A note on Lagrangian intersections and Legendrian Cobordism
AA NOTE ON LAGRANGIAN INTERSECTIONSAND LEGENDRIAN COBORDISM.
LARA SIMONE SU ´AREZ
Abstract.
Let Λ , Λ (cid:48) be a pair of closed Legendrian submanifolds in a closed contact manifold( Y, ξ = Ker ( α )) related by a Legendrian cobordism W ⊂ ( C × Y, ˜ ξ = Ker ( − y d. x + α )). In thisnote, we show that in the hypertight setting, if Λ intersects a closed, weakly exact or monotonepre-Lagrangian P ⊂ Y for reasons of Floer homology, then so does Λ (cid:48) . Introduction
Let (
Y, ξ ) be a co-oriented closed contact manifold. In this paper we consider pairs ( P, Λ)consisting of a closed Legendrian submanifold Λ ⊂ Y and a closed pre-Lagrangian P ⊂ Y (see Definition 4). A usual question in symplectic topology is that of the displaceability ofLagrangian submanifolds by Hamiltonian isotopy. In contact topology, this question can beinterpreted as the search for pairs ( P, Λ) with the so called intersection property as defined byEliashberg-Polterovich in [10]. A pair ( P, Λ) of (
Y, ξ ) has the intersection property , if for everycontactomorphism φ ∈ Cont ( Y, ξ ) the intersection P ∩ φ (Λ) is non-empty. A tool to findsuch pairs was introduced by Eliashberg-Hofer-Salamon in [9]. It is a Floer homology group HF ( P, Λ; Z ) for pairs ( P, Λ), that is invariant under Legendrian isotopy. In some special cases,this group is isomorphic to the homology of the Legendrian Λ. This is the case in the followingexamples from [9], which are the first examples of pairs with the intersection property:(1) Let X be a closed manifold and P + ( T ∗ X ) the space of co-oriented contact elementswith the canonical contact structure. If there is a non-singular closed one form β on X then there is a pre-Lagrangian P associated to graph( β ). If moreover, P is foliated byclosed Legendrians and Λ is any Legendrian leaf, then the pair ( P, Λ) has the intersectionproperty.(2) Let (
M, ω ) be a closed symplectic manifold with [ ω ] ∈ H ( M ; Z ). In a prequantizationspace QM of ( M, ω ), if there is a closed Lagrangian submanifold L ⊂ ( M, ω ) thatsatisfies the Bohr-Sommerfeld condition and π ( M, L ) = 0 then the corresponding pre-Lagrangian P in QM and a flat Legendrian lift Λ of L is a pair with the intersectionproperty. This last example was independently discovered by Ono [17].In [10] Eliashberg-Polterovich also defined the stable intersection property of a pair, meaningthat in the stabilized setting ( T ∗ S × Y, Ker ( rdt + α )) the pair ( S × { } × P, S × { } × Λ) hasthe intersection property. They showed that the previous examples have the stable intersectionproperty. The (stable) intersection property is related to the orderability of the contact manifold.In fact in [10, Theorem 2.3.A] they proved that having a pair with this property implies thatthe manifold is orderable. The 0 here stands for the connected component of the identity. a r X i v : . [ m a t h . S G ] J un LARA SIMONE SU ´AREZ
From another perspective, we can say that in the previous examples, the stabilization preservesthe intersection property. In the same way, in this note we remark, in particular for the aboveexamples, that the intersection property is preserved by Legendrian cobordism with the followingproperty: there is a contact form with no contractible Reeb orbits or chords with boundary onthe cobordism. This is the hypertight property defined below in Definition 2.The notion of Legendrian cobordism was introduced by Arnol’d ([2], [3]) who computed theimmersed Legendrian cobordism group (oriented and not-oriented) when the immersed Legen-drian has dimension n = 1, the cases n ≥ J ( R × N ), projects to an immersed exact Lagrangiancobordism in T ∗ N . Biran-Cornea [5] showed that monotone embedded Lagrangian cobordismpreserves Floer-theoretical Lagrangian invariants after Chekanov [6] showed how these preservecertain counts of Maslov two disks.In general, embedded Legendrian cobordism does not preserve holomorphic curves type invari-ants. This is the case, for instance of the linearized contact homology, a Legendrian analogueof the Lagrangian Floer homology. In ( R , dz − pdq ) for example, it follows from the work ofArnol’d that two Legendrian knots are oriented Legendrian cobordant if and only if they havethe same Maslov index. Hence any two knots with same Maslov index but different linearizedcontact homology are still Legendrian cobordant.However, under additional restrictions, we show that Legendrian cobordism does preserve theFloer homology group of Eliashberg-Hofer-Salamon. More precisely we prove the following: Theorem 1.
Let ( Y, ξ ) be a closed hypertight contact manifold and Λ , Λ (cid:48) ⊂ Y a pair of hypertightLegendrian submanifolds related by a connected hypertight Legendrian cobordism ( W ; Λ , Λ (cid:48) ) ⊂ ( C × Y, ˜ ξ ) . If P ⊂ Y is a closed weakly exact or monotone with N P ≥ pre-Lagrangiansubmanifold then HF ( P, Λ) ∼ = HF ( P, Λ (cid:48) ) . The proof of Theorem 1 is an adaptation to the cobordism setting of [9, Theorem 3.7.3]. Wedefine a Floer complex for the Lagrangian lift of a Legendrian cobordism ( W ; Λ , Λ (cid:48) ) and asuitable cylindrical Lagrangian ˜ P obtained from P . We then observe that the homology of thiscomplex is isomorphic to the homology of the Floer complex of the Legendrian boundary Λ (orΛ (cid:48) ) and the pre-Lagrangian P . This proof uses a similar strategy to the one used by Biran-Cornea [5] to show that monotone Lagrangian cobordism preserves Floer homology. CombiningTheorem 1 with [9, Theorem 2.5.1-2.5.4] (see section 2.2 Theorem 11) we get the followingcorollary. Corollary 2.
Under assumptions in Theorem 1, consider a pair ( P, ( W ; Λ , Λ (cid:48) )) where P isweakly exact and Λ ⊂ P . If moreover boundary homomorphism π ( Y, P ) → π ( P ) is trivial,then if { φ t } ≤ t ≤ is a contact isotopy such that P (cid:116) φ (Λ (cid:48) ) . Then φ (Λ (cid:48) ) ∩ P ≥ rank ( H ∗ (Λ , Z )) . Acknowledgements.
I thank Egor Shelukhin for suggesting this project and helpful discussions.This research is supported by the Floer Centre of Geometry at Ruhr-University Bochum and
NOTE ON LAGRANGIAN INTERSECTIONS AND LEGENDRIAN COBORDISM 3 is part of a project in the SFB/TRR 191
Symplectic Structures in Geometry, Algebra andDynamics , funded by the DFG. 2.
Setting
Let (
Y, ξ ) be a closed co-oriented contact 2 n +1-manifold. Denote by Cont + ( ξ ) the set of contactforms defining the same co-orientation on ξ .For a choice of α ∈ Cont + ( ξ ), the symplectic manifold ( S α ( Y ) , ω ) := ( R × Y, d( e θ α )) is calledthe symplectization of ( Y, α ).Given η ∈ Cont + ( ξ ), its Reeb vector field is the unique vector field R η ∈ Γ( T Y ) satisfying thetwo equations: ι R η η = 1 ,ι R η d η = 0 . Its associated flow is denoted by { φ tR η } and it is called the Reeb flow . Definition 3.
A contact manifold ( Y, ξ ) that admits a contact form with no contractible periodicReeb orbits is called hypertight . Such a contact form is called a hypertight contact form .A Legendrian submanifold Λ ⊂ Y for which there is a hypertight contact form such that anyReeb chord with boundary on Λ represents a non-trivial class in π ( Y, Λ) is called a hypertightLegendrian submanifold. Some examples of hypertight contact manifolds are jet spaces, certain pre-quantization spacesand certain unit cotangent bundles, with their corresponding standard contact structures.In this note all contact manifolds and Legendrians are assumed to be hypertight.
Definition 4. A pre-Lagrangian is a ( n + 1) -dimensional submanifold P ⊂ Y n +1 for whichthere exists a contact form α ∈ Cont + ( ξ ) such that d( α | P ) = 0 i.e α | P is closed. If there is a function f : P → R such that α | P = d f then P is called a exact pre-Lagrangian and f is called contact potential on P . An exact Lagrangian lift ˆ P of an exact pre-Lagrangianis given by ˆ P = { } × P ⊂ R × Y. Note that ˆ f : ˆ P → R , ˆ f (0 , p ) = f ( p ) is a primitive for e θ α | ˆ P .A pre-Lagrangian P is called weakly exact when d α | π ( Y,P ) = 0 and monotone if there is apositive constant K such that for all u ∈ π ( Y, P ) we have (cid:90) ∂u α = Kµ ( u )where µ denotes the Maslov class µ : π ( Y, P ) → Z . The positive generator of the image of thehomomorphism defined by µ is denoted N P and is called the minimal Maslov number of P . Example 5.
Let ( Y, ξ ) = ( S ( T ∗ X ) , Ker ( p d q | S ( T ∗ X ) )) where the right hand side is the unitcotangent bundle of ( X, g ) with respect to some choice of a Riemannian metric g . Let η be anowhere vanishing closed 1-form. If Γ η ⊂ T ∗ X denotes the graph of η , then Γ η/ || η || ⊂ S ( T ∗ X ) is a pre-Lagrangian for the contact form || η || p d q . LARA SIMONE SU ´AREZ
Example 6.
Let ( Y, ξ ) = QX be the prequantization of a symplectic manifold ( X, ω ) . Topolog-ically Y is an S -principal bundle over X with projection map p : Y → X . Given a Lagrangiansubmanifold L ⊂ X its lift p − ( L ) ⊂ ( Y, ξ ) is a pre-Lagrangian. Legendrian cobordism.
Let ( x, y ) denote coordinates on C . Then ˜ α = − y d x + α is acontact form on C × Y . Denote the resulting contact manifold by( ˜ Y , ˜ ξ ) = ( C × Y, Ker ( ˜ α )) . Let π C : C × Y → C denote the projection map. Given Λ , Λ (cid:48) ⊂ Y two closed Legendriansubmanifolds, in [2] and [3] Arnol’d introduced the notion of Legendrian cobordism on whichthe following definition is based. Definition 7. A Legendrian cobordism ( W ; Λ , Λ (cid:48) ) is an embedded Legendrian submanifold W ⊂ ( ˜ Y , ˜ ξ ) such that for some < (cid:15) ∈ R and R ≥ we have that W ∩ π − C ([ (cid:15), R − (cid:15) ] × R ) is a smooth compact manifold with boundary Λ (cid:116) Λ (cid:48) and W ∩ π − C ( C \ ([ (cid:15), R − (cid:15) ] × R )) = ( −∞ , (cid:15) ) × { } × Λ (cid:116) ( R − (cid:15), ∞ ) × { } × Λ (cid:48) . Example 8. (Legendrian suspension) [10]
Let H : R × Y → R be a contact Hamiltonian with H t ≡ for t ≤ and t ≥ . The associated contact vector field X H is given by the twoconditions: ι X H α = H and ι X H d α = − d H + ( ι R α d H ) α . Denote by { ψ t } the contact isotopyassociated to X H . Let Λ ⊂ M be a compact Legendrian. The map: Φ : R × Λ → C × Y ( t, p ) (cid:55)→ ( t, H ( t, ψ t ( p )) , ψ t ( p )) , is a Legendrian embedding into ( C × M, Ker ( ˜ α )) defining a Legendrian cobordism (Φ( R × Λ); Λ , ψ (Λ)) . Remark . Any Legendrian isotopy defines a Legendrian cobordism given by the Legendriansuspension of a contact Hamiltonian generating the isotopy.
Example 9.
The trace of surgery: Elementary Legendrian cobordisms can be constructed us-ing the Lagrangian handle in Haug [14] building on the work of Dimitroglou Rizell [7] . Thisconstruction produces an embedded Legendrian cobordism in J ( R n +1 ) .The Legendrian k -handle: Consider the non compact Legendrian W (cid:15),k ⊂ J ( R n +1 ) given by W (cid:15),k = { ( x , x , ± d F ( x , x ) , ± F ( x , x )) ∈ J ( R n +1 ) | ( x , x ) ∈ U } for x = ( x , ..., x n ) ∈ R n where F ( x , x ) = ( f ( x , x )) , f ( x , x ) = k (cid:88) i =1 x i − n (cid:88) i = k +1 x i + σ ( k (cid:88) i =1 x i ) ρ ( x ) − and U = { ( x , x ) | f ( x , x ) ≥ } , where the functions σ and ρ look like: NOTE ON LAGRANGIAN INTERSECTIONS AND LEGENDRIAN COBORDISM 5
In the case a Legendrian Λ contains a l -sphere S l ⊂ Λ which bounds an isotropic l +1 -disk D l +1 ⊂ Y \ Λ , compatible with the Legendrian in some sense made precise in [14] , after surgery a newLegendrian is obtained, and an embedded Legendrian cobordism between the initial Legendrianand the one produced after the surgery exists. Example 10.
Let ( M, λ ) be a Liouville domain. The Legendrian lift of an immersed exactLagrangian cobordism ( W ; L , L ) ⊂ ( C × M, d( − y d x + λ )) to the contactization of C × M ,defined by C × C ( M, λ ) = ( C × M × R z , − y d x + λ + d z ) . More precisely, if f : W → R isa potential for the Lagrangian W , the Lagrangian ˜ W := { ( w, f ( w )) | w ∈ W } is an embeddedLegendrian cobordism in C × C ( M, λ ) . Previous results.
We denote by HF ( P, Λ; Z ) the homology of the complex defined byEliashberg-Hofer-Salamon in [9]. In [9, Theorem 3.7.3] they showed that in the hypertightsetting, this homology group is well defined and invariant under compactly supported Hamil-tonian isotopy. Let ( Y, ξ ) be as in Examples (1) or (2) in the introduction, namely, either (1)a sphere cotangent bundle of a compact manifold or (2) a prequantization space. Assume thatthe contact manifold and pre-Lagrangian submanifold satisfy the topological condition that theboundary map π ( Y, P ) → π ( P ) is trivial. In this setting, when the pair ( P, Λ) is either thegraph of a non-zero closed form foliated by closed Legendrians and a Legendrian leaf of it in thefirst case, or the pre-Lagrangian lift of a Lagrangian satisfying the Bohr-Sommerfeld conditionwith the property that π ( Y, P ) = 0 and a Legendrian lift of it, then the following theoremholds.
Theorem 11. [9] [1995] Let { φ t } ≤ t ≤ be a contact isotopy and Λ a Legendrian such that P (cid:116) φ (Λ) . Then φ (Λ) ∩ P ≥ rank ( H ∗ (Λ , Z )) . In the same direction Akaho [1], proved a version of the previous theorem replacing the hyper-tight condition by a small energy condition. Results about displaceability of pre-Lagrangianscan also be found in the work of Marinkovic-Pabiniak [16].3.
A Floer complex for the pair ( P, W ) . The version of Floer complex associated to the pair (
P, W ), where P ⊂ Y is a weakly exact ormonotone pre-Lagrangian (with N P ≥
2) and ( W, Λ , Λ (cid:48) ) ⊂ C × Y is a Legendrian cobordism, isan adaptation to the cobordism setting (following Biran and Cornea in [5]) of the Floer complexas defined in [9].Let ˜ α ∈ Cont + ( ˜ ξ ) be a contact form on C × Y that can be written as ˜ α = − y d x + α where α ∈ Cont + ( ξ ). We assume that ˜ α satisfies: LARA SIMONE SU ´AREZ • P is a pre-Lagrangian for α , i.e dα | P = 0. • W is hypertight for ˜ α .To the pair ( P, W ) we associate the pair ( L P , L W ) in ( S ( ˜ Y ) , ˜ ω = d( e θ ˜ α )), where L P = { } × R × { } × P denotes the Lagrangian lift of the pre-Lagrangian ˜ P = R × { } × P ⊂ C × Y and L W = R × W is the Lagrangian lift of W .3.1. Choices.
The Floer complex will depend on the choice of triples (
H, f, J ) where:(1) Denote by π C : C × Y → C the projection. Let us fix a compact set B ⊂ C of the form[ R − , R + ] × [ R − , R + ] for R ± ∈ R big enough, such that π C ( W ) ∩ C \ B is( −∞ , R − ] × { } ∪ [ R + , ∞ ) × { } . (2) A Hamiltoninan H : [0 , × S ( ˜ Y ) → R compactly supported in R × π − C ( B ) ⊂ S ( ˜ Y ).We use the Hamiltonian flow of H to perturb L P to make it transverse to L W on R × π − C ( B ) ⊂ S ( ˜ Y ).(3) A smooth function f : C → R with the property that f ( x, y ) = a + x + b + for x < R − − (cid:15) and f ( x, y ) = a − x + b − for x > R + + (cid:15) where a ± , b ± ∈ R and (cid:15) >
0. We consider the Hamiltonian ˜ f = e θ ( f ◦ π C ) on S ( ˜ Y ). Itis the Hamiltonian lift of the contact Hamiltonian f ◦ π C to S ( ˜ Y ). The correspondingHamiltonian flow ψ t ˜ f : S ( ˜ Y ) → S ( ˜ Y ) is given by( θ, x, y, p ) ∈ R × C × Y (cid:55)→ ψ t ˜ f ( θ, x, t, p ) = ( θ, x, y + a ± t, φ a ± xtR α ( p )) , for ( θ, x, y, p ) outside R × B × Y . This Hamiltonian flow perturbs the cylindrical endsof L P in such a way that the time one map ψ f ( L P ) ∩ L W = ∅ outside R × B × Y . Allthis while keeping the contact form invariant at infinity.(4) J is an admissible ˜ ω -compatible almost complex structure on S ( ˜ Y ). Admissible means acylindrical and d ˜ α -compatible almost complex structure for which dπ is ( J, j )-holomorphicwhere π : ( S ( ˜ Y ) , J ) → ( C , j ) denotes the projection and j denotes the complex structureon C . We define admissible in more detail in the next section.Let Λ = { ∞ (cid:88) k =0 a k T λ k | a k ∈ Z , λ k ∈ R , lim k →∞ λ k = ∞} . The (cid:63) -Floer complex of the pair ( L P , L W ) is denoted by(1) CF (cid:63) ( L P , L W ; H, f, J ) := ( Λ (cid:104) ˆ I (cid:63) ( ψ f ( L P ) , L W ; H ) (cid:105) , d J ) . The set ˆ I (cid:63) ( ψ f ( L P ) , L W ; H ) is constructed as follows:Consider the set P ( ψ f ( L P ) , L W ) := { γ ∈ C ([0 , S ( ˜ Y )) | γ (0) ∈ ψ f ( L P ) , γ (1) ∈ L W } of pathsbetween ψ f ( L P ) and L W . For a fix ∗ ∈ ψ f ( L P ) ∩ L W denote by (cid:63) = [ ∗ ] ∈ π ( P ( ψ f ( L P ) , L W ))and let P (cid:63) ( ψ f ( L P ) , L W ) be the connected component corresponding to the class (cid:63) . When ψ f ( L P ) ∩ L W is not transverse, we use the Hamiltonian H . Denote by I (cid:63) ( ψ f ( L P ) , L W ; H ) := { x ∈ ψ H ( ψ f ( L P )) ∩ L W | x ∈ P (cid:63) ( ψ H ( ψ f ( L P )) , L W ) } NOTE ON LAGRANGIAN INTERSECTIONS AND LEGENDRIAN COBORDISM 7 the intersection points in the connected component of (cid:63) . Thenˆ I (cid:63) ( ψ f ( L P ) , L W ; H ) := { ( x, ˜ x ) ∈ I (cid:63) ( φ f ( L P ) , L W ; H ) × C ([0 , , P (cid:63) (cid:16) ψ H ( ψ f ( L P )) , L W (cid:17) ) | ˜ x (0) = x, ˜ x (1) = ∗} / ∼ where ( x, ˜ x ) ∼ ( x (cid:48) , ˜ x (cid:48) ) iff x = x (cid:48) and µ (˜ x x (cid:48) )) = 0 where µ denotes the Maslov index and ˜ x (cid:48) denotes the same map with opposite orientation. Then the elements of ˆ I (cid:63) ( ψ f ( L P ) , L W ; H ) havea well defined index | [( x, ˜ x )] | = µ (˜ x ) ∈ Z .By choosing suitable coefficients a ± for the perturbation f we can ensure that ψ f ( L P ) ∩ L W iscontained in a bounded region. Then perturbing by a generic, compactly supported Hamiltonian H will ensure that ψ H ( ψ f ( L P )) ∩ L W is a finite set and the intersection is transverse makingthe set I (cid:63) ( ψ f ( L P ) , L W ; H ) finite.Let J = { J t } t ∈ [0 , be a time dependent almost complex structure and let ˆ x ± = [( x ± , ˜ x ± )] ∈ ˆ I (cid:63) ( ψ f ( L P ) , L W ; H ). We define moduli spaces M (ˆ x − , ˆ x + ; J ) := (cid:110) u ∈ C ([0 , × R ; S ( ˜ Y )) | ∂ J u =0 ,E ( u ) < ∞ lim s →±∞ u ( t,s )= x ± ( t ) ,u (0 ,s ) ∈ ψ H ( ψ f ( L P )) ,u (1 ,s ) ∈ L W µ (˜ x − )= µ ( u x + ) (cid:111) / R . Admissible almost complex structure.
Definition 12.
An almost complex structure J on S ( ˜ Y ) = R × ˜ Y is called cylindrical if J is R -invariant and J ∂ θ ∈ T ˜ Y . Let J be an almost complex structure on ˜ ξ compatible with the two form d ˜ α . By setting J ∂ θ = R ˜ α , J extends to a cylindrical almost complex structure ˜ J ˜ α on R × ˜ Y . For (ˆ a, ˆ m ) ∈ T ( a,m ) R × ˜ Y ,˜ J ˜ α is given by(2) ( ˜ J ) ( a,m ) (ˆ a, ˆ m ) = ( − ˜ α ( a,m ) ( ˆ m ) , J m ( π ˜ ξ ( ˆ m )) + ˆ aR ˜ α ) , here π ˜ ξ : T ( R × ˜ Y ) → ˜ ξ is the projection to ˜ ξ parallel to R ˜ α . Such an almost complex structureis called compatible with the contact form ˜ α . Definition 13.
An almost complex structure J on S ( ˜ Y ) is called admissible if:C1 J is cylindrical and restricts to a d ˜ α -compatible complex structure on the plane bundle ˜ ξ .C2 There is a compact set D ⊂ R × C such that D = D × D with B ⊂ D for B thefixed compact set on section 3.1, choice 1. Moreover, J | (( R × C ) \ D ) × Y ) = ˜ J ˜ α where the righthand side is ˜ α -compatible as defined above.C3 On (( R × C ) \ D ) × Y ) the almost complex structure J satisfies that the projection π : ( R × C × Y, J ) → ( C , i ) , is ( J, j ) -holomorphic.We denote the space of admissible almost complex structures by J ad . Notice that any J compatible almost complex structure on ξ extends to an admissible one.Indeed, since ˜ ξ ( x,y,p ) = R (cid:104) ∂ x + yR α (cid:105)⊕ R (cid:104) ∂ y (cid:105)⊕ ξ p then J extends to ˜ ξ by setting J ( x,y,p ) ( ∂ x + yR α ) = ∂ y and J ( x,y,p ) ( ∂ y ) = − ( ∂ x + yR α ) and then ˜ J ˜ α ∈ J ad is admissible. LARA SIMONE SU ´AREZ
Proposition 14.
Let J = { J t } t ∈ [0 , be a time dependent family of generic and admissiblealmost complex structures and let ˆ x ± = [( x ± , ˜ x ± )] ∈ ˆ I (cid:63) ( ψ f ( L P ) , L W ; H ) . The moduli space M (ˆ x − , ˆ x + ; J ) is a manifold of dimension ( | x − | − | x + | − . If ( | x − | − | x + | −
1) = 0 then M (ˆ x − , ˆ x + ; J ) is a compact manifold. If ( | x − | − | x + | −
1) = 1 then M (ˆ x − , ˆ x + ; J ) admits acompactification by gluing broken strips.Proof. The first claim about transversality of the moduli spaces M (ˆ x − , ˆ x + ; J ) for generic choiceof J goes back to Floer and can be found in Floer [13, Theorem 4a]. The proof of compactnessis a combination of arguments in Biran-Cornea [5, Lemma 4.2.1] and Eliashberg-Hofer-Salamon[9, section 3.9]. In [9] the compactness of moduli spaces when the contact manifold is compactis treated. We consider the non-compact manifold C × Y . If we show that for J admissible allholomorphic curves of finite energy are contained in the interior of S ( K × Y ) where K ⊂ C isa compact set, then we are in the same situation as in [9] and the compactness follows from [9,Theorem 3.9.1].To see that the holomorphic curves under consideration here are contained in S ( K × Y ) it isenough to take K = D from C2 in Definition 13 and use that the map π : ( S ( ˜ Y ) , ˜ J ) → ( C , i ) is( ˜ J , j )-holomorphic on S (( C \ K ) × Y ). This implies that for any ˜ J -holomorphic curve u : Σ → S ( ˜ Y ) the map π ◦ u is holomorphic and then its image is contained in some compact set K by[5, Lemma 4.2.1].The condition C1 of a cylindrical almost complex structure guarantees that no sequence ofholomorphic strips can escape to the convex end, this in addition to the hypertight setting,guarantees no escape to the concave end. Moreover, no sphere bubbling can happen since thesymplectic manifold is exact, and disk bubbling is impossible on L W because it is exact and on L P because it is weakly exact or monotone with N L ≥
2. Then the only configurations that canappear in the compactified moduli spaces are broken trajectories. (cid:3)
The differential is defined by(3) d (ˆ x − ) = (cid:88) | ˆ x + | = | ˆ x − |− (cid:88) u ∈M (ˆ x − , ˆ x + ; J ) T ω ( u ) ˆ x + Once the compactness of the moduli spaces is ensured, it is standard to show that d = 0. Thereader can check for example [12, Lema 3.2]. Finally, we set CF ( L P , L W ; H, f, J ) := ⊕ (cid:63) ∈ π ( P ( ψ H ( ψ f ( L P )) ,L W )) CF (cid:63) ( L P , L W ; H, f, J ) Remark . In a similar way, the Floer complex CF ( ˆ P , L Λ , H, J ) for a pair ( P, Λ) consisting of aclosed weakly exact or monotone pre-Lagrangian and a closed hypertight Legendrian of (
Y, ξ ),can be defined. In this case ˆ
P , L Λ ⊂ ( S ( Y ) , ω ), and since Y is compact, the Floer complexconsider here is a minor modification from the one defined in [9]. The difference being that herewe decided to consider the Novikov ring and all connected components of the path space. Wewill denote the homology of this complex by HF ( P, Λ).
Proposition 15.
The homology of the complex CF ( L P , L W ; H, f, J ) is well defined and it isindependent on the choice of compactly supported H and generic and admissible J and de-pends on the sign of the function f outside the compact set B . We denote this homology by HF ( L P , L W ; [ f ]) . NOTE ON LAGRANGIAN INTERSECTIONS AND LEGENDRIAN COBORDISM 9
Proof.
The proof of this proposition follows the standard arguments. The only difference here isthe non-compactness of ˜ Y and then we only need to justify the compactness of the moduli spacesinvolved in the proofs. For the invariance under choice of compactly supported H a chain mapcan be constructed using moving boundary conditions so that the moduli spaces of pseudo-holomorphic curves with moving boundary conditions will be compact by the choice of J inDefinition 13; C1 guarantees that no curves escape to the convex end, C3 that no curves escapeto the C direction and the hypertight condition guaranties that no curve escape to the concaveend. The invariance under change of almost complex structure follows from [9, Theorem 3.7.3].For the invariance under change of function f , let f be another function satisfying choice 2 in3.1 and with the same sign as f outside B . To prove the invariance we use an auxiliary function f so that f and f coincide inside the compact B and so that f and f coincide outside a biggercompact, say 2 B , additionally f should not create any new intersection points. Then one canuse a homotopy g τ = τ f + (1 − τ ) f and construct a chain map between the correspondingcomplexes using moving boundary conditions. And do similarly for f and f . The coincidenceof f and f outside 2 B means that the moving boundary argument applies (because the endsassociated to g τ remain constant to those of f outside of 2 B ). The coincidence of f and f inside B means that, by taking B big enough, the complexes of f and f coincide. (cid:3) Invariance under non-compactly supported Hamiltonian perturbations.
Admissible Hamiltonian isotopies.
Let H : [0 , × S ( ˜ Y ) → R be a time dependent Hamil-tonian and denote by ψ t its Hamiltonian flow. We call H admissible if: • There is a constant
K > H has support on [0 , × [ − K, K ] × ˜ Y . • There exist a compact set B (cid:48) containing the fixed compact set B from 3.1, B ⊂ B (cid:48) , suchthat for all t ∈ [0 ,
1] and q ∈ L V ∩ π − ( C \ B ) we have ψ t ( q ) ∈ L V ∩ π − ( C \ B (cid:48) ), where V ∈ { P, W } . Remark . An example of such a Hamiltonian isotopy is the Hamiltonian isotopy obtained bylifting the contact isotopy associated to a translation along the x -axis, by means of a suitablecutoff. Proposition 16.
The homology of the complex CF ( L P , L W ; H, f, J ) is invariant under admis-sible Hamiltonian isotopies { ψ t } t ∈ [0 , : HF ( L P , L W ; [ f ]) ∼ = HF ( ψ ( L P ) , L W ; [ f ]) . Proof.
The proof of this statement is similar to the proof of the analogous statements for La-grangian cobordisms in [5, Proposition 4.3.1]. It consists in constructing a chain map usingmoving boundary conditions: c : CF ( L P , L W ; H, f, J ) → CF ( ψ ( L P ) , L W ; ψ ◦ H ◦ ( ψ ) − , ψ ◦ f ◦ ( ψ ) − , J ) , ˆ x (cid:55)→ c (ˆ x ) = (cid:88) ˜ y (cid:88) u ∈M β (ˆ x, ˆ y ; J ) T ω ( u ) − (cid:82) R ×{ } H β ( s ) ( u ( s, ds ˆ y, where M β (ˆ x, ˆ y ; J ) denotes the following moduli space of J -holomorphic maps with movingboundary condition. Let β : R → [0 ,
1] be a smooth function with β ( s ) = 0 for s ≤ β ( s ) = 1for s ≥ β ( s ) > , u ∈ M β (ˆ x, ˆ y ; J ) if: (1) ∂ J u = 0 , (2) u ( s, ∈ ψ β ( s )(1 − t ) ( ψ H ( ψ f (( L P )) , (3) u ( s, ∈ L W , (4) it has finite energy,(5) µ (˜ x ) = µ ( u y ) . The only difference with the setting in [5, Proposition 4.3.1] is the possibility of losing com-pactness along the θ direction of the symplectization. Notice that by conditions C1, C2 foradmissible almost complex structure J , no J -holomorphic curve can escape to ±∞ . There-fore the moduli spaces with moving boundary condition M β (ˆ x, ˆ y ; J ) are compact. Once this isensured, the proof of [5, Proposition 4.3.1] adapts to this setting. (cid:3) Proof of Theorem 1
Theorem 17.
Let ( Y, ξ ) be a closed hypertight contact manifold and Λ , Λ (cid:48) ⊂ Y a pair of hyper-tight Legendrian submanifolds related by a connected hypertight Legendrian cobordism ( W ; Λ , Λ (cid:48) ) ⊂ ( C × Y, ˜ ξ ) . If P ⊂ Y is a closed weakly exact or monotone with N P ≥ pre-Lagrangian sub-manifold then HF ( L P , L R ×{ }× Λ ; [ f ]) ∼ = HF ( L P , L R ×{ }× Λ (cid:48) ; [ f ]) . Proof.
The proof consists in first choosing special data (
H, f, J ) to define the Floer complex CF ( L P , L W ; H, f, J ) and then finding suitable admissible Hamiltonian isotopy. Let B ⊂ C besuch that W is cylindrical outside π − C ( B ), and H a Hamiltonian with support on [ − K, K ] × B × Y for K >
0. Let be f ( x, y ) = β ( x ) ax where a > R > B ⊂ [ − R, R ] × [ − R, R ] the smooth function β : R → R satisfies β ( x ) = 1 on ( −∞ , − R ], β ( x ) = − R, ∞ ) and ˙ β ( x ) <
0. Such an f can be chosen such that π C ( L P ) looks likeThe result follow from the fact that the translation on x , T t : C → C , ( x, y ) (cid:55)→ ( x + t, y ), inducesa contact Hamiltonian on C × Y that lifts to an admissible Hamiltonian isotopy, denoted by ψ t ,defined on a neighborhood of L P by the lift of the contact isotopy ψ T t ◦ π C and with support on NOTE ON LAGRANGIAN INTERSECTIONS AND LEGENDRIAN COBORDISM 11 a slightly bigger neighborhood. Then from Proposition 16 HF ( L P , L W ; [ f ]) ∼ = HF ( ψ ( L P ) , L W ; [ f ]) . The right hand side of the equality is isomorphic to HF ( P, Λ) and the left hand side to HF ( P, Λ (cid:48) ). (cid:3) References [1] M. Akaho. Hofer’s symplectic energy and lagrangian intersections in contact geometry.
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