Dehn-Seidel twist, C^0 symplectic topology and barcodes
DDehn-Seidel Twist, C symplectic topology and barcodes Alexandre JannaudJanuary 2021
Abstract
We initiate the study of the C symplectic mapping class group, i.e. the groupof isotopy classes of symplectic homeomorphisms. We prove that the square of theDehn-Seidel twist does not belong to the connected component of the identity of thegroup of symplectic homeomorphisms of some Liouville domain. This generalizes to C settings a celebrated result of Seidel. As a consequence, we obtain the non-trivialityof the C symplectic mapping class group in these domains.For that purpose, we develop a method coming from Floer theory and barcodestheory. This builds on the recent developments of C -symplectic topology. In particu-lar, we adapt and generalize to our context results by Buhovsky-Humilière-Seyfaddiniand Kislev-Shelukhin. Contents
Introduction 11 Preliminaries 132 Barcodes and action selectors in symplectic topology 183 Continuity of the barcode 354 The Dehn-Seidel twist in C -symplectic geometry 52 Introduction
We will work with a symplectic manifold ( M n , ω ) . The group of symplectomorphismswill be denoted Symp(
M, ω ) , the group of symplectomorphisms isotopic to the identityin Symp(
M, ω ) will be denoted by Symp ( M, ω ) and the group of Hamiltonian diffeomor-phisms Ham(
M, ω ) . 1 a r X i v : . [ m a t h . S G ] J a n symplectic topology C symplectic topology was born with the famous Gromov-Eliashberg theorem [20] statingthat given a symplectic manifold ( M, ω ) , if a sequence of symplectomorphisms C -convergesto a diffeomorphism, then this diffeomorphism is a symplectomorphism as well.Considering this theorem, symplectic homeomorphisms were naturally defined as the C -closure of symplectomorphisms. Definition 0.1.
Let ( M, ω ) be a symplectic manifold. A homeomorphism ϕ of M is calleda symplectic homeomorphism if it is the uniform limit of a sequence of symplectic diffeo-morphisms. The main goal in C -symplectic topology is then to understand whether it is possibleor not to do symplectic topology with continuous objects. By C -topology we mean thetopology induced by, for ϕ and ψ homeomorphisms, d ( ϕ, ψ ) = max (cid:40) sup p ∈ M d ( ϕ ( p ) , ψ ( p )) , sup p ∈ M d ( ϕ − ( p ) , ψ − ( p )) (cid:41) , for an arbitrary Riemannian metric d in M .Laudenbach and Sikorav [38] proved an analogue of the Gromov-Eliashberg theorem,but with Lagrangian submanifolds replacing symplectomorphisms.More than a decade later, C -symplectic topology took a step forward, when Oh andMüller [49] introduced a notion of Hamiltonian homeomorphisms, which they called hameo-morphisms. These maps have the property of being generated in some sense by continuousHamiltonians, hence appearing as good C generalizations of Hamiltonian diffeomorphisms.This notion renewed the interest for C symplectic topology.More recently, C symplectic topology took a second step forward. Humilière-Leclercq-Seyfaddini proved a result of coisotropic rigidity in [31] and a reduction result in [32], bothpapers proving that, on many aspects, symplectic homeomorphisms tend to behave assymplectic diffeomorphisms. At the same time, Buhovsky-Opshtein [11] exhibited, amongother rigidity results, the first flexibility behaviour for symplectic homeomorphisms: asymplectic homeomorphism leaving invariant a smooth symplectic submanifold V , andwhose restriction to V is smooth but not symplectic. It was shortly followed by thecounter-example to the Arnold conjecture by Buhovsky-Humilière-Seyfaddini [9], which isanother beautiful example of C -symplectic flexibility.In parallel, much progress has been made regarding the barcodes and action selec-tors which are the main tools used to study these homeomorphisms. The main resultsconcern the C -continuity for action selectors, started by Seyfaddini [58] with his ε -shifttrick, and followed by Buhovsky-Humilière-Seyfaddini [10]. Seyfaddini [58], Buhovsky-Humilière-Seyfaddini [10], Kawamoto [33], Shelukhin [59, 60] proved the C -continuity ofthe action selectors in various settings. Using a result of Kislev-Shelukhin [36], this implies2he C -continuity of barcodes in the same settings. Le Roux-Seyfaddini-Viterbo [40] provedthe continuity of barcodes for Hamiltonians on surfaces, without using Kislev-Shelukhin’sresult. Dehn twists and mapping class groups
Figure 1: Dehn Twist in T ∗ S .The red curve represents the image by the Dehn twist of a fiber of T ∗ S .Dehn twists are diffeomorphisms supported in the neighbourhood of a simple loop insurfaces.Let us first describe the local model. We consider the annulus S × [ − ,
1] = T ∗ S .We denote τ : T ∗ S → T ∗ S the map given by τ ( θ, t ) = ( θ + 2 πf ( t ) , t ) , where f : [ − , → R + is a smooth function equal to near − and equal to near . Thismap is called a twist map. Now that we have our model, we can describe the Dehn twistfor surfaces. It consists of a map which agrees with our local model on the neighbourhoodof a given loop l and is equal to the identity away from this loop. It is called the Dehn twistalong l and it is denoted τ l . One can prove that the isotopy class of τ l only depends onthe isotopy class of l . If the loop along which the Dehn twist is defined is not contractible,then the Dehn twist is not isotopic to the identity.The Dehn twists are of particular interest when studying the mapping class group ofsurfaces. Let us recall that the mapping class group is defined, in the case of a smoothoriented manifold M by MCG( M ) = π (Diff + ( M )) . Let Σ be an oriented smooth surface and denote ω an associated symplectic form on Σ .3e denote by MCG ω (Σ) the mapping class group for area-preserving diffeomorphisms.This MCG ω (Σ) is nothing but π (Symp(Σ , ω )) . Let us also denote by MCG(Σ , C ) = π (Homeo + (Σ)) the mapping class group for homeomorphisms and by MCG ω (Σ , C ) = π (Homeo + ,ω (Σ)) the mapping class group for area-preserving homeomorphisms.One can prove that the mapping class group MCG(Σ) is generated by Dehn twists. Weactually have the following isomorphisms:
MCG ω (Σ) ∼ = MCG(Σ) ∼ = MCG(Σ , C ) ∼ = MCG ω (Σ , C ) . (1)The first isomorphism is a consequence of Moser’s trick. The surjectivity of the secondone comes from the fact that any homeomorphism is a limit of diffeomorphisms, which isfor example proven in [40], together with the fact that the group of homeomorphisms islocally contractible [14]. Its injectivity comes from the local contractibility of the group ofdiffeomorphisms [39]. Finally, the third isomorphism is due to Fathi [23].In symplectic geometry, the mapping class group we are interested in is of course relatedto symplectomorphisms: MCG ω ( M ) = π (Symp( M, ω )) . These Dehn twists have been generalized to higher dimensions by Arnold [3] and theyhave been then intensively studied by Seidel in his PhD thesis [56] and in [52, 53, 54]. Wecall these higher dimensional maps generalized Dehn twist , or
Dehn-Seidel twists . Theyare defined in the neighbourhood of a Lagrangian sphere L , and thus will be denoted τ L .Let us briefly present Seidel’s description of these maps. As in dimension , we start bydescribing a local model in the cotangent bundle of a sphere. We denote T ∗ S n = { ξ ∈ T ∗ S n , | ξ | ≤ } , where | · | denotes the dual of the standard round metric on S n . In coordinates we have T ∗ S n = { ( u, v ) ∈ R n +1 × R n +1 , | u | ≤ , | v | = 1 , (cid:104) u, v (cid:105) = 0 } , and ω T ∗ S n = (cid:80) i du i ∧ dv i . We set σ t ( u, v ) = (cid:18) cos(2 πt ) u − sin(2 πt ) v | u | , cos(2 πt ) v + sin(2 πt ) u | u | (cid:19) , for t ∈ [0 , , and ( u, v ) ∈ T ∗ S n \ S n . When t = 1 / , σ corresponds to the antipodalmap: σ / ( u, v ) = ( − u, − v ) . Note that this antipodal map extends continuously to thezero-section. We choose a cut-off function ρ : [0 , → R such that ρ is equal to near and equal to near . We can now define τ by τ ( ξ ) = σ ρ ( | ξ | ) ( ξ ) . T ∗ S n . When n = 1 , it is isotopic to the modelDehn twist on surfaces described above.We now want to embed our local model into a symplectic manifold, matching the zero-section with a Lagrangian sphere. Let ( M, ω ) be a symplectic manifold with boundary,together with a Lagrangian embedding l : S n → M . Using Weinstein’s neighbourhoodtheorem, we may implant this local model in the neighbourhood of the Lagrangian sphere l ( S n ) = L . The isotopy class in Symp(
M, ω ) of the resulting map τ l only depends on l .This map is called the generalized Dehn , or Dehn-Seidel twist along l .In his PhD thesis [56], Seidel proved that in dimension , the square of a Dehn-Seideltwist is isotopic to the identity through smooth diffeomorphisms but is not through sym-plectomorphisms. He later generalized the last part of this result to higher dimensionsusing the technology of Lagrangian Floer cohomology in [53].Using Seidel’s notations, we start by describing what an ( A k ) -configuration is. Let M be a n -dimensional compact symplectic manifold. Definition 0.2. An ( A k ) -configuration in M is a family of Lagrangian spheres ( l , ...l k ) with images ( L , ...L k ) such that • they are pairwise transverse • for ≤ j ≤ k − , | L i ∩ L j | = 1 if i = j ± and | L i ∩ L j | = ∅ else. Seidel proved [52] that the affine hypersurface ( H, ω ) in C n +1 equipped with the stan-dard symplectic form satisfying the equation z + z + · · · + z n = z m +1 n +1 + 12 contains an ( A m ) -configuration of Lagrangian n -spheres. The name comes from the factthat these hypersurfaces are the Milnor fibres of type ( A m ) -singularities.Following Seidel’s paper [52], we briefly describe these Lagrangian spheres for n = 2 .Let us denote π : H → C the projection onto the ( z , z ) complex plane and σ the mapdefined by σ ( z , z , z ) = ( z , z , e iπ/ ( m +1) z ) . The projection is an ( m + 1) -fold coveringbranched along C = { z + z = , ( z , z ) ∈ C } whose covering group is generated by σ . We now consider the map f : S ⊂ R → C defined by f ( x , x , x ) = ( x (1 + ix ) , x (1 + ix )) . For all x ∈ S , we have f ( x ) ∈ C \ C . This map is an immersion with one double point: f (1 , ,
0) = f ( − , , . Let us denote ˜ f : S → H a lift of f . One can show that ˜ f ( S ) and σ ◦ ˜ f ( S ) have only one intersection point, at ˜ f (1 , , . In the same way, the family ( ˜ f ( S ) , σ ˜ f ( S ) , ..., σ m − ˜ f ( S )) -form ω on H , diffeomorphic to ω , such that these spheres are Lagrangians, and thus ( H, ω ) admits a ( A m ) -configuration. The fact that these two -forms are diffeomorphictells that ( H, ω ) contains such a configuration as well.This allows us to have such configurations inside a Liouville domain. These objectswere intensively studied by Khovanov-Seidel [35], Seidel-Thomas [57], Seidel [55], Keating[34]...The theorem of Seidel which interests us in this paper is the following ([53]) Theorem 1 (Seidel [53]) . Let ( M n , ω ) be a compact symplectic manifold with contact typeboundary, with n even, which satisfies [ ω ] = 0 and c ( M, ω ) = 0 . Assume that M containsan A -configuration ( l ∞ , l (cid:48) , l ) of Lagrangian spheres. Then M contains infinitely manysymplectically knotted Lagrangian spheres. More precisely, if one defines L (cid:48) ( k ) = τ kl ( L (cid:48) ) for k ∈ Z , then all the L (cid:48) ( k ) are isotopic as smooth submanifolds of M, but no two of themare isotopic as Lagrangian submanifolds. Here, c ( M, ω ) denotes the first Chern class of the tangent bundle T M . This theoremimmediately implies that τ L is not isotopic to the identity in Symp(
M, ω ) . Historically,this is the first higher dimensional result on the symplectic mapping class group. Remark 2.
In Seidel’s theorem, it is assumed that n is even. Indeed, for n odd, one canprove that the square of the Dehn-Seidel twist acts non-trivially on homology making theprevious result irrelevant. However, in the same paper [53], Seidel also proved an odd-dimensional counterpart of this theorem in which one should consider a composition ofnon-isotopic Dehn-Seidel twists.Seidel’s result is deeply related to Picard-Lefschetz theory and thus to homological mir-ror symmetry. Nevertheless, this is an entirely different subject that will not be addressedhere. However, many progress have been made on more related topics. For exampleEvans [22] and Li-Li-Wu [43] showed that the symplectic mapping class group of somespecific blow-ups of CP is generated by Dehn-Seidel twists. Khovanov-Seidel [35] andSeidel-Thomas [57] proved that if two Lagrangian spheres intersect transversely at a singlepoint, their associated Dehn twists satisfy a braid relation. This result was generalizedin [34] by Keating for more general pairs of Lagrangians. In some specific cases Evans[22] and Wu [66] proved that there is a weak homotopy equivalence between the groupof compactly supported symplectomorphisms and a braid group on the disk. Moreover,Dimitroglou-Rizell and Evans [17] constructed from Dehn twists non-contractible familiesof symplectomorphisms.As shown by Seidel’s result, these questions are closely related to Lagrangian iso-topy questions. For instance Coffey [15] showed that under specific conditions, on a -dimensional manifold M together with a (very) specific Lagrangian submanifold L , Symp( M ) is homotopy equivalent to the space of Lagrangian embeddings of L .6 ehn-Seidel twist and C symplectic mapping class group We now turn our attention to the core of this paper. Inspired by the pioneering workof Seidel on the group of symplectomorphisms, we would like to study the topology ofthe group
Symp(
M, ω ) of symplectic homeomorphisms. In particular, we would like tounderstand the C symplectic mapping class group, i.e. the group π (Symp( M, ω )) . There is a priori no reason for this group to be non trivial. Indeed, the flexibilityresults such as the C -counter example to the Arnold conjecture ([9]) show that sometimessymplectic homeomorphisms behave very differently than their smooth counter parts. Thisled Ivan Smith to ask the following question. Question 1.
Is the square of the Dehn-Seidel twist connected to the identity in
Symp(
M, ω ) ,where ( M, ω ) is a symplectic manifold as in Seidel’s Theorem 1? Answering this question would help to understand the relation between the symplecticmapping class group and the C symplectic mapping class group. It would show that thenatural map induced by the inclusion π (Symp( M, ω )) J −→ π (Symp( M, ω )) (2)is non-trivial. Here, Symp(
M, ω ) , which denotes the set of symplectic homeomorphisms,is equipped with the C -topology, whereas Symp(
M, ω ) is equipped with C ∞ -topology.The main result of this paper is to answer Question 1, which is achieved by provingthe following theorems. Theorem A.
Let ( M n , ω ) be a n -dimensionnal Liouville domain with n even, n ≥ and c ( M, ω ) = 0 . Assume that M contains an A -configuration of Lagrangian spheres ( l, l (cid:48) ) .Then, τ l is not in the connected component of the identity in Symp(
M, ω ) . Unlike in Seidel’s theorem, we only assume that M contains an A -configuration. Itwas probably known that Seidel’s Theorem 1 holds for an A -configuration as well, butwe were not able to find an appropriate reference. This theorem implies that the group π (Symp( M, ω )) is not trivial. Of course, an immediate consequence of the previous the-orem is the following corollary answering Question 1. Corollary B.
Under the same assumptions as Theorem A, the map τ l is not isotopic tothe identity in Symp(
M, ω ) . We have to discuss the relation between Theorem A and its Corollary B. In smoothsymplectic geometry, the two results would be equivalent. However, in C -symplecticgeometry, there is no reason for this equivalence to hold and it is actually related to animportant. It is the question of the local path-connectedness of Ham , the C closure ofHamiltonian diffeomorphisms, or Symp , which can be formulated in the following way. in a private discussion with V. Humilière uestion 2. Given an arbitrary neighbourhood U of the identity in Ham(
M, ω ) or in Symp(
M, ω ) , is there a neighbourhood V contained in U such that every element in V canbe connected to the identity using a path in V ? Consequently, whether
Symp(
M, ω ) is locally path-connected in dimension greater orequal to remains an open question. It is unknown whether the connected componentof the identity is equal to the path-connected component of the identity in Symp(
M, ω ) .It is known that a positive answer to Question 2 would, for example, solve the C -fluxconjecture.This question is particularly complex. One could think of it as an analogue of theNearby Lagrangian conjecture, but for symplectomorphism isotopies instead of Lagrangianisotopies.The nearby Lagrangian conjecture was proposed by Arnold. It states that given acotangent bundle T ∗ L , any closed exact Lagrangian submanifold L (cid:48) ⊂ T ∗ L is Hamiltonianisotopic to the zero-section. This conjecture is exceptionally difficult to prove. However,important progress has been made. It was proved for T ∗ S by Hind [29] and T ∗ T byGoodman-Ivrii-Rizell [18]. On general cotangent bundles, a series of works by Fukaya-Seidel-Smith [28], Abouzaid [1], Kragh [37] and Abouzaid-Kragh [2] led to the fact that forany closed exact Lagrangian L (cid:48) , the projection of L (cid:48) onto the zero section L is a homotopyequivalence. Even if this conjecture is not proven in its full generality, those results havealready been used. For instance, in Shelukhin’s proof of the Viterbo conjecture [60], itallows him to extend his results to all exact Lagrangian submanifolds.Moreover, note that Theorem A also implies the following corollary since Ham(
M, ω ) is connected (as the closure of a connected space). Corollary C.
Under the hypothesis of Theorem A, τ l does not belong to Ham(
M, ω ) . Denoting
Symp c ( M, ω ) the set of compactly supported symplectomorphisms in ( M, ω ) ,the most explicit corollary may be the following one: Corollary D.
Let L be a n -dimensional manifold with n > . Then the Dehn-Seidel-twistalong L is not in the connected component of the identity in Symp c ( T ∗ L, ω ) . By Weinstein’s neighbourhood theorem, the C ∞ -counterpart of this corollary was aconsequence of Seidel’s Theorem 1.As we will see in Section 4.4, we also have results for n = 2 with stronger assumptionson the Lagrangian configuration. Theorem E.
Let ( M , ω ) be a -dimensional Liouville domain, such that c ( M, ω ) = 0 .Assume that M contains an A -configuration of Lagrangian spheres ( l, l (cid:48) , l ∞ ) .Then, τ l is not isotopic to the identity in Symp(
M, ω ) . J defined by (2). This map is very poorly understoodand we have the following open question. Question 3.
For a general symplectic manifold ( M, ω ) , is the map J injective? Is itsurjective? Note that Theorem A implies that, at least, this map is non-trivial on some manifolds.Moreover, a positive answer to Question 2 would imply the surjectivity of the map J .However, in some specific cases, some results exist. As mentioned earlier in (1), weknow that, for surfaces, this map is an isomorphism.The case of the n -ball is also very interesting. Let us denote Symp c ( B n , ω ) the groupof compactly supported symplectomorphisms of B n ⊂ R n . Using Alexander’s trick, i.e.conjugating by x (cid:55)→ t · x , one gets that Symp c ( B n , ω ) , the group of compactly supportedsymplectic homeomorphisms, is contractible. Consequently, we have that MCG ω ( B n , C ) is trivial and so is the map J . On the other hand, it is not known whether the group Symp c ( B n , ω ) is connected, except when n = 1 or . Indeed, in this case, this group iscontractible. The case n = 2 was proven by Gromov.As this example shows, it could well be that the C symplectic mapping class groupturns out to be simpler to study in general than the smooth symplectic mapping classgroup. Techniques involved
Seidel’s proof cannot directly be adapted to symplectic homeomorphisms. Indeed, it isbased on Floer homology which only applies to smooth objects. However, we will see thatbarcodes form a rich enough invariant that can be defined for symplectic homeomorphismsand that offers a good substitute to Floer homology.
Floer homology
Since the introduction of Floer Homology by Floer in [24], many other Floer (co)homologieshave been defined. The one we are particularly interested in is the Lagrangian intersectionFloer cohomology.We will be working with exact Lagrangian submanifolds. In an exact symplectic man-ifold ( M, ω = dλ ) , an exact Lagrangian submanifold L is a Lagrangian submanifold suchthat the restriction λ | L of the -form λ is exact.Let L, L (cid:48) be two closed exact Lagrangian submanifolds in an exact symplectic man-ifold ( M, ω ) . We assume that their intersections are transverse. The Floer complex isgenerated by the intersection points χ ( L, L (cid:48) ) of the two Lagrangian submanifolds L and L (cid:48) . To define the differential, we have to count J -holomorphic strips, for a chosen almostcomplex structure J , between two intersection points, with boundaries on both Lagrangian9ubmanifolds. Of course, for all the objects at stake to be well-defined, some perturbationsare required. Once this Floer cohomology is defined, we have the following theorem. Theorem 3 (Floer [25]) . Let ( M, ω ) be a symplectically aspherical symplectic manifold,together with a closed weakly-exact Lagrangian submanifold L . Then, HF ∗ ( L, L ; Z / ∼ = H ∗ ( L, Z / . Many generalizations have been proved since then by Oh [46], Fukaya-Oh-Ohta-Ono[27]...One of the many interesting properties of this cohomology is its Hamiltonian invariance,i.e. let φ be a Hamiltonian diffeomorphism on M , then HF ∗ ( L, L (cid:48) ) ∼ = HF ∗ ( L, φ ( L (cid:48) )) . When L = L (cid:48) , we denote HF ( L, H ) = HF ( L, ϕ H ( L )) and for all K, H ∈ Ham(
M, ω ) ,we have HF ( L, H ) ∼ = HF ( L, K ) . The invariance property makes this cohomology a greattool to study Hamiltonian diffeomorphisms.Moreover, the structure of this cohomology is very rich. Indeed, given three closedexact Lagrangian submanifolds L , L , L in ( M, ω ) , counting pseudo-holomorphic curvesbetween three intersection points, one can define a product structure µ : HF ( L , L ) ⊗ HF ( L , L ) → HF ( L , L ) . This product equips HF ( L, L ) with a ring structure and the isomorphism of Theorem 3is a ring isomorphism. Given more Lagrangian submanifolds, we can also define higherproducts µ k , k ∈ N . Action selectors and Barcodes
Action selectors were introduced by Viterbo [65] for Lagrangian submanifolds in a cotan-gent bundle using generating functions theory. After this construction, it was adapted tomany contexts by Oh [47], Schwarz [51], Leclercq [41] and others... They contributed tothe definition of many useful tools, such as the spectral norm [65], or the study of otherones such as the Hofer norm, defined for Hamiltonian diffeomorphisms [30]. These actionselectors are fundamental symplectic invariants and are thus deeply studied. Since theseobjects will be discussed in much more detail later on, we will be brief here.Given a non-zero homology class α ∈ HF ( H ) (respectively a cohomology class in HF ( L, H ) ), the associated action selector l ( α, H ) is the minimal action above (respectivelymaximal action under) which this class is represented in homology. These action selectorshave been subject to a lot of works and have been shown to satisfy many interesting10roperties. For instance, one relevant result for us is that Buhovsky-Humilière-Seyfaddini[10] proved that they are locally C -Lipschitz in the Hamiltonian Floer homology case.Thanks to Theorem 3, one can define the spectral norm γ by γ ( L, H ) = l ([ L ] , H ) − l ([ pt ] , H ) . This spectral norm is continuous with respect to a certain distance, the Hofer distance.Barcodes come from a totally different area of mathematics: topological data analysis.A barcode is a collection of intervals (called bars) used to represent certain algebraicstructures called persistence modules. They were introduced by Edelsbrunner et al. [19]and, for example, found applications in image recognition with the work of Carlson et al.[12].The terminology of barcodes was brought into symplectic topology by Polterovich andShelukhin [50] although germs of this theory were already present in the work of Barannikov[5] and Usher [62, 63]. Indeed, they observed that Floer theories carry natural persistencemodule structures, coming from the action filtration.The space of barcodes may be equipped with a distance, called the bottleneck distance .One can associate a barcode to a Morse function, and this barcode is C -continuous withrespect to the Morse function. They satisfy many more properties that will be discussedin much more details later.Barcodes are of particular interest since they carry the information on the action filtra-tion in Floer (co)homology. Given two exact Lagrangian submanifolds L, L (cid:48) in a symplecticmanifold ( M, ω ) , this filtration is given by the cohomology of the following subcomplexes.For all κ ∈ R , we define CF ∗ ,κ ( L, L (cid:48) ) = span Z / (cid:8) z ∈ χ ( L, L (cid:48) ) , A L,L (cid:48) ( z ) < κ (cid:9) ⊂ CF ∗ ( L, L (cid:48) ) , where χ ( L, L (cid:48) ) denotes the generators of the Floer complex CF ( L, L (cid:48) ) , and A L,L (cid:48) the actionfunctional associated to the pair of Lagrangians. When the parameter κ increases, someclasses appear while some other ones vanish. The bars of the associated barcode encodethe levels at which classes appear and disappear.But maybe the most interesting property of the space of barcodes is that, beingequipped with a distance, it has a topology. This allows us to state our following maintool-theorem, giving a local C -Lipschitz continuity of the barcodes. Theorem F.
Let M be a Liouville domain. Let L and L (cid:48) be two closed exact Lagrangiansubmanifolds, and assume that H ( L (cid:48) , R ) = 0 . The map ϕ ∈ Symp(
M, ω ) (cid:55)→ ˆ B ( ϕ ( L (cid:48) ) , L ) , here ˆ B ( L, L (cid:48) ) denotes the barcodes associated to the exact Lagrangian submanifolds L and L (cid:48) , is locally Lipschitz continuous with respect to the C -distance and extends continuouslyto Symp(
M, ω ) . To prove this theorem, we will adapt the proof of some recent continuity results toour context. The first useful result comes from a work of Kislev-Shelukhin. In [36], theyproved that, in the case of a Lagrangian submanifold together with a Hamiltonian function,the aforementioned barcodes are continuous with respect to the Lagrangian spectral norm γ ( L, H ) .The second result is the one we mentioned before: Buhovsky-Humilière-Seyfaddini[10] proved that action selectors (in the Hamiltonian case) are locally C -Lipschitz. Thisallows to extend these objects and the different spectral invariants to the C -closure, i.e.to Hamiltonian homeomorphisms. This provides invariants that will be used to study theseobjects. Organisation
In the first section, we give some notations and conventions that will be used in the rest ofthis paper. The second section is a short presentation of the theory of persistence modulesand barcodes focusing on the properties we are interested in. We also prove some smalltopological observations on this set, both completeness and connectivity results. We thengive the definition the barcodes for Lagrangian Floer cohomology. We then prove that theproduct operations in Floer cohomology respects the filtration. We also present the actionselectors for a pair of Lagrangian and define the spectral distance in the case of a pairexact Lagrangians non-necessarily Hamiltonian isotopic, together with some properties.Note that the same definition also appears in Shelukhin’s work [60].The third section is the proof of our main tool-theorem used to get our results on theDehn-Seidel twist. We prove Theorem F. Using this theorem, we also prove the two pointsof Theorem 3.3. The first point is a connectivity result while the second one associates acontinuous path of barcodes to a continuous path in
Symp .Finally, in the last section, we state and prove our main results, Theorem A and itscorollaries, along with their counterparts in dimension . Acknowledgments
This paper results from the author’s PhD thesis at the Institut Mathématiques de Jussieu-Paris Rive Gauche (IMJ-PRG) and is grateful for the partial support by the ANR project“Microlocal" ANR-15-CE40-0007. It was finished during a post-doc position supported bythe FNS and so the author thanks Université de Neuchâtel for its hospitality. The authoralso thanks Sobhan Seyfaddini and Felix Schlenk for beautiful insights, Ailsa Keating foruseful communications, Michael Usher and Jean-François Barraud for their so precious12dvice and Côme Dattin for many hours of deeply profitable conversations. He is alsoutterly grateful to his wonderful PhD advisors, Vincent Humilière and Alexandru Oancea,for all their support and mentoring.
All the following notions in this section are originally due to Floer [25]. One can also referto e.g. Auroux [4], Oh [46], Seidel [55]...Let ( M, ω ) be a Liouville domain, with dλ = ω , and let L and L (cid:48) be two closedconnected exact Lagrangian submanifolds in M . We denote f L : L → R and f L (cid:48) : L (cid:48) → R the functions satisfying df L = λ | L and df L (cid:48) = λ | L (cid:48) . We recall that these functions arewell-defined up to a constant. Definition 1.1.
In our context, the action functional on the space of paths from L to L (cid:48) P ( L, L (cid:48) ) is the map A L,L (cid:48) : P ( L, L (cid:48) ) → R defined by the expression A L,L (cid:48) ( γ ) = (cid:90) γ ∗ λ + f L ( γ (0)) − f L (cid:48) ( γ (1)) , with γ ∈ P ( L, L (cid:48) ) . Remark 1.2.
This definition of the action presents an unusual choice regarding the clas-sical conventions used in cohomology. Indeed the differential in cohomology decreases thisaction. This choice does not fundamentally matter but it makes the definitions of persis-tence modules and barcodes easier as our setting thus matches with the usual definitionsof these objects.The critical points of A L,L (cid:48) are the intersection points between L and L (cid:48) . At such apoint p , we have A L,L (cid:48) ( p ) = f L ( p ) − f L (cid:48) ( p ) . We denote
Spec(
L, L (cid:48) ) the set of critical values of A L,L (cid:48) and χ ( L, L (cid:48) ) , the intersection pointsbetween L and L (cid:48) . These are the generators of the chain complex of Floer cohomology.We can associate an action to a set of intersection points. Let p , ...p k , for k ∈ N be pointsin χ ( L, L (cid:48) ) . The action of the formal sum of these points is the maximum of the differentactions, i.e. A L,L (cid:48) ( p + ... + p k ) = max {A L,L (cid:48) ( p ) , ..., A L,L (cid:48) ( p k ) } . Since the energy of a Floer strip connecting p to q is always strictly positive, thedifferential strictly decreases the action, i.e. A L,L (cid:48) ( p ) > A L,L (cid:48) ( ∂p ) , for all p in χ ( L, L (cid:48) ) , with ∂ denoting the Floer differential.13o achieve the transversality and compactness of the moduli spaces as well as thetransversality of the intersections required to define Floer cohomology, we need to considerHamiltonian and (time dependant) almost-complex structure perturbations, which we willdenote by the pair ( H, J t ) or simply ( H, J ) . The generators of the Floer complex are thenthe flow lines γ : [0; 1] → M such that ˙ γ ( t ) = X H ( t, γ ( t )) ,γ (0) ∈ L, γ (1) ∈ L (cid:48) . We will denote χ H ( L, L (cid:48) ) these generators of the Floer complex. When we define theaction in this context, we have to take into account the Hamiltonian perturbation. TheHamiltonian action of a path γ from L to L (cid:48) is then defined as A HL,L (cid:48) ( γ ) = (cid:90) γ ∗ λ − H ( γ ) dt + f L ( γ (0)) − f L (cid:48) ( γ (1)) . (3)We denote by Spec(
L, L (cid:48) ; H ) the set of critical values of this action functional. The criticalpoints are the above mentioned generators of the Floer complex. We now get the Floercomplex CF ( L, L (cid:48) ; H, J ) , and then the Floer cohomology HF ( L, L (cid:48) ; H, J ) . Remark 1.3.
If the Lagrangian submanifolds L and L (cid:48) were transverse, we could ofcourse choose H = 0 . Moreover, for two given Lagrangian submanifolds, the Hamiltonianperturbation to achieve transversality can be chosen as small as desired. Remark 1.4.
The generators can be seen as intersection points between φ H ( L ) and L (cid:48) andso we have an identification of the Floer complexes CF ( L, L (cid:48) ; H ) and CF ( φ H ( L ) , L (cid:48) ; 0) .Using for example Proposition 9.3.1 in [44], one can show that up to a constant, the actiondefined with Equation (3) corresponds to f φ H ( L ) − f L (cid:48) .In the following Sections, we will either consider the Lagrangian submanifolds L and L (cid:48) as Lagrangian submanifolds in M or as Lagrangian submanifolds in T ∗ L . We will denote HF ( L, L (cid:48) ; H, J, M ) when the Floer cohomology is computed in M and HF ( L, L (cid:48) ; H, J, T ∗ L ) when the Floer cohomology is computed in T ∗ L .Let ( M, ω ) and ( M (cid:48) , ω (cid:48) ) be two Liouville domains, together with two pairs of closedexact Lagrangian submanifolds ( L , L ) ⊂ M and ( L (cid:48) , L (cid:48) ) ⊂ M (cid:48) . Let us recall that weare working with Z / -coefficients. Then, there is a Künneth-type formula HF ( L , L ; H, J ) ⊗ HF ( L (cid:48) , L (cid:48) ; H (cid:48) , J (cid:48) ) ∼ = HF ( L × L (cid:48) , L × L (cid:48) ; H ⊕ H (cid:48) , J ⊕ J (cid:48) ) . (4)14his isomorphism is natural, resulting from the fact that a pseudo holomorphic curve v in ( M × M (cid:48) , J ⊕ J (cid:48) ) can be written as v = ( u, u (cid:48) ) , where u is pseudo-holomorphic curve in M and u (cid:48) in M (cid:48) . At the chain level, for ( p, p (cid:48) ) ∈ χ H ( L , L ) × χ H (cid:48) ( L (cid:48) , L (cid:48) ) , the isomorphismis simply defined by ( p, p (cid:48) ) (cid:55)→ ( p, p (cid:48) ) ∈ χ ( L × L (cid:48) , L × L (cid:48) ) . To choose some conventions, we point out that the case when L and L (cid:48) and actuallyassume that L (cid:48) = L . This choice is not restrictive thanks to the Hamiltonian invariance ofthe Floer cohomology.In this case, it is indeed easier to work with more general conditions on the Lagrangiansubmanifolds considered. Due to Weinstein’s neighbourhood theorem and energy esti-mates, choosing to work in the cotangent bundle T ∗ L of the Lagrangian L will not berestrictive. A longer and more detailed discussion on this subject will be held in Section3.2.Let ε > and choose a ε -small Morse function f : L → R . We extend this function to T ∗ L by setting H = f ◦ π : T ∗ L → R , (5)where π : T ∗ L → L is the natural projection. The exact Lagrangian submanifold φ H ( L ) is the graph of df and intersects L transversely. Note that if we work in a symplecticmanifold M instead of T ∗ L , the cotangent bundle of L , we have to multiply H by a cut-offfunction equal to near L .With this perturbation, a critical point p of f is exactly an intersection point between L and φ H ( L ) . We then obtain A HL,φ H ( L ) ( p ) = − H ( p ) . (6)For a good choice of almost-complex structure J and of shift in the definition of thedegree of the intersection points, the matching associates a generator of the Floer cochaincomplex CF ( L, L ; H, J ) of degree i to a critical point of Morse index n − i , i.e a generatorof the Morse cochain complex CM ( L, H ) of index i [26].This identification is associated to a correspondence between the moduli spaces. TheFloer cochain complex CF ( L, L ; H, J ) is then identified with the Morse cochain complex CM ( L, H ) . Together with the Hamiltonian invariance of Floer cohomology, it implies thefollowing proposition. Proposition 1.5 (Floer [25]) . Let L and L (cid:48) be two Lagrangian submanifolds which areHamiltonian isotopic to each other, such that [ ω ] · π ( M, L ) = [ ω ] · π ( M, L (cid:48) ) = 0 , then HF ∗ ( L, L (cid:48) ) ∼ = HF ( L, L ) ∼ = H ∗ ( L ; Z / .
15e assume in this statement that the choice of shift in the definition of the degree forthe generators of the Floer complexes make the degree equal to the Morse index.
Remark 1.6.
Both the Floer cochain complex and the Morse cochain complex carry anatural filtration that will be discussed in details in Section 2. The filtration for the Floercomplex is given by the action functional. The filtration for the Morse complex is givenby the Morse function f .However, with our choice of action for Floer cohomology, the identification betweenthese two complexes does not respect these natural filtrations. Indeed the differentialdecreases the action functional in Floer cohomology while the differential increases theaction in Morse cohomology. Consequently we have to consider the filtration given by − f .We denote CF ( L, L ; H, J ; A HL,L ) the Floer cochain complex with the filtration given by A HL,L and CM ( L, f ; − f ) the Morse cochain complex with the filtration given by − f .Together with the formula (6), this leads, for the ε -small Hamiltonian defined in theformula 5, to CF ( L, L ; H, J ; A HL,L ) ∼ = CM ( L, H ; − H ) . Remark 1.7.
We can choose the Morse function f to have a unique maximum and aunique minimum on L . This implies that there is a unique generator of CM ( L, H ) and aunique generator of CM n ( L, H ) . With the previously mentioned good choice of grading,this implies that there is also a unique generator of CF ( L, L ; H, J ) and a unique generatorof CF n ( L, L ; H, J ) .In the following sections, we will not be interested in the Hamiltonian or almost-complexstructure perturbation, we will just want these Hamiltonian perturbations to be ε -small,for a given ε > . Thus, using Kislev-Shelukhin’s notations [36], we will denote the Floercomplex of L and L (cid:48) CF ∗ ( L, L (cid:48) ; D ) , where D denotes the data perturbation, i.e. the pair ( H, J ) . The set of generators willthen be denoted χ D ( L, L (cid:48) ) . The perturbation data is said to be ε -small if the Hamiltonianis ε -small. When not needed, we will just write CF ∗ ( L, L (cid:48) ) , and assume that there is asuitable perturbation data implied.We now have to make some remarks concerning the relation between action and energy,when there is a data perturbation D . As we will later only be concerned about C -smallperturbations, we will only describe this situation here. However, if one wants to computeFloer cohomology for a particular Hamiltonian H , this Hamiltonian term has to be takeninto account when defining the action of the generators of the Floer complex. We canchoose a perturbation data to achieve transversality everywhere and conduct the sameargument as the following. 16et p, q be two perturbed intersection points in χ D ( L, L (cid:48) ) together with u , a J -holomorphicstrip from p to q . When computing the energy E ( u ) , one has to take into account the per-turbation data. So the energy writes as E ( u ) = A L,L (cid:48) ( p ) − A L,L (cid:48) ( q ) + f D ( p, q ) , where f D is a function depending smoothly on D and such that f D converges to zero whenthe Hamiltonian part of the perturbation data D goes to zero. According to Remark 1.3,this perturbation data can be chosen as small as wished, so that, for all ε > , we can find D such that E ( u ) ≤ A L,L (cid:48) ( p ) − A L,L (cid:48) ( q ) + ε, (7)and thus A L,L (cid:48) ( q ) ≤ A L,L (cid:48) ( p ) + ε. This last remark will one of the key arguments in Section 2.3 to define persistence modulesand barcodes associated to Lagrangian Floer cohomology.The Floer cochain complex can be equipped with product operations. We will onlygive the basic ideas, for more details see e.g. Auroux presentation in [4] or the books ofOh [48] and Seidel [55].Let L , L and L be three Lagrangian submanifolds of a symplectic manifold ( M, ω ) .Under suitable assumptions, we define a product operation from the Floer complexes CF ( L , L , D ) and CF ( L , L , D (cid:48) ) to CF ( L , L , D (cid:48)(cid:48) ) for a suitable choice of perturba-tion data collection, i.e. a linear map CF ( L , L , D ) ⊗ CF ( L , L , D ) → CF ( L , L , D ) , which induces a well-defined product HF ( L , L , D ) ⊗ HF ( L , L , D ) → HF ( L , L , D ) . We recall that we can in fact define, given k + 1 exact Lagrangian submanifolds L , ..., L k in a Liouville domain ( M, ω ) , a map µ k : CF ( L k − , L k , D ) ⊗ · · · ⊗ CF ( L , L , D ) → CF ( L , L k , D ) . This map is (2 − k ) -graded when it is possible to define a grading on the Floer complexes.Moreover, we have the following property [55]. Proposition 1.8.
Let L and L (cid:48) be two closed exact Lagrangian submanifolds in M . Theproduct CF ( L (cid:48) , L, D ) ⊗ CF ( L (cid:48) , L (cid:48) , D ) → CF ( L (cid:48) , L, D ) s cohomologically unital. This unit is given by the image of the fundamental class [ L (cid:48) ] of L (cid:48) in HF ( L (cid:48) , L (cid:48) ) . Adequate choices of Hamiltonian perturbations together with Morse-Bott theory makeit possible to have an isomorphism on the level of cochain complexes which leads to thefollowing proposition (see [7]).
Proposition 1.9.
Let L and L (cid:48) be two closed exact Lagrangian submanifolds in M and ε > . Let f a Hamiltonian perturbation for ( L (cid:48) , L (cid:48) ) defined as in Remark 1.7 and H aHamiltonian perturbation for ( L (cid:48) , L ) . Assume that f and H are ε -small. Then for ε smallenough, the following map is an isomorphism: µ ( · , z ) : CF ( L (cid:48) , L ; H ) → CF ( L (cid:48) , L ; H f ) , where z is the unique representative of the image (Proposition 1.5) of the fundamental class [ L (cid:48) ] in CF ( L (cid:48) , L (cid:48) ; f ) and H f = f (cid:93)H , the perturbation of H by the Hamiltonian f . Persistence module over a field K is a family ( V t ) t ∈ R of finite dimensional vector spacesover K equipped with a doubly-indexed family of linear maps, called structure maps, i st : V s → V t , for all s ≤ t ∈ R satisfying:1. V t = 0 for t (cid:28) ,2. for all s, t, r ∈ R , such that r ≤ s ≤ t , we have i st ◦ i rs = i rt and i ss = Id V s ,3. for all r ∈ R , there is ε > such that i st are isomorphisms for all r − ε < s ≤ t ≤ r ,4. there is a finite set of points S ( V ) ⊂ R such that for all r ∈ R \ S ( V ) , there exists ε > such that i st are isomorphisms for all r − ε < s ≤ t < r + ε .We will denote the persistence module V or ( V, i ) . The set S ( V ) is called the spectrum of V . We will denote by V ∞ the direct limit V ∞ = lim −→ t → + ∞ V t , together with i s : V s → V ∞ the natural map.Let ( V, i ) and ( V (cid:48) , i (cid:48) ) be two persistence modules. A morphism of persistence modules h : ( V, i ) → ( V (cid:48) , i (cid:48) ) is a family of morphisms h t : V t → V (cid:48) t , t ∈ R such that h t i st = i (cid:48) st h s for s < t .Let ( V, i ) be a persistence module, and δ ≥ . The δ -shifted persistence module ( V [ δ ] , i [ δ ]) is the persistence module with vector spaces V [ δ ] t = V t + δ and maps i [ δ ] st = i s + δt + δ .18e will denote sh ( δ ) V : V → V [ δ ] the natural shift morphism of persistence modules givenby sh ( δ ) tV = i tt + δ : V t → V t + δ . A morphism of persistence modules h : V → V (cid:48) naturally induces a shifted morphism ofshifted persistence modules h [ δ ] : V [ δ ] → V (cid:48) [ δ ] . For δ ≤ , we denote V [ δ ] the persistencemodule such that V [ δ ][ − δ ] ∼ = V .Given V and V (cid:48) be two persistence modules together with δ, ε ≥ , we say that theyare ( δ, ε ) -interleaved if there exist two morphisms of persistence modules f : V → V (cid:48) [ δ ] and g : V (cid:48) → V (cid:48) [ ε ] such that g [ δ ] ◦ f = sh ( δ + ε ) V and f [ ε ] ◦ g = sh ( δ + ε ) V (cid:48) . The pair ( f, g ) is called a ( δ, ε ) -interleaving. If ε = δ , it is a δ -interleaving , and V, V (cid:48) are δ -interleaved. The interleaving distance between V and V (cid:48) is then d inter ( V, V (cid:48) ) = inf { δ | V, V (cid:48) are δ -interleaved } . This interleaving distance satisfies the triangle inequality. Let
U, V and W be three per-sistence modules, then d inter ( U, W ) ≤ d inter ( U, V ) + d inter ( V, W ) . (8)Let us now introduce the closely related notion of barcodes.Let J be a non-empty interval in R of the form ( a, b ] or ( a, + ∞ ) , with a and b in R .The interval module I = K J is the persistence module with vector spaces I t = K , if t ∈ J , otherwise,and structure maps i st = Id , if s, t ∈ J, , otherwise.The following structure theorem, proven in [16], relates barcodes and persistence mod-ules. Theorem 2.1.
For any persistence module V , there is a unique collection of pairwisedistinct intervals ( J i ) i ∈I of the form ( a i , b i ] or ( a i , + ∞ ) , with a i , b i ∈ S ( V ) , and multiplicity i ∈ N such that V ∼ = (cid:77) i ∈I ( K J i ) m i . From this theorem, we can associate a barcode associated to V . A multiset is a pair B = ( S, m ) where S is a set and m : S → N ∪ { + ∞} is the multiplicity function. Thisfunction tells how many times each s ∈ S occurs in B . Definition 2.2.
We denote by B ( V ) the multiset containing m J copies of each interval J appearing in the structure theorem, and I ( B ( V )) the set of intervals J i without multiplicity. B ( V ) is called the barcode associated to V , and the intervals J i are called bars. We willdenote B ( V ) = (cid:77) J ∈I ( B ( V )) J m J . We can equip the set of barcodes with a distance, which is called the bottleneck distance.
Definition 2.3.
Let I be a non-empty interval of the form ( a, b ] or ( a, + ∞ ) , and δ ∈ R such that δ < b − a . We denote I − δ the interval ( a − δ, b + δ ] or ( a − δ, + ∞ ) . Let B and B (cid:48) be two barcodes, and δ ≥ . They admit a δ -matching if we can delete in both ofthem some bars of length smaller than δ to get two barcodes ¯ B and ¯ B (cid:48) and find a bijection φ : ¯ B → ¯ B (cid:48) such that if φ ( I ) = J , then I ⊂ J − δ and J ⊂ I − δ . As it was the case for persistence modules, the bottleneck-distance between the barcodes B and B (cid:48) is then defined as d bottle ( B, B (cid:48) ) = inf { δ | B and B (cid:48) admit a δ -matching } . The bottleneck distance is non-degenerate: if B and B (cid:48) are two barcodes such that d bottle ( B, B (cid:48) ) = 0 , then B = B (cid:48) .The two notions of interleaving and bottleneck distance are closely related: an isometrytheorem [6] states that for V, V (cid:48) two persistence modules, d inter ( V, V (cid:48) ) = d bottle ( B ( V ) , B ( V (cid:48) )) . As for persistence modules, given a barcode B and δ ∈ R , we will denote B [ δ ] the barcodeobtained from B by an overall shift of δ . If B is a barcode associated to a persistencemodule V , then B [ δ ] is the barcode associated with the persistence module V [ δ ] .20 .2 A bit of topology One of the main objectives of this work is to obtain new information concerning C -symplectic topology using the technology of barcodes applied to Floer homology. We willbe working on cases where the number of generators of the chain complex is finite. Definition 2.4.
A barcode is said to be finite if it contains finitely many intervals countedwith multiplicity. We will denote B f the set of finite barcodes. Since we study C objects in a world of smoothness, we need, at some point, to takelimits, and hence limits of finite barcodes which are not necessarily finite. Thus, thequestion of closedness and completeness naturally arise. This will be achieved through theset-up given in the following definition. Definition 2.5.
We denote by B the set of barcodes satisfying the following condition: forall ε > , the number of bars of length greater or equal to ε is finite. Remark 2.6.
In [13, 8] such barcodes are referred to as “ q -tame barcodes".The following proposition, proved by Bubenik and Vergili [8], justifies the introductionof the set B . Proposition 2.7.
The space B is complete. Aside from completeness, the following proposition explains why the set B is of par-ticular interest for us, as we will be working with limits of sequences of finite barcodes. Proposition 2.8.
The set B f is dense in B for the topology induced by the bottleneckdistance.Proof. Let us pick B ∈ B . We set ( B n ) n ∈ N a sequence of barcodes defined by B n = (cid:77) I ∈I ( B ) l ( I ) ≥ n I m I , where l ( I ) is the length of the interval I , and m I its multiplicity in B . By definition of B ,for all n ∈ N , B n is a finite barcode, and for all n ∈ N , B n satisfies d bottle ( B n , B ) = 12 n . This implies that ( B n ) n ∈ N ⊂ B f converges to B for the bottleneck distance.When studying homology or cohomology, the presence of a Z -grading is important.We can easily incorporate this notion to obtain those of persistence modules of Z -graded21ector spaces such that the structure maps respect the grading. For instance, if we have afamily of persistence modules V r indexed by the integers, the persistence module ( ⊕ r ∈ Z V r , ⊕ r ∈ Z i r ) has such a structure. We can then define an interleaving distance as d inter ( V, V (cid:48) ) = max r ∈ Z { d inter ( V r , V (cid:48) r ) } , where V = ⊕ r V r and V (cid:48) = ⊕ r V (cid:48) r are two Z -graded persistence modules.We can incorporate this notion in the same way for barcodes. A Z -graded barcode isa family of barcodes ( B r ) r ∈ Z . We denote B = (cid:77) r ∈ Z B r , the Z -graded barcode B associated to the family ( B r ) r ∈ Z . Then, as for persistence modules,the bottleneck distance for graded barcodes is defined by d bottle ( B, B (cid:48) ) = max r ∈ Z { d bottle ( B r , B (cid:48) r ) } , where B = (cid:76) r ∈ Z B r and B (cid:48) = (cid:76) r ∈ Z B (cid:48) r . Remark 2.9.
Let B = (cid:76) r ∈ Z B r be a Z -graded barcode and let I be a bar in B . We call index of I , denoted Ind( I ) , the integer r ∈ Z such that I is a bar of B r .A finite graded barcode B = ( B r ) r ∈ Z is a graded barcode such that there is finitelymany bars in the whole family ( B r ) r ∈ Z . Since we will always consider graded barcodes, wealso denote B f the set of finite graded barcodes. By abuse of notations, we also denote B the set of graded barcodes B = ( B r ) r ∈ Z such that for all ε > there is a finite number ofbars of length greater than ε in the whole family ( B r ) r ∈ Z . From now on, when using thenotation B or B f , we will always refer to their graded version. Remark 2.10.
Let B = ( B r ) r ∈ Z , if B is finite, then B r has more than bars for onlyfinitely many r ∈ Z . In the same way, if B ∈ B , then for all ε > , B r has more than bars of length greater than for only finitely many r ∈ Z .With these graded barcodes, we still have B f = B , and for the same reason as in the non-graded case, B is complete.22efore moving on and defining barcodes for objects of real interest, we have to makesome observations regarding the connectedness of B .First of all, let us introduce the map that counts the number of semi-infinite bars ineach degree. Definition 2.11.
We define σ ∞ : B → N Z by σ ∞ ( B ) = ( σ n ) n ∈ Z with ∀ n ∈ Z , σ n = (cid:88) I ∈I ( B ) l ( I )=+ ∞ Ind( I )= n m I . This map will be very useful. Indeed the following property shows that its relationwith the bottleneck distance is quite straightforward.
Proposition 2.12.
For all
B, B (cid:48) ∈ B , d bottle ( B, B (cid:48) ) < + ∞ ⇐⇒ σ ∞ ( B ) = σ ∞ ( B (cid:48) ) . Since the proof is straight-forward, we leave it to the reader to check that this lemmaindeed holds. For a complete proof, one can refer to the author’s thesis.
Remark 2.13.
The barcode B ∞ introduced in the proof strongly relates to what wedefined as V ∞ at the beginning of Subsection 2.1. The number of bars in each degree r ∈ Z is equal to the dimension of the degree r component of V ∞ .This proposition immediately implies the following corollary, which is topologicallyreally useful. Corollary 2.14. σ ∞ is locally constant. Thanks to this corollary and Definition 2.11 of σ ∞ , we can now state the followingproposition. Proposition 2.15.
The connected components of B are indexed by the graded numberof semi-infinite bars, i.e. two barcodes belong to the same connected component of B ifand only if they have the same number of semi-infinite bars in each degree. Moreover theconnected components are path-connected.Proof. Since the map σ ∞ is locally constant, it is constant on the connected componentsof B . This means that if two barcodes B, C ∈ B are in the same connected component,then σ ∞ ( B ) = σ ∞ ( C ) , i.e. B and C have the same number of semi-infinite bars in eachdegree.Conversely, let B be a barcode in B . With r ∈ Z denoting the degree, we write B = (cid:76) r ∈ Z B r and denote B r = (cid:77) i ∈I rB ( a i , + ∞ ) ⊕ (cid:77) i ∈J r ( a j , b j ] .
23e define for all t ∈ [0 , B rt = (cid:77) i ∈I rB ((1 − t ) a i , + ∞ ) ⊕ (cid:77) i ∈J rB ((1 − t ) a j , (1 − t ) b j ] , and B t = (cid:76) r ∈ Z B rt . The path ( B t ) t ∈ [0 , is a continuous path of barcodes from B to B ( B ) = (cid:77) r ∈ Z (cid:77) i ∈I rB (0 , + ∞ ) . Let B and C be two barcodes in B such that they have the same number of semi-infinitebars in each degree. Then for all r ∈ Z , I rB = I rC so B ( B ) = B ( C ) .This implies that the two barcodes B and C are isotopic and thus in the same connectedcomponent of B which concludes the proof of this proposition.The following corollary is a direct and obvious consequence of the previous Proposi-tion 2.15, but its formulation will be useful later. Corollary 2.16.
Let ( B t ) t ∈ [0;1] be a continuous path of graded barcodes. Then for all t ∈ [0; 1] and for all k , the number of semi-infinite bars of B tk is constant with respect tothe parameter t . Let us now introduce another space of barcodes which will allow us to get our desiredresults.
Definition 2.17.
We define ˆ B as the set of barcodes B quotiented by the action by overallshift of R on B , i.e. B and B (cid:48) represent the same class in ˆ B if and only if there is c ∈ R such that B = B (cid:48) [ c ] . Since the action of R by an overall shift on B is free and proper, all the above mentionedtopological properties also hold for ˆ B .The only remaining question is the completeness of ˆ B . The distance on ˆ B is given bythe Hausdorff distance between the equivalence classes which will be denoted δ . Lemma 2.18.
The set ˆ B is complete for the distance δ .Proof. Let (ˆ b n ) n ∈ N be a Cauchy sequence in ˆ B . There is a strictly increasing sequence ( N p ) p ∈ N such that ∀ k ∈ N , δ (ˆ b N p − ˆ b N p + k ) ≤ p . Let us choose b ∈ B a representative of ˆ b N and b (cid:48) ∈ B a representative of ˆ b N . Then, bydefinition of the equivalence classes, there exists c ∈ R such that d bottle ( b , b (cid:48) [ c ]) ≤ . c ∈ R and all b, b (cid:48) ∈ B , we have d bottle ( b, b (cid:48) ) = d bottle ( b [ c ] , b (cid:48) [ c ]) . Now set b = b (cid:48) [ c ] . We will inductively construct a sequence ( b p ) p ∈ N such that for all p , thebarcode b p is a representative of ˆ b N p and d bottle ( b p , b p +1 ) ≤ p +1 . Let p ∈ N and assumethat for all p ∈ { , ..., p } , the barcode b p is constructed.The barcode b p represents the class of ˆ b N p . Let us fix b (cid:48) p +1 representing the class of ˆ b N p +1 . Since δ (ˆ b N p , ˆ b N p +1 ) ≤ p , there exists c p +1 such that d bottle ( b p , b (cid:48) p +1 [ c p +1 ]) ≤ p +1 . We define b p +1 = b (cid:48) p +1 [ c p +1 ] . And thus we obtain our sequence ( b p ) p ∈ N inductively.By the triangle inequality 8 and a classical high school result, we obtain for all p, k ∈ N d bottle ( b p , b p + k ) ≤ p . Consequently ( b p ) p ∈ N is a Cauchy sequence which converges to a barcode b ∈ B since B iscomplete. This straightforwardly implies that (ˆ b n ) n ∈ N converges to ˆ b , the equivalence classof b , and so ˆ B is complete. Let ( M, ω = dλ ) be a Liouville domain, and L, L (cid:48) two closed exact Lagrangian submanifoldsintersecting transversely, together with two primitive functions f L : L → R and f L (cid:48) : L (cid:48) → R such that df L = λ | L and df L (cid:48) = λ | L (cid:48) . We assume that the Floer cohomologyis well defined for some Hamiltonian perturbation H and some almost complex structure J t . This is generically satisfied. We assume throughout this subsection that all the Floercohomologies are well-defined. For all κ ∈ R , we define CF ∗ ,κ ( L, L (cid:48) ; J t , H ) = span Z / (cid:8) z ∈ χ ( L, L (cid:48) ) , A HL,L (cid:48) ( z ) < κ (cid:9) ⊂ CF ∗ ( L, L (cid:48) ; J t , H ) . Let us recall that, for all x ∈ CF ∗ ,κ ( L, L (cid:48) ; J t , H ) , we have A HL,L (cid:48) ( ∂x ) < A HL,L (cid:48) ( x ) < κ. This means that CF ∗ ,κ ( L, L (cid:48) ; J t , H ) is in fact a subcomplex of CF ∗ ( L, L (cid:48) ; J t , H ) , andconsequently we can define: HF ∗ ,κ ( L, L (cid:48) ; J t , H ) = H ∗ ( CF ∗ ,κ ( L, L (cid:48) ; J t , H )) . Moreover, the inclusions of cochain complexes, i.e. ∀ κ (cid:48) < κ ∈ R , CF ∗ ,κ (cid:48) ( L, L (cid:48) ; J t , H ) ⊂ CF ∗ ,κ ( L, L (cid:48) ; J t , H ) i κ (cid:48) κ in cohomology which commute for κ < κ < κ , thus satisfying the prop-erty required for structure maps. Finally, (( HF ∗ ,κ ( L, L (cid:48) ; J t , H )) κ ∈ R , i ) has the structure ofa finite Z -graded persistence module. We denote its associated graded barcode B ( L, L (cid:48) ; J t , H ) = B (cid:0) ( HF ∗ ,κ ( L, L (cid:48) ; J t , H )) κ ∈ R , i (cid:1) . Since χ H ( L, L (cid:48) ) is finite, B ( L, L (cid:48) ; J t , H ) is a finite barcode. We will denote ˆ B ( L, L (cid:48) ; J t , H ) its image in ˆ B .It is easy to recover the cohomology from the barcode. Indeed, by definition lim → κ →∞ CF ∗ ,κ ( L, L (cid:48) ; J t , H ) = CF ∗ ( L, L (cid:48) ; J t , H ) and then lim → κ →∞ HF ∗ ,κ ( L, L (cid:48) ; J t , H ) = HF ∗ ( L, L (cid:48) ; J t , H ) . This means that HF ∗ ( L, L (cid:48) ; J t , H ) corresponds to the bars that survive when κ goes toinfinity, i.e. Remark 2.19.
The graded rank of HF ( L, L (cid:48) ; J t , H ) is equal to the graded number ofsemi-infinite bars in B ( L, L (cid:48) ; J t , H ) .We can now recall the definition of selectors. This selector, denoted by l ( · , L, L (cid:48) ; J t , H ) ,can be understood as the action selector of the persistence module HF κ ( L, L (cid:48) ; J t , H ) . Letus give an explicit definition. Definition 2.20.
To any α ∈ HF ∗ ( L, L (cid:48) ; J t , H ) \ { } , we associate l ( α, L, L (cid:48) ; J t , H ) = inf { κ ∈ R , α ∈ Im i κ : HF ∗ ,κ ( L, L (cid:48) ; J t , H ) → HF ∗ ( L, L (cid:48) ; J t , H ) } . (9)These numbers are exactly all the different starting points of the semi-infinite bars,i.e. each semi-infinite bar corresponds to some non-zero α ∈ HF ∗ ( L, L (cid:48) ; J t , H ) , and thestarting point of this particular semi-infinite bar is given by l ( α, L, L (cid:48) ; J t , H ) .The following proposition gives classical properties of these action selectors as found in[65, 51, 47, 41]. Proposition 2.21.
For every pair of closed exact Lagrangian submanifolds in a Liouvilledomain, and every non-zero class α ∈ HF ( L, L (cid:48) ; J t , H ) , the action selector l ( α, L, L (cid:48) ; J t , H ) satisfies: • l ( α, L, L (cid:48) ; J t , H ) < + ∞ , • l ( α, L, L (cid:48) ; J t , H ) ∈ Spec ( L, L (cid:48) ; H ) , l ( α, L, L (cid:48) ; J t , H ) does not depend on J t hence will be denoted l ( α, L, L (cid:48) ; H ) , • | l ( α, L, L (cid:48) ; H ) − l ( α, L, L (cid:48) ; H (cid:48) ) | ≤ (cid:107) H − H (cid:48) (cid:107) , where (cid:107) · (cid:107) denotes the Hofer norm. The second property is called the spectrality property, and the fourth one the Lipschitzcontinuity property. These are classical results when studying action selectors and thuswe will not prove them here. However, we can say that the first three properties directlyfollow from the definition. The fourth one is a direct consequence of the construction ofcontinuation maps used to prove that the cohomology does not depend on the choice ofthe Hamiltonian perturbation.These action selectors satisfy the so-called Lagrangian splitting formula which is adirect consequence of the Künneth formula (4); see for example [21] or [32].
Proposition 2.22.
Let ( M, ω ) and ( M (cid:48) , ω (cid:48) ) be symplectic manifolds as before, and ( L , L ) ⊂ M , ( L (cid:48) , L (cid:48) ) ⊂ M (cid:48) two pairs of closed exact Lagrangian submanifolds. Let H and H (cid:48) betwo Hamiltonian perturbations to achieve transversality. Then, for α ∈ HF ( L , L ; J t , H ) and α (cid:48) ∈ HF ( L (cid:48) , L (cid:48) ; J (cid:48) t , H (cid:48) ) two non-zero cohomology classes, l ( α ⊗ α (cid:48) ; L × L (cid:48) , L × L (cid:48) ; H ⊕ H (cid:48) ) = l ( α, L , L ; H ) + l ( α (cid:48) , L (cid:48) , L (cid:48) ; H (cid:48) ) , where α ⊗ α (cid:48) is defined by the Künneth formula (4). The continuation maps in Floer cochain complexes give the continuity of the barcodeswith respect to the Hofer distance:
Proposition 2.23.
Let
L, L (cid:48) be two closed exact Lagrangian submanifolds in a Liouvilledomain, and let
H, K be two Hamiltonians together with time dependent almost-complexstructure J and J (cid:48) such that the graded barcodes B ∗ ( L, L (cid:48) ; J t , H ) and B ∗ ( L, L (cid:48) ; J (cid:48) t , K ) arewell-defined. Then, d bottle ( B ( L, L (cid:48) ; J t , H ) , B ( L, L (cid:48) ; J (cid:48) t , K )) ≤ (cid:107) H − K (cid:107) , where (cid:107) . (cid:107) denotes the Hofer distance. Note that this bound does not depend on choice of the almost complex structures J and J (cid:48) .The proof of this proposition is a straightforward translation to our context of a well-known result proven by Polterovich-Shelukhin [50] and Usher-Zhang [64] in full generality.In the following sections, we do not really care about the Hamiltonian perturbation.The fact that given any two closed exact Lagrangian submanifolds, the Hamiltonian per-turbation can be made as small as one wishes by Remark 1.3, together with the Proposi-tion 2.21 allows us to define, for a a non-zero class in HF ( L, L (cid:48) ; J t , H ) l ( a ; L, L (cid:48) ) = lim H ∈H→ l ( a ; L, L (cid:48) ; H ) , H is the set of Hamiltonians satisfying the transversality requirements. If L and L (cid:48) intersect transversely, it equals the action selector defined in Definition 2.20.We can also use for barcodes the perturbation data notation as in Section 1, i.e. denot-ing D the pair ( H, J ) where H is the Hamiltonian perturbation and J the regular almostcomplex structure, the barcode can be written B ( L, L (cid:48) ; D ) . We will denote ˆ B ( L, L (cid:48) ; D ) its image in ˆ B .Following Proposition 2.23, given two closed exact Lagrangian submanifolds L, L (cid:48) in aLiouville domain ( M, ω = dλ ) with two primitive functions f L : L → R and f L (cid:48) : L (cid:48) → R such that df L = λ | L and df L (cid:48) = λ | L (cid:48) , the map H (cid:55)→ B ( L, L (cid:48) ; H, J ) is continuous with respect to the Hofer distance. Since the space of barcodes is completeby Proposition 2.7, we can take the limit of B ( L, L (cid:48) ; D ) as the Hamiltonian part of theperturbation goes to zero and thus define B ( L, L (cid:48) ) = lim H → B ( L, L (cid:48) ; H, J ) . For two exact Lagrangian submanifolds L and L (cid:48) , we denote ˆ B ( L, L (cid:48) ) the image of ˆ B ( L, L (cid:48) ) in ˆ B . In this section, we focus on the action for a product on Floer complexes. Regarding thedegree, results are the same as those in non-filtered Floer cohomology. However we need tounderstand precisely how we can bound the shift of action in order to define this structureon filtered Floer cohomology.Let ( M, ω ) be a n -dimensional exact symplectic manifold. Let L , L , L be threepairwise transverse closed exact Lagrangian submanifolds in M . We assume that theproduct is well defined.Since these Lagrangian submanifolds are exact, they come with three primitive func-tions (defined up to a constant) f i : L i (cid:55)→ R , such that df i = λ | L i for i ∈ { , , } . Let p ∈ χ ( L , L ) , p ∈ χ ( L , L ) and z ∈ CF ∗ ( L , L ) such that µ ( p , p ) = z . Note that z is a formal sum of ( q j ) j ∈ χ ( L , L ) .Let us recall that A L ,L ( p ) = f ( p ) − f ( p ) , A L ,L ( p ) = f ( p ) − f ( p ) , A L ,L ( q i ) = f ( q i ) − f ( q i ) . Let u : Σ → ( M ; L , L , L ) be a pseudo-holomorphic curve with punctures asymptoticto ( p , p , q j ) as classically defined for the product in Lagrangian Floer cohomology. Letus denote for i ∈ {
0; 1; 2 } the paths γ i : [0; 1] → L i such that γ i ([0; 1]) = u ( D , ∂ D ) ∩ L i .We set the orientations of the γ i for i ∈ {
0; 1; 2 } such that their concatenation γ (cid:93)γ (cid:93)γ turns counterclockwise as in Figure 2.Since ω is exact, equal to dλ , Stokes’ theorem gives Area( u ) = (cid:90) D u ∗ ω = (cid:90) γ λ L + (cid:90) γ λ L + (cid:90) γ λ L . Moreover, all the L i being exact Lagrangian submanifolds, with associated functions f i ,we get: ∀ i ∈ { , , } , (cid:90) γ i λ i = f i ( γ i (1)) − f i ( γ i (0)) . Then,
Area( u ) = f ( p ) − f ( q j ) + f ( p ) − f ( p ) + f ( q j ) − f ( p )= A L ,L ( p ) + A L ,L ( p ) − A L ,L ( q j ) . Since the area of u is positive, we have A L ,L ( q j ) < A L ,L ( p ) + A L ,L ( p ) . A L ,L ( z ) = max j {A L ,L ( q j ) } . We immediately get A L ,L ( z ) < A L ,L ( p ) + A L ,L ( p ) . As done in previous sections, we now have to discuss the case where we do not assumethe transversality properties, and hence where we need a perturbation data D . The argu-ment is exactly the same as for Inequality (7), as the perturbation data has to be takeninto account in the same way when computing E ( u ) . Since the perturbation data D canbe chosen as small as desired, as before, for all ε > , we can find D such that all ourcohomologies are well-defined and E ( u ) ≤ A L ,L ( p ) + A L ,L ( p ) − A L ,L ( z ) + ε. We then straightforwardly obtain A L ,L ( z ) ≤ A L ,L ( p ) + A L ,L ( p ) + ε, for p ∈ χ D ( L , L ) , p ∈ χ D ( L , L ) and z ∈ CF ∗ ( L , L ; D ) with µ ( p , p ) = z .This means that the product preserves the filtration and immediately implies the fol-lowing lemma which will be essential for the upcoming discussions. Lemma 2.24.
Let L , L , L be three closed exact Lagrangian submanifolds in ( M, ω ) exact, together with a ε -small perturbation data collection D , and let p ∈ CF k ( L , L ; D ) ,with action b . Let us assume that the product µ ( p , · ) : CF ∗ ( L , L ; D ) (cid:55)→ CF ∗ + k ( L , L ; D ) is well defined.Then, we have a morphism of persistence modules: µ ( p , · ) : CF ∗ ,t ( L , L ; D ) → CF ∗ + k,t + b + ε ( L , L ; D ) , ∀ t ∈ R µ ( p , · ) : CF ∗ ( L , L ; D ) → CF ∗ + k ( L , L ; D )[ b + ε ] . Lemma 2.25.
Let L , L , L be three closed exact Lagrangian submanifolds in ( M, ω ) exact together with a perturbation data ε -small D . Let p ∈ CF k ( L , L ; D ) , with action a , p ∈ CF k ( L , L ; D ) , with action b . The following maps obtained by composition µ ( p , µ ( p , · )) : CF ( L , L ; D ) → CF ( L , L ; D )[ a + b + 3 ε ] ,µ ( p , µ ( p , · )) : CF ( L , L ; D ) → CF ( L , L ; D )[ a + b + 3 ε ] , are well-defined and filtered chain homotopic to the maps µ ( µ ( p , p ) , · ) : CF ( L , L ; D ) → CF ( L , L ; D )[ a + b + 3 ε ] , ( µ ( p , p ) , · ) : CF ( L , L ; D ) → CF ( L , L ; D )[ a + b + 3 ε ] . Proof.
The composition maps are well-defined and filtered by the preceding lemma. Sincethe product in Lagrangian Floer cohomology is associative (following from the next equal-ity), we only have to check that the associator behaves correctly with respect to the filtra-tion. Let us recall that for our chain complexes we have µ ( µ ( p , p ) , q ) + µ ( p , µ ( p , q )) = ∂µ ( p , p , q ) + µ ( ∂p , p , q )+ µ ( p , ∂p , q ) + µ ( p , p , ∂q ) , where q is an element of CF ( L , L ) . Then, the exact same computation as for µ givesus A L ,L ( µ ( p , p , q )) ≤ A L ,L ( p ) + A L ,L ( p ) + A L ,L ( q ) + 3 ε. Moreover, the differential decreases the action, so that max {A L ,L ( ∂µ ( p , p , q )) , A L ,L ( µ ( ∂p , p , q )) , A L ,L ( µ ( p , ∂p , q )) , A L ,L ( µ ( p , p , ∂q )) }≤ A L ,L ( p ) + A L ,L ( p ) + A L ,L ( q ) + 3 ε. This means that the homotopy defined from µ between the two different compositionspreserves the filtration, which concludes the proof of this lemma.The following lemma will be a key argument in the proof of Section 3.3. Lemma 2.26.
Let
L, L (cid:48) be two closed exact Lagrangian submanifolds in ( M, ω ) togetherwith ε -small perturbation data f and let H behave as in Proposition 1.9. Denote H f = f (cid:93)H .Let z ∈ CF ( L (cid:48) , L (cid:48) ; f, J ) be as in the same proposition. The multiplication map m ( · , z ) : CF ∗ ( L (cid:48) , L ; H ) → CF ∗ ( L (cid:48) , L ; H f )[2 ε ] are filtered chain-homotopic to the standard inclusion and hence induce ε -shift maps onthe persistence modules.Proof. Let us recall that Proposition 1.9 tells us the multiplication by z is an isomorphismof cochain complexes and hence induces the standard inclusion of persistence modules. Wenow just need the energy estimate.Since the Hamiltonian part of the perturbations are ε -small, the action of z is smallerthan ε and H f is ε close to H . Consequently, using the same argument as the one implyingLemma 2.24, this map induces a ε + ε = 2 ε -shift of action. This concludes the proof ofthis lemma. 31 .5 Spectral norm and exact Lagrangians in a cotangent bundle Given a closed exact Lagrangian submanifold L together with a non-degenerate Hamilto-nian H , the spectral norm γ L ( H ) is defined as γ L ( H ) = l ([ L ] , L, L ; H ) + l ([ L ] , L, L ; H ) , where [ L ] denotes the image of the fundamental class [ L ] through the isomorphism ofProposition 1.5. It is equal to the diameter of the spectrum Spec(
L, L ; H ) . This is calledthe Lagrangian spectral norm or Viterbo norm as its first version was introduced by Viterboin [65]. A similar version also exists in Hamiltonian Floer homology.Let L and L (cid:48) be two closed exact Lagrangian submanifolds in a symplectic manifold M as before, together with a Hamiltonian perturbation H . Then, in the same spirit, we set γ ( L, L (cid:48) ; H ) = Diam(Spec ∗ ( L, L (cid:48) ; H )) , where Spec ∗ ( L, L (cid:48) ; H ) is the set of action selectors for HF ( L, L (cid:48) ; H, J ) . We denote by Diam( · ) the diameter (i.e. max − min ). Note that this definition is only interesting whenthe cohomology HF ( L, L (cid:48) ; H, J ) has rank at least 2. By Proposition 2.22, and with thesame notations we immediately get γ ( L × L (cid:48) , L × L (cid:48) ; H ⊕ H (cid:48) ) = γ ( L , L ; H ) + γ ( L (cid:48) , L (cid:48) ; H (cid:48) ) . (10)Consequently, if M = M (cid:48) , L = L (cid:48) , L = L (cid:48) and H = H (cid:48) , γ ( L × L , L × L ; H ⊕ H ) = 2 γ ( L , L ; H ) . (11) Remark 2.27.
Let L and L (cid:48) be two closed exact Lagrangian submanifolds in a Liouvilledomain ( M, ω ) together with a Hamiltonian H and a function f : M → R . The fourthpoint of Proposition 2.21 together with the definition of γ tells us that | γ ( L, L (cid:48) ; H + f ) − γ ( L, L (cid:48) ; H ) | ≤ f − min f ) . Note that we do not need any transversality assumptions for the intersections between L and L (cid:48) . Indeed, by the continuity of the spectral invariants, γ ( L, L (cid:48) ; H ) is defined for all H . So we can set γ ( L, L (cid:48) ) = lim H ∈H→ Diam(Spec ∗ ( L, L (cid:48) ; H )) , where H is the set of Hamiltonians satisfying the transversality requirements. Remark 2.28.
Given two Lagrangian submanifolds L and L (cid:48) , we can actually define γ directly since the spectrum is defined without any transversality assumptions.The question of the continuity of γ with respect to the C -distance is a fundamental32ne. This has been proved for specific symplectic manifolds in [65, 58, 10, 33, 60]. We willalso prove its continuity in our context in Section 3.4, and thus we will not discuss it morehere.An important question is the relation between two C -close Lagrangian submanifolds.It is related to Arnold’s famous Nearby Lagrangian Conjecture. The results on this con-jecture will be useful for both another definition of γ and for the arguments of Section 3.3.Let us start by stating this conjecture. Conjecture 2.29.
Let M be a closed connected manifold. Any closed exact connectedLagrangian submanifold in T ∗ M is Hamiltonian isotopic to the zero section. Together with Weinstein’s theorem, this conjecture implies that in a symplectic mani-fold M together with a closed connected Lagrangian submanifold L ⊂ M , any exact closedconnected Lagrangian submanifold L (cid:48) ⊂ M is Hamiltonian isotopic to L if L (cid:48) is C -closeenough to L .For most cases, this conjecture is still open and subject to a lot of research. It has beenfully proved in special cases. Hind [29] proved the following theorem: Theorem 2.30.
The Nearby Lagrangian Conjecture is true in T ∗ S . For T ∗ S , there is not much to discuss and it is also true. Dimitroglou-Rizell, Goodmanand Ivrii [18] proved it for T ∗ T .In more general context, important progress has been made by Fukaya, Seidel andSmith [28], proving that when the Maslov class vanishes, the projection π : L (cid:48) → L ⊂ T ∗ L induces an isomorphism on homology. This result was improved later by Abouzaid-Kraghin the following theorem [2]. Theorem 2.31.
Let L be a closed connected manifold together with L (cid:48) an exact closedconnected Lagrangian submanifold in T ∗ L . Then, there exists an integer i = i L (cid:48) ∈ Z such that for every exact closed Lagrangian submanifold K in T ∗ L , there are chain-levelquasi-isomorphisms in both directions between CF ∗ ( L (cid:48) , K ) and CF ∗ + i ( L, K ) and between CF ∗ ( K, L (cid:48) ) and CF ∗− i ( K, L ) . These quasi-isomorphisms are compatible with the productstructure in Floer cohomology. Let L and L be two closed exact Lagrangian submanifolds in ( T ∗ L, ω = dλ ) exact,together with two primitives functions f L and f L such that df L i = λ | L i , for i ∈ { , } .As mentioned before, this theorem allows us to state another definition of γ ( L , L ) . In-deed, previous Theorem 2.31 and Proposition 1.5 respectively tell us that we have the twofollowing isomorphisms HF ∗ ( L, L ) ∼ −→ σ HF ∗ ( L , L ) , ∗ ( L ) ∼ −→ θ HF n −∗ ( L, L ) . By abuse of notation, we denote [ L ] = σ ◦ θ ([ L ]) ∈ HF ( L , L ) , [ pt ] = σ ◦ θ ([ pt ]) ∈ HF n ( L , L ) . It is known that for such Lagrangian submanifolds L and L , the quantity γ ( L , L ) admits the following alternative definition. This definition is actually the standard defini-tion of γ ( L , L ) ; see [45, 42]. Definition 2.32. γ ( L , L ) = l ([ L ] , L , L ) − l ([ pt ] , L , L ) . Let us give the basic properties of γ . Following Definition 1.1, we have A L ,L = −A L ,L . Together with the fact that the two complexes CF ( L , L ) and CF ( L , L ) aredual to each other, we have l ([ pt ] , L , L ) = − l (cid:0) σ (cid:48) ◦ θ ([ L ]) , L , L (cid:1) , where σ (cid:48) is the isomorphism from HF ( L, L ) to HF ( L , L ) given by Theorem 2.31. Wethus obtain γ ( L , L ) = l ([ L ] , L , L ) + l (cid:0) σ (cid:48) ◦ θ ([ L ]) , L , L (cid:1) . Consequently, for all L and L exact in T ∗ L , γ ( L , L ) = γ ( L , L ) . (12)Moreover, for all L , L , L closed exact Lagrangian submanifolds in T ∗ L with primitivefunctions f L , f L , f L , it also satisfies the triangle inequality γ ( L , L ) ≤ γ ( L , L ) + γ ( L , L ) . (13)Indeed, if x ∈ CF ( L , L ) and y ∈ CF ( L , L ) both represent the fundamental class intheir respective homology, so does µ ( x, y ) in CF ( L , L ) (see Section 3.3). Together withLemma 2.24, we immediately obtain this triangle inequality.34 Continuity of the barcode
The object of this section is to prove the following theorem, corresponding to Theorem Fin the Introduction which will be the key to prove our results concerning the Dehn-Seideltwist. It shows a certain local Lipschitz continuity on barcodes associated to Lagrangiansubmanifolds. We will always assume that the considered Lagrangian submanifolds areconnected.
Theorem 3.1.
Let M be a Liouville domain. Let L and L (cid:48) be two closed exact Lagrangiansubmanifolds, and assume that H ( L (cid:48) , R ) = 0 . Then there exist K ≥ and l > such thatfor all ϕ and ψ in Symp(
M, ω ) , if d C ( ϕ, ψ ) ≤ l , we have d bottle ( ˆ B ( ϕ ( L (cid:48) ) , L ) , ˆ B ( ψ ( L (cid:48) ) , L )) ≤ Kd C ( ϕ, ψ ) . The fact that we have a uniform Lipschitz continuity with respect to the C distanceimmediately implies the following corollary. Corollary 3.2.
The map ϕ (cid:55)→ ˆ B ( ϕ ( L (cid:48) ) , L ) continuously extends to a map Symp(
M, ω ) → ˆ B . Since L and L (cid:48) are closed, the number of semi-infinite bars of B ( ϕ ( L (cid:48) ) , L ) stays finitefor all ϕ ∈ Symp(
M, ω ) . This extension to the closure requires to work with B as definedin Definition 2.5 which is the completion of the space of barcodes B f by Proposition 2.7.As we will see in the proof, we will then have to work in the space of barcodes up to shift ˆ B . From Theorem 3.1, we obtain the following theorem. It is direct consequence of thecontinuity of barcodes together with its Corollary 3.2. Theorem 3.3.
Let M be a Liouville domain. Let L and L (cid:48) be two exact compact La-grangian submanifolds, and assume that H ( L (cid:48) , R ) = 0 . Consider two symplectomorphisms ϕ and ψ . • If these two symplectomorphisms are in the same connected component of
Symp(
M, ω ) ,then the two barcodes ˆ B ( ϕ ( L (cid:48) ) , L ) and ˆ B ( ψ ( L (cid:48) ) , L ) are in the same connected com-ponent of ˆ B . • If these two symplectomorphisms are isotopic in
Symp(
M, ω ) , then there is a contin-uous path of barcodes from ˆ B ( ϕ ( L (cid:48) ) , L ) to ˆ B ( ψ ( L (cid:48) ) , L ) . This continuous path can also be directly constructed in the following way from Corol-lary 3.2. Let us denote ( φ t ) t ∈ [0 , the path in Symp(
M, ω ) from ϕ to ψ . For each t ∈ [0 , ,35orollary 3.2 allows to associate a barcode ˆ B t to φ t . The path of barcodes is then the path ( ˆ B t ) t ∈ [0 , .The second point of Theorem 3.3 can also be understood as a consequence of the firstpoint as ˆ B is locally path-connected. Remark 3.4.
In smooth symplectic topology, the two points in Theorem 3.3 would beequivalent. However, in C symplectic topology we do not know whether Symp(
M, ω ) is locally path-connected, thus it is not known whether the connected components of Symp(
M, ω ) are path-connected. Consequently the first point implies the second one butthe reciprocal implication is far from clear. In order to prove Theorem 3.1, we prove the two following propositions. The first onebounds the bottleneck distance by the spectral norm γ . Proposition 3.5.
Let L and L (cid:48) be two closed exact Lagrangian submanifolds in a Liou-ville domain ( M, ω ) . There exists δ > , independant of L , such that for all ϕ and ψ ∈ Symp(
M, ω ) satisfying d C ( ϕ, ψ ) ≤ δ , then d bottle ( ˆ B ( ϕ ( L (cid:48) ) , L ) , ˆ B ( ψ ( L (cid:48) ) , L )) ≤ γ ( L (cid:48) , ψ − ◦ ϕ ( L (cid:48) )) . This proposition is an adaptation to our context of a similar statement proved by Kislevand Shelukhin [36]. In their case, L = L (cid:48) is a weakly monotone Lagrangian submanifoldin a closed symplectic manifold and ϕ and ψ are Hamiltonian diffeomorphisms.This proposition will be proven in Subsection 3.3. The second proposition asserts that γ ( L (cid:48) , ϕ ( L (cid:48) )) goes to zero, as ϕ goes to identity and will be proven in Subsection 3.4. Proposition 3.6.
There exist constants l ≥ and κ ≥ such that and for all ϕ ∈ Symp(
M, ω ) satisfying d C ( ϕ, Id M ) ≤ l , we have γ ( L (cid:48) , ϕ ( L (cid:48) )) ≤ κd C ( ϕ, Id M ) . This proposition is an adaptation to our context of a lemma of Buhovsky-Humilière-Seyfaddini [10]. In their paper, they proved the same result for a Lagrangian submanifoldHamiltonian isotopic to the zero section in a cotangent bundle.
Proof of Theorem 3.1.
Let ϕ and ψ be in Symp(
M, ω ) such that d C ( ϕ, ψ ) ≤ l . We canassume without loss of generality that l ≤ δ . (See the choice of l in Section 3.4.)36ndeed we have d bottle ( ˆ B ( ϕ ( L (cid:48) ) , L ) , ˆ B ( ψ ( L (cid:48) ) , L )) ≤ γ ( L (cid:48) , ψ − ◦ ϕ ( L (cid:48) )) ≤ κd C ( ψ − ◦ ϕ, Id M )= κ sup x ∈ M d ( ψ − ( x ) , ϕ − ( x )) ≤ κd C ( ψ, ϕ ) . Setting K = κ , this proves Theorem 3.1.Let us now briefly sketch the proof of Proposition 3.5 and Proposition 3.6 and set upsome conventions. Proposition 3.5 will be implied by the case where ψ = Id M .Let us fix ε > , ε (cid:48) (cid:28) ε and assume that all the Hamiltonian parts of the perturbationdata at stake in this proof are of C -norm smaller than ε (cid:48) .Let us fix such a perturbation data collection D such that HF t ( ϕ ( L (cid:48) ) , L ; D ) , HF t ( L (cid:48) , L ; D ) , HF t ( ϕ ( L (cid:48) ) , L (cid:48) ; D ) , HF t ( L (cid:48) , L (cid:48) ; D ) and HF t ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) ); D ) are well defined. Remark 3.7.
In the case of HF ( L (cid:48) , L (cid:48) ; D ) , we require that the Hamiltonian perturbationis defined in the following way (see also Remark 1.7). Let f be a ε (cid:48) / -small Morse functiondefined on L (cid:48) with a unique maximum and a unique minimum. We extend it to a Hamilto-nian H which is supported on a ε -small tubular neighbourhood of L (cid:48) . This constructionimplies that there is only one element in CF n ( L (cid:48) , L (cid:48) ; D ) and only one in CF ( L (cid:48) , L (cid:48) ; D ) .We perform the same construction in the case of HF ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) ); D ) .We aim to find two morphisms of persistence modules A = { A t } t ∈ R and B = { B t } t ∈ R together with δ, δ (cid:48) ∈ R : A t : CF t ( ϕ ( L (cid:48) ) , L ; D ) (cid:55)−→ CF t + δ ( L (cid:48) , L ; D ) ,B t : CF t ( L (cid:48) , L ; D ) (cid:55)−→ CF t + δ (cid:48) ( ϕ ( L (cid:48) ) , L ; D ) , such that these maps are filtered and their compositions are chain homotopic to shifts ofpersistence modules: sh ϕ ( L (cid:48) ) : B ( ϕ ( L (cid:48) ) , L ; D ) (cid:55)−→ B ( ϕ ( L (cid:48) ) , L ; D )[ δ + δ (cid:48) + ε (cid:48) ] sh L (cid:48) : B ( L (cid:48) , L ) (cid:55)−→ B ( L (cid:48) , L )[ δ + δ (cid:48) + ε (cid:48) ] . If they indeed satisfy the above conditions, these maps A and B provide a δ + δ (cid:48) + ε (cid:48) -matching. Then, to achieve the proof, we will only have to bound the shift δ + δ (cid:48) + ε (cid:48) by the C distance between ϕ and Id M . We will prove that this shift is in fact equal to37 γ ( L (cid:48) ; ϕ ( L (cid:48) ); D ) + ε (cid:48) , and use this to get the bound. This is the purpose of Section 3.3.Proving that this bound goes to zero when ϕ C -converges to the identity is the purposeof the last Section 3.4.Following Kislev and Shelukhin’s idea [36], these maps A and B will come from themultiplication in Floer cohomology:• A corresponds to the multiplication by a specific class [ x ] in HF ( L (cid:48) , ϕ ( L (cid:48) ); D ) .• B corresponds to the multiplication by a specific class [ y ] in HF ( ϕ ( L (cid:48) ) , L (cid:48) ; D ) .These choices will be achieved using Abouzaid-Kragh’s Theorem 2.31 [2]. This result re-quires the Lagrangian submanifolds to be in a cotangent space. To obtain this requirement,we will consider a symplectomorphism ϕ C -close enough to the identity so that ϕ ( L (cid:48) ) isincluded in a Weinstein neighbourhood of L (cid:48) . We thus obtain two cohomologies whichcould be different: the one computed in M and the one computed in T ∗ L (cid:48) . Consequently,for the sake of our argument, we will first prove that we have the isomorphisms HF ( L (cid:48) , ϕ ( L (cid:48) ); D , M ) ∼ = HF ( L (cid:48) , ϕ ( L (cid:48) ); D , T ∗ L (cid:48) ) ,HF ( ϕ ( L (cid:48) ) , L (cid:48) ; D , M ) ∼ = HF ( ϕ ( L (cid:48) ) , L (cid:48) ; D , T ∗ L (cid:48) ) . Of course we will also prove that these isomorphisms respect the filtration. We will in factonly give the details for one of these isomorphisms since the proofs are identical for both.By abuse of notation, we denote by D both the perturbation data in M and its image in T ∗ L . This is the purpose of the following Section 3.2. Remark 3.8.
Now that the proof is sketched, we can explain the conditions required forthe two Lagrangian submanifolds L (cid:48) and L in Theorem 3.1. These are both exactnessconditions. In the previous chapters, to define Lagrangian Floer cohomology, the productand the action filtration, we require the considered Lagrangian submanifolds to be exact.This exactness condition is also required for Theorem 2.31 that will be used to constructthe maps A and B .The condition H ( L (cid:48) , R ) = 0 guarantees that, for any symplectomorphism ϕ ∈ Symp(
M, ω ) , ϕ ( L (cid:48) ) is an exact Lagrangian submanifold as well. With these conditions, we are sure thatall the above mentioned objects used in the following proof will be well defined.This implies that, when working with ϕ ∈ Ham(
M, ω ) , we can drop the condition H ( L (cid:48) , R ) = 0 for the weaker condition that L (cid:48) is exact. Indeed, the image of an exactLagrangian submanifold by a Hamiltonian diffeomorphism is always exact.The following Subsections 3.2 and 3.3 are dedicated to the proof of Proposition 3.5 andSubsection 3.4 to the proof of Proposition 3.6.38 .2 Equality of the barcodes in M and in T ∗ L (cid:48) If the symplectomorphism ϕ is C -close enough to the identity, then ϕ ( L (cid:48) ) is included ina Weinstein neighbourhood of L (cid:48) . This will contribute to the definition of the constants δ and l ∈ R at stake in Proposition 3.5 and Proposition 3.6. By abuse of notation, wealso denote L (cid:48) , ϕ ( L (cid:48) ) their respective images in T ∗ L (cid:48) by a Weinstein embedding. Denoting D a perturbation data in T ∗ L (cid:48) , we also denote D its pull-back by the chosen Weinsteinembedding. Let us recall that HF s ( ϕ ( L (cid:48) ) , L (cid:48) ; D , M ) is the filtered cohomology computedin M and HF s ( ϕ ( L (cid:48) ) , L (cid:48) ; D , T ∗ L (cid:48) ) the filtered cohomology computed in T ∗ L (cid:48) . We aim toprove that these two cohomologies are isomorphic and that this isomorphism respects thefiltration.This section is thus dedicated to the proof of the following Lemma 3.9. The ideafor this is to localize the Floer trajectories near L (cid:48) . Indeed, this will imply that the Floertrajectories in M and T ∗ L (cid:48) are in correspondence, and thus the two cochain complexesare isomorphic. Lemma 3.9. If ϕ is C -close to the identity, then for an arbitrary choice of data thereexists C ∈ R such that for all s ∈ R HF s ( ϕ ( L (cid:48) ) , L (cid:48) ; D , M ) ∼ = HF s ( ϕ ( L (cid:48) ) , L (cid:48) ; D , T ∗ L (cid:48) )[ C ] . We can actually choose the primitive functions of the -forms λ M and λ T ∗ L (cid:48) restricted tothe Lagrangian submanifolds such that the shift C is equal to .Proof. The idea here is to retract the Lagrangian submanifold ϕ ( L (cid:48) ) by the negative Li-ouville flow. This will decrease the diameter of the spectrum and thus allow to have smallenough energy estimates on the moduli spaces.Let us choose two Weinstein’s tubular neighbourhoods U and W of L (cid:48) such that U (cid:98) W .We denote ψ : W → T ∗ L (cid:48) , the symplectic embedding provided by Weinstein’s theorem.Let us suppose that ϕ is close to Id M , so that we have ϕ ( L (cid:48) ) included in U . We have twoLiouville forms on W . The first one is the Liouville form λ M restricted to W . The secondone is the Liouville form obtained from the Liouville form λ T ∗ L (cid:48) on T ∗ L (cid:48) : ψ ∗ λ T ∗ L (cid:48) . Letus recall that ψ ∗ λ T ∗ L (cid:48) − λ M is closed on W . Since H ( L (cid:48) , R ) = 0 we have H ( W, R ) = 0 .Then ψ ∗ λ T ∗ L (cid:48) − λ M is exact on W and consequently there exists a function F : W → R such that ψ ∗ λ T ∗ L (cid:48) = ( λ M ) | W + dF .Let us pick a cut-off function β : W → R such that β is constant, equal to on U and equal to near the boundary of W . By abuse of notation, we denote F the functiondefined on M equal to βF on W and continuously extended by outside of W . The -form ( λ M + dF ) is a Liouville form on M equal to ψ ∗ λ T ∗ L (cid:48) on U . We thus obtain a globallydefined negative Liouville flow (i.e. the flow of the negative Liouville vector field) on M which preserves U and matches with the negative Liouville flow on T ∗ L (cid:48) .39n T ∗ L (cid:48) , let us denote ϕ t −L the negative Liouville flow. When we apply this flow to ψ ( ϕ ( L (cid:48) )) for t ∈ R + , we obtain a smooth path ( L (cid:48) t ) t ∈ R + of Lagrangian submanifolds in T ∗ L (cid:48) . We can now consider the smooth path of Lagrangian submanifolds in M given by ( L t ) t ∈ R + = ( ψ − ( L (cid:48) t )) t ∈ R + . Lemma 3.10.
For all t ∈ R + we have Spec( L (cid:48) t , L (cid:48) ; D , T ∗ L (cid:48) ) = Spec( L t , L (cid:48) ; D , M ) + C t ,where C t ∈ R . Moreover, we can choose the primitive functions of the -forms λ M and λ T ∗ L (cid:48) restricted to the Lagrangian submanifolds such that C t is equal to for all t . From now on, assume that the primitives on the Lagrangian submanifolds are chosenso that for all t ∈ R , C t = 0 . Since in T ∗ L (cid:48) we have ( ϕ t −L ) ∗ ω = e − t ω , and ( ϕ t −L ) is equalto the identity on L (cid:48) , we get Spec( L (cid:48) t , L (cid:48) ; D , T ∗ L (cid:48) ) = e − t Spec( L (cid:48) , L (cid:48) ; D , T ∗ L (cid:48) ) . (14)Figure 3: Evolution of the barcode during the Liouville retraction Lemma 3.11.
For T large enough, there is a canonical identification between the cochaincomplexes CF ( L (cid:48) T , L (cid:48) ; D , T ∗ L (cid:48) ) and CF ( L (cid:48) T , L (cid:48) ; D , M ) given by the Weinstein’s neighbour-hood embedding. Corollary 3.12.
For T large enough there is a canonical isomorphism HF s ( L (cid:48) T , L (cid:48) ; D , T ∗ L (cid:48) ) ∼ = HF s ( L (cid:48) T , L (cid:48) ; D , M ) holding for all s ∈ R . emark 3.13. Let us recall that if we are working in an exact symplectic manifold ( M, dλ ) and the path ( L t ) t ∈ [0 , is a smooth path of exact Lagrangian submanifolds, there is asmooth path ( φ t ) t ∈ [0 , in Ham(
M, dλ ) such that ∀ t ∈ [0 , , φ t ( L ) = L t . (15)Consequently, since the paths ( L t ) t ∈ R and ( L (cid:48) t ) t ∈ R are smooth, the associated barcodepaths are continuous according to the previous expression (15) and Proposition 2.23. Letus denote B t = B ( L t , L (cid:48) ; D , M ) and B (cid:48) t = B ( L (cid:48) t , L (cid:48) ; D , T ∗ L (cid:48) ) . The previous lemma tellsthat for T large enough, B T = B (cid:48) T .Moreover let ( B t ) t ∈ [0;1] be a continuous path of barcodes such that there is a positivecontinuous function f : R → R which satisfies ∀ t ∈ [0; 1] , Spec( B t ) = f ( t )Spec( B ) . Since there is no bifurcation in the spectrum, B t is a dilation by f ( t ) of B . Lemma 3.14.
Let L be a closed exact Lagrangian submanifold in a Weinstein neighbour-hood U of L (cid:48) with associated embedding ψ . For all t ∈ [ − T, let us denote L (cid:48) t = ϕ t −L ◦ ψ ( L ) .Assume that for all t ∈ [ − T, L (cid:48) t ⊂ U . Let us denote L t = ψ − ( L (cid:48) t ) . Then for all s ∈ R HF s ( L (cid:48) t , L (cid:48) ; ( ϕ t −L ) ∗ D , T ∗ L (cid:48) ) ∼ = HF se − t ( ψ ( L ) , L (cid:48) ; D , T ∗ L (cid:48) ) ,HF s ( L t , L (cid:48) ; ( ϕ t −L ) ∗ D , M ) ∼ = HF se − t ( L, L (cid:48) ; D , M ) . Applying this lemma to L T together with Lemma 3.11 and the fact that ϕ − T −L ( L (cid:48) T ) = ϕ ( L (cid:48) ) , we finally obtain HF s ( ϕ ( L (cid:48) ) , L (cid:48) ; D , M ) ∼ = HF s ( ϕ ( L (cid:48) ) , L (cid:48) ; D , T ∗ L (cid:48) ) for all s ∈ R . This concludes the proof of Lemma 3.9.Let us now prove the Lemmas used in the proof of Lemma 3.9. Proof of Lemma 3.10.
Remark 1.4 tells us that the complexes CF ( L (cid:48) t , L (cid:48) ; D , T ∗ L (cid:48) ) and CF ( L t , L (cid:48) ; D , M ) can respectively be seen as the complexes CF ( L (cid:48)D t , L (cid:48) ; T ∗ L (cid:48) ) and CF ( L D t , L (cid:48) ; M ) where L (cid:48)D t and L D t respectively denote the images of L (cid:48) t and L t by the time- of the Hamiltonian perturbation as explained in Remark 1.4. The actions of the originalcomplexes and those of the new ones are equal up to a shift by constants respectively c and c (cid:48) . We assume, without any loss of generality, that we have chosen a good almost complexstructure.Fix t ∈ R + . Let x be in χ ( L D t , L (cid:48) ) ⊂ M , with action A L D t ,L (cid:48) ( x ) . Denote x (cid:48) = ψ ( x ) which is consequently in χ ( L (cid:48)D t , L (cid:48) ) ⊂ T ∗ L (cid:48) with action A L (cid:48)D t ,L (cid:48) ( x (cid:48) ) . Set C t = A L (cid:48)D t ,L (cid:48) ( x (cid:48) ) − L D t ,L (cid:48) ( x ) .For any other y ∈ χ ( L D t , L (cid:48) ) together with y (cid:48) = ψ ( y ) ∈ χ ( L (cid:48)D t , L (cid:48) ) , let us denote γ a path from x to y in L D t and γ a path from y to x in L (cid:48) . We denote γ (cid:48) and γ (cid:48) theirrespective images by ψ . From Definition 1.1 and the fact that the differential decreasesthe action, we have A L D t ,L (cid:48) ( y ) − A L D t ,L (cid:48) ( x ) = (cid:90) γ λ M + (cid:90) γ λ M , A L (cid:48)D t ,L (cid:48) ( y (cid:48) ) − A L (cid:48)D t ,L (cid:48) ( x (cid:48) ) = (cid:90) γ (cid:48) λ T ∗ L (cid:48) + (cid:90) γ (cid:48) λ T ∗ L (cid:48) . Denoting γ (cid:93)γ the concatenation of γ and γ we get A L (cid:48)D t ,L (cid:48) ( y (cid:48) ) = A L (cid:48)D t ,L (cid:48) ( x (cid:48) ) + (cid:90) γ (cid:48) (cid:93)γ (cid:48) λ T ∗ L (cid:48) = A L (cid:48)D t ,L (cid:48) ( x (cid:48) ) + (cid:90) γ (cid:93)γ ψ ∗ λ T ∗ L (cid:48) = A L (cid:48)D t ,L (cid:48) ( x (cid:48) ) + (cid:90) γ (cid:93)γ λ M + dF = A L (cid:48)D t ,L (cid:48) ( x (cid:48) ) + (cid:90) γ (cid:93)γ λ M since γ (cid:93)γ is a loop = A L (cid:48)D t ,L (cid:48) ( x (cid:48) ) + A L D t ,L (cid:48) ( y ) − A L D t ,L (cid:48) ( x )= A L D t ,L (cid:48) ( y ) + C t . Since this is true for any t ∈ R + and any pair ( y, y (cid:48) ) such as before, we can conclude that ∀ t ∈ R + , Spec( L (cid:48)D t , L (cid:48) ; T ∗ L (cid:48) ) = Spec( L D t , L (cid:48) ; M ) + C t . So ∀ t ∈ R + , Spec( L (cid:48) t , L (cid:48) ; D , T ∗ L (cid:48) ) = Spec( L t , L (cid:48) ; D , M ) + C t + c (cid:48) − c. Now, for all t , choosing two primitive functions on L (cid:48) t and L t such that A L (cid:48)D t ,L (cid:48) ( x (cid:48) ) − c = A L D t ,L (cid:48) ( x ) − c (cid:48) gives C t + c (cid:48) − c = 0 , which finishes this proof. Proof of Lemma 3.11.
These two cochain complexes are generated by the perturbed inter-section points, which are identified by Weinstein’s neighbourhood embedding. To provethis lemma, we thus have to show that for T large enough, the differential is the same,i.e. that the J -holomorphic curves between two intersection points agree. To do so we willshow that if T is large enough, no such J -holomorphic curves can go outside of W .Since we are working with a Liouville domain, which is always tame, Sikorav’s proposi-tion . . and its corollary in [61] are verified. Consequently, there exists a constant κ ∈ R ,42uch that for any compact subset K , any compact connected J -holomorphic curve u suchthat u ∩ K (cid:54) = ∅ , and ∂u ⊂ K satisfies u ⊂ U ( K, κ A ( u )) , (16)where U ( K, κ A ( u )) is the κ A ( u ) -neighbourhood of K . Let us fix δ > small enough suchthat we can find a compact neighbourhood K of L (cid:48) such that U ( K, δ ) ⊂ W .Let us denote Γ t , the diameter of the spectrum Spec( L (cid:48) t , L (cid:48) ; D , T ∗ L (cid:48) ) , which is equalby Lemma 3.10 to the diameter of the spectrum Spec( L (cid:48) t , L (cid:48) ; D , M ) . From the previousequality 14, we have Γ t = e − t Γ . Set t δ = ln( Γ κδ ) . We then have ∀ t ≥ t δ , Γ t ≤ δκ . Let us recall that, the area of a J -holomorphic strip between two intersection points isequal to the difference of action between the two intersection points. This area is thusbounded by the diameter of the spectrum Γ t . Let us fix T > t δ . A J -holomorphic strip u between two generators of CF ( L (cid:48) T , L (cid:48) ; D , M ) satisfies A ( u ) ≤ δκ . Inclusion 16 then becomes u ⊂ U ( K, κ A ( u )) = U ( K, δ ) ⊂ W. This means that the J -holomorphic strips defining the differential of the chain complex CF ( L (cid:48) T , L (cid:48) ; D , M ) stay in W . They are identified by the embedding ψ to the J -holomorphicstrips defining the differential of the chain complex CF ( L (cid:48) T , L (cid:48) ; D , T ∗ L (cid:48) ) . Consequently thedifferential of the two chain complexes behave well with respect to the embedding ψ . Thisconcludes the proof of this lemma. Remark 3.15.
In this proof, we only dealt with J -holomorphic curves computing thedifferential. However, we can conduct the exact same proof with other moduli spaces. Thisimplies that the µ k -operations are preserved by the isomorphism given by Lemma 3.11. Proof of Lemma 3.14.
The symplectic invariance of Floer cohomology tells that there is afunction f : R → R such that HF s ( L (cid:48) t , ϕ t −L ( L (cid:48) ); ( ϕ − t −L ) ∗ D , T ∗ L (cid:48) ) ∼ = HF f ( s ) ( ψ ( L ) , L (cid:48) ; D , T ∗ L (cid:48) ) ,HF s ( L t , ψ − ◦ ϕ t −L ( L (cid:48) ); ( ϕ t −L ) ∗ D , M ) ∼ = HF f ( s ) ( L, L (cid:48) ; D , M ) . We can indeed write the second isomorphism since the negative Liouville flow has beenglobally defined on M . Since ϕ t −L ( L (cid:48) ) = L (cid:48) for all t , we have HF s ( L (cid:48) t , L (cid:48) ; ( ϕ t −L ) ∗ D , T ∗ L (cid:48) ) ∼ = HF f ( s ) ( ψ ( L ) , L (cid:48) ; D , T ∗ L (cid:48) ) , F s ( L t , L (cid:48) ; ( ϕ t −L ) ∗ D , M ) ∼ = HF f ( s ) ( L, L (cid:48) ; D , M ) . Moreover Equality 14 tells that f ( s ) = se − t . We then have HF s ( L (cid:48) t , L (cid:48) ; ( ϕ t −L ) ∗ D , T ∗ L (cid:48) ) ∼ = HF se − t ( ψ ( L ) , L (cid:48) ; D , T ∗ L (cid:48) ) . The same computation as in Lemma 3.10 gives HF s ( L t , L (cid:48) ; ( ϕ t −L ) ∗ D , M ) ∼ = HF se − t ( L, L (cid:48) ; D , M ) . Let us assume, once and for all that ϕ is sufficiently close to identity, so that ϕ ( L (cid:48) ) isinside the Weinstein neighbourhood of L (cid:48) and reciprocally. In this section, we will bound the bottleneck distance by the spectral norm γ . Proposi-tion 3.5 is a consequence of the following proposition. Proposition 3.16.
Let L and L (cid:48) be two closed exact Lagrangian submanifolds in a Liouvilledomain ( M, ω ) . There exists δ > , independant of L , such that for all ϕ ∈ Symp(
M, ω ) satisfying d C ( ϕ, Id M ) ≤ δ , then there exists C ∈ R such that d bottle ( B ( L (cid:48) , L ) , B ( ϕ ( L (cid:48) ) , L )[ C ]) ≤ γ ( L (cid:48) , ϕ ( L (cid:48) )) . In [36], Kislev and Shelukhin proved a similar statement in a different setting. Intheir case, L = L (cid:48) is a weakly monotone Lagrangian submanifold in a closed symplecticmanifold and ϕ is a Hamiltonian diffeomorphism. The following proof of Proposition 3.16is an adaptation of their proof to our setting.We choose δ > so that, for all ϕ ∈ Symp(
M, ω ) , if d C ( ϕ, Id M ) ≤ δ then ϕ ( L (cid:48) ) isincluded in a Weinstein neighbourhood of L (cid:48) . We will denote this Weinstein neighbourhood W ( L (cid:48) ) .We can now prove Proposition 3.5 required to prove Theorem 3.1. Proof of Proposition 3.5.
To prove this proposition, we will apply Proposition 3.16 to thesymplectomorphism ψ − ◦ ϕ . As in Proposition 3.16, we choose δ > so that, for all φ ∈ Symp(
M, ω ) , if d C ( φ, Id M ) ≤ δ then φ ( L (cid:48) ) is included in W ( L (cid:48) ) , a Weinstein neigh-44ourhood of L (cid:48) . Let us assume that d C ( ϕ, ψ ) ≤ δ . d C ( ϕ, ψ ) = max (cid:26) sup x ∈ M d ( ϕ ( x ) , ψ ( x )) , sup x ∈ M d ( ϕ − ( x ) , ψ − ( x )) (cid:27) ≥ sup x ∈ M d ( ϕ − ( x ) , ψ − ( x ))= sup x ∈ M d ( ψ − ◦ ϕ ( x ) , x )= d C ( ψ − ◦ ϕ, Id M ) . So we get d C ( ψ − ϕ, Id M ) ≤ δ. We introduced the set of barcodes quotiented by an overall shift ˆ B to get rid of the shiftin the inequality of Proposition 3.16. Indeed, when working with the barcodes in ˆ B , thisinequality becomes d bottle ( ˆ B ( L (cid:48) , L ) , ˆ B ( ϕ ( L (cid:48) ) , L )) ≤ γ ( L (cid:48) , ϕ ( L (cid:48) )) . By invariance of the barcode under the action of a symplectomorphism, we have d bottle ( ˆ B ( ϕ ( L (cid:48) ) , L ) , ˆ B ( ψ ( L (cid:48) ) , L )) = d bottle ( ˆ B ( L (cid:48) , ψ − ( L )) , ˆ B ( ψ − ◦ ϕ ( L (cid:48) ) , ψ − ( L ))) . By the previous inequality and Proposition 3.16, we then have d bottle ( ˆ B ( L (cid:48) , ψ − ( L )) , ˆ B ( ψ − ◦ ϕ ( L (cid:48) ) , ψ − ( L ))) ≤ γ ( L (cid:48) , ψ − ◦ ϕ ( L (cid:48) )) , which concludes the proof of this proposition.Let us now prove Proposition 3.16 and the desired bound. We start by introducing theinterleaving maps. Set up to define the interleaving maps
As explained in Remark 3.8, the condition H ( L (cid:48) , R ) = 0 guarantees that for all ϕ ∈ Symp(
M, ω ) , ϕ ( L (cid:48) ) is an exact Lagrangian submanifold. Hence, we can apply Abouzaid-Kragh’s Theorem 2.31 [2], thus obtaining two isomorphisms HF ( L (cid:48) , L (cid:48) ; D , T ∗ L (cid:48) ) ∼ −→ σ HF ( L (cid:48) , ϕ ( L (cid:48) ); D , T ∗ L (cid:48) ) ,HF ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) ); D , T ∗ L (cid:48) ) ∼ −→ σ (cid:48) HF ( ϕ ( L (cid:48) ) , L (cid:48) ; D , T ∗ L (cid:48) ) . Moreover, by Proposition 1.5 applied to L (cid:48) , together Poincaré duality there is an isomor-45hism θ : H ∗ ( L (cid:48) ) → HF n −∗ ( L (cid:48) , L (cid:48) ; D , T ∗ L (cid:48) ) . By symplectic invariance of Floer cohomology, we have HF ∗ ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) ); φ ∗ D , T ∗ L (cid:48) ) ∼ = HF ∗ ( L (cid:48) , L (cid:48) ; D , T ∗ L (cid:48) ) . As above, we also have an isomorphism θ (cid:48) : H ∗ ( L (cid:48) ) → HF n −∗ ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) ); D , T ∗ L (cid:48) ) . Let us choose c ∈ HF ( L (cid:48) , L (cid:48) ; D , T ∗ L (cid:48) ) to be the class θ ([ L (cid:48) ]) , and c (cid:48) ∈ HF ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) ); D , T ∗ L (cid:48) ) the class θ (cid:48) ([ L (cid:48) ]) . Moreover assume that the gradings are chosen so that c and c (cid:48) are bothof degree .Lemma 3.9 provides two isomorphisms ζ and ζ (cid:48) between HF ( L (cid:48) , ϕ ( L (cid:48) ); T ∗ L (cid:48) ) and HF ( L (cid:48) , ϕ ( L (cid:48) ); M ) and between HF ( ϕ ( L (cid:48) ) , L (cid:48) ; T ∗ L (cid:48) ) and HF ( ϕ ( L (cid:48) ) , L (cid:48) ; M ) . We can nowchoose two cycles x ∈ CF ( L (cid:48) , ϕ ( L (cid:48) ); M ) and y ∈ CF ( ϕ ( L (cid:48) ) , L (cid:48) ; M ) such that [ x ] = ζ ( σ ( c ))[ y ] = ζ (cid:48) ( σ (cid:48) ( c (cid:48) )) . Let us choose two primitive functions f (cid:48) : L (cid:48) → R and g : ϕ ( L (cid:48) ) → R such that df (cid:48) = λ | L (cid:48) , dg = λ | ϕ ( L (cid:48) ) and such that we can find• z such that [ z ] = c ∈ HF ( L (cid:48) , L (cid:48) ; D ) with A ( z ) ≤ ε (cid:48) / (cid:28) ε • z (cid:48) such that [ z (cid:48) ] = c (cid:48) ∈ HF ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) ); D ) with A ( z (cid:48) ) ≤ ε (cid:48) / (cid:28) ε . According to the previous choices of degree, we actually have [ z ] ∈ HF ( L (cid:48) , L (cid:48) ; D ) and [ z (cid:48) ] ∈ HF ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) ); D ) . Lemma 3.17.
The multiplication maps m ( · , z ) : CF ∗ ( L (cid:48) , L ; D ) → CF ∗ ( L (cid:48) , L ; D )[ ε (cid:48) ] m ( · , z (cid:48) ) : CF ∗ ( ϕ ( L (cid:48) ) , L ; D ) → CF ∗ ( ϕ ( L (cid:48) ) , L ; D )[ ε (cid:48) ] are filtered chain-homotopic to the standard inclusions and hence induce the ε (cid:48) -shift mapson the persistence modules.Proof. This lemma is an immediate consequence of Lemma 2.26.By abuse of notation, to make the following expressions clearer, we denote [ L (cid:48) ] = ζ ◦ σ ◦ θ ([ L (cid:48) ]) ∈ HF ( L (cid:48) , ϕ ( L (cid:48) ); D , M ) and [ ϕ ( L (cid:48) )] = ζ (cid:48) ◦ σ (cid:48) ◦ θ (cid:48) ([ L (cid:48) ]) ∈ HF ( ϕ ( L (cid:48) ) , L (cid:48) ; D , M ) .46ow, let us choose x ∈ CF ( L (cid:48) , ϕ ( L (cid:48) ); D ) and y ∈ CF ( ϕ ( L (cid:48) ) , L (cid:48) ; D ) as above such that: l ([ L (cid:48) ]; L (cid:48) , ϕ ( L (cid:48) ); D ) ≤ A ( x ) = a ≤ l ([ L (cid:48) ]; L (cid:48) , ϕ ( L (cid:48) ); D ) + ε (cid:48) ,l ([ ϕ ( L (cid:48) )]; ϕ ( L (cid:48) ) , L (cid:48) ; D ) ≤ A ( y ) = b ≤ l ([ ϕ ( L (cid:48) )]; ϕ ( L (cid:48) ) , L (cid:48) ; D ) + ε (cid:48) , which is possible by definition of l ([ L (cid:48) ]; L (cid:48) , ϕ ( L (cid:48) ); D ) and l ([ ϕ ( L (cid:48) )]; ϕ ( L (cid:48) ) , L (cid:48) ; D ) .Moreover, by definition of x, y , we have [ µ ( y, x )] = [ z ] ∈ HF ( L (cid:48) , L (cid:48) ; D ) . Indeed, upto the appropriate isomorphisms, the cycles x, y, z all represent the same class [ L ] in theirrespective cochain complexes. With our particular choice of perturbation data for the pair ( L (cid:48) , L (cid:48) ) as explained in Remark 3.7, the cycle z is the only representative of his class. Thesame argument holds for z (cid:48) . Consequently, we have the following lemma. Lemma 3.18. µ ( y, x ) = z ∈ CF ( L (cid:48) , L (cid:48) ; D ) ,µ ( x, y ) = z (cid:48) ∈ CF ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) ); D ) . Remark 3.19.
If we choose to work with ϕ being a Hamiltonian diffeomorphism and notonly a symplectomorphism, the definition of x and y is much easier. In this case, it isachieved without Abouzaid-Kragh’s result [2] of Theorem 2.31.Indeed, continuation morphisms give the isomorphisms: HF ∗ ( ϕ ( L (cid:48) ) , L (cid:48) ) ∼ = HF ∗ ( L (cid:48) , L (cid:48) ) ∼ = HF ∗ ( ϕ ( L (cid:48) ) , ϕ ( L (cid:48) )) ∼ = HF ∗ ( L (cid:48) , ϕ ( L (cid:48) )) . Since these continuations morphisms are compatible with the product structure onLagrangian Floer cohomology, we can directly define x and y , and it is easy to see thatthe product by these elements will not be constant equal to . Indeed we easily have [ µ ( y, x )] = [ z ][ µ ( x, y )] = [ z (cid:48) ] . Moreover, the two multiplication operators m ( · , z ) and m ( · , z (cid:48) ) are still filtered chain-homotopic to the standard inclusion. Bounding the bottleneck distance
Now that our objects are defined, we can adapt the Kislev-Shelukhin method [36] toour context. Except for the context, we do not claim anything new here. The point hereis to carefully study the shifts of action induced by the the multiplication by the elementsintroduced above. Let us start with the two following lemmas.47 emma 3.20.
The maps µ ( · , x ) : CF ∗ ( ϕ ( L (cid:48) ) , L ; D ) → CF ∗ ( L (cid:48) , L ; D )[ a + ε (cid:48) ] µ ( · , y ) : CF ∗ ( L (cid:48) , L ; D ) → CF ∗ ( ϕ ( L (cid:48) ) , L ; D )[ b + ε (cid:48) ] are well-defined and induce filtered maps of chain complexes. Lemma 3.21.
The maps µ ( µ ( · , y ) , x ) : CF ∗ ( L (cid:48) , L ; D ) → CF ∗ ( L (cid:48) , L ; D ))[ a + b + 3 ε (cid:48) ] µ ( µ ( · , x ) , y ) : CF ∗ ( ϕ ( L (cid:48) ) , L ; D ) → CF ∗ ( ϕ ( L (cid:48) ) , L ; D )[ a + b + 3 ε (cid:48) ] are well-defined and filtered chain homotopic to the multiplication operators: µ ( · , µ ( y, x )) : CF ∗ ( L (cid:48) , L ; D ) → CF ∗ ( L (cid:48) , L ; D )[ a + b + 3 ε (cid:48) ] µ ( · , µ ( x, y )) : CF ∗ ( ϕ ( L (cid:48) ) , L ; D ) → CF ∗ ( ϕ ( L (cid:48) ) , L ; D )[ a + b + 3 ε (cid:48) ] Proof.
These two lemmas directly follow from the discussion on the product structure.Lemma 2.24 gives the first one and Lemma 2.25 the second one.
Remark 3.22.
In Kislev and Shelukhin’s paper [36], there is another term in the previousequality which is a boundary. This additional term induces a shift in action by a constant β which vanishes in our case.We now have the relation between the previous maps and the multiplication by z or z (cid:48) : the maps µ ( · , µ ( y, x )) : CF ∗ ( L (cid:48) , L ; D ) → CF ∗ ( L (cid:48) , L (cid:48) ; D )[ a + b + 3 ε (cid:48) ] µ ( · , µ ( x, y )) : CF ∗ ( ϕ ( L (cid:48) ) , L ; D ) → CF ∗ ( ϕ ( L (cid:48) ) , L ; D )[ a + b + 3 ε (cid:48) ] are equal to the multiplication operators: µ ( · , z ) : CF ∗ ( L (cid:48) , L ; D ) → CF ∗ ( L (cid:48) , L ; D ) a + b + 3 ε (cid:48) ] µ ( · , z (cid:48) ) : CF ∗ ( ϕ ( L (cid:48) ) , L ; D ) → CF ∗ ( ϕ ( L (cid:48) ) , L ; D )[ a + b + 3 ε (cid:48) ] Following Lemma 3.17, we obtain on the level of filtered homology the shifts of persis-tence modules morphisms sh L (cid:48) : CF ∗ ( L (cid:48) , L ; D ) → CF ∗ ( L (cid:48) , L ; D )[ a + b + 4 ε (cid:48) ] sh ϕ ( L (cid:48) ) : CF ∗ ( ϕ ( L (cid:48) ) , L ; D ) → CF ∗ ( ϕ ( L (cid:48) ) , L ; D )[ a + b + 4 ε (cid:48) ] . Let us recall that the barcodes are C -continuous by Proposition 2.23. We can take the48imit as the Hamiltonian part of the perturbation data goes to zero as explained afterProposition 2.23 and assume that a < l ([ L (cid:48) ]; L (cid:48) , ϕ ( L (cid:48) )) + 2 ε (cid:48) ,b < l ([ L (cid:48) ]; ϕ ( L (cid:48) ) , L (cid:48) ) + 2 ε (cid:48) . Consequently we have shift maps of barcodes without the perturbation data: sh L (cid:48) = µ ( · , x ) ◦ µ ( · , y ) : B ( L (cid:48) , L ) → B ( L (cid:48) , L )[ γ ( L (cid:48) , ϕ ( L (cid:48) )) + 6 ε (cid:48) ] sh ϕ ( L (cid:48) ) = µ ( · , y ) ◦ µ ( · , x ) : B ( ϕ ( L (cid:48) ) , L ) → B ( ϕ ( L (cid:48) ) , L )[ γ ( L (cid:48) , ϕ ( L (cid:48) )) + 6 ε (cid:48) ] . Indeed, as discussed in Section 2.5, l ([ L (cid:48) ]; L (cid:48) , ϕ ( L (cid:48) )) + l ([ L (cid:48) ]; ϕ ( L (cid:48) ) , L (cid:48) ) = γ ( L (cid:48) , ϕ ( L (cid:48) )) ≥ . For readability reasons, we denote α = l ([ L (cid:48) ]; L (cid:48) , ϕ ( L (cid:48) )) and ¯ α = l ([ L (cid:48) ]; ϕ ( L (cid:48) ) , L (cid:48) ) . With this expression, the multiplication operators appear as maps between persistencemodules: µ ( · , x ) : B ( ϕ ( L (cid:48) ) , L ) → B ( L (cid:48) , L )[ α + 3 ε (cid:48) ] µ ( · , y ) : B ( L (cid:48) , L ) → B ( ϕ ( L (cid:48) ) , L )[ ¯ α + 3 ε (cid:48) ] . Let us recall that, by Definition 2.32, we have γ ( L (cid:48) , ϕ ( L (cid:48) )) = α + ¯ α . Consequently, theprevious multiplication operators can be written as µ ( · , x ) : B ( ϕ ( L (cid:48) ) , L ) → B ( L (cid:48) , L )[ ( α − ¯ α )][ γ ( L (cid:48) , ϕ ( L (cid:48) )) + 3 ε (cid:48) ] µ ( · , y ) : B ( L (cid:48) , L )[ ( α − ¯ α )] → B ( ϕ ( L (cid:48) ) , L )[ γ ( L (cid:48) , ϕ ( L (cid:48) )) + 3 ε (cid:48) ] Together with the previous identity of persistence modules, this is the exact definition ofthe fact that B ( L (cid:48) , L ) and B ( ϕ ( L (cid:48) ) , L )[ ( α − ¯ α )] are γ ( L (cid:48) , ϕ ( L (cid:48) ))+3 ε (cid:48) -interleaved. Takingthe limit as ε (cid:48) goes to zero, we get d bottle ( B ( L (cid:48) , L ) , B ( ϕ ( L (cid:48) ) , L )[ ( α − ¯ α )]) ≤ γ ( L (cid:48) , ϕ ( L (cid:48) )) . (17)Setting C = ( α − ¯ α ) , this concludes the proof of Proposition 3.16.49 .4 Bounding the spectral norm by the C -distance We will now prove Proposition 3.6. This proof is an adaptation to our context of a lemmaand a proof of Buhovsky-Humilière-Seyfaddini [10]. In their paper, they proved the sameresult for a Lagrangian submanifold Hamiltonian isotopic to the zero section in a cotangentbundle.Here we are working with L (cid:48) being a closed exact Lagrangian submanifold in M . Let usdenote W ( L (cid:48) ) a Weinstein neighbourhood of L (cid:48) . By definition, if ϕ ∈ Symp(
M, ω ) is closeenough to Id M , then ϕ ( L (cid:48) ) ⊂ W ( L (cid:48) ) . By abuse of notation, we also respectively denote B and ϕ ( L (cid:48) ) the images by a Weinstein embedding of respectively B and ϕ ( L (cid:48) ) in T ∗ L (cid:48) ,where B is a ball in W ( L (cid:48) ) .We will start by stating two key lemmas without any proof. Indeed these lemmas areadaptations to our particular context of Buhovsky-Humilière-Seyfaddini’ s lemmas and theproofs they give apply verbatim to our situation. We will then apply these lemmas and dosome basic computations to prove Proposition 3.6. For more details, one can also refer tothe author’s thesis. Lemma 3.23.
Let B be a ball in L (cid:48) . Let Symp B ( M, ω ) := { ϕ ∈ Symp(
M, ω ) | ϕ ( L (cid:48) ) ∩ T ∗ B = 0 B ) } . There exist δ > and C > such that for any ϕ ∈ Symp B ( M, ω ) , if d C (Id M , ϕ ) ≤ δ , then γ ( L (cid:48) , ϕ ( L (cid:48) )) ≤ Cd C (Id M , ϕ ) . In order to finish the proof of Proposition 3.6, we need to reduce to Lemma 3.23. Indeedin the hypothesis, we do not have such a ball B . To do so, we will use and adapt to ourcontext a trick from [10]. This trick consists in doubling the coordinates and introducingthe following auxiliary map: Φ : ϕ × ϕ − = M × M → M × M, where M × M is equipped with the natural symplectic form ω ⊕ ω .Buhovsky-Humilière-Seyfaddini [10] also gives the following lemma: Lemma 3.24.
For any ball B in M , there is a smaller ball B (cid:48) ⊂ B with the followingproperty. There exists ∆ > such that for any ϕ ∈ Symp(
M, ω ) with d C ( ϕ, Id M ) < ∆ ,we can find a symplectomorphism Ψ ∈ Symp( M × M, ω ⊕ ω ) satisfying:1. supp (Ψ) ⊂ B × B and supp (Φ ◦ Ψ) ⊂ M × M \ B (cid:48) × B (cid:48) ,2. d C (Ψ , Id M × M ) < C B d C ( ϕ, Id M ) and d C (Φ ◦ Ψ , Id M × M ) < C (cid:48) B d C ( ϕ, Id M ) , where C B and C (cid:48) B do not depend on ϕ . This Lemma 3.24 together with Lemma 3.23 will allow to conclude the proof of Proposi-tion 3.6 and thus Theorem 3.1. Indeed, we have proven that B ( L (cid:48) , L ) and B ( ϕ ( L (cid:48) ) , L )[ ( α − ¯ α )] are γ ( L (cid:48) , ϕ ( L (cid:48) )) -interleaved. We now just have to find two constants κ > and l > such that if d C ( ϕ, Id M ) ≤ l , then γ ( L (cid:48) , ϕ ( L (cid:48) )) ≤ κd C ( ϕ, Id M ) .50et us pick a point x ∈ L (cid:48) and a ball B x centered on x . Lemma 3.24 provides a smallerball B (cid:48) also centered on x . Pick a smaller ball B , centered on x and whose closure isincluded in B (cid:48) . The same lemma also provides l > and a symplectomorphism Ψ such that l < ∆ and if d C ( ϕ, Id M ) < l , then Φ ◦ Ψ( L (cid:48) ∩ B × L (cid:48) ∩ B ) ∩ T ∗ ( B (cid:48) × B (cid:48) ) = L (cid:48) ∩ B × L (cid:48) ∩ B .Then, let us pick another ball B centered on y ∈ L (cid:48) × L (cid:48) whose closure is included in M \ B × B . Since d C (Ψ , Id M × M ) ≤ C B d C ( ϕ, Id M ) and supp (Ψ) ⊂ B × B , we can find l > such that if d C ( ϕ, Id M ) < l , then Ψ and B satisfy the conditions of Lemma 3.24.Let us choose l > { δ, l , l } . Then we have, using successively Proposition 2.22and its consequence of Equality (10) and the triangle Inequality (13) and the symmetry of γ (12), for all ϕ such that d C ( ϕ, Id M ) < l : γ ( L (cid:48) , ϕ ( L (cid:48) )) = 12 γ ( L (cid:48) × L (cid:48) , Φ( L (cid:48) × L (cid:48) ))= 12 γ (Ψ − Φ − ( L (cid:48) × L (cid:48) ) , Ψ − ( L (cid:48) × L (cid:48) )) ≤ γ ( L (cid:48) × L (cid:48) , Ψ − ( L (cid:48) × L (cid:48) )) + 12 γ (Ψ − Φ − ( L (cid:48) × L (cid:48) ) , L (cid:48) × L (cid:48) )= 12 γ ( L (cid:48) × L (cid:48) , Ψ − ( L (cid:48) × L (cid:48) )) + 12 γ ( L (cid:48) × L (cid:48) , ΦΨ( L (cid:48) , L (cid:48) )) . For the second equality, the same argument as in Lemma 3.10 indeed tells that γ ( L (cid:48) × L (cid:48) , Φ( L (cid:48) × L (cid:48) )) = γ (Ψ − Φ − ( L (cid:48) × L (cid:48) ) , Ψ − ( L (cid:48) × L (cid:48) )) , when composing by Ψ − Φ − . Asimilar argument holds for the first equality and for the last one.Choosing B × B for the ball in Lemma 3.23, we can apply it to Φ ◦ Ψ for all ϕ suchthat d C ( ϕ, Id M ) < l . We then get that there is a constant C > such that for all these ϕ , we have: γ ( L (cid:48) × L (cid:48) , Φ ◦ Ψ( L (cid:48) × L (cid:48) )) ≤ C d C (Φ ◦ Ψ , Id M × M ) ≤ C C (cid:48) B d C ( ϕ, Id M ) . Moreover, for all such ϕ , Lemma 3.23 gives for Ψ a constant C : γ ( L (cid:48) × L (cid:48) , Ψ − ( L (cid:48) × L (cid:48) )) ≤ C d C (Ψ − , Id M × M ) ≤ C C B d C ( ϕ, Id M ) . Putting all this together, we get: γ ( L (cid:48) , ϕ ( L (cid:48) )) ≤
12 ( C C (cid:48) B + C C B ) d C ( ϕ, Id M ) . By setting κ = ( C C (cid:48) B + C C B ) , we get that for all ϕ such that d C ( ϕ, Id M ) ≤ l , then γ ( L (cid:48) , ϕ ( L (cid:48) )) ≤ κd C ( ϕ, Id M ) .Taking into account the discussions in Subsection 3.1.2, the proof of Theorem 3.1 is51ow complete, and this Section 3 finished. C -symplectic geometry Now that all our main tools have been introduced, we can prove our theorems regardingthe Dehn-Seidel twist.
Since the square of the Dehn-Seidel twist has been proved to be isotopic to the identity in
Diff( M n ) when n = 2 [56], it is a natural question to ask whether this is true in higherdimensions. Since this map is symplectic, it is also natural to wonder whether this alsoholds in Symp(
M, ω ) , or whether this is a purely smooth (non-symplectic) result. Even ifthe answer to the first question is still unknown, regarding the second one, Seidel proved in[53] a stronger result, by considering images of Lagrangian submanifolds instead of directlyconsidering the Dehn twist. For the reader’s convenience, let us recall Seidel’s theorem. Theorem 4.1 (Seidel [53]) . Let ( M n , ω ) be a compact symplectic manifold with contacttype boundary, with n even, which satisfies [ ω ] = 0 and c ( M, ω ) = 0 . Assume that M contains an A -configuration ( l ∞ , l (cid:48) , l ) of Lagrangian spheres. Then M contains infinitelymany symplectically knotted Lagrangian spheres. More precisely, if one defines L (cid:48) ( k ) = τ kl ( L (cid:48) ) for k ∈ Z , then all the L (cid:48) ( k ) are isotopic as smooth submanifolds of M, but no twoof them are isotopic as Lagrangian submanifolds. Since no two of these Lagrangian submanifolds are isotopic as Lagrangian submanifolds,the following corollary is immediate.
Corollary 4.2. τ l is not in the identity component of Symp c ( T ∗ S n ) , the compactly sup-ported symplectomorphisms of T ∗ S n . Indeed, if this symplectomorphismmorphism was in the identity component of
Symp c ( T ∗ S n ) ,its conjugation by the embedding j would also be in the identity component of Symp c ( M, ω ) ,and thus, all the Lagrangian spheres in Theorem 4.1 would be isotopic as Lagrangian sub-manifolds. Remark 4.3.
Let L be a Lagrangian sphere in M . Then for any Lagrangian sphere L (cid:48) in M , τ l ( L (cid:48) ) is a Lagrangian sphere as well.It can be checked that the Dehn-Seidel twist is in fact an exact symplectomorphism.The proof of this theorem deeply relies on the isotopy invariance of Floer homologytogether with the action of the Dehn-Seidel twist on the Maslov index. The proof wewill give for the analogous result in C symplectic topology also relies on barcodes andconsequently on Floer cohomology. However there are some technical difficulties to adaptSeidel’s proof to our context. 52 emark 4.4. A similar result holds when working with odd n . However, one shouldnot consider the square of the Dehn-Seidel twist, but the cube of the composition of twoDehn-Seidel twists defined along different but intersecting Lagrangian submanifolds [53].We introduced the notion of Milnor fibres after Definition 0.2 as these are examples ofmanifolds satisfying the conditions required for Theorem 4.1.We state here the following technical lemma, which was a key argument in Seidel’sproof [53] and which will be useful in the following computations. Lemma 4.5.
There is a unique L ∞ grading ˜ τ l of τ l which acts trivially on the part of L ∞ which lies over M \ Im( i ) . It satisfies ˜ τ l ˜ L = ˜ L [1 − n ] for any grading ˜ L of L . We recall that here i is the Weinstein embedding at stake in the definition of theDehn-Seidel twist as presented in the Introduction.Let us briefly recall the property of this grading we will need for the following section.For more detailled explanations, see [53].Denoting ˜ L [ k ] the graded Lagrangian ˜ L whose grading has been shifted by k ∈ Z /N ,we have the following useful property, where ˜ L and ˜ L are two Z /N -graded Lagrangiansubmanifolds in a symplectic manifold ( M, ω ) . HF ∗ ( ˜ L [ k ] , ˜ L [ l ]) ∼ = HF ∗− k + l ( ˜ L [ k ] , ˜ L [ l ]) . (18)In addition we have the invariance under the action of a graded symplectomorphisms ˜ ϕ : HF ∗ ( ˜ ϕ ( ˜ L ) , ˜ ϕ ( ˜ L )) ∼ = HF ∗ ( ˜ L , ˜ L ) , (19)the Poincaré duality HF ∗ ( ˜ L , ˜ L ) ∼ = HF n −∗ ( ˜ L , ˜ L ) . (20)Finally, when Proposition 1.5 holds, we have a graded counterpart: HF ∗ ( ˜ L, ˜ L ) ∼ = (cid:77) i ∈ Z H ∗ + iN ( L, Z / . (21) As mentioned in the previous section, Floer cohomology will be essential to our proof. It isactually possible to compute the action of the Dehn-Seidel twist on the Floer cohomologyof certain exact Lagrangian submanifolds. It is the object of the following theorem, alsoproved by Seidel [54].
Theorem 4.6.
Let l : S n → M be a Lagrangian sphere in ( M n , ω ) with image L . Forany two exact Lagrangian submanifolds L , L ∈ M , there is a long exact sequence of Floer ohomology groups: HF ( τ l ( L ) , L ) (cid:47) (cid:47) HF ( L , L ) n (cid:117) (cid:117) HF ( L, L ) ⊗ HF ( L , L ) − n (cid:106) (cid:106) Now that this theorem is stated, we make some computations of this long exact se-quence, in order to use it in our context.
Proposition 4.7.
Let ( M n , ω ) be a connected Liouville domain, n > , c ( M, ω ) = 0 .Assume that M contains an A -configuration of Lagrangian spheres ( l, l (cid:48) ) . Let x = L ∩ L (cid:48) . Choose an ∞ -Maslov covering on M and L ∞ -gradings ˜ L, ˜ L (cid:48) of L and L (cid:48) such that ˜ I ( x, ˜ L (cid:48) , ˜ L ) = 0 .Then, for all k ∈ Z , HF ∗ ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) (cid:29) HF ∗ + k ( ˜ L (cid:48) , ˜ L (cid:48) ) . Proof.
Since our Lagrangian submanifolds are spheres of dimension > , they are exact.We can thus apply Seidel’s Theorem 4.6 to L = L = L (cid:48) to get an exact triangle in Floercohomology: HF ( τ l ( L (cid:48) ) , L (cid:48) ) (cid:47) (cid:47) HF ( L (cid:48) , L (cid:48) ) n (cid:117) (cid:117) HF ( L, L (cid:48) ) ⊗ HF ( L (cid:48) , L ) − n (cid:106) (cid:106) Seidel’s exact triangle in our particular case leads to a long exact sequence in Floer coho-mology: 54 F ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) HF ( ˜ L (cid:48) , ˜ L (cid:48) ) (cid:76) k + l = n HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ L (cid:48) , ˜ L ) HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) HF ( ˜ L (cid:48) , ˜ L (cid:48) ) (cid:76) k + l = n +1 HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ L (cid:48) , ˜ L ) HF n − ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) HF n − ( ˜ L (cid:48) , ˜ L (cid:48) ) (cid:76) k + l =2 n − HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ L (cid:48) , ˜ L ) HF n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) HF n ( ˜ L (cid:48) , ˜ L (cid:48) ) (cid:76) k + l =2 n HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ L (cid:48) , ˜ L ) · · · (cid:63) Following Equality (21) regarding the properties of the cohomology of graded La-grangian submanifolds, we obtain HF ( ˜ L (cid:48) , ˜ L (cid:48) ) = Z / [0] ⊕ Z / [ n ] . Moreover, we can choose particular gradings for L and L (cid:48) such that, together with thePoincaré duality (20) we have: HF ( ˜ L (cid:48) , ˜ L ) = Z / [0] and HF ( ˜ L, ˜ L (cid:48) ) = Z / [ n ] . Consequently we have (cid:77) k + l = j HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ L (cid:48) , ˜ L ) = (cid:40) Z / if j = n elseWe now have to discuss the arrow (cid:63) . Since HF ( ˜ L (cid:48) , ˜ L (cid:48) ) and (cid:76) k + l = n HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ L (cid:48) , ˜ L ) are both of dimension , this arrow is either a bijection or zero. We will check both cases.Let us start with the case when the arrow (cid:63) is a bijection. Our long exact sequencethen becomes 55 HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) Z / Z / HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 0 HF n − ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 0 HF n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) Z / · · · We conclude that HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) = Z / [ n ] . We now consider the three Lagrangian submanifolds:
L, L (cid:48) and τ l ( L (cid:48) ) . The subman-ifolds L and L (cid:48) are still two Lagrangian spheres in ( M, ω ) . Moreover τ l ( L (cid:48) ) is an exactLagrangian submanifold as well according to Remark 4.3.We can now apply Seidel’s Theorem 4.6 to L = L , L = τ l ( L (cid:48) ) and L = L (cid:48) . We thusget an exact triangle in Floer cohomology: HF ( τ l ( L (cid:48) ) , L (cid:48) ) (cid:47) (cid:47) HF ( τ l ( L (cid:48) ) , L (cid:48) ) n (cid:116) (cid:116) HF ( L, L (cid:48) ) ⊗ HF ( τ l ( L (cid:48) ) , L ) − n (cid:106) (cid:106) This exact triangle leads to a long exact sequence in Floer cohomology:56 F − n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) HF − n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) (cid:76) k + l =1 HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L ) HF − n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) HF − n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) (cid:76) k + l =2 HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L ) HF n − ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) HF n − ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) (cid:76) k + l =2 n − HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L ) HF n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) HF n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) (cid:76) k + l =2 n HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L ) · · · Using Equality (19) and Lemma 4.5 together with the fact that HF ( ˜ L (cid:48) , ˜ L ) = Z / [0] , wecan compute HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L ) : HF ∗ ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L ) = HF ∗ ( ˜ L (cid:48) , ˜ τ l − ( ˜ L ))= HF ∗ ( ˜ L (cid:48) , ˜ L [ n − HF ∗ + n − ( ˜ L (cid:48) , ˜ L ) We thus obtain HF ∗ ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L ) = (cid:40) Z / if ∗ = 1 − n else . We also have (cid:77) k + l = j HF k ( ˜ L, ˜ L (cid:48) ) ⊗ HF l ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L ) = (cid:40) Z / if j = 10 else . Our long exact sequence then becomes: 57 HF − n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 Z / HF − n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 0 HF n − ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 0 HF n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) Z / · · · So finally, we get: HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) = Z / [ n ] ⊕ Z / [2 − n ] . We now have to check the case where the arrow (cid:63) is zero. In this case our long exactsequence then becomes: HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) Z / Z / HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 0 HF n − ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 0 HF n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) Z / · · · We conclude that HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) = Z / [0] ⊕ Z / [1] ⊕ Z / [ n ] . As above, we apply Seidel’s Theorem 4.6 to L = L , L = τ l ( L (cid:48) ) and L = L (cid:48) . The exacttriangle and the computation of the previous case lead to the following exact sequence:58 HF − n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 Z / HF − n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 0 HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) Z / HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) Z / HF n − ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) 0 0 HF n ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) Z / · · · So finally, we obtain HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) ) = Z / [ n ] ⊕ Z / [1] ⊕ Z / [0] ⊕ Z / [2 − n ] . This concludes the proof of Proposition 4.7.
The computation of Floer cohomology of Proposition 4.7 indicates that we may have tosplit the cases for n = 2 and n ≥ . We will discuss in this section the case when n ≥ .The case n = 2 will be discussed in the following Section 4.4. We will now prove Theorem Astated in the introduction. For the reader’s convenience, we repeat it here. Theorem 4.8.
Let ( M n , ω ) be a n -dimensionnal Liouville domain, n even, n ≥ , c ( M, ω ) = 0 . Assume that M contains an A -configuration of Lagrangian spheres ( l, l (cid:48) ) .Then, τ l is not in the connected component of the identity in Symp(
M, ω ) .Proof. Let us assume that τ l is in the connected component of the identity in Symp(
M, ω ) .The first point of Theorem 3.3 implies then that ˆ B ( L (cid:48) , L (cid:48) ) and ˆ B ( τ l ( L (cid:48) ) , L (cid:48) ) are in the sameconnected component in ˆ B , i.e. up to an overall shift, they have their semi-infinite bars inthe same degree.Moreover, since HF ( L (cid:48) , L (cid:48) ) = Z / [0] ⊕ Z / [ n ] , Remark 2.19 implies that B = ˆ B ( L (cid:48) , L (cid:48) ) has only two semi-infinite bars, one in degree and one in degree n . Consequently, thebarcode ˆ B ( τ l ( L (cid:48) ) , L (cid:48) ) has two semi-infinite bars, one in degree and one in degree n .59owever, let us recall that Proposition 4.7 gives, for all k ∈ Z , HF ( ˜ τ l ( ˜ L (cid:48) ) , ˜ L (cid:48) [ k ]) (cid:29) Z / [0] ⊕ Z / [ n ] . This means that ˆ B ( τ l ( L (cid:48) ) , L (cid:48) ) cannot have the semi-infinite bars in the same degreeas for ˆ B ( L (cid:48) , L (cid:48) ) . This contradicts the fact that they should be in the same connectedcomponent and thus concludes the proof of this theorem.The following statements correspond to Corollary B and Corollary C stated in the in-troduction. For the reader’s convenience, we repeat them here. The first one is a straight-forward consequence of Theorem 4.8. Corollary 4.9.
Let ( M n , ω ) be a n -dimensional Liouville domain, n even, n ≥ , c ( M, ω ) = 0 . Assume that M contains an A -configuration of Lagrangian spheres ( l, l (cid:48) ) .Then, τ l is not isotopic to the identity in Symp(
M, ω ) . In particular, MCG ω ( M, C ) is non-trivial. Let us recall that, according to the discussion held in the introduction, Corollary 4.9does not imply Theorem 4.8. Indeed, whether
Symp(
M, ω ) is locally path-connected re-mains an open question.The following theorem is also a consequence of Theorem 4.8. Indeed the subspace Ham(
M, ω ) ⊂ Symp(
M, ω ) is connected as it is the closure of the connected space Ham(
M, ω ) . Theorem 4.10.
Let ( M n , ω ) be a n -dimensional Liouville domain, n even, n ≥ , c ( M, ω ) = 0 . Assume that M contains an A -configuration of Lagrangian spheres ( l, l (cid:48) ) .Then, τ l does not belong to Ham(
M, ω ) . When working in dimension , we cannot use the computation of Proposition 4.7. However,we can apply Hind’s Theorem 2.30 on the nearby Lagrangian conjecture [29]. Nevertheless,we also have to use Seidel’s Theorem 4.1. Consequently, in dimension , we require an A -configuration instead of an A -configuration as in higher dimensions. Theorem 4.11.
Let ( M , ω ) be a compact connected -dimensional submanifold with con-tact type boundary, [ ω ] = 0 , c ( M, ω ) = 0 . Assume that M contains an A -configurationof Lagrangian spheres ( l, l (cid:48) , l ∞ ) .Then, τ l does not belong to Ham(
M, ω ) . Using Hind’s proof of the Nearby Lagrangian Conjecture of Theorem 2.30 in the caseof T ∗ S [29], the proof is quite straightforward and does not require the use of barcodes.60 roof. Let us assume that there is a sequence of Hamiltonian diffeomorphisms ( ϕ n ) n ∈ N of M which C -converges to τ l . Then, for N large enough, we have that ϕ N ( L (cid:48) ) is includedin a Weinstein neighbourhood of τ l ( L (cid:48) ) .Moreover, τ l ( L (cid:48) ) is a Lagrangian sphere and so its Weinstein neighbourhood is, bydefinition, symplectomorphic to a neighbourhood of the zero section in T ∗ S . Conse-quently, we are under the condition of application of the Nearby Lagrangian Conjectureas in Theorem 2.30, in the case of T ∗ S .We get that ϕ N ( L (cid:48) ) is Lagrangian isotopic to τ l ( L (cid:48) ) , which contradicts Seidel’s resultin Theorem 4.1.For the reader’s convenience, we repeat here the statement of Theorem E. It is thecounterpart in dimension of Corollary 4.9. Theorem 4.12.
Let ( M , ω ) be a compact connected -dimensionnal submanifold with con-tact type boundary, [ ω ] = 0 , c ( M, ω ) = 0 . Assume that M contains an A -configurationof Lagrangian spheres ( l, l (cid:48) , l ∞ ) .Then, τ l is not isotopic to the identity in Symp(
M, ω ) . Note that none of Theorem 4.11 and Theorem 4.12 imply the other.
Proof.
As above for the case n = 2 , this proof heavily relies on the proof of the NearbyLagrangian conjecture for T ∗ S as in Theorem 2.30 [29].Let us assume that τ l is connected to identity in Symp(
M, ω ) . This means that wecan find a continuous path ( ϕ t ) t ∈ [0;1] ⊂ Symp(
M, ω ) such that ϕ = Id M and ϕ = τ l .Since for all t ∈ [0; 1] , ϕ t is in Symp(
M, ω ) , we can find sequences ϕ tn ∈ Symp(
M, ω ) such that ∀ t ∈ (0; 1) , lim n →∞ ϕ tn = ϕ t . Let us choose a Weinstein neighbourhood W ( L (cid:48) ) of L (cid:48) together with ε > such that,for all ϕ ∈ Symp(
M, ω ) , if d C ( ϕ, Id M ) < ε , then ϕ ( L (cid:48) ) ⊂ W ( L (cid:48) ) .The path ϕ t being continuous, we can find a finite sequence ( ϕ t i ) i ∈ (cid:74) N (cid:75) ⊂ Symp(
M, ω ) such that ϕ t = Id M , ϕ t N = τ l and ∀ i ∈ (cid:74) N (cid:75) , d C ( ϕ t i − , ϕ t i ) < ε . Moreover, for each ( t i ) i ∈ (cid:74) N − (cid:75) , we can find n i such that d C ( ϕ t i , ϕ t i n i ) < ε . We choose ϕ t = Id M and ϕ t N N = τ l .Consequently, we get a sequence ( ϕ t i n i ) i ∈ (cid:74) N (cid:75) ⊂ Symp(
M, ω ) such that ϕ t n = Id M ,61 t N n N = τ l which satisfies ∀ i ∈ (cid:74) N (cid:75) , d C (( ϕ t i − n i − ) − ◦ ϕ t i n i , Id M ) ≤ d C ( ϕ t i − n i − , ϕ t i n i ) ≤ d C ( ϕ t i − n i − , ϕ t i − ) + d C ( ϕ t i − , ϕ t i ) + d C ( ϕ t i , ϕ t i n i ) < ε + ε + ε = ε. Then ϕ t i n i ( L (cid:48) ) = ϕ t i − n i − ◦ ( ϕ t i − n i − ) − ◦ ϕ t i n i ( L (cid:48) ) ⊂ ϕ t i − n i − ( W ( L (cid:48) )) ∼ = W ( ϕ t i n i ( L (cid:48) )) , where W ( ϕ t i n i ( L (cid:48) )) denotes a Weinstein neighbourhood of ϕ t i n i ( L (cid:48) ) .Applying now Hind’s Theorem 2.30, we obtain that for all i ∈ (cid:74) N (cid:75) , ϕ n i ,t i ( L (cid:48) ) isLagrangian isotopic to ϕ n i − ,t i − ( L (cid:48) ) . Gluing these paths together, we finally get that L (cid:48) is Lagrangian isotopic to τ l ( L (cid:48) ) . This contradicts Seidel’s result of Theorem 4.1 andconcludes this proof. Remark 4.13.
One can directly prove Corollary 4.9 without the use of barcodes. Indeed,an argument similar to the one for the case n = 2 holds for the proof of this corollary.The idea is to find a sequence ( ϕ n ) n ∈ N such that the cohomology HF ( ϕ ( L (cid:48) ) , L (cid:48) ) remainsconstant. We can then conclude in the same way.We could have also proved this result using the second point of Theorem 3.3 to constructa continuous path of barcodes between ˆ B ( L (cid:48) , L (cid:48) ) and ˆ B ( τ l ( L (cid:48) ) , L (cid:48) ) . This path togetherwith Corollary 2.16 telling that the degree of the semi-infinite bars cannot change along acontinuous path leads to a contradiction. References [1]
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