Now define˜ σ ip = Q v ( p );˜ v ( p ) σ p for p ∈ D . Then { ˜ σ , . . . , ˜ σ n − } defines a Lagrangian subbundle of thesymplectic vector bundle (˜ v ∗ ( ξ ) , dλ ). Hence it induces a unique trivialization of ˜ v ∗ ξ , whichrestricts on the boundary to a trivialization of ˜ γ ∗ ξ .We now turn our attention to the Reeb dynamics near N . Recall from page 76 that N is contained in ( Y − V, λ ), which is strictly contactomorphic to ( S × D ∗ S n − , α V ), where α V = a (1 − k p k ) dθ + λ std . Lemma 9.12. Let q = ( q , . . . , q n − ) be Riemannian normal coordinates in some open set U ⊂ ( S n − , g ) and let p = ( p , . . . , p n ) be the dual coordinates. The Reeb vector field of α V isgiven by R α V = 11 + k p k a∂ θ + 2 X i,j g ij p i ∂ q j − X i,j,k p i p j ∂ k g ij ∂ p k on S × D ∗ U .Proof. A direct computation using the formulas α V = 1 a − X i,j p i p j g ij dθ + X i p i dq i dα V = − a X i,j g ij p i dp j ∧ dθ − a X i,j,k p i p j ∂ k g ij dq k ∧ dθ + X i dp i ∧ dq i shows that α V ( R α V ) = 1 and dα V ( R α V , − ) = 0. (cid:3) Lemma 9.13. Consider the map π : Y − V → S given by π ( s, θ, q, p ) = θ . Then the pair ( V, π ) defines an open book decomposition of Y . Moreover, λ is a Giroux form for the contactstructure ξ = ker λ .Proof. It is clear that ( V, π ) defines an open book decomposition of Y . To verify that λ is aGiroux form, observe by Lemma 9.12 that the Reeb vector field is transverse to the pages of π . The claim then follows by combining Definition 3.26 and Remark 3.27. (cid:3) By Lemma 9.12, the map γ : R / Z → S × D ∗ U given by the formula γ ( t ) = (2 πt, , 0) de-fines a simple Reeb orbit in Y . Let γ k denote its k -fold cover. There is an obvious trivialization τ of ξ | γ k given by(9.8) τ = { ∂ p , . . . , ∂ p n − , ∂ q , . . . , ∂ q n − } . Let τ be the trivialization of ξ | γ k defined as follows:(9.9) τ = { sin(2 πkt ) ∂ q + cos(2 πkt ) ∂ p , ∂ p , . . . ∂ p n , cos(2 πkt ) ∂ q − sin(2 πkt ) ∂ p , ∂ q , . . . , ∂ q n } . Observe that τ extends to a disk spanning γ in Y . Lemma 9.14. With respect to the trivialization τ , the linearized Reeb flow along γ k is givenby the matrix (9.10) (cid:18) t (cid:19) , where each entry of this matrix should be viewed as an ( n − × ( n − diagonal matrix.Proof. Note that τ can be extended to a trivialization ˜ τ of ker( α V ) over S × D ∗ U , where˜ τ = (cid:26) ∂ p , . . . , ∂ p n − , ∂ q − ap − k p k ∂ θ , . . . , ∂ q n − − ap n − − k p k ∂ θ (cid:27) . Using the formula for R α V given in Lemma 9.12, one can easily compute L ∂ pi R α V (cid:12)(cid:12) p =0 , q =0 = 2 ∂ q i L ∂ qi − api −k p k ∂ θ R α V (cid:12)(cid:12)(cid:12) p =0 , q =0 = 0Hence, the matrix A ( t ) representing the linearized Reeb flow ξ γ k (0) → ξ γ k ( t ) with respect tothe trivialization τ is given by A ( t ) = exp (cid:18) t (cid:18) (cid:19)(cid:19) = (cid:18) t (cid:19) where each entry should be interpreted as a multiple of the ( n − × ( n − 1) identity matrix. (cid:3) Corollary 9.15. The Robbin-Salamon index satisfies: µ τ RS ( γ k ) = ( n − / . Hence, (9.11) µ τRS ( γ k ) = ( n − / k. Proof. The first computation follows from [Gut12, Prop. 54] (there is a sign change due to thefact that the matrix we are considering is the transpose of that considered in [Gut12, Prop.54], but the proof is entirely analogous). The second computation follows from the fact (seethe proof of Lemma 57 in [Gut12]) that the Robbin-Salamon index satisfies the so-called “loopproperty”, i.e. given a path of symplectic matrices φ : [0 , → Sp(2 n, R ) with φ (0) = φ (1) = id,and given a path ψ : [0 , → Sp(2 n, R ), we have(9.12) µ RS ( φψ ) = µ RS ( ψ ) + 2 µ ( φ ) , where φ is the Maslov index of the path. (cid:3) By Lemma 9.12, N = {k p k = 0 } is preserved by the Reeb flow and is foliated by Reeb orbitsin a Morse-Bott family.Given ǫ > U ǫ = {k p k < ǫ } ∩ Y . This is a neighborhood of N , which we identify with S × D ∗ ǫ S n − via the contactomorphism defined on page 76. Let f : U ǫ → R be the function corresponding to S × D ∗ ǫ S n − → R ( θ, q , p ) ρ ( k p k ) g ( q ) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 81 under this identification, where g is a perfect Morse function on S n − and ρ : R → [0 , 1] is asmooth bump function with ρ ( x ) = 1 for x near 0 and ρ ( x ) = 0 for x > ǫ/ Lemma 9.16. Fix T > . If ǫ is small enough, all closed Reeb orbits of ( Y , λ ) which arecontained in U ǫ − N have action at least T . (cid:3) We now consider a perturbed contact form λ δ := (1+ δf ) λ . Since f is compactly-supportedin U ǫ , the form λ δ can be viewed as a contact form both on U ǫ and on Y . Lemma 9.17. Fix T > . If ǫ, δ are small enough, then there are exactly two simple Reeborbits in U ǫ with action < T . We label them γ a and γ b , and they correspond respectively to theminimum and maximum of f .Proof. Combine Lemma 9.16 with the argument of [Bou02, Lem. 2.3]. (cid:3) Lemma 9.18. Let T > be as in Lemma 9.17. After possibly further shrinking ǫ, δ , wemay assume that any Reeb orbit of ( Y , λ δ ) contained in U ǫ and having Conley-Zehnder index(measured with respect to a trivialization which extends to a spanning disk) less than T /a is amultiple of γ a or γ b . In addition, we have (9.13) CZ τ ( γ ka ) = µ τRS ( γ k ) − ( n − / a ( δf ) = 2 k, and (9.14) CZ τ ( γ kb ) = µ τRS ( γ k ) − ( n − / b ( δf ) = ( n − 1) + 2 k. Proof. First of all, observe by Lemma 9.12 that the boundary of U ǫ is preserved by the Reebflow of λ . It follows that the Reeb flow of λ has “bounded return time”, in the terminologyof [Bou02, Def. 2.5].Next, it follows from (9.11) that the Robbin-Salamon index of any Reeb orbit γ containedin the Morse-Bott submanifold N = {k p k = 0 } ⊂ Y satisfies µ RS ( γ ) = ( n − / γ ) = ( n − / T γ a, where T γ is the length of γ . It follows that these orbits satisfy “index positivity” (with constant2 /a ), in the terminology of [Bou02, Def. 2.6].The first claim now follows from [Bou02, Lem. 2.7]. The index computations follow bycombining (9.11) with [Bou02, Lem. 2.4]. (cid:3) We now put together the above results. For any integer N > 0, let us defineΣ N = { k ∈ Z | < k < N, k even } and let Σ N = { k ∈ Z | k < N, k = n − j, j ≥ } . Proposition 9.19. Given any N > , there exists A > so that (9.15) CH U =0 k − ( n − ( Y , ξ , V ; r ) = g CH k − ( n − ( Y , ξ , V ; r ) = Q ⊕ Q if k ∈ Σ N ∩ Σ N , Q if k ∈ Σ N ∪ Σ N − (Σ N ∩ Σ N ) , if k / ∈ Σ N ∪ Σ N , k < N whenever a > A . Proof. According to Corollary 9.10, we may fix A > Y , λ ) in the complement of N ⊂ Y have index at least N . We now choose ǫ, δ small enoughso that the conclusions of Lemma 9.18 hold with T = N . Since f δ is compactly-supportedin U ǫ , we find that the only Reeb orbits of ( Y , λ δ ) having index less than N are multiples of γ a , γ b .According to (9.13) and (9.14), it is now enough to check that the differential vanishes onthe set Ω N = { γ k a a , γ k b b | CZ τ ( γ k a a ) < N, CZ τ ( γ k b b ) < N } . To see this, observe that for γ ∈ Ω N we have(9.16) CZ τ ( γ ) = 2 wind( γ ) , mod( n − β of curves of index 1 with ˆ V ∗ β = 0. Then thelinking number of the positive puncture equals the sum of the linking numbers of the negativepunctures. Hence, by (9.16), the index of the positive puncture equals the sum of the indicesof the negative punctures, mod( n − β has index 1, this means that 1 = 0 , mod( n − n > (cid:3) Corollary* 9.20. Let N > be as in Proposition 9.19. Then for all integers k < N we have g CH ˜ ǫ k − ( n − ( Y , ξ , V ; r ) = g CH k − ( n − ( Y , ξ , V ; r ) , where the right-hand side was computed in Proposition 9.19. (cid:3) We now turn out attention to computing certain Legendrian invariants. Let Λ ⊂ ( Y , ξ )be defined as above (see (9.5)). Recall that the relative symplectic filling ( ˆ W , ˆ λ , H ) gives anaugmentation ˜ ǫ : ˜ A ( Y , ξ , V ; r ) → Q .It follows from Corollary* 8.14 and Lemma 9.8 that e L ˜ ǫ ( Y , ξ , V, Λ; r )is a ( Z × Z )-bigraded algebra with differential of bidegree ( − , HC • , • ( e L ˜ ǫ ( Y, ξ, V ; r ))is a ( Z × Z )-bigraded Q -vector space.We now have the following computation. Proposition* 9.21. Given a positive integer N ≫ , let ⊕ j ≤ N e L ˜ ǫ • ,j ( Y , ξ , V, Λ; r ) ⊂ e L ˜ ǫ ( Y , ξ , V, Λ; r ) be the bigraded sub-module of elements of winding number at most N . Then this sub-module canbe generated by products of total winding number ≤ N of Reeb chords { a k } k ∈ N + and { b k } k ∈ N + ,where | a k | = 2 k − and | b k | = n − k . (Note that we do not say anything about thedifferential).Proof. Since ( Y − V, λ ) is strictly contactomorphic to ( S × D ∗ S n − , α V ), we have that( Y − V, λ δ ) is strictly contactomorphic to ( S × D ∗ S n − , α δ := (1 + δf ) α V ).Recall that f depends on a positive real parameter ǫ > f to the Legendrian Λ = { } × S n − is equal to g , a Morsefunction with exactly two critical points: one minimum a and one maximum b .Let c denote either a or b . As in lemma 9.18, we let γ c denote the simple Reeb chord (withis also a Reeb orbit) passing through c and let γ kc denote its k -fold cover. Observe first of allthat all Reeb chords for Λ are contained in U ǫ – this follows from the fact that f is compactlysupported in U ǫ (see page 80). Hence, as in Lemma 9.16, if we assume that ǫ > OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 83 enough, then there exists T > T have winding number greater than N . By a routine adaptation of Lemma 9.17 (or rather theproof of [Bou02, Lem. 2.3]), one concludes that the only Reeb chords of winding number lessthan or equal to N are the γ ka and γ kb .We can assume without loss of generality that there are normal coordinates q = ( q , . . . , q n − )defined in a neighborhood U c ⊂ Λ of c in which g is given by g = g ( c ) + ǫ n − X i =1 q i , where ǫ = 1 if c = a and ǫ = − c = b . The Reeb vector field of α δ is given by R α δ = 11 + δf R α + 2 ǫδ (1 + δf ) X i q i − p i k p k X j,k g jk p k q j ∂ p i on S × D ∗ U c for k p k sufficiently small (i.e. satisfying ρ ( k p k ) = 1). We will now show thatfor every k ≥ 1, the indices of γ ka and γ kb as Reeb chords are given byCZ + ( γ ka ) = 2 k CZ + ( γ kb ) = 2 k + n − a k = γ ka and b k = γ kb , we have | a k | = CZ + ( a k ) − k − | b k | =CZ + ( b k ) − k + n − 2, as desired.To compute CZ + ( γ kc ), we start by computing the linearized Reeb flow along γ kc with respectto the trivialization τ (see (9.8)). We proceed as in Lemma 9.14: we have L ∂ pi R α δ (cid:12)(cid:12) p =0 , q =0 = 21 + δg ( c ) ∂ q i L ∂ qi − api −k p k ∂ θ R α δ (cid:12)(cid:12)(cid:12) p =0 , q =0 = ǫ δ (1 + δg ( c )) ∂ p i Hence, the matrix A ( t ) representing the linearized Reeb flow ξ γ kc (0) → ξ γ kc ( t ) with respect tothe trivialization τ satisfies A ′ ( t ) = SA ( t ) with S = ǫ δ (1+ δg ( c )) δg ( c ) ! , where each entry should be interpreted as a multiple of the ( n − × ( n − 1) identity matrix.Setting µ = δ (1+ δg ( c )) and ν = δg ( c ) for ease of notation, it follows that A ( t ) = exp( tS ) = cosh( t √ µν ) p µ/ν sinh( t √ µν ) p ν/µ sinh( t √ µν ) cosh( t √ µν ) ! if ǫ = 1 cos( t √ µν ) − µ/ν sin( t √ µν ) ν/µ sin( t √ µν ) cos( t √ µν ) ! if ǫ = − µ, ν > δ is sufficiently small).Let L ( t ) ⊂ ξ γ kc ( t ) be the path of Lagrangian subspaces obtained by applying the linearizedReeb flow to the tangent space T c Λ ⊂ ξ c and let ˜ L ( t ) be the loop obtained by closing up L ( t ) by a positive rotation. Since T c Λ is represented by (cid:18) I n − (cid:19) in the trivialization τ , L ( t ) is represented by A ( t ) (cid:18) I n − (cid:19) . In the two-dimensional case (i.e. n − µ τ ( ˜ L ( t )) = ( ǫ = 11 if ǫ = − L ( t ) splits as a direct sum of n − µ τ ( ˜ L ( t )) = ( ǫ = 1 n − ǫ = − + ( γ kc ) = µ τ (Λ n − C ˜Λ) = µ τ ( ˜ L ( t )), where τ is a trivialization of the contact structurealong γ kc which extends to a spanning disk. For example, we can take τ to be the trivializationdefined in equation (9.9). The difference µ τ ( ˜ L ( t )) − µ τ ( ˜ L ( t )) is equal to twice the Maslovindex of the loop of symplectic matrices relating τ and τ , i.e. µ τ ( ˜ L ( t )) − µ τ ( ˜ L ( t )) = 2 µ (cid:18) cos(2 πkt ) − sin(2 πkt )sin(2 πkt ) cos(2 πkt ) (cid:19) = 2 k. It follows that CZ + ( γ ka ) = 2 k CZ + ( γ kb ) = 2 k + n − (cid:3) It will be useful to record the following consequence of the above computation. Corollary* 9.22. Suppose that n ≥ is even . Then we have rk HC n, ( e L ˜ ǫ ( Y , ξ , V, Λ; r )) = 1 . Proof. Indeed, note that the generators described in Proposition* 9.21 satisfy link( a k ) =link( b k ) = k . It thus follows that CC n − , ( e L ǫ ( Y , ξ , V ; r )) = CC n +1 , ( e L ǫ ( Y , ξ , V ; r )) = 0 . On the other hand, CC n, ( e L ǫ ( Y , ξ , V ; r )) is generated by the word b b . (cid:3) Applications to contact topology Contact and Legendrian embeddings. We begin by introducing some standard def-initions in the theory of contact and Legendrian embeddings. Definition 10.1. Given a smooth manifold Y n − , a formal contact structure (or almost-contact structure ) is the data of a pair ( η, ω ), where η ⊂ T Y is a codimension 1 distributionand ω ∈ Ω ( Y ) is a 2-form whose restriction to η is non-degenerate. A formal contact structureis said to be genuine if it is induced by a contact structure.If Y n − is orientable, then a formal contact structure is the same thing as a lift of theclassifying map Y → BSO (2 n + 1) to a map Y → B ( U ( n ) × id) = BU ( n ). Definition 10.2 (see Def. 2.2 in [CE20]) . Let ( Y n − , ξ = ker α ) be a contact manifold. Givena formal contact manifold ( V m − , η, ω ) where 1 ≤ m ≤ n − 1, a formal (iso)contact embeddingis a pair ( f, F s ) where OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 85 • F s is a fiberwise injective bundle map T V → T Y defined for s ∈ [0 , • f : V → Y is a smooth map and df = F , • F defines a fiberwise conformally symplectic map ( η, ω ) → ( ξ, dα ).Observe that the above properties are independent of the choice of contact form α .Two formal contact embeddings i , i : ( V, ζ, ω ) → ( Y, ξ ) are said to be formally isotopic ifthey can be connected by a family { i t } t ∈ [0 , of formal contact embeddings.A (genuine) contact embedding ( V, ζ ) → ( Y, ξ ) is simply a smooth embedding φ : V → Y such that φ ∗ ( ζ ) = ξ | φ ( V ) . In particular, every contact embedding induces a formal contactembedding by taking F s = F = df . Definition 10.3 (see Def. 2.1 in [CE20]) . Let ( Y n − , ξ ) be a contact manifold. Given asmooth n -dimensional manifold Λ, a formal Legendrian embedding is a pair ( f, F s ) where • F s is a fiberwise injective bundle map T V → T Y defined for s ∈ [0 , • df = F • im( F ) ⊂ ξ .Two formal Legendrian embeddings are said to be formally isotopic if they can be connectedby a family of Legendrian embeddings. A (genuine) Legendrian embedding Λ → ( Y, ξ ) is asmooth embedding φ : Λ → Y such that dφ ( T Λ) ⊂ ξ ⊂ T Y . In particular, a Legendrianembedding canonically induces a formal Legendrian embedding.We now review some foundational facts about loose Legendrians. Recall that a LegendrianΛ in a (possibly non-compact) contact manifold ( Y, ξ ) of dimension at last five is defined to beloose if it admits a loose chart. For concreteness, we adopt as our definition of a loose chartthe one given in [CE12, Sec. 7.7].Loose Legendrians satisfy the following h-principle due to Murphy [Mur19, Thm. 1.2]: givena pair of loose Legendrian embeddings f , f : Λ → ( Y, ξ ) which are formally isotopic, then f , f are genuinely isotopic, i.e. isotopic through Legendrian embeddings.Given an arbitrary Legendrian submanifold Λ in a contact manifold ( Y , ξ ) of dimensionat least five, one can perform a local modification called stabilization which makes Λ loosewithout changing the formal isotopy class of the tautological embedding Λ −→ Λ . Thismodification can be realized in multiple essentially equivalent ways. In this paper, we will takeas our definition of stabilization any construction which satisfies the properties stated in thefollowing lemma. Lemma 10.4. Given a Legendrian submanifold Λ ⊂ ( Y , ξ ) and an open set U ⊂ Y suchthat U ∩ Λ is nonempty, there exists a Lagrangian embedding f : Λ → Y which is formallyisotopic to the tautological embedding Λ −→ Λ via a family of formal Legendrian embeddings { ( f t , F ts ) } t ∈ [0 , which are independent of t on (Λ ∩ ( Y − U )) .We put Λ := f (Λ ) and say that Λ is the stabilization of Λ inside U .Proof. To construct Λ, we follow the procedure described in [CE12, Sec. 7.4]. As the reader mayverify, this construction can be assumed to happen entirely inside a suitably chosen Darbouxchart U ⊂ U . In addition, the construction depends on the choice of a function f ; usingthe fact that Y has dimension at least five, we may (and do) assume that χ ( { f ≥ } ) = 0.To construct the formal isotopy, we simply follow the proof of [CE12, Prop. 7.23] (using theassumption that χ ( { f ≥ } ) = 0). The reader may verify that the argument there is entirelylocal, so that the isotopy can indeed be assumed to be fixed outside of U (and in particularoutside of U ). (cid:3) Embeddings into overtwisted contact manifolds. Given an overtwisted contactmanifold ( Y, ξ ), we will say in light of Definition 9.1 that a contact embedding ( V, ζ ) → ( Y, ξ )is loose if its image has overtwisted complement. The following proposition states that everyisotopy class of formal contact embeddings can be represented by a loose contact embedding. Proposition 10.5. Suppose that ( Y, ξ ) is an overtwisted contact manifold and let i : ( V, ζ ) → ( Y, ξ ) be a formal contact embedding. Then there exists an open subset Ω ⊂ Y such that Y − Ω isovertwisted and a formal contact embedding j : ( V, ζ ) → Ω ⊂ ( Y, ξ ) such that i and j are formally contact isotopic.Proof. We will assume for simplicity that V is connected but the proof can easily be generalized.Let D ot ⊂ ( Y, ξ ) be an overtwisted disk. Let f t be a family of formal contact embeddings suchthat f is the underlying smooth map induced by i , and Im( f ) ∩ D ot = ∅ . Let Ω ⊂ Y be aconnected open subset such that Im( f ) ⊂ Ω ⊂ Ω ⊂ Y − D ot . According to [BEM15, Prop.3.8], we can assume by choosing Ω large enough that (Ω , ξ ) is overtwised.For purely algebro-topological reasons, there exists a family ξ t of formal contact structureson Y with the following properties: • ξ = ξ , • ξ t is constant in the complement of Ω, • ξ is a genuine contact structure in a neighborhood V ⊂ Ω of Im( f ) and f is a genuinecontact embedding with respect to ξ .Since ξ is genuine on V ∪ ( Y − Ω), it follows from the relative h-principle for overtwistedcontact structures [BEM15, Thm. 1.2] that ξ is homotopic to a genuine overwisted contactstructure through a homotopy fixed on V ∪ ( Y − Ω). Thus we may as well assume in the thirdbullet above that ξ is genuine everywhere.Since (Ω , ξ ) is overtwisted, it follows from the [BEM15, Thm. 1.2] that there exists a homo-topy ˜ ξ t of genuine contact structures such that ˜ ξ = ξ = ξ , ˜ ξ = ξ , and ˜ ξ t is independent of t on Y − Ω.By Gray’s theorem, there is an ambient isotopy ψ t : Y → Y which is fixed on Y − Ω andhas the property that ψ ∗ t ˜ ξ t = ˜ ξ = ξ . The composition ψ − ◦ f is in the same class of formalcontact embeddings as f and gives the desired genuine embedding. (cid:3) We now describe a procedure for constructing pairs of codimension 2 contact embeddings inovertwisted manifolds which are formally isotopic but fail to be isotopic as contact embeddings. Construction 10.6. Let ( Y n − , ξ ) be a closed, overtwisted contact manifold and let ( Y , B, π )be an open book decomposition which supports ξ . Let i : ( B, ξ | B ) → ( Y , ξ ) be the tauto-logical embedding of the binding and let j : ( B, ξ | B ) → ( Y , ξ ) be a loose contact embeddingformally isotopic to i , whose existence follows from Proposition 10.5. Let D ot ⊂ Y be anovertwisted disk which is disjoint from j ( B ).Choose an open subset U ⊂ Y whose closure is disjoint from i ( B ) ∪ j ( B ) ∪ D ot , andsuch that i , j are formally isotopic in the complement of U . Now let ( Y, ξ ) be obtainedby attaching handles of arbitrary index along isotropic submanifolds contained inside U (seeConstruction 8.15). We let ( ˆ X, ˆ λ ) denote the resulting Weinstein cobordism with positive end( Y, ξ ) and negative end ( Y , ξ ). OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 87 Observe that ( Y, ξ ) is still overtwisted and that i , j can also be viewed as codimension 2contact embeddings into ( Y, ξ ). We denote these latter embeddings by i, j : ( B, ξ | B ) → ( Y, ξ ).By construction, the embeddings i, j are formally isotopic. Theorem 10.7. The embeddings i, j which arise from Construction 10.6 are not genuinelyisotopic. In fact, i is not genuinely isotopic to any reparametrization of j in the source, meaningthat the codimension submanifolds ( i ( B ) , ξ | i ( B ) ) , ( j ( B ) , ξ | j ( B ) ) are not contact isotopic.Proof. According to Corollary 6.8, the cobordism ( ˆ X, ˆ λ ) induces a map of unital Q -algebras g CH • ( Y, ξ, B ; r ) → g CH • ( Y , ξ , B ; r ) , for any element r ∈ R + ( Y , ξ , B ) ≡ R + ( Y, ξ, B ). Moreover, Theorem 9.4 implies that g CH • ( Y, ξ, B ; r ) = 0for appropriate r ∈ R + ( Y, ξ, B ). It follows that g CH • ( Y, ξ, B ; r ) = 0.If we assume that ( i ( B ) , ξ | i ( B ) ) , ( j ( B ) , ξ | j ( B ) ) are isotopic as codimension 2 contact subman-ifolds, then g CH • ( Y, ξ, B ; r ) = g CH • ( Y, ξ, j ( B ); r ′ )for some datum r ′ ∈ R + ( Y, ξ, j ( B )).On the other hand, observe that ( Y − j ( B ) , ξ ) is overtwisted by construction. Hence Theo-rem 9.2 implies that g CH • ( Y, ξ, j ( B ); r ′ ) = 0. (cid:3) Example 10.8. By a well-known theorem of Giroux and Mohsen [Gei08, Thm. 7.3.5], anycontact manifold ( Y, ξ ) admits an open book decomposition ( Y, B, π ) which supports ξ . HenceConstruction 10.6 and Theorem 10.7 can be applied to any overtwisted contact manifold.We also consider the following modification of Construction 10.6. Construction 10.9. Let ( Y n − , ξ ) be a closed, overtwisted contact manifold and let ( Y , B, π )be an open book decomposition which supports ξ . Suppose that there exists a Legendriansubmanifold Λ ⊂ Y such that B = τ (Λ) is a contact pushoff of Λ. Let D ot ⊂ Y be anovertwisted disk.Let U ⊂ Y − B − D ot be an open ball which intersects Λ. Let Λ ′ ⊂ ( Y , ξ ) be obtainedby stabilizing Λ inside U (see Lemma 10.4). Let U be the union of U with a tubularneighborhood of Λ. Let V ′ = τ (Λ ′ ) ⊂ U be a choice of contact pushoff for Λ ′ .Let i : ( B, ξ | B ) → ( Y , ξ ) be the tautological embedding. [CE20, Lem. 3.4] implies that i is formally isotopic to some codimension 2 contact embedding j : ( B, ξ | B ) → ( Y , ξ ) where j ( B ) = B ′ . Choose such a formal isotopy and let T ⊂ Y be its trace.Let U ⊂ Y be an open set whose closure is disjoint from T ∪U ∪ D ot . As in Construction 10.6,let ( Y, ξ ) be obtained by attaching handles of arbitrary index along some collection of isotropicsinside U . Let ( ˆ X, ˆ λ ) denote the resulting Weinstein cobordism with positive end ( Y, ξ ) andnegative end ( Y , ξ ).It follows from our choice of U that ( Y, ξ ) is overwisted and that Λ , Λ ′ , V, V ′ can be viewedas submanifolds of ( Y, ξ ). It also follows that Λ ′ is the stabilization of Λ as submanifolds of( Y, ξ ), and that V (resp. V ′ ) is the contact pushoff of Λ (resp. Λ ′ ). Corollary 10.10. The submanifolds ( V, ξ | V ) and ( V ′ , ξ | V ) are not isotopic through codimen-sion contact submanifolds. Hence the Legendrian submanifolds Λ , Λ ′ ⊂ ( Y, ξ ) are not isotopicthrough Legendrian submanifolds.Proof. The proof of the first statement is identical to that of Theorem 10.7. The secondstatement follows from the fact that V, V ′ are respectively the contact pushoff of Λ , Λ ′ . (cid:3) Example 10.11. Let ( Y , ξ ) = obd( T ∗ S n − , τ − ), where τ − is a left-handed Dehn twist.Note by [CMP19, Thm. 1.1] that ( Y , ξ ) is overtwisted (in fact, ( Y, ξ ) is contactomorphic to( S n − , ξ ot )). Let P = T ∗ S n − ⊂ Y be a page of the open book and let Λ ⊂ ( Y , ξ ) be theLegendrian which corresponds to the zero section of P . Then the binding of the open bookdecomposition is also a contact pushoff of Λ. We may therefore apply Construction 10.9 tothis data. Remark . Consider the special case of Construction 10.6 and Construction 10.9 where U is empty, i.e. one does not attach any handles. In this case, Theorem 10.7 and Corollary 10.10are essentially equivalent to the statement that the binding of an open book decomposition istight (i.e. must intersect any overtwisted disk). This statement was proved in dimension 3 byEtnyre and Vela-Vick [EVV10, Thm. 1.2], and in general by Klukas [Klu18, Cor. 3].10.3. Contact embeddings into the standard contact sphere. In this section, we exhibitexamples of pairs of codimension 2 contact embeddings into tight contact manifolds which areformally isotopic but are not isotopic through genuine contact embeddings.We begin with the following construction. Construction 10.13. Let ( Y n − , ξ ) be a contact manifold for n ≥ 3. Let ( V, ξ | V ) bea codimension 2 contact submanifold and let Λ ⊂ ( Y , ξ ) be a loose Legendrian such thatΛ ∩ V = ∅ .Choose an open ball U ⊂ Y such that ( U , U ∩ Λ) is a loose chart for Λ. Next, choose anopen ball O ⊂ Y − V − U (By definition of a loose chart, U ∩ Λ is a proper subset of Λ, so itis clear that such choices exist).Let Λ ′ be obtained by stabilizing Λ inside O . It follows from Lemma 10.4 that Λ and Λ ′ areformally isotopic via a formal isotopy fixed outside of O .According to Lemma 10.14 below, we can (and do) fix a contactomorphism f : ( Y , ξ ) → ( Y , ξ ) with the following properties:(1) f is isotopic to the identity,(2) f (Λ) = Λ ′ ,(3) the tautological contact embedding i ′ : ( V, ξ | V ) → ( Y − Λ ′ , ξ ) is formally isotopicto the embedding i ′ := f ◦ i ′ : ( V, ξ | V ) → ( Y − Λ ′ , ξ ) (we emphasize here that theformal isotopy is contained in the open contact manifold ( Y − Λ ′ , ξ )).Finally, let ( Y, ξ ) be obtained by attaching a Weinstein n -handle along Λ ′ ⊂ ( Y , ξ ) asdescribed in Construction 8.15. We assume without loss of generality that the attachingregion Λ ′ ⊂ V disjoint from V and f ( V ), and that i ′ and i ′ are formally isotopic in Y − V .We let ι : Y − V ֒ → Y be the canonical inclusion.Let i = ι ◦ i ′ : ( V, ξ | V ) → ( Y, ξ )be the tautological contact embedding and define i := ι ◦ i ′ : ( V, ξ | V ) → ( Y, ξ ) . It is an immediate consequence of (3) and our choice of V that i and i are formally isotopic. Lemma 10.14. With the notation of Construction 10.13, there exists a contactomorphism f : ( Y , ξ ) → ( Y , ξ ) satisfying the properties (1-3) stated in Construction 10.13.Proof. Recall that U is disjoint from O . Recall also that ( U , U ∩ Λ) is a loose chart for Λ, whichmeans in particular that U deformation retracts onto U ∩ Λ. Using these two facts, it is nothard to verify that there exists a family of formal contact embeddings j t : ( V, ξ | V ) → ( Y , ξ ),for t ∈ [0 , OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 89 • j = i ′ , • j t ( V ) is disjoint from O ∪ Λ for all t ∈ [0 , • j ( V ) is disjoint from U ∪ O ∪ Λ.By the h -principle for loose Legendrian embeddings [Mur19, Thm. 1.2], there exists a globalcontact isotopy φ t , for t ∈ [0 , φ = Id and φ (Λ) = Λ ′ . By the Legendrian isotopyextension theorem [Gei08, Thm. 2.6.2], this isotopy can be assumed to be compactly-supportedand constant in a neighborhood W of j ( V ), where W is disjoint from U ∪ O ∪ Λ.Let f := φ and observe that f satisfies (1-2). Observe that f ◦ j t defines a formal contactisotopy from f ◦ i ′ = i ′ to j in the complement of Λ ′ = f (Λ). Since i ′ is formally isotopic to j in the complement of Λ ′ ⊂ Λ ∪ O , we find that f satisfies (3). (cid:3) It will be useful to record the following basic observation, which is a consequence of the factstated in Definition 2.1 that an isotopy of contactomorphisms induces a Hamiltonian isotopyof symplectizations. Lemma 10.15 (cf. Definition 2.1) . Let ( ˆ X, ˆ λ ) be a relative cobordism from ( Y + , λ + ) to ( Y − , λ − ) . Given contactomorphisms f ± : ( Y ± , λ ± ) → ( Y ± , λ ± ) contact isotopic to the iden-tity, there is a symplectomorphism F : ( ˆ X, ˆ λ ) → ( ˆ X, ˆ λ ) which agrees near infinity with the lifts ˜ f ± : ( SY ± , λ Y ± ) → ( SY ± , λ Y ± ) . (cid:3) Let us now return to the geometric setup considered in Section 9.3. In particular, we let( ˆ W , ˆ λ a ) := ( D × T ∗ S n − , a r dθ + λ std ) , where a > W , λ a ) , ( Y , ξ = ker λ ), V ⊂ Y , Λ ⊂ Y , and H = { } × T ∗ S n − be defined as inSection 9.3. Note that Λ is a loose Legendrian according to [CM19, Prop. 2.9].Construction 10.13 applied to the above data produces a contactomorphism f : ( Y , ξ ) → ( Y , ξ ), a Liouville domain ( X, λ ) with positive contact boundary ( Y, ξ = ker λ ), and a pair offormally isotopic contact embeddings i , i : ( V, ξ | V ) → ( Y, ξ ).We let r = ( α, τ, a ) ∈ R + ( Y , ξ , V ), where α := ( λ ) | V and τ is the trivialization of ξ | V /T V which is unique since H ( V ; Z ) = 0 (see Corollary 10.17). We let r ′ = (( i ′ ) ∗ α, τ, a ) ∈ R + ( Y , ξ , V ), where i ′ is defined as in Construction 10.13 and τ is again unique. Sincethe surgery resulting from Construction 10.13 away from V and i ′ ( V ), we may identify R ( Y, ξ, V ) = R ( Y , ξ , V ) and R ( Y, ξ, i ( V )) = R ( Y , ξ , i ′ ( V )).As in Section 9.3, let e : R × Y → ( ˆ W , ˆ λ , H ) be the canonical marking furnished by theLiouville flow and let ˜ ǫ : e A ( Y , ξ , V ; r ) → Q be the associated augmentation.By Lemma 10.15, there is a symplectomorphism ψ : ( ˆ W , ˆ λ a ) → ( ˆ W , ˆ λ a ) which coincidesnear infinity with the lift ˜ f : SY → SY . Let H ′ ⊂ ˆ W be a symplectic submanifold which iscylindrical at infinity and coincides with the symplectization of f ( V ) = i ′ ( V ) on [0 , ∞ ) × Y .Such a surface can be constructed by taking the backwards Liouville flow of ψ ( H ).Let ˜ ǫ ′ : e A ( Y , ξ , i ′ ( V ); r ′ ) → Q be the augmentation induced by the relative symplecticcobordism (( ˆ W , ˆ λ a , H ′ ) , ˜ ǫ ).Observe that ( Y , ξ , V ) ∈ G and hence also ( Y , f ∗ ξ , f ( V )) = ( Y , ξ , i ′ ( V )) ∈ G (seeDefinition 3.28). The following lemma shows that we also have ( Y, ξ, i ( V )) ∈ G . Lemma 10.16. Up to contactomorphism, ( Y, ξ ) = ob( T ∗ S n − , τ S ) = ( S n − , ξ std ) , where τ S denotes a right-handed Dehn twist. Moreover, the first contactomorphism can be assumed totake i ( V ) to the binding of the open book decomposition ob( T ∗ S n − , τ S ) . Proof. By construction, there is an open book decomposition of ( Y , ξ ) agreeing (up to con-tactomorphism) with ob( T ∗ S n − , id), such that i ( V ) is the binding and Λ ′ is the zero sectionof a page. Note now that attaching a handle to the zero section of a page of ( Y , ξ ) =ob( T ∗ S n − , id) simply changes the open book by a positive stabilization [vK17, Thm. 4.6].Hence, i ( V ) is the binding of ob( T ∗ S n − , τ S ) = ( S n − , ξ std ). (cid:3) Corollary 10.17. The manifolds Y , W , Y, W have vanishing first and second homology andcohomology with Z -coefficients. Hence the same is also true for the pairs ( W , Y ) and ( W, Y ) .Finally, we have H ( V ; Z ) = 0 .Proof. By construction, W is obtained by attaching a handle of index n to W . The unionof the core and co-core of this handle has codimension n . Hence, for i ≤ n − 2, we have H i ( W ; Z ) = H i ( W ; Z ) and H i ( Y ; Z ) = H i ( Y ; Z ). Now, W is homotopy equivalent to S n − by definition, while Y is homeomorphic to S n − by Lemma 10.16. Since n ≥ 4, it follows that Y , W , Y, W have vanishing first and second homology. The vanishing of cohomology in thesame degrees follows by the universal coefficients theorem for cohomology.The vanishing of H ( V ; Z ) was proved in Lemma 9.8. (cid:3) As a result of Corollary 10.17, Definition 6.12, Lemma 6.13 and Definition* 8.12, the in-variants considered in the proof of Theorem* 10.18 below, as well as the maps between theseinvariants, are all canonically ( Z × Z )-bigraded. Theorem* 10.18. For n ≥ and a ≫ large enough, the contact embeddings i , i :( V, ξ | V ) → ( Y, ξ ) = ( S n − , ξ std ) are not isotopic through contact embeddings.Proof. We suppose for contradiction that i and i are genuinely isotopic. This implies thatthere exists a contactomorphism g : ( Y, ξ, V ) → ( Y, ξ, i ( V )). It follows by Lemma 10.16 that( Y, ξ, V ) ∈ G .According to Corollary* 9.20 (and the description of the generators in Proposition 9.19), wemay (and do) fix a ≫ g CH ˜ ǫ n − ( n − , ( Y , ξ , V ; r ) = g CH ˜ ǫ n +1 − ( n − , ( Y , ξ , V ; r ) = 0 . Since ( Y , ξ , V ) ∈ G , it follows by Corollary* 7.18 that(10.2) g CH ˜ ǫ • , • ( Y , ξ , V ; r ) = g CH ˜ ǫ ′ • , • ( Y , ξ , i ( V ); r ′ ) . Similarly, it follows by Definition/Assumption* 8.6 that(10.3) e L ˜ ǫ • , • ( Y , ξ , V, Λ; r ) = e L ˜ ǫ ′ • , • ( Y , ξ , i ( V ) , Λ ′ ; r ′ ) . Let e : R × Y → ˆ W be the canonical marking and consider the resulting relative filling(( ˆ W , ˆ λ, H ) , e ). Let ˜ ǫ : e A ( Y, ξ, V ; r ) → Q be the induced augmentation.Let φ : ( ˆ W , ˆ λ, H ) → ( ˆ W , ˆ λ, H ) be a symplectomorphism which agrees with the lift of g nearinfinity. Let ˜ ǫ ′ : e A ( Y, ξ, i ( V ); r ′ ) → Q be the augmentation induced by (( ˆ W , ˆ λ, H ′ ) , e ).Then according to Lemma* 7.17 and Corollary* 7.18, we have(10.4) g CH ˜ ǫ • , • ( Y, ξ, V ; r ) = g CH ˜ ǫ ′ • , • ( Y, ξ, i ( V ); r ′ ) . It then follows by Lemma* 10.19 that g CH ˜ ǫ n − ( n − , ( Y, ξ, V ; r ) = 0. Hence Lemma* 10.20implies that g CH ˜ ǫ n − ( n − , ( Y , ξ , V ; r ) = 0 . This contradicts (10.1). (cid:3) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 91 Lemma* 10.19. We have g CH ˜ ǫ ′ n − ( n − , ( Y, ξ, i ( V ); r ′ ) = 0 . Proof. On the one hand, Corollary* 9.22 and (10.3) imply thatrk HC n, ( e L ˜ ǫ ′ ( Y , ξ , i ( V ) , Λ ′ ; r ′ )) = 1 . On the other hand, by (10.1) and (10.2), we have that(10.5) g CH ˜ ǫ ′ n − ( n − , ( Y , ξ , i ( V ); r ′ ) = g CH ˜ ǫ ′ n +1 − ( n − , ( Y , ξ , i ( V ); r ′ ) = 0 . It then follows by Theorem* 8.17 and Remark 8.18 that(10.6) g CH ˜ ǫ ′ n − ( n − , ( Y, ξ, i ( V ); r ′ ) ≃ HC n, ( e L ˜ ǫ ′ ( Y , ξ , i ( V ) , Λ ′ ; r ′ )) . This proves the claim. (cid:3) Lemma* 10.20. The natural map g CH ˜ ǫ n − ( n − , ( Y, ξ, V ; r ) → g CH ˜ ǫ n − ( n − , ( Y , ξ , V ; r ) is injective.Proof. Since Λ ′ is loose in Y − V , it follows by Proposition* 9.3 and Lemma 7.9 that HC k ( L ˜ ǫ ( Y , ξ , V, Λ; r )) = 0for all k ∈ Z . The lemma thus follows from Theorem* 8.17 and Remark 8.18. (cid:3) Remark . One can slightly tweak Construction 10.13 so that Λ , Λ ′ are disjoint and Λ ∪ Λ ′ is a loose Legendrian link. One can then upgrade Lemma 10.14 to require that f (Λ) = Λ ′ and f (Λ ′ ) = Λ in (2) of Construction 10.13. In particular, this means that Λ ′ is a stabilization ofΛ and Λ is a stabilization of Λ ′ .Let us apply this tweaked construction to the setup considered in Construction 10.13, where( Y , ξ ) = ob( T ∗ S n − , id), for n ≥ V ⊂ ( Y , ξ ) is the binding and Λ ⊂ ( Y , ξ ) be the zerosection of a page. It is well known that the zero section of a page in ob( T ∗ S n − , id) is thestandard Legendrian unknot. Hence Lemma 10.16 implies that i ( V ) is the pushoff of thestandard unknot. By construction, it now also follows that i ( V ) is the contact pushoff ofa stabilization of the unknot. Theorem* 10.18 thus recovers (for n ≥ 3) the basic exampleconstructed by Casals and Etnyre in [CE20, Sec. 5].10.4. Relative symplectic and Lagrangian cobordisms. Recall that an exact relativesymplectic cobordism from ( Y + , ξ + , V − ) to ( Y − , ξ − , V − ) is an exact symplectic cobordism( ˆ X, ˆ λ ) from ( Y + , ξ + ) to ( Y − , ξ − ) along with a codimension 2 symplectic submanifold H ⊂ ˆ X which agrees with the symplectization of V ± near the ends; see Definition 2.18. In the specialcase where ( ˆ X, ˆ λ ) is the symplectization of ( Y ± , ξ ± ) and H is diffeomorphic to R × V ± , wespeak of a symplectic concordance from V + to V − . These notions were first considered by[Bow10] in his PhD thesis.It is straightforward to show that isotopic contact submanifolds are concordant. Given anycontact manifold ( Y, ξ ), the notion of symplectic concordance thus induces a binary relation ≺ s on the set of isotopy classes of codimension 2 contact submanifolds. We write V ≺ s V ′ toindicate that there exists a symplectic concordance from V to V ′ , and we write V ⊀ s V ′ toindicate that such a concordance does not exist.It is natural to consider the following two basic questions about the binary relation ≺ s . (1) Is (cid:22) s a nontrivial relation? i.e. given a contact manifold ( Y, ξ ), do there exist codi-mension 2 contact submanifolds V, V ′ such that V, V ′ are not isotopic as contact sub-manifolds but V ≺ s V ′ ?(2) Is this relation non-symmetric? i.e. given a contact manifold ( Y, ξ ), do there codimen-sion 2 contact submanifolds V, V ′ such that V ≺ s V ′ but V ′ ⊀ s V ?For transverse links in ( S , ξ std ), both of these questions were answered affirmatively byBowden using gauge theory [Bow10, Sec. 7]. The following theorem gives a positive answer to(2) in certain overtwisted contact manifolds of arbitrary dimension. Theorem 10.22. Let V = i ( B ) , V ′ = j ( B ) ⊂ ( Y, ξ ) be the codimension contact submanifoldsdescribed in Construction 10.6. Recall that ( Y − V ′ , ξ ) and (a fortiori) ( Y, ξ ) are overtwisted.There does not exist a relative symplectic cobordism ( ˆ X, ˆ λ, H ) from ( Y, ξ, V ′ ) to ( Y, ξ, V ) suchthat H ( H ; Z ) = H ( H ; Z ) = 0 . In particular, V ′ ⊀ s V .Proof. Suppose for contradiction that such a relative symplectic cobordism exists. Accordingto Theorem 9.4, we have g CH • ( Y, ξ, V ; r ) = 0 for some r = ( α, τ, r ) ∈ R ( Y, ξ, V ) which wenow view as fixed. According to Theorem 9.2, we also have g CH • ( Y, ξ, V ′ ; r ′ ) = 0 for all r ′ = ( α ′ , τ ′ , r ′ ) ∈ R ( Y, ξ, V ′ ). Choose r ′ depending on our previous choice of r so that r ′ ≥ e E (( H,λ H ) α ′ α ) r . Then Corollary 6.8 along with our topological assumptions on H furnishes aunital Q -algebra map g CH • ( Y, ξ, V ′ ; r ′ ) → g CH • ( Y, ξ, V ; r ) . This gives the desired contradiction. (cid:3) Remark . To the best of our knowledge, Theorem 10.22 is the first result about symplecticcobordisms and concordances in dimensions > 3. We note that we could as well have provedTheorem 10.22 using the full invariant CH • ( − ; − ) instead of its reduced counterpart.Turning to question (1), we expect that there does exist a concordance from V to V ′ , andthat this should be provable using the techniques of [EM17]. However, we do not discuss thisquestion further here.We now consider Lagrangian cobordisms and concordances between Legendrian subman-ifolds. Recall from Definition 2.21 that an exact Lagrangian cobordism from Λ + to Λ − issimply a triple ( ˆ X, ˆ λ, L ), where ( ˆ X, ˆ λ ) is an exact symplectic cobordism with ends ( Y ± , ξ ± )and L is a cylindrical Lagrangian with ends Λ ± ⊂ ( Y ± , ξ ± ). In the special case where ( ˆ X, ˆ λ ) isthe symplectization of ( Y ± , ξ ± ) and where L is diffeomorphic to R × Λ ± , we speak instead ofa Lagrangian concordance . Lagrangian cobordisms have been studied in symplectic topologysince the beginnings of the subject. The more restrictive notion of a Lagrangian concordancewas introduced by Chantraine [Cha10].It is again straightforward to show that isotopic Lagrangian submanifolds are concordant,and we obtain a binary relation ≺ l on the set of Legendrian isotopy classes of a given contactmanifold ( Y, ξ ). One can then ask, as above, whether ≺ l is non-trivial and non-symmetric.Proposition 10.24 and Corollary 10.25 give obstructions to the existence of certain exactLagrangian cobordisms in overtwisted contact manifolds of arbitrary dimension. To the bestof our knowledge, such statements cannot be proved using invariants which are currently inthe literature. Proposition 10.24. Let ( Y, ξ ) be a contact manifold. Let Λ , Λ ′ be Legendrian knots such that H ( τ (Λ ′ ); Z ) = H ( τ (Λ ′ ); Z ) = 0 . Suppose that ( SY, λ Y , L ) is a Lagrangian concordance from Λ ′ to Λ . Given r = ( α, τ, r ) ∈ R ( Y, ξ, τ (Λ)) , there a map of Q -algebras (10.7) g CH • ( Y, ξ, τ (Λ ′ ); r ′ ) → g CH • ( Y, ξ, τ (Λ); r ) OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 93 for some r ′ = ( α ′ , τ ′ , r ′ ) ∈ R ( Y, ξ, τ (Λ ′ )) . (A similar statement holds for the non-reducedinvariants CH • ( − ) ).Proof. Observe that the trivial Lagrangian cobordism L = R × Λ ⊂ R × Y admits a “symplecticpush-off” τ ( L ) := R × τ (Λ) ⊂ R × Y . It follows by the Lagrangian neighborhood theorem thatany Lagrangian concordance ( SY, λ Y , L ) also admits a symplectic push-off ( SY, λ Y , H ), whichis a relative symplectic cobordism from ( Y, ξ, τ (Λ ′ )) to ( Y, ξ, τ (Λ)).Fix α ′ arbitrarily and choose r ′ so that r ′ ≥ e E (( H,λ H ) α ′ α ) r (note that τ ′ is unique since H ( τ (Λ ′ ); Z ) = 0). The claim now follows from Corollary 6.8. (cid:3) Corollary 10.25. Suppose that Λ , Λ ′ ⊂ ( Y, ξ ) are as in Construction 10.9. Then Λ ′ ⊀ l Λ . Incontrast, a result of Eliashberg–Murphy [EM13, Thm. 2.2] implies that Λ ≺ Λ ′ .Proof. Suppose for contradiction that Λ ′ ≺ l Λ. As in the proof of Corollary 10.10, we have g CH • ( Y, ξ, τ (Λ); r ) = 0for a suitable choice r ∈ R ( Y, ξ, τ (Λ)). On the other hand, we have g CH • ( Y, ξ, τ (Λ ′ ); r ′ ) = 0 forall r ′ ∈ R ( Y, ξ, τ (Λ ′ )). This gives a contradiction in view of Proposition 10.24. (cid:3) Remark . To the best of our knowledge, the above results on Lagrangiancobordisms cannot be obtained using invariants which are currently in the literature. Whileinvariants such as Legendrian contact homology provide a lot of information about Lagrangiancobordisms and concordances in certain tight contact manifolds, these invariants are blind inthe overtwisted setting.We end by considering Lagrangian cobordisms and concordances in certain (not necessarilyovertwisted) contact manifolds. In particular, we consider the question of when a Lagrangiancobordism can be displaced from a codimension 2 symplectic submanifold. Construction 10.27. Let ( Y , ξ ) = obd( T ∗ S n − , id) for n ≥ 3. Let ( V, ξ ) ⊂ ( Y , ξ ) be thebinding and let Λ ⊂ ( Y , ξ ) be the zero section of a page, which is a loose Legendrian by[CM19, Prop. 2.9]. Let U ⊂ Y − V be a small ball which intersects Λ in an ( n − ′ be obtained by stabilizing Λ inside U .Let U ⊂ Y − ( V ∪ Λ ∪ U ) be an open subdomain. Let ( Y, ξ ) be obtained by attaching asequence of handles along isotropics contained in U . Observe that V, Λ , Λ ′ can be viewed assubmanifolds of both Y and Y ; we will not distinguish these embeddings in our notation. Welet ( ˆ X, ˆ λ, ˆ V ) be the associated relative symplectic cobordism from ( Y, ξ, V ) to ( Y , ξ , V ).Observe that Λ , Λ ′ are Lagrangian isotopic: indeed, Λ is loose and Λ ′ is a stabilization of Λ.It follows of course that Λ ≺ Λ ′ and Λ ′ ≺ Λ. However, Theorem* 10.28 gives an obstruction toremoving intersection points between Lagrangian concordances and a codimension 2 symplecticsubmanifold. Theorem* 10.28. With the setup and notation of Construction 10.27, any concordance from Λ ′ to Λ must intersect ˆ V non-trivially. In contrast, [EM13, Thm. 2.2] implies that there existsa concordance from Λ to Λ ′ in the complement of ˆ V .Proof. Note that we can identify R ( Y , ξ , V ) = R ( Y, ξ, V ). According to Theorem* 9.7, L ( Y , ξ , V, Λ; r ) = 0 for suitable r ∈ R ( Y , ξ , V ). In constrast, Proposition* 9.3 implies that L ( Y, ξ, V, Λ ′ ; r ) = 0 since Λ ′ is loose in Y − V by construction. It now follows by Proposition* 8.7that there is a unital map of Q [ U ]-algebras L ( Y, ξ, V, Λ ′ ; r ) → L ( Y , ξ , V, Λ; r ) . This gives a contradiction. (cid:3) We remark that Construction 10.27 could be generalized in various directions without af-fecting the validity of Theorem* 10.28, but we do not pursue this here. Appendix A. Connected sums of almost-contact manifolds Let G be a connected subgroup of SO( n ). An almost G -structure on a smooth orientedmanifold M is a homotopy class of maps M → BG lifting the classifying map of the tangentbundle of M : BGM B SO( n ) T M An almost G -manifold is a manifold equipped with an almost G -structure. Example A.1. Taking G = U( n ) ⊂ SO(2 n ) yields the usual notion of an almost complexmanifold. Almost-contact manifolds correspond to G = U( n ) ⊂ SO(2 n + 1).If the n -dimensional sphere S n admits an almost G -structure, a result of Kahn [Kah69,Theorem 2] implies that for any two n -dimensional almost G -manifolds M and N , there existsan almost G -structure on M N which is compatible with the given ones on M and N in thecomplement of the disks used to form the connected sum. In general, this structure is notunique, so the connected sum M N is not well-defined as an almost G -manifold. However,we will show in A.1 that a choice of almost G -structure β on S n induces a canonical almost G -structure on the connected sum of any two almost G -manifolds. Hence, any such β givesrise to a connected sum operation ( M, N ) M β N for almost G -manifolds. Moreover, theset of almost G -structures on S n forms a group under this operation (with identity β ). Inappendix A.2, we will show that this group acts on the set of almost G -structures of any n -dimensional almost G -manifold.A.1. Connected sums of almost G -manifolds. Let S n be the unit sphere in R n +1 , equippedwith its standard orientation as the boundary of the unit disk D n +1 . We will write its pointsas pairs ( x, z ) ∈ R n × R . Define D − = { ( x, z ) ∈ S n | z < / } ,D + = { ( x, z ) ∈ S n | z > − / } ,A = D − ∩ D + ,C ± = D ± \ A. Note that D − and D + are open disks, C − and C + are closed disks, A is an open annulus, and S n = D − ∪ D + = C − ⊔ A ⊔ C + .Let M and N be smooth connected oriented n -dimensional manifolds. Choose orientationpreserving embeddings i + : D + → M and i − : D − → N . We define the connected sum M N = M i + ,i − N by M N = (cid:0) M \ i + ( C + ) ⊔ N \ i − ( C − ) (cid:1) / ∼ where i + ( x ) ∼ i − ( x ) for every x ∈ A .We will now explain how to construct a classifying map for the tangent bundle of M N .The following elementary fact from topology will be useful. This assumption is used in the proof of Proposition A.6. OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 95 Proposition A.2. Let i : A → X be a cofibration. Assume A is contractible. Then forany connected space Y and continuous maps F : X → Y , f : A → Y , there exists a map F ′ : X → Y homotopic to F such that F ′ ◦ i = f . Let τ S : S n → B SO( n ) be a classifying map for T S n . Let τ M and τ N be classifying mapsfor T M and T N such that τ M ◦ i + = τ S | D + and τ N ◦ i − = τ S | D − (such maps always exist byProposition A.2). Define τ M N to be the unique map M N → B SO( n ) which coincides with τ M on M \ i + ( C + ) and with τ N on N \ i − ( C − ). Proposition A.3. τ M N is a classifying map for T ( M N ) . We start with an easy topological lemma. Lemma A.4. Let E be an oriented vector bundle over a manifold M n and let i : D n → M n be an embedding. Then any automorphism of i ∗ E can be extended to an automorphism of E .Proof. Let φ be an automorphism of i ∗ E . Since D n is contractible, we can trivialize i ∗ E and think of φ as a map D n → GL + ( n ). Clearly φ | ∂D n is nullhomotopic, and since GL + ( n )is connected, we can extend φ to a map ˜ φ : D n → GL + ( n ) which is constant with valueId ∈ GL + ( n ) near ∂D n . Using a tubular neighborhood of i ( ∂D n ) ⊂ M , we can also extend i to an embedding ˜ i : D n → M n . Then ˜ φ gives us an automorphism of ˜ i ∗ E which is equal to theidentity over a neighborhood of ∂D n ⊂ D n and hence extends trivially to an automorphism of E . (cid:3) Proof of Proposition A.3. Let ˜ γ n → B SO( n ) be the universal bundle over B SO( n ). We wantto show that T ( M N ) is isomorphic to τ ∗ M N ˜ γ n .The tangent bundle T ( M N ) of the connected sum is obtained by gluing T ( M \ i + ( C + ))and T ( N \ i − ( C − )) along the maps di + : T A → T ( M \ i + ( C + )) and di − : T A → T ( N \ i − ( C − )).Because of our assumption that τ M ◦ i + = τ S | D + and τ N ◦ i − = τ S | D − , we also have that τ ∗ M N ˜ γ n is obtained by gluing ( τ M | M \ i + ( C + ) ) ∗ ˜ γ n and ( τ N | N \ i − ( C − ) ) ∗ ˜ γ n along bundle maps ( τ S | A ) ∗ ˜ γ n → ( τ M | M \ i + ( C + ) ) ∗ ˜ γ n and ( τ S | A ) ∗ ˜ γ n → ( τ N | N \ i − ( C − ) ) ∗ ˜ γ n covering i + : A → M \ i + ( C + ) and i − : A → N \ i − ( C − ) respectively. Hence, in order to show that T ( M N ) is isomorphic to τ ∗ M N ˜ γ n , it suffices to construct a commutative diagram T ( M \ i + ( C + )) ( τ M | M \ i + ( C + ) ) ∗ ˜ γ n T A ( τ S | A ) ∗ ˜ γ n T ( N \ i − ( C − )) ( τ N | N \ i − ( C − ) ) ∗ ˜ γ ndi + di − where the horizontal arrows are bundle isomorphisms.Start by fixing an isomorphism φ : T S n → τ ∗ S ˜ γ n , and let the middle arrow of the diagrambe the restriction of φ to T A . To get the top and bottom arrows, it suffices to find bundleisomorphisms completing the following commutative squares: T M τ ∗ M ˜ γ n T D + ( τ S | D + ) ∗ ˜ γ ndi + φ T D − ( τ S | D − ) ∗ ˜ γ n T N τ ∗ N ˜ γ ndi − φ This is possible by Lemma A.4. (cid:3) We are now ready to define the connected sum of two almost G -manifolds. Definition A.5. Suppose that S n admits an almost G -structure, and fix a choice β of onesuch structure. Let β M and β N be almost G -structures on M and N respectively. We definean almost G -structure β M β β N on M N as follows.Pick maps ˜ τ S : S n → BG , ˜ τ M : M → BG and ˜ τ N : N → BG representing β , β M and β N respectively. By Proposition A.2, we can assume that ˜ τ M ◦ i + = ˜ τ S | D + and ˜ τ N ◦ i − = ˜ τ S | D − .Hence, there is a unique map ˜ τ M N = ˜ τ M ˜ τ S ˜ τ N : M N → BG which coincides with ˜ τ M on M \ i + ( C + ) and with ˜ τ N on N \ i − ( C − ). By Proposition A.3, thecomposition M N BG B SO( n ) ˜ τ M N is a classifying map for T ( M N ). Hence, we can (and do) define β M β β N to be the homotopyclass of ˜ τ M N . Proposition A.6. The almost G -structure β M β β N is well-defined, i.e. independent of thechoice of ˜ τ S , ˜ τ M and ˜ τ N .Proof. Let ˜ τ jS , ˜ τ jM , and ˜ τ jN represent β , β M and β N respectively, where j ∈ { , } . As inDefinition A.5, we assume that ˜ τ jM ◦ i + = ˜ τ jS | D + and ˜ τ jN ◦ i − = ˜ τ jS | D − .Fix a homotopy ˜ τ tS between ˜ τ S and ˜ τ S . We will show that there exist homotopies ˜ τ tM and˜ τ tN such that ˜ τ tM ◦ i + = ˜ τ tS | D + and ˜ τ tN ◦ i − = ˜ τ tS | D − . This implies that ˜ τ M N is homotopic to˜ τ M N and hence that β M β β N is well-defined.Pick an arbitrary homotopy h : M × I → BG between ˜ τ M and ˜ τ M and define a map g : D + × ∂I → BG by g ( x, t, 0) = h ( i + ( x ) , t ), g ( x, , s ) = ˜ τ S ( x ), g ( x, , s ) = ˜ τ S ( x ) and g ( x, t, 1) = ˜ τ tS ( x ). We canextend g to a map ˆ g : D + × I → BG since the obstruction to doing so lies in H ( D + × I , D + × ∂I ; π ( BG )) ∼ = π ( BG ) ∼ = π ( G ) , which is trivial by our assumption that G is connected.Let f : (cid:0) M × ( I × { } ∪ { } × I ∪ { } × I ) (cid:1) ∪ (cid:0) i + ( D + ) × I (cid:1) → BG be defined by • f ( x, t, 0) = h ( x, t ), f ( x, , s ) = ˜ τ M ( x ) and f ( x, , s ) = ˜ τ M ( x ) for x ∈ M ; • f ( x, t, s ) = ˆ g ( i − ( x ) , t, s ) for x ∈ i + ( D + ).Since i + : D + → M is a cofibration, the domain of f is a retract of M × I . We can thereforeextend f to a map ˆ f : M × I → BG . Restricting ˆ f to M × I × { } then provides us with ahomotopy ˜ τ tM such that ˜ τ tM ◦ i + = ˜ τ tS | D + .The same argument gives us a homotopy ˜ τ tN such that ˜ τ tN ◦ i − = ˜ τ tS | D − , so this completesthe proof. (cid:3) Definition A.7. If M = ( M, β M ) and N = ( N, β N ) are almost G -manifolds, their connectedsum (with respect to β ) is the almost G -manifold M β N := ( M N, β M β β N ). OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 97 As usual, there is an ambiguity in the notation M β N since the construction of the con-nected sum involves a choice of embeddings i + : D + → M , i − : D − → N . However, the resultis independent of these choices up to the appropriate notion of equivalence, as one wouldexpect. Definition A.8. A diffeomorphism of almost G -manifolds f : ( M, β M ) → ( N, β N ) consists ofa smooth diffeomorphism f : M → N such that f ∗ β N = β M . Proposition A.9. The connected sum M β N is well-defined up to diffeomorphism of almost G -manifolds. More precisely, given any orientation preserving embeddings i + , j + : D + → M and i − , j − : D − → N , there exists an orientation preserving diffeomorphism φ : M i + ,i − N → M j + ,j − N such that φ ∗ ( β M j + ,j − ,β β N ) = β M i + ,i − ,β β N for any almost G -structures β M , β N on M , N .Proof. This follows from the isotopy extension theorem as in the smooth case. (cid:3) Remark A.10 (Connected sums of contact manifolds) . Suppose that ( M , α ) , ( M , α ) arecontact manifolds. Then one can form the connected sum ( M M , α α ), which is also acontact manifold. The connected sum is obtained by choosing Darboux balls in M , M andconnecting them by a “neck”. This operation can also be understood as a contact surgeryalong a 0-sphere. We refer to [BvK10, Sec. 6.2] and [vK17, Sec. 3] for more details.Let β ∈ alm U ( n ) ( S n − ) be the almost-contact structure induced by the standard contactstructure on the sphere. Then the operation of connected sum (with respect to β ) of almostU( n − − G -manifolds: Proposition A.11. Let M , N , P be connected almost G -manifolds of dimension n and let β , β ′ be almost G -structures on S n . (1) M β ( S n , β ) ∼ = M . (2) ( M β N ) β ′ P ∼ = M β ( N β ′ P ) .Proof. If one takes i − : D − → S n to be the inclusion map, then the connected sum M S n is canonically identified with M as a smooth manifold. If one further takes ˜ τ N = ˜ τ S inDefinition A.5, then this identification is compatible with the almost G -structures on M S n and M . This proves that M β ( S n , β ) ∼ = M .To prove that ( M β N ) β P ∼ = M β ( N β P ), choose embeddings i + : D + → M , i − : D − → N , j + : D + → N and j − : D − → P . If we assume that i − and j + have disjoint images,then i − induces an embedding D − → N j + ,j − P , j + induces an embedding D + → M i + ,i − N ,and there is a canonical identification of smooth manifolds( M i + ,i − N ) j + ,j − P ∼ = M i + ,i − ( N j + ,j − P ) . Moreover, this identification is compatible with the almost G -structures in the sense that forany choice of maps ˜ τ S , ˜ τ ′ S , ˜ τ M , ˜ τ N , ˜ τ P , the following diagram commutes: ( M i + ,i − N ) j + ,j − P BGM i + ,i − ( N j + ,j − P ) ∼ = (˜ τ M ˜ τS ˜ τ N ) ˜ τ ′ S ˜ τ P ˜ τ M ˜ τS (˜ τ N ˜ τ ′ S ˜ τ P ) (cid:3) A.2. The group of almost G -structures on the sphere. We will denote the set of almost G -structures on a manifold M by alm G ( M ). More generally, if A ⊂ M is a closed subset and β is an almost G -structure on some open neighborhood of A , then alm G ( M, A ; β ) will denotethe set of almost G -structures on M which agree with β near A .In this section, we will show that β is a group operation on alm G ( S n ), with β as identityelement. The resulting group will be denoted by alm βG ( S n ). We will then show that alm βG ( S n )acts on alm G ( M ), and more generally on alm G ( M, A ; β ) if M \ A is connected. Proposition A.12. Given any β ∈ alm G ( S n ) , there exists a β ∈ alm G ( S n ) such that β β β = β .Proof. Recall the decomposition S n = C − ∪ A ∪ C + introduced at the beginning of section A.1.We will use the notation h τ − , τ A , τ + i to denote the unique (assuming it exists) map S n → BG which coincides with the given maps τ − : C − → BG , τ A : A → BG and τ + : C + → BG on C − , A and C + respectively.Let ˜ τ S = h τ − S , τ AS , τ + S i be a representative for β . Given β and β in alm G ( S n ), we can chooserepresentatives of the form h τ − , τ AS , τ + S i and h τ − S , τ AS , τ +2 i by Proposition A.2. Then β β β is represented by h τ − , τ AS , τ +2 i . Hence, all we need to show is that for any τ − : C − → BG ,there exists τ +2 : C + → BG such that h τ − , τ AS , τ +2 i is homotopic to ˜ τ S . This again follows fromProposition A.2. (cid:3) Corollary A.13. ( alm G ( S n ) , β ) is a group with identity β .Proof. This follows from Proposition A.11 and Proposition A.12. (cid:3) Remark A.14 . The group ( alm G ( S n ) , β ) is independent of β up to isomorphism. Indeed,given any x, y, β, β ′ ∈ alm G ( S n ), it follows from Proposition A.11 that( x β β ′ ) β ′ ( y β β ′ ) = ( x β ( β ′ β ′ y )) β β ′ = ( x β y ) β β ′ , which implies that the map ( alm G ( S n ) , β ) → ( alm G ( S n ) , β ′ ) x x β β ′ is a group isomorphism.Given orientation preserving embeddings i + : D + → M and i − : D − → S n , the results ofsection A.1 give us a well-defined map alm G ( M ) × alm βG ( S n ) → alm G ( M i + ,i − S n ) . For the remainder of this section, we will take i − to be the inclusion map D − ֒ → S n . Then M i + ,i − S n is canonically identified with M (regardless of what i + is) and we get a map(A.1) alm G ( M ) × alm βG ( S n ) → alm G ( M ) . OMOLOGICAL INVARIANTS OF CODIMENSION 2 CONTACT SUBMANIFOLDS 99 By Proposition A.11, this is a group action. 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