Bounds on the Lagrangian spectral metric in cotangent bundles
aa r X i v : . [ m a t h . S G ] A ug BOUNDS ON THE LAGRANGIAN SPECTRAL METRIC INCOTANGENT BUNDLES
PAUL BIRAN AND OCTAV CORNEA
Abstract.
Let N be a closed manifold and U ⊂ T ∗ ( N ) a bounded domain in the cotangentbundle of N , containing the zero-section. A conjecture due to Viterbo asserts that the spectralmetric for Lagrangian submanifolds that are exact-isotopic to the zero-section is bounded. Inthis paper we establish an upper bound on the spectral distance between two such Lagrangians L , L , which depends linearly on the boundary depth of the Floer complexes of ( L , F ) and( L , F ), where F is a fiber of the cotangent bundle. Introduction and main results
Let N be a closed manifold and T ∗ ( N ) its cotangent bundle, endowed with its standardsymplectic structure. A domain U ⊂ T ∗ ( N ) is called bounded if there is a Riemannian metric g on N such that U is contained inside the unit-ball cotangent bundle of T ∗ ( N ) with respect tothe metric associated to g on the fibers of T ∗ ( N ). More specifically, U ⊂ { v ∈ T ∗ ( N ) | | v | ≤ } ,where |·| is the norm on the fibers of T ∗ ( N ) corresponding to the metric g via the isomorphism T ∗ ( N ) ∼ = T ( N ) induced by g . Since N is compact, the boundedness of U is independent ofthe choice of g .For a domain W ⊂ T ∗ ( N ) we denote by L ex ( W ) the collection of closed exact Lagrangiansubmanifolds of W (exactness is considered here with respect to the canonical Liouville form)and by L ex ,N ( W ) ⊂ L ex ( W ) the collection of Lagrangians that are exact isotopic (within T ∗ ( N )) to the zero section N ⊂ T ∗ ( N ).There are several Ham-invariant metrics on L ex ,N ( T ∗ ( N )). For example, the Hofer metric onHam( T ∗ ( N )) descends to a non-degenerate metric d Hof on L ex ,N ( T ∗ ( N )). Another importantmetric, due to Viterbo, is the spectral metric. This was originally defined for L ex ,N ( T ∗ ( N )),but thanks to more recent developments can be extended to the entire of L ex ( T ∗ ( N )). (SeeRemark 2.2.3 - (3) and Remark 2.2.2 - (2) for more on this.) The spectral distance γ ( L , L )between two elements L , L ∈ L ex ( T ∗ ( N )) is define as: γ ( L , L ) = c ([ N ]; L , L ) − c ([pt]; L , L ) , (1)where c ([ N ]; L , L ), c ([pt]; L , L ) stand for the spectral invariants associated to ( L , L ), forthe fundamental class [ N ] ∈ H n ( N ) and for the class of a point [pt] ∈ H ( N ), correspondingly.See § § § Date : August 12, 2020.The second author was supported by an individual NSERC Discovery grant.
It is well known that γ ( L , L ) ≤ d Hof ( L , L ) for all L , L ∈ L ex ,N ( T ∗ ( N )). Howeverbeyond this inequality, little is known about the relation between these two metrics.Let U ⊂ T ∗ ( N ) be a bounded domain. It is also known that, at least for some N ’s, theHofer metric on L ex ,N ( U ) is unbounded. This has been proved for several cases like N = S by Khanevsky [Kha] and is conjectured to hold for all N ’s.In contrast to the Hofer metric, there is the following conjecture regarding the spectralmetric: Conjecture (Viterbo) . The spectral metric on L ex ,N ( U ) is bounded. This was conjectured by Viterbo in [Vit1] for the case N = T n , and is expected to holdfor all closed manifolds N . Recently Shelukhin [She2, She1] proved this conjecture for severalclasses of manifolds N (including T n ).Our main result, which applies to all closed manifolds N , is the following. Theorem A.
Let N be any closed manifold and U ⊂ T ∗ ( N ) a bounded domain. There existconstants A, B > that depend only on U such that for every L , L ∈ L ex ,N ( U ) we have: γ ( L , L ) ≤ B (cid:0) β ( CF ( L , F q )) + β ( CF ( L , F q )) (cid:1) + A. (2) Here F q = T ∗ q ( N ) is the fiber of the cotangent bundle at an arbitrary point q ∈ N , viewedas a Lagrangian submanifold of T ∗ ( N ) , and β ( CF ( L i , F q )) is the boundary depth of the Floercomplex of the pair ( L i , F q ) , i = 0 , , defined with coefficients in Z . See § . (1) Clearly, the above conjecture of Viterbo would follow from Theorem Aif we can show that the boundary depth β ( CF ( L, F q )) is uniformly bounded in L ∈L ex ,N ( U ).(2) The converse to the statement made at point (1) above turns out to be also true.Namely, if the conjecture of Viterbo holds true then the boundary depth β ( CF ( L, F q ))is uniformly bounded in L ∈ L ex ,N ( U ). This follows by a relatively simple argumentthat we summarize in § CF ( F q , L i ) depends on the point q ∈ N and so does its boundarydepth β ( CF ( L i , F q )). However, as we will see in §
6, Lemma 6.1.1, the difference | β ( CF ( L, F q ′ )) − β ( CF ( L, F q ′′ )) | is bounded, uniformly in q ′ , q ′′ ∈ N , L ∈ L ex ,N ( U ).Therefore the formulation of inequality (2) with constants A, B that do not depend on q , makes sense.(4) While the chain complex CF ( L, F q ) might be complicated and have arbitrary largerank, its homology is very simple: HF ( L, F q ) ∼ = Z for every q ∈ N , L ∈ L ex ,N ( U ).1.1. Strategy and main ideas in the proof.
The starting point of the proof is borrowedfrom [FSS1] - we embed a tubular neighborhood U of the zero section of T ∗ ( N ) into a real affinealgebraic manifold E which also serves as the total space of a Lefschetz fibration π : E −→ C endowed with a real structure. The embedding can be arranged such that the zero section issent to (one of the components of) the real part of E . OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 3
The 2’nd step appeals to our previous work [BC4] which establishes canonical presentationsof Lagrangians K in Lefschetz fibrations as iterated cone decompositions with standard factors.These iterated cone decompositions take place in the category of modules over the Fukayacategory of E and hold up to quasi-isomorphisms. The factors in the decomposition of K consist of the Yoneda modules of certain Lefschetz thimbles emanating from the critical pointsof π along N , as well as some factors that involve the Floer complexes of pairs of thimbles andpairs of the type (Thimble , K ). This makes it possible to express CF ( L, K ) for every exactLagrangians L , as an iterated cone involving chain complexes of the types CF ( L, Thimble), CF (Thimble , K ) and CF of pairs of thimbles. Note that the 2’nd and 3’rd types do notinvolve L .By specializing to the case K = N and taking the L to correspond to a Lagrangian in theneighborhood U of the zero-section, the previous cone decomposition of CF ( L, N ) reduces nowto terms of the type CF ( L, F q ), for different critical points q ∈ N of π , and some other fixedchain complexes that do not depend on L . The terms F q appear here because the previouslymentioned thimbles coincide within U with the fibers F q of the cotangent bundle. A “local toglobal” argument in Floer theory shows that replacing the thimble emanating from a criticalpoint q ∈ N of π by F q does not change the respective Floer complexes.The next step is to analyze the spectral metric using the above cone decompositions. Thisrequires a refinement of the cone decomposition in the realm of filtered Floer theory. It turnsout that the above cone decomposition continues to hold in the filtered sense up to a boundedaction shift. Therefore, in principle once can recover (up to a bounded shift) the filtered Floerhomology of ( L, N ) from the filtered Floer homology of the factors mentioned above and theknowledge of the chain maps between the factors which form the cones. In practice this notso effective, as these chain maps are in general hard to describe explicitly. Fortunately, thisobstacle can be overcome by algebraic means which are described next.The next step in the proof is purely algebraic. Here we obtain a coarse uniform upperbounds on the spectral range of filtered mapping cones C = Cone( C ′ f −→ C ′′ ) between twofiltered chain complexes C ′ and C ′′ . By “spectral range” of a filtered chain complex we meanthe difference between the highest and the lowest spectral invariants of that complex. It turnsout that one can derive such a bound on the spectral range of C which involves only thefollowing pieces of data: the spectral ranges of C ′ and C ′′ , the boundary depths of C ′ and C ′′ and the amount of filtration shift in the map f . A crucial point here is that our bound isuniform in f in the sense that it does not involve specific information on the map f , exceptof the extent by which it shifts the filtrations. We also establish an analogous upper boundfor the boundary depth of C . Having these two algebraic ingredients at hand, we can derivesimilar upper bounds for the spectral range and boundary depth of iterated cones.The final step puts the geometry and algebra together. We apply the algebraic estimateson the spectral range to the previously mentioned cone decomposition of CF ( L, N ). Whileit is possible to describe relatively precisely the chain maps between the terms in this de-composition, this is delicate. Fortunately, this is not needed here as we can easily bound the
PAUL BIRAN AND OCTAV CORNEA amount by which these maps shift filtrations. Consequently we obtain an upper bound on thespectral range of CF ( L, N ) as the sum of two terms: one of them is a constant A that comesfrom the spectral ranges of the factors in our cone decomposition (these are straightforwardto determine) and some uniformly bounded errors that come from our coarse estimates. Thisconstant depends on U but not on L since the only appearance of L in the cone decompo-sition of CF ( L, N ) is in terms of the type CF ( L, F q ). However, the spectral range of suchterms is 0 because HF ( L, F q ) is 1-dimensional. The second summand in our bound looks like Bβ ( CF ( L, F q )), where B is a constant and β ( CF ( L, F q )) is the boundary depth of CF ( L, F q ).Our main result now easily follows from these bounds.The above is only an outline of the main ideas in the proof. Along the way there are severaladditional ingredients required for the proof to work. These have to do with technicalities inFloer theory, Lefschetz fibrations and filtered homological algebra.1.2. Organization of the paper.
The rest of the paper is organized as follows. Section 2reviews necessary preliminaries on filtered Floer theory in the framework of exact Lagrangiansubmanifolds in Liouville manifolds. We also prove in Section 2.4 a general “local vs. global”result, comparing the Lagrangian Floer persistent homologies in a Liouville subdomain withthe same type of homology in the entire Liouville manifold.Section 3 is devoted to Lefschetz fibrations and their relevance to our problem. We go overreal Lefschetz fibrations in general and then review a construction from [FSS1] which gives anembedding of a neighborhood of the zero-section in T ∗ ( N ) into a real Lefschetz fibration E .We then go over a construction coming from [BC4] which alters the Lefschetz fibration E intoan an extended Lefschetz fibration E ′ containing a collection of matching spheres that will beuseful for our purposes. Part of this section is devoted to showing that the construction of E ′ can be made while preserving a geometric setting amenable to Floer theory like exactness etc.Section 4 is dedicated to comparison between the filtered Floer theory inside E and thesame theory viewed in E ′ . In particular we show there that the matching spheres from E ′ , constructed in Section 3, correspond in E to some Lefschetz thimbles emanating from N . These in turn coincide near N with cotangent fibers of T ∗ ( N ). We show that thesecorrespondences hold also in a Floer-theoretic sense.Section 5 is central for the proof of the main theorem. There we discuss iterated conedecompositions in the Fukaya categories of E and E ′ . In particular we show how to representLagrangian submanifolds in E ′ as iterated cones with standard terms coming from the match-ing spheres from Section 3. Moreover, in Section 5.2 and 5.3 we extend these decompositionsto the realm of Fukaya categories endowed with action filtrations. In particular we also derivea filtered version of the Seidel exact triangle associated to a Dehn-twist.Section 6 combines the geometric contents of the previous sections together with some fil-tered homological algebra (developed in Section 7) to conclude the proof of the main theorem.We also sketch the argument for the converse.The algebraic ingredients necessary for the paper are concentrated in Section 7. Thisis a purely algebraic section in which we study spectral invariants and boundary depth of OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 5 filtered chain complexes. Special attention is given to filtered mapping cones and we establishestimates on the spectral range and boundary depth in that case.The paper can be read linearly, with the exception of Section 7 which is the last one,but is being referred to at many instances along the paper. At the same time, Section 7 isindependent of the rest the paper and can be read separately.1.3.
Acknowledgments.
We thank Egor Shelukhin for suggesting this project to us andalso pointed out the relevance of [FSS1] in this context. This work was initiated and partiallycarried out during our two weeks stay at the Mathematical Research Institute of Oberwolfachin May 2017, in the framework of the Research in Pairs program. We would like to thank theOberwolfach Institute for the wonderful hospitality and working conditions during our visit.We would like to thank Sobhan Seyfaddini for useful discussions related to § Contents
1. Introduction and main results 12. Lagrangian Floer theory and spectral invariants 53. Cotangent bundles and real Lefschetz fibration 174. Floer theory in E versus E ′ Lagrangian Floer theory and spectral invariants
Here we briefly recall the definitions of spectral invariants, boundary depth and the spectralmetric on the space of Lagrangian submanifolds. We refer the reader to [PRSZ, PSS, Oh1,DKM, KMN, Lec, LZ, UZ, Ush1, Ush2, Vit2] for more details on the general theory of theseconcepts.2.1.
Filtered chain complexes and their invariants.
Fix a unital ring R and let C bea chain complex of R -modules. By a filtration on C we mean an increasing filtration ofsubcomplexes of R -modules, indexed by the real numbers. More specifically, for every α ∈ R we are given a subcomplex C ≤ α ⊂ C of R -modules and for every α ≤ β we have C ≤ α ⊂ C ≤ β . For simplicity we will assume from now on that the filtration on C is exhaustive, i.e. ∪ α ∈ R C ≤ α = C .The inclusions C ≤ α ⊂ C ≤ β , α ≤ β , and C ≤ α ⊂ C induce maps in homology which wedenote by: i β,α : H ∗ ( C ≤ α ) −→ H ∗ ( C ≤ β ) , i α : H ∗ ( C ≤ α ) −→ H ∗ ( C ) . PAUL BIRAN AND OCTAV CORNEA
Given a homology class a ∈ H ∗ ( C ) we define its spectral invariant σ ( a ) ∈ R ∪ {−∞} to be σ ( a ) := inf { α ∈ R | a ∈ image i α } . (3)Note that σ (0) = −∞ .Another important measurement for our purposes is the boundary depth β ( C ) of a filteredchain complex C , which is defined as follows: β ( C ) := inf { r ≥ | ∀ α, ∀ c ∈ C ≤ α which is a boundary in C, ∃ b ∈ C ≤ α + r s.t. c = d ( b ) } . We will elaborate more on spectral invariants, boundary depth and other measurements offiltered chain complexes in § Filtered Lagrangian Floer theory.
In what follows all symplectic manifolds and theirLagrangian submanifolds will be implicitly assumed to be connected, unless otherwise men-tioned. And all Hamiltonian functions [0 , × W −→ R will be implicitly assumed to becompactly supported.2.2.1. Liouville and Stein manifolds.
In the following we will be mainly concerned with sym-plectic manifolds of two types: Liouville domains and manifolds that are Stein at infinity. Werefer the reader to [CE] for the foundations of the theory of such manifolds and much more.Below we briefly recall the basic notions needed for our purposes.A compact Liouville domain (
W, ω = dλ ) consists of a compact manifold W with boundary ∂W and an exact symplectic structure ω , with a given primitive 1-form λ (called the Liouvilleform) such that the following holds: the Liouville vector field X λ , defined by i X λ ω = λ , isoutward transverse to ∂W . Under this assumption the restriction λ ∂W := λ | ∂W is a contactform and we denote by ξ λ := ker λ ∂W the contact structure defined by λ ∂W on ∂W . We write ψ t : W −→ W , t ≤
0, for the flow of X λ (which exists for all t ≤ ψ ∗ t λ = e t λ and ψ ∗ t ω = e t ω .For a Liouville domain ( W, ω = dλ ), consider the embedding Ψ : ( −∞ , × ∂W −→ W ,( s, x ) ψ s ( x ). We have: Ψ ∗ λ = e s λ ∂W , Ψ ∗ ω = d ( e s λ ∂W ). Define an almost complexstructure J λ on ( −∞ , × ∂W as follows. Fix an almost complex structure J ξ λ on ξ λ whichis compatible with ω | ξ λ . Denote by R λ ∂W ∈ T ( ∂W ) the Reeb vector field corresponding to λ ∂W . Define J λ | ξ λ := J ξ λ and J λ ( ∂∂s ) := R . Note J λ is compatible with Ψ ∗ ω and moreover thefunction φ : ( −∞ , × ∂W −→ R , φ ( s, x ) := e s , is a potential for Ψ ∗ ω , i.e. Ψ ∗ ω = − dd J λ φ (infact we have d J λ φ = − e s λ ). In particular, φ is J -plurisubharmonic (or J -convex). Using themap Ψ we can endow image(Ψ) with the almost complex structure Ψ ∗ ( J λ ) which, by abuse ofnotation, will also be denoted by J λ . (Note that in general J λ does not extend from image(Ψ)to the entire of W .)Sometimes it will be useful to work with the completion ( c W , b ω = d b λ ) of a compact Liouvilledomain ( W, ω = dλ ). More precisely, set c W := W ∪ Ψ (cid:0) [ − ǫ, ∞ ) × ∂W (cid:1) , OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 7 where the gluing identifies [ − ǫ, × ∂W with a collar neighborhood of ∂W in W via the map Ψ.The Liouville form b λ is defined by extending λ from W to the cylindrical part [0 , ∞ ) × ∂W by b λ = e s λ ∂W , where s ∈ [0 , ∞ ). We denote the corresponding symplectic structure by b ω := d b λ .All the previous structures, like X λ , ψ t , φ and J λ , extend in an obvious way to thecompletion. More specifically, the Liouville vector field X b λ (defined by i X b λ b ω = b λ ) ex-tends X λ by ∂∂s along the cylindrical part. We denote the flow of X b λ by b ψ t . Note thatthis flow is complete (i.e. exists for all times t , both positive and negative). Next, we ex-tend the almost complex structure J λ from image Ψ to an almost complex structure b J λ on(image Ψ) ∪ Ψ (cid:0) [ − ǫ, ∞ ) × ∂W (cid:1) ⊂ c W by the same recipe defining J λ , namely: b J λ := J λ onimage Ψ, and b J λ ( s,x ) | ξ λ := J ξ λ , J λ ( s,x ) ( ∂∂s ) := R λ ∂W , for every ( s, x ) ∈ [0 , ∞ ) × ∂W (where here weview ξ λ ⊂ T ( s,x ) ( s × ∂W )) . Finally note that the plurisubharmonic function φ : image Ψ −→ R extends to the cylindrical part [0 , ∞ ) × ∂W by b φ ( s, x ) = e s and b λ = − d b J λ b φ , b ω = − dd b J λ b φ .Another type of symplectic manifolds that we will encounter are Stein manifolds, whichare very much related to the above. By a Stein manifold we mean a triple ( V, J V , ϕ ), where( V, J V ) is an open complex manifold (with integrable J V ) and ϕ : V −→ R is an exhaus-tion plurisubharmonic function. Exhaustion means that ϕ is proper and bounded from be-low, and plurisubharmonic means that the 2-form ω ϕ := − dd J V ϕ is compatible with J V (i.e. ω ϕ ( u, J V u ) > ∀ u and ω ϕ ( J V u, J V v ) = ω ϕ ( u, v ), ∀ u, v ). Denote λ ϕ := − d J V φ and for R ∈ R , V ϕ ≤ R := { x ∈ V | ϕ ( x ) ≤ R } . (Similarly we have V ϕ We will call ( V, J V , R , ϕ R , b ω R ) a completion of ( V, J V , R , ϕ, ω ).Finally, we will also need the notion of Liouville manifolds that are Stein at infinity. Theseare symplectic manifolds that are Stein at infinity, ( V, J V , ϕ, R , ω = dλ ), but now we assumein addition that the symplectic structure ω is globally exact with a prescribed primitive λ .Moreover, λ is assumed to satisfy λ = − d J V ϕ along V ϕ ≥ R .Note that, as for the case of Stein manifolds, if R ≥ R is a regular value of ϕ then( V ϕ ≤ R , ω = dλ ) is a compact Liouville domain.Note also that for the completion of Liouville manifolds that are Stein at infinity, theLiouville vector field X b λ R is defined all over V and moreover, its flow exists for all t ∈ R .2.2.2. Floer theory. We will work here with Floer homology and singular homology, both takenwith coefficients in Z . We will generally omit the Z from the notation (e.g. writing H ∗ ( L ) for H ∗ ( L ; Z )). Our setting is almost identical to [Sei2, Chapter III, Section 8], with two slightdifferences. Firstly, we work with homological conventions rather than with cohomologicalones. Secondly, we work in an ungraded setting.Let ( V, ω = dλ ) be an exact symplectic manifold with a given primitive λ for the symplecticstructure. We assume further that this symplectic manifold is of one of the following threetypes:(1) ( V, dλ ) is a compact Liouville domain.(2) ( V, dλ ) is the completion ( c V ′ , b ω ′ = d b λ ′ ) of a compact Liouville domain ( V ′ , ω ′ = dλ ′ ).(3) ( V, dλ ) can be endowed with a structure ( V, J V , R , ϕ, ω = dλ ) of a Liouville manifoldwhich is Stein at infinity. In that case we also fix the additional structures J V , ϕ, R .We denote by Int V the interior of V . (Note that only in case (1), we have Int V $ V .) Denoteby J V the space of ω -compatible almost complex structures on V which coincide with, J λ nearthe boundary of V in case (1), or with b J λ at infinity in case (2), or coincide with J V on V ϕ ≥ R for some R ≥ R in case (3).Let L , L ⊂ Int V be two closed exact Lagrangian submanifolds. (Exactness of a La-grangian L will be generally considered with respect to the given Liouville form λ . In casewe want to emphasize the form with respect to which L is exact we will call L a λ -exactLagrangian.) We fix primitive functions h L i : L i −→ R to λ | L i , i = 0 , H : [0 , × V −→ R be a Hamiltonian function. Write H t ( x ) = H ( t, x ). Henceforth wewill implicitly assume that there exists a compact subset K ⊂ Int V such that for all t ∈ [0 , H t is constant outside of K . The Hamiltonian vector field X Ht = X H t of H isgiven by ω ( X Ht , · ) = − dH t ( · ).Denote by P L ,L = (cid:8) γ : [0 , −→ V | γ (0) ∈ L , γ (1) ∈ L (cid:9) the space of paths with endpoints on L , L . The action functional A H : P L ,L −→ R is defined as follows: A H ( γ ) := Z H ( t, γ ( t )) dt − Z λ ( ˙ γ ( t )) dt + h L ( γ (1)) − h L ( γ (0)) . (4)Denote by O ( H ) = O L ,L ( H ) ⊂ P L ,L the set of Hamiltonian chords with endpoints on( L , L ), namely the set of orbits γ : [0 , −→ W of X Ht with γ (0) ∈ L , γ (1) ∈ L . OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 9 Let D = ( H, J ) be a regular Floer datum, consisting of a Hamiltonian function H : [0 , × W −→ R and a time-dependent almost complex structure J = { J t } t ∈ [0 , , with J t ∈ J V forevery t . Sometimes we will write O L ,L ( D ) (or O ( D )) for O L ,L ( H ).The negative gradient flow of A H (with respect to a metric on P L ,L induced by J ) givesrise to the Floer equation associated to D : u : R × [0 , −→ M, u ( R × ⊂ L , u ( R × ⊂ L ,∂ s u + J t ( u ) ∂ t u = J t X Ht ( u ) ,E ( u ) := Z ∞−∞ Z | ∂ s u | dtds < ∞ . (5)where ( s, t ) ∈ R × [0 , E ( u ) in the last line of (5) is the energy of a solution u and we consider only finite energy solutions. (Note also that the norm | ∂ s u | in the definitionof E ( u ) is calculated with respect to the Riemannian metric associated to ω and J t .) Solutions u of (5) are also called Floer trajectories.For γ − , γ + ∈ O ( H ) we have the space of parametrized Floer trajectories u connecting γ − to γ + : M ( γ − , γ + ; D ) = n u | u solves (5) and lim s →±∞ u ( s, t ) = γ ± ( t ) o . (6)Note that R acts on this space by translations along the s -coordinate. This action is generallyfree, with the only exception being γ − = γ + and the stationary solution u ( s, t ) = γ − ( t ) at γ − .Whenever, γ − = γ + we denote by M ∗ ( γ − , γ + ; D ) := M ( γ − , γ + ; D ) (cid:14) R (7)the quotient space (i.e. the space of non-parametrized solutions).For a generic choice of Floer datum D the space M ∗ ( γ − , γ + ; D ) is a smooth manifold(possibly with several components having different dimensions). Moreover, its 0-dimensionalcomponent M ∗ ( γ − , γ + ; D ) is compact hence a finite set.The Floer complex CF ( L , L ; D ) is the vector space, over Z , with a basis formed by theset O ( H ): CF ( L , L ; D ) = M γ ∈O ( H ) Z γ. (8)Its differential d : CF ( L , L ; D ) −→ CF ( L , L ; D ) is defined by counting solutions of theFloer equation: d ( γ − ) := X γ + ∈O ( H ) Z M ∗ ( γ − , γ + ; D ) γ + , ∀ γ − ∈ O ( H ) , (9)and extending linearly over Z . The homology of CF ( L , L ; D ) is denoted by HF ( L , L ; D )- the Floer homology of ( L , L ).The Floer homology is independent of the choice of the Floer datum in the sense that for ev-ery two regular choices of Floer data D , D ′ there is a quasi-isomorphism, canonical up to chainhomotopy, ψ D , D ′ : CF ( L , L ; D ) −→ CF ( L , L ; D ′ ), called a continuation map. The (now canonical) isomorphisms induced in homology H ( ψ D , D ′ ) : HF ( L , L ; D ) −→ HF ( L , L ; D ′ )form a directed system and we can regard the collection of vector spaces HF ( L , L ; D ),parametrized by regular Floer data D , as one vector space and denote it by HF ( L , L ).2.2.3. PSS and naturality. Given a Hamiltonian function F : [0 , × V −→ R , denote by F ( t, x ) := − F ( t, φ Ft ( x )) and b F ( t, x ) = − F (1 − t, x ). The flows of these functions are φ Ft =( φ Ft ) − and φ b Ft = φ F − t ◦ ( φ F ) − respectively. Note that both these flows have the sametime-1 map: φ F = φ b F = ( φ F ) − . For two Hamiltonian functions F, G : [0 , × V −→ R ,denote by G F : [0 , × V −→ R the function ( G F )( t, x ) = G ( t, x ) + F ( t, ( φ Gt ) − ( x )). ItsHamiltonian flow is φ G Ft = φ Gt ◦ φ Ft . Given a Floer datum D = ( F, J ) and a Hamiltonianflow φ Gt generated by G we denote by φ G ∗ D := ( G F, φ G ∗ J ) the push-forward Floer datum,where ( φ G ∗ J ) t := Dφ Gt ◦ J t ◦ ( Dφ Gt ) − .Let L , L ⊂ Int V be two exact Lagrangians and assume that the Floer datum D = ( F, J )is regular. Let G be another Hamiltonian function. There is a naturality map N G : CF ( L , L ; D ) −→ CF ( L , φ G ( L ); φ G ∗ D ) , N G ( γ )( t ) := φ Gt γ ( t ) , ∀ γ ∈ O L ,L ( F ) . (10)The map N G is a chain isomorphism.Consider now a Lagrangian L ′ which is exact isotopic to L . Fix a Hamiltonian function G such that φ G ( L ) = L ′ . The map induced in homology by N G is compatible with thehomological maps induced by continuation. Therefore N G induces as well defined isomorphism HF ( L , L ) −→ HF ( L , L ′ ). Moreover, this isomorphism is independent of the choice of G (among Hamiltonian functions G with φ G ( L ) = L ′ ). We thus obtain a system of canonicalisomorphisms N L L ′ ,L : HF ( L , L ) −→ HF ( L , L ′ ), defined for every pair of exact isotopicLagrangians L , L ′ . Moreover, N L L ,L = id , N L L ′′ ,L ′ ◦ N L L ′ ,L = N L L ′′ ,L . Remarks . (1) For the latter statement to hold it is important that the Lagrangiansare exact, or more generally weakly exact. Indeed, in the presence of holomorphicdisks (e.g. for monotone Lagrangians) the isomorphisms N L L ′ ,L might depend on thehomotopy class of the path between L and L ′ inside the space of exact Lagrangians.(2) Denote by ∗ : HF ( L , L ) ⊗ HF ( L , L ′ ) −→ HF ( L , L ′ ) the product induced by thechain level µ -operation. Then there exists a class c L ,L ′ ∈ HF ( L , L ′ ) such that N L L ′ ,L ( a ) = a ∗ c L ,L ′ for every a ∈ HF ( L , L ). In fact, c L ,L ′ = N L L ′ ,L ( e L ), where e L ∈ HF ( L , L ) is the unity.Similarly to the maps N L L ′ ,L we also have canonical isomorphisms N L ′ ,L L : HF ( L , L ) −→ HF ( L ′ , L ), defined in an analogous way.We now turn to the PSS isomorphism. Let L ⊂ Int V be an exact Lagrangian. Let m = ( f, ρ ) be a Morse datum, consisting of a Morse function f : L −→ R and a Riemannian OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 11 metric ρ on L . Denote by C ( L ; m ) the Morse complex associated to m . Let D = ( H, J ) be aregular Floer datum for the pair ( L, L ). The PSS map is a quasi-isomorphism P SS m , D : C ( L ; m ) −→ CF ( L, L ; D ) (11)canonical up to chain homotopy. Moreover, the maps P SS m , D , defined for different m , D , arecompatible with the corresponding continuation maps up to chain homotopy. Consequently,the isomorphism induced by P SS in homology P SS : H ∗ ( L ) −→ HF ( L, L ) , which we also denote by P SS , is independent of the data m , D . Moreover, this map ismultiplicative (with respect to the intersection product on H ∗ ( L ) and the triangle productinduced by µ on HF ( L, L )) and it sends the fundamental class [ L ] to the unit e L ∈ HF ( L, L ).We refer the reader to [KM, Alb] for the definition and properties of this map. Remarks . (1) Let L , L ⊂ Int V be two exact Lagrangians that are exact isotopic.Choose any exact isotopy φ t : L −→ Int V , t ∈ [0 , φ = inclusion of L ⊂ V and φ ( L ) = L . By a result of Hu-Lalonde-Leclercq [HLL] the map φ ∗ : H ∗ ( L ; Z ) −→ H ∗ ( L ; Z ), induced in homology by φ , is independent of the choice ofthe isotopy { φ t } . Therefore there is a canonical map φ ∗ : H ∗ ( L ; Z ) −→ H ∗ ( L ; Z )between any two exact isotopic exact Lagrangians in Int V . The map φ ∗ is compatiblewith Floer theory in the following sense. First note that if { φ t } is an exact isotopyas above its time-1 map induces a map in Floer homology φ HF : HF ( L , L ) −→ HF ( L , L ). Moreover, this map is independent of the choice of the isotopy (in fact, φ HF = N L ,L L ◦ N L L ,L ). Write φ HF := φ HF . Standard arguments then show that φ ∗ equals the composition H ∗ ( L ; Z ) P SS −−→ HF ( L , L ) φ HF −−→ HF ( L , L ) P SS − −−−−→ H ∗ ( L ; Z ) . (2) In general the space of exact Lagrangians in V might be disconnected (and even con-tain Lagrangians of different topological types). However, in certain situation this isnot expected to be so. For example, a version of the nearby Lagrangian conjecture as-serts that if V = T ∗ ( N ) is the cotangent bundle of a closed manifold N then all exactLagrangians are exact isotopic to the zero-section. While this is still open in gen-eral, a result of Fukaya-Seidel-Smith [FSS1, FSS2] and independently of Nadler [Nad],says that under mild topological assumptions on N the following holds. Every exactLagrangian L ⊂ T ∗ ( N ) is canonically isomorphic, when viewed as an objects in the(compact) derived Fukaya category of T ∗ ( N ), to the zero-section. Moreover, this iso-morphism induces the same map HF ( L, L ) −→ HF ( N, N ) as the one induced by theprojection pr : T ∗ ( N ) −→ N on homology H ∗ ( L ) −→ H ∗ ( N ), under the canonicalidentifications HF ( L, L ) ∼ = H ∗ ( L ) and HF ( N, N ) ∼ = H ∗ ( N ). Action filtrations and Floer persistent homology. We begin by recalling the fundamen-tals of filtered Lagrangian Floer theory in the exact setting. Much of the general theory hasbeen developed in [Oh1, Oh2, Lec, LZ, DKM, KMN], though in somewhat different frame-works like monotone (and weakly exact) Lagrangians. The essence however remains the sameand a considerable part of these papers applies with minor changes to the exact case too.In order to define the action functional and its induced filtrations in Floer theory we needto endow each exact Lagrangian L with a primitive h L : L −→ R of the exact form λ | L . Wewill refer to h L as a marking of L and to the pair ( L, h L ) as a marked Lagrangian . However,for simplicity of notation we will often continue to denote marked Lagrangians by a singleletter, e.g. L , with the understanding that the primitive h L has been fixed.Let L , L ⊂ Int V be two marked Lagrangians. Let D = ( H, J ) be a regular Floer datumfor ( L , L ). For α ∈ R denote CF ≤ α ( L , L ; D ) := M γ ∈O ( H ) , A H ( γ ) ≤ α Z γ. (12)For convenience we extend A H to all elements of CF ( L , L ; D ) by defining it on λ = P ki =1 a i γ i , a i ∈ Z , to be: A H ( λ ) = max {A H ( γ i ) | a i = 0 } = inf (cid:8) α | λ ∈ CF ≤ α ( L , L ; D ) (cid:9) . Here we use the convention that max ∅ = −∞ , so that A H (0) = −∞ .The subspaces CF ≤ α ⊂ CF are in fact subcomplexes. This is so because for every Floer tra-jectory u ∈ M ( γ − , γ + ; D ) we have the following action-energy relation: A H ( γ + ) = A H ( γ − ) − E ( u ) ≤ A H ( γ − ). Therefore A H ( dγ ) ≤ A H ( γ ), hence d ( CF ≤ α ( L , L ; D )) ⊂ CF ≤ α ( L , L ; D ).We write HF ≤ α ( L , L ; D ) := H ∗ ( CF ≤ α ( L , L ; D )) and for α ≤ β ≤ ∞ we denote by i β,α : HF ≤ α ( L , L ; D ) −→ HF ≤ β ( L , L ; D ) the map induced by the inclusion CF ≤ α ( L , L ; D ) ⊂ CF ≤ β ( L , L ; D ). For β = ∞ we abbreviate i α := i ∞ ,α .The homologies HF ≤ α ( L , L ; D ), α ∈ R , and the maps i β,α , α ≤ β , fit together into a per-sistence module which we denote by HF ≤• ( L , L ; D ) and call the Floer persistent homology.Next, we briefly discuss to what extent the Floer persistent homology depends on the Floerdata. The continuation maps ψ D ′ , D do not preserve action-filtrations in general, hence thereis no meaning to write H ( CF ≤ α ( L , L )) without specifying the Floer datum. Nevertheless, if D ′ = ( H, J ′ ) and D ′′ = ( H, J ′′ ) are two regular Floer data with the same Hamiltonian function H , then one can choose the continuation map ψ D ′′ , D ′ : CF ( L , L ; D ′ ) −→ CF ( L , L ; D ′′ )to be action preserving. Moreover, for such Floer data, the chain homotopies between ψ D ′ , D ′′ ◦ ψ D ′′ , D ′ and id can be also chosen to preserve action. It follows that ψ D ′′ , D ′ inducesan isomorphism between the persistence modules HF ≤• ( L , L ; D ′ ) and HF ≤• ( L , L ; D ′′ ).Moreover, standard arguments imply that this isomorphism is canonical (in the sense thatthere is a preferred such isomorphism). Thus the Floer persistent homology of ( L , L ) de-pends only on the Hamiltonian function in the Floer data, hence will sometimes be denotedby HF ≤• ( L , L ; H ). In case L ⋔ L we can take the Hamiltonian function to be 0, and theFloer persistent homology using this choice will be abbreviated as HF ≤• ( L , L ). OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 13 The persistence modules HF ≤• ( L , L ; D ) give rise to a variety of numerical invariants. Themost important for us will be spectral invariants and boundary depth.Given a ∈ HF ( L , L ; D ) we denote by σ ( a ; L , L ; D ) the spectral invariant of a , defined bythe recipe in (3) of § CF ( L , L ; D ). By the preceding discussion thespectral invariants σ ( a ; L , L ; ( H, J )) as well as boundary depth β ( CF ( L , L ; ( H, J )) do notdepend on J , hence we will sometimes denote them by σ ( a ; L , L ; H ) and β ( CF ( L , L ; H ))respectively.Next we discuss the version of spectral invariants involved in the definition of the spectralmetric, namely c ( a ; L , L ), where L ⊂ Int V is a marked exact Lagrangian, a ∈ H ∗ ( L ), and L ⊂ Int V is another marked Lagrangian which is exact isotopic to L . (Here the markingon L is arbitrary and is not assumed to be related in any way to the given marking of L viaany isotopy going from L to L .) Consider the following composition of isomorphisms H ∗ ( L ) P SS −−−→ HF ( L , L ) N L L ,L −−−−−→ HF ( L , L ) . Assume first that L intersects L transversely. Choose an almost complex structure J suchthat the Floer datum (0 , J ) is regular. Consider the chain complex CF ( L , L ; (0 , J )) endowedwith the action filtration, as defined at (12). Consider also the class N L L ,L ◦ P SS ( a ) viewedas an element of H ∗ ( CF ( L , L ; (0 , J ))) = HF ( L , L ). We then define c ( a ; L , L ) := σ (cid:0) N L L ,L ◦ P SS ( a ); L , L ; 0 (cid:1) . (13)In case L and L do not intersect transversely, we define c ( a ; L , L ) = lim k H k→ σ (cid:0) N L L ,L ◦ P SS ( a ); L , L ; H (cid:1) , where k H k := R (cid:0) max x ∈ V H ( t, x ) − min x ∈ V H ( t, x ) (cid:1) dt , and k H k → φ H ( L ) ⋔ L . The fact that the limit exists and is finite follows fromLipschitz continuity of the spectral invariants σ with respect to the Hofer norm (see e.g. [Lec]).Finally, given an exact Lagrangian L ⊂ Int V we define the spectral distance γ on the space L ex ,L (Int V ) of Lagrangians in Int V which are exact isotopic to L by γ ( L , L ) = c ([ L ]; L , L ) − c ([pt]; L , L ) , ∀ L , L ∈ L ex ,L (Int V ) . (14) Remarks . (1) The primitives h L i : L i −→ R for the exact 1-forms λ | L i , i = 1 , H a (time dependent) constant C ( t ). Different such choiceshave no effect on Floer complex CF ( L , L ; D ) and its homology, but they do add aconstant to the action functional, hence shift the filtration on CF ( L , L ; D ) by anoverall constant. Consequently, the spectral numbers σ ( a ; L , L ; H ) and c ( a ; L , L )get shifted by a constant which is independent of a . Let a ′ , a ′′ ∈ HF ( L , L ), b ′ , b ′′ ∈ H ∗ ( L ). It follows that each of the differences σ ( a ′′ ; L , L ; H ) − σ ( a ′ ; L , L ; H ) , c ( b ′′ ; L , L ) − c ( b ′ ; L , L ) is independent of the preceding choices. In particular, the spectral distance γ ( L , L )is independent of any choice of marking on L and L .(2) The action functional and the spectral invariants depend on the choice λ of the Liouvilleform. However, altering λ by an exact 1-form has no effect on these quantities. Morespecifically, let f : V −→ R be a smooth function and consider λ ′ = λ + df . The latteris also a primitive of the symplectic form ω .Clearly, a Lagrangian in V is λ -exact if and only if it is λ ′ -exact. Let L , L ⊂ V be two λ -exact Lagrangians and fix primitives h L i : L i −→ R for λ | L i , i = 0 , 1. Then h ′ L i := h L i + f | L i is a primitive for λ ′ | L i . Denote by A ′ H : P L ,L −→ R the actionfunctional defined using λ ′ and the primitives h ′ L i , and by A H the one defined using λ and the h L i ’s. A simple calculation shows that A ′ H = A H . It follows that thespectral invariants σ and c remain the same when replacing λ by λ ′ (provided we usethe primitives h ′ L i as above). Consequently, the spectral metric γ remains unchangedtoo (the latter does not even depend on the choices of the primitive functions h L i or h ′ L i ).(3) In case V = T ∗ ( N ) is the cotangent bundle of a closed manifold N , one can extend thedefinition of the spectral invariants c ( a ; L , L ) as well as the spectral metric γ ( L , L )to arbitrary pairs of exact Lagrangians (i.e. including also pairs that are, hypothetically,not isotopic one to the other). This follows from point (2) of Remark 2.2.2.Another source of numerical invariants comes from the barcode B ( HF ≤• ( L , L ; D )) of thepersistence module HF ≤• ( L , L ; D ), see [PRSZ] for the definition. Of main interest for ourconsiderations is the boundary depth β ( L , L ; D ), which by definition is the length of thelongest finite bar in the barcode B ( HF ≤• ( L , L ; D )). We will discuss this invariant in moredetail in § Weakly filtered Fukaya categories. Occasionally it will be convenient to view allexact Lagrangian submanifold as objects in a Fukaya category, taking into account actionfiltrations.Denote by F uk ( V ) the Fukaya category whose objects are the closed marked Lagrangiansubmanifolds L ⊂ V (see the beginning of § F uk ( V ) is an A ∞ -categorywhose realization requires additional auxiliary structures, namely Floer data for all pairs ofobjects as well as coherent perturbation data for every tuple of objects. We will suppressthese choices from the notation, whenever these choices are clear (or irrelevant). We referto [Sei2] for the foundations of Fukaya categories. In contrast to this (and most) referenceson the subject, our Fukaya categories (and all Floer complexes in general) will be ungraded.The Fukaya category F uk ( V ) has the structure of a so called weakly filtered A ∞ -category.This means that hom F uk ( V ) ( L , L ) = CF ( L , L ) between every pair of objects ( L , L ) is afiltered chain complex, and moreover each of the higher order operations µ d , d ≥ 2, preservesthese filtrations up to a uniformly bounded error (i.e. the error for µ d depends only on d , and OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 15 not on the objects involved in it). We refer the reader to [BCS, § 2] for more details on thistheory.2.4. Local and global Floer theory. Let ( V, J V , ϕ, R , ω = dλ ) be a Liouville manifoldwhich is Stein at infinity. Let W ⊂ V be a compact Liouville subdomain, endowed withthe structures λ and ω coming from V . Let L , L ⊂ Int W be two closed marked λ -exactLagrangian submanifolds. Consider Hamiltonian functions H : [0 , × W −→ R , compactlysupported in [0 , × Int W , such that φ H ( L ) ⋔ L . We will view these also as Hamiltonianfunctions on V by extending them to be 0 outside W .The following proposition compares the local and global Floer invariants of ( L , L ). It saysthat the Floer homologies as well as filtered numerical invariants of ( L , L ; H ), when viewedeither in W (“local”) or in V (“global”), coincide. Proposition 2.4.1. There exist isomorphisms of persistence modules j ≤• : HF ≤• (cid:0) L , L ; H ; ( W , ω = dλ ) (cid:1) −→ HF ≤• (cid:0) L , L ; H ; ( V, ω = dλ ) (cid:1) defined for every pair of closed marked Lagrangians ( L , L ) and H as above. Moreover, thecorresponding isomorphisms j := j ≤∞ : HF (cid:0) L , L ; ( W , ω = dλ ) (cid:1) −→ HF (cid:0) L , L ; ( V, ω = dλ ) (cid:1) on the total homologies are independent of H and have the following further properties:(1) They are compatible with the triangle products.(2) They are compatible with the naturality maps N L L ′ ,L from § L ′ and L are exact-isotopic) as well as with PSS (in case L = L ).(3) They preserve spectral invariants, namely σ ( j ( a ); L , L ; H ; ( V, λ )) = σ ( a ; L , L ; H ; ( W , λ )) , ∀ a ∈ HF ( L , L ; ( W , ω )) . Remark . Proposition 2.4.1 does not hold without the assumption that L , L are ex-act. For example, take L = L to be a circle in V = R endowed with the standardsymplectic structure ω std , and let W be a small tubular neighborhood of this circle. Then HF ( L , L ; W , ω std ) ∼ = H ∗ ( S ) but HF ( L , L ; V, ω std ) = 0. Proof of Proposition 2.4.1. The main idea in the proof is based on a rescaling (or shrinking)argument from [FSS1, Section 5] which we adapt here to our setting.We will assume without loss of generality that L ⋔ L and that H ≡ 0. This simplifiesnotation and the proof of the general case is very similar to the one we will present below.Fix R > R such that R > max W ϕ (so that W ⊂ V ϕ 0, consider thespace J ( T ) of almost complex structures J on V that have the following properties:(1) J is compatible with b ω R .(2) J = b J λ on N ([ − δ, T ]).(3) J = J V at infinity.Denote the space of time-dependent almost complex structure J = { J t } t ∈ [0 , with J t ∈ J ( T ) for every t , by J [0 , T ) . Lemma 2.4.3. There exists T > such that the following holds for every T ≥ T : for everyregular Floer datum D = (0 , J ) with J ∈ J [0 , T ) and every Floer strip u : R × [0 , −→ V corresponding to ( L , L ; D ) we have image u ⊂ W .Proof of Lemma 2.4.3. Consider the Lagrangian submanifolds L − T := ψ − T ( L ), L − T := ψ − T ( L )of W . Note that L − T , L − T are both λ -exact and L − T ⋔ L − T . For x ∈ L ∩ L write x − T := ψ − T ( x ). Denote by A ( L ,L ) and by A ( L − T ,L − T ) the action functionals of ( L , L ) andof ( L − T , L − T ) respectively, both defined with the Hamiltonian perturbation term H ≡ 0. Asimple calculation shows that A ( L − T ,L − T ) ( x − T ) = e − T A ( L ,L ) ( x ) . (15)Let u : R × [0 , −→ R be a Floer strip associated to ( L , L ; (0 , J )) with J ∈ J [0 , T ) . Put v − T := b ψ − T ◦ u . Then v − T is a Floer strip corresponding to ( L − T , L − T ; (0 , ( b ψ − T ) ∗ J ). Notethat ( b ψ − T ) ∗ J is compatible with b ω R . Moreover, by the definition of J ( T ) we have ( b ψ T ) ∗ J = J on N ([ − δ − T, J ≡ J λ on N ([ δ, E . We have: E ( v − T ) = e − T E ( u ) ≤ e − T (cid:0) max x ∈ L ∩ L A ( L ,L ) ( x ) − min y ∈ L ∩ L A ( L ,L ) (cid:1) . (16)By a standard energy-length (a.k.a. monotonicity) estimate for pseudo-holomorphic curves(see e.g. [FSS1, Section 5.a]) we have that image v − T ⊂ W provided that the right-hand sideof (16) is small enough, which in turn can be assured by taking T to be large enough.Now L − T , L − T ⊂ W \ N ([ − δ − T, J λ -convex function φ : N ( R ) −→ R , φ ( s, x ) = e s ) we in fact have:image v − T ⊂ W \ N ([ − δ − T, . It follows that u = b ψ T ◦ v − T has its image inside W \ N ([ − δ, ⊂ W . This concludes theproof of Lemma 2.4.3. (cid:3) We proceed now with the proof of Proposition 2.4.1. Fix J λ on N ([ − δ, D = ( H ≡ , J ) with J ∈ J [0 , T ) . By standard transversality arguments, forevery T > J as above which makes D regular. Lemma 2.4.3 implies that there OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 17 exists T > T ≥ T and every J ∈ J [0 , T ) with (0 , J ) regular, the identitymap i : CF (cid:0) L , L ; (0 , J | W ); ( W , ω = dλ ) (cid:1) −→ CF (cid:0) L , L ; (0 , J ); ( V, b ω R = d b λ R ) (cid:1) (17)is a chain map. Clearly i preserves action, hence induces an isomorphism of persistencemodules i ≤• : HF ≤• ( L , L ; H = 0; ( W , ω )) −→ HF ≤• ( L , L ; H = 0; ( V, b ω R )) . Finally, note that by the maximum principle the persistence modules HF ≤• ( L , L ; H =0; ( V, b ω R = d b λ R )) and HF ≤• ( L , L ; H = 0; ( V, ω = dλ )) coincide. Thus the isomorphism i ≤• induces the isomorphism j ≤• claimed by the proposition. It implies also the statement atpoint (3).As mentioned at the beginning of the proof, the arguments above can be easily adaptedto the case of Floer data of the type D = ( H, J ) with J ∈ J [0 , T ) and H : [0 , × V −→ R compactly supported inside [0 , × ( W \ N ([ − δ, L , · · · , L d ⊂ W \N ([ − δ, T > S (with ( d + 1) boundarypunctures) and for every choice of perturbation data D L ,...,L d = ( K, J ) with Hamiltonian term K such that K z is compactly supported in W \ N ([ − δ, z ∈ S and with almostcomplex structure J = { J } z ∈ S such that J z ∈ J ( T ) for every z ∈ S , the following holds: everyFloer polygon u : S −→ V corresponding to ( L , . . . , L d ; ( K, J )) satisfies image u ⊂ W .The statements at points (1) and (2) readily follow. (cid:3) Cotangent bundles and real Lefschetz fibration Real Lefschetz fibrations. In this paper we will adopt the following definition of Lef-schetz fibrations, essentially as in [FSS1]. By a Lefschetz fibration π : E −→ C we mean asymplectic manifold E , endowed with a symplectic structure ω E as well as an ω E -compatiblealmost complex structure J E such that the following holds:(1) π is ( J E , i )-holomorphic and has a finite number of critical points. Moreover, weassume that every critical value of π corresponds to precisely one critical point of π .We denote the set of critical points of π by Crit( π ) and by Critv( π ) ⊂ C the set ofcritical values of π . For every z ∈ C we denote by E z = π − ( z ) the fiber over z .(2) All the critical point of π are ordinary double points in the following sense. For every p ∈ Crit( π ) there exist a J E -holomorphic chart around p (hence J E is integrable onthis chart) with respect to which π is a holomorphic Morse function.(3) There exists and exhaustion function ϕ E : E −→ R and R ∈ R such that ( E, J E , R , ϕ E , ω E )is a symplectic manifold which is Stein at infinity. (See § K ⊂ C there exists R K ≥ R such that eachlevel set ϕ − E ( R ), R ≥ R K , intersects each fiber E z , z ∈ K , transversely. Note that this implies that for every z ∈ K , Crit( ϕ E | E z ) ⊂ E ϕ E ≤ R . Thus ( E z , J E | E z , R K , ϕ E | E z , ω E | E z )is a symplectic manifold which is Stein at infinity, for every z ∈ K \ Critv( π ).(5) Denote by Γ the symplectic connection on E \ Crit( π ), associated to ω E . (Recall thatthe horizontal distribution of this connection is the ω E -complement of the tangentspaces of the fibers of π .) Let γ : [0 , −→ C be a smooth curve. Then the paralleltransport Π γ : E γ (0) −→ E γ (1) along γ is well defined at infinity.We now turn to real Lefschetz fibrations. By a real structure on a Lefschetz fibration π : E −→ C we mean an involution c E : E −→ E which is anti ω E -symplectic and covers(with respect to π ) the standard complex conjugation c C : C −→ C . We will assume inaddition that c E is anti J E -holomorphic. We denote by E R ⊂ E the fixed locus of c E and callit the real part of E . Note that E R is automatically a smooth Lagrangian submanifold of E (of course, it might be void).It turns out that every smooth connected closed manifold can be realized as the real partof a Lefschetz fibration. This is proved in [FSS1, Section 3]. More precisely, in that paper thefollowing is proved. Given a connected closed n -manifold N and a Morse function f : N −→ R with the property that the level set of each critical value contains precisely one critical value,there exist the following:(1) A smooth affine variety E , endowed with a complex structure denote by J E .(2) A proper holomorphic function π : E −→ C .(3) A plurisubharmonic function ϕ : E −→ R which is proper and bounded below. Denoteby ω E = − dd C ϕ the associated symplectic structure on E . Put also λ E = − d C ϕ , sothat ω E = dλ E . (Here and in what follows, for a real valued function ϕ on a complexmanifold with complex structure J we denote by d C ϕ the 1-form dh ◦ J .)(4) An anti- J E -holomorphic involution c E : E −→ E .with the following properties:(1) The function ϕ is c E -invariant. In particular c E is anti- ω E -symplectic.(2) π ◦ c E = c C ◦ π , i.e. c E covers the standard complex conjugation c C .(3) π : E −→ C is a Lefschetz fibration (in the sense of the definition from the beginningof § ω E and J E . Moreover, when endowed with c E , π : E −→ C is a real Lefschetz fibration according to the preceding definition.(4) The real part E R ⊂ E (with respect to c E ) is diffeomorphic to N .Moreover, E and its associated structures above can be chosen such that there is a diffeomor-phism ϑ : N −→ E R with π | E R ◦ ϑ : N −→ R arbitrarily close to f in the C -topology.Note that Critv( π ) is invariant under the conjugation c C , hence the points of Critv( π ) \ R come in pairs of conjugate points. Further, we have π ( E R ) ⊂ R and Critv( π | R ) = Critv( π ) ∩ R .For a point x ∈ Critv( π ) ∩ R denote by T ↑ x ⊂ E the Lefschetz thimble associated to thecurve [0 , ∞ ) ∋ t itx ∈ C .3.2. Embedding the ball cotangent bundle into a real Lefschetz fibration. A simplecalculation shows that λ E | E R = 0, hence E R ⊂ E is a λ E -exact Lagrangian submanifold. OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 19 Fix a Riemannian metric on N and denote by | · | the norm on the fibers of T ∗ ( N ) corre-sponding to the Riemannian metric via the isomorphism T ∗ ( N ) ∼ = T ( N ) induced by the samemetric. We denote T ∗≤ r ( N ) = { v ∈ T ∗ ( N ) | | v | ≤ r } the radius- r ball cotangent bundle. Similarly we have T ∗ There exist isomorphisms of persistence modules j ≤• : HF ≤• ( L, T ↑ x j ; ( U ≤ r , ω E )) −→ HF ≤• ( L, T ↑ x j ; ( E, ω E )) defined for all closed marked λ E -exact Lagrangians L ⊂ U ≤ r . Moreover, the correspond-ing isomorphisms j := j ≤∞ : HF ( L, T ↑ x j ; ( U ≤ r , ω E )) −→ HF ( L, T ↑ x j ; ( E, ω E )) on the totalhomologies have the following properties:(1) They are compatible with the triangle products (among closed Lagrangians).(2) They are compatible with the naturality maps N L ′ ,L T ↑ xj from § L ′ and L are exact-isotopic).(3) They preserve spectral invariants, namely σ ( j ( a ); L, T ↑ x j ; ( U ≤ r , λ E )) = σ ( a ; L, T ↑ x j ; ( E, λ E )) , ∀ a ∈ HF (( L, T ↑ x j ); ( U ≤ r , ω E )) . Completely analogous statements to the above continue to hold also for pairs of the type ( T ↑ x j , L ) with L ⊂ U ≤ r closed λ E -exact Lagrangians. We will omit the proof, as it is based on very similar ideas as the proof of Proposition 2.4.1.3.3. The extended Lefschetz fibration. In order to use the theory developed in [BC4] weconsider yet another Lefschetz fibration π ′ : E ′ −→ C , which we call the extended fibration of E . The construction is taken from [BC4] and goes as follows. Write the critical values of π as Critv( π ) = { x , . . . , x k , z , z , . . . , z l , z l } , where x i ∈ R are the real critical values and z j , z j are pairs of non-real complex conjugate critical values of π . Let p i ∈ E x i be the critical pointcorresponding to x i . Let ν > ν > | Im z j | for every j . Proposition 3.3.1. There exists a Lefschetz fibration π ′ : E ′ −→ C with the following prop-erties:(1) ( E ′ , π ′ , J E ′ , ω E ′ ) coincides with ( E, π, J E , ω E ) over { z ∈ C | − ν < Im z } . Moreover, Critv( π ′ ) = { x , . . . , x k , x ′ , . . . , x ′ k , z , z , . . . , z l , z l } , namely every real critical value x i has now a corresponding critical value x ′ i (which is not assumed to be real anymore).The new critical values x ′ i have Im x ′ i < − ν , and they are placed as depicted in Figure 1.(2) Denote by γ i ⊂ C , i = 1 , . . . , k , the paths connecting x i with x ′ i , as in figure 1 anddenote by p ′ i ∈ E ′ x ′ i the critical point corresponding to x ′ i . The Lefschetz thimblesemanating from p i and from p ′ i along the two opposite ends of γ i form a matchingsphere S i ⊂ E ′ , lying over γ i . (Put in different words, the vanishing cycles emanatingfrom p i along γ i converge over the other end of γ i to the point p ′ i and their union formsa smooth Lagrangian sphere S i .)(3) The symplectic structure ω E ′ is exact. Moreover, it admits a primitive λ E ′ whichcoincides with λ E over E | − ν< Im z .(4) There exists an exhaustion function ϕ ′ : E ′ −→ R and R ∈ R such that ( E ′ , J E ′ , ϕ ′ , R , ω E ′ ) is a symplectic manifold which is Stein at infinity.(5) The matching spheres S i from (2) are λ E ′ -exact.Remark . We do not require that the exact 1-form λ E ′ from point (3) of the propositioncoincides with − d J E ′ ϕ ′ at infinity. While it seems that this can be arranged, we will not needsuch a statement in the following. Proof of Proposition 3.3.1. Statements (1), (2) and (4) follow from the theory developedin [Sei2, Sections 15d, 16e].To prove (3) we begin by showing that ω E ′ is exact. Denote E + := E | {− ν< Im z } . Let γ ′ i ⊂ C be the path obtained from γ i by chopping a little neighborhood of its second end near x ′ i , namely γ ′ i = γ i \ D ′ i , where D ′ i is a little open disk around x ′ i . Fix also another point y i ∈ γ i ∩ E + which is different from x i .Denote by T x ′ i ⊂ E ′ the Lefschetz thimble emanating from p i along the path γ ′ i and by T y i ⊂ T x ′ i the part of that thimble lying over γ ′ i , between x i and y i . Denote by ∂T x ′ i and ∂T y i OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 21 Figure 1. The extended Lefschetz fibration E ′ and the matching spheres S j ,projected to C .the boundaries of these “partial” thimbles. These are Lagrangian spheres in the fibers of E ′ over x ′ i and y i respectively.By standard topological arguments there is a canonical isomorphism κ : H ( E + , ∪ ki =1 ∂T y i ) −→ H ( E ′ ) , (18)where the homologies are taken with any given coefficient group. This isomorphism is inducedfrom the following chain-level map. Let C be a relative cycle of ( E + , ∪ ki =1 ∂T y i ). For w ∈ γ i denote by Π y i ,wγ i the parallel transport (with respect to the connection induced by ω E ′ ) along γ i from E ′ y i = E y i to E ′ w . Take the part of ∂C lying in ∂T y i and consider its trail under thisparallel transport from y i till x ′ i , namely the union of Π y i ,wγ i ( ∂C ∩ ∂T y i ), where w runs along γ i between y i and x ′ i . Note that while Π y i ,wγ i is in general not defined for the end point w = x ′ i ,here we apply Π y i ,x ′ i γ i to ∂C ∩ ∂T y i which yields the point p ′ i . Therefore the trail of ∂C ∩ ∂T y i along γ i between y i and x ′ i is well defined and gives another relative cycle in ( E ′ , ∂T y i ), whichwe denote by Tr y i ,x ′ i ( ∂C ). Note that ∂ Tr y i ,x ′ i ( ∂C ) = − ( ∂C ∩ ∂T y i ).We can now cap the trails Tr y i ,x ′ i ( ∂C ), i = 1 , . . . , k , to C along ∂C ∩ ∂T y i , and obtain atthe end an absolute cycle C ′ in E ′ . The map κ is induced by the chain level map C C ′ .In order to show that ω E ′ is exact, we will use the isomorphism κ , with coefficients in R .It is enough to prove that h [ ω E ′ ] , κ ( A ) i = 0 for every A ∈ H ( E + , ∪ ki =1 ∂T y i ; R ). For this end,note that ω E ′ vanishes over each of the trails Tr y i ,x ′ i ( ∂C ), hence h [ ω E ′ ] , κ ( A ) i = h [ ω E ′ ] , A i = h [ ω E ] , A i , where the last equality holds because ω E ′ | E + = ω E | E + . Now ω E = dλ E , hence h [ ω E ] , A i = k X i =1 h [ λ E | ∂T yi ] , ∂ i A i , (19)where ∂ i A is the component of ∂A corresponding to H ( ∂T y i ; R ). But T y i is clearly a λ E -exactLagrangian submanifold, thus the right-hand side of (19) vanishes. This completes the proofthat ω E ′ is exact.Next, we prove that ω E ′ admits a primitive λ E ′ that extends λ E | E + . We claim that this wouldfollow from the assertion that the map induced by inclusion i ∗ : H ( E + ; R ) −→ H ( E ′ ; R )is injective. Indeed, fix a small ǫ > x ′ j < − ( ν + ǫ ) for all j , and write E + ǫ = E | − ( ν + ǫ ) < Im z . Denote by i ǫ : E + ǫ −→ E ′ the inclusion. Clearly i ∗ is injective iff i ǫ ∗ : H ( E + ǫ ; R ) −→ H ( E ′ ) is injective. Fix any primitive λ ′ of ω E ′ and consider the 1-form λ E | E + − λ ′ | E + . This form is closed because ω E | E + = ω E ′ | E + . Since i ǫ ∗ is injective, the restrictionmap ( i ǫ ) ∗ : H ( E ′ ; R ) −→ H ( E + ǫ ; R ) is surjective, hence there exists a closed 1-form α ′ on E ′ and a smooth function f : E + ǫ −→ R such that α ′ | E + ǫ = λ E | E + ǫ − λ ′ | E + ǫ + df . Now cut off thefunction f in between E + and E + ǫ to obtain another function f ′ : E ′ −→ R which coincideswith f on E + and vanishes outside of E + ǫ . The desired 1-form λ E ′ is then given by λ E ′ := α ′ + λ ′ − df ′ . To complete the proof it remains to show that i ∗ : H ( E + ; R ) −→ H ( E ′ ; R ) (20)is injective. To this end, denote by F = π − ( w ) the fiber of π : E −→ R over a regular value w of π with w ∈ { z ∈ C | Im z > − ν } .Assume first that dim F > 0. By standard arguments, the inclusions F ⊂ E + and F ⊂ E ′ induce isomorphisms H ( F ) ∼ = H ( E + ) and H ( F ) ∼ = H ( E ′ ), where the homologies are takenwith arbitrary coefficients. Therefore i ∗ : H ( E + ) −→ H ( E ′ ) is an isomorphism.Assume now that dim F = 0. Choose a small ǫ > π are in { Im z > − ν + ǫ } and write E ′− = E ′ | Im z< − ν + ǫ . Note that E + ∩ E ′− is homotopyequivalent to F which is discrete, hence H ( E + ∩ E ′− ; R ) = 0. By the Mayer-Vietoris sequencefor E ′ = E + ∪ E ′− it follows that i ∗ : H ( E + ; R ) −→ H ( E ′ ; R ) is injective.This completes the proof of the injectivity of i ∗ in (20) for all possible values of dim F ,hence also the proof of point (3) of the proposition.Point (5) is obvious if dim F > S i ) ≥ F = 0.In this case N ≈ S , and without loss of generality we may assume that the number of realcritical values of π is k = 2. (This is not really essential for the rest of the proof, it justsimplifies a bit the notation.) Let λ E ′ be a 1-form from point (3), whose existence we havejust proved. In the course of the argument below we will need to alter this 1-form, so we willdenote it by λ ′ .Let E + ǫ be as earlier in the proof. Denote by j ǫ ∗ : H ( E + ǫ ; R ) −→ H ( E, ∂T x ′ ∪ ∂T x ′ ; R ), i ǫ ∗ : H ( E + ǫ ; R ) −→ H ( E ′ ; R ) the maps induced by the inclusion E + ǫ ⊂ E ′ . Similarly to the OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 23 isomorphism from (18) we have also an isomorphism κ : H ( E, ∂T x ′ ∪ ∂T x ′ ; R ) −→ H ( E ′ ; R )which we continue denoting by κ and which is defined by exactly the same means.Consider the homology classes [ S ] , [ S ] ∈ H ( E ′ ; R ) as well as the subspace image i ǫ ∗ ⊂ H ( E ′ ; R ). We claim that no non-trivial linear combination of [ S ] , [ S ] belongs to image i ǫ ∗ .This can be easily seen by looking at the images of κ − [ S ] = [ T x ′ ], κ − [ S ] = [ T x ′ ] under thethe connecting homomorphism ∂ ∗ : H ( E, ∂T x ′ ∪ ∂T x ′ ; R ) −→ H ( ∂T x ′ ∪ ∂T x ′ ; R ) = H ( ∂T x ′ ; R ) ⊕ H ( ∂T x ′ ; R )and noting that κ − (image i ǫ ∗ ) = image j ǫ ∗ is sent to 0 by ∂ ∗ .In view of the preceding claim we can find a closed 1-form θ on E ′ such that:(1) [ θ ] ∈ H ( E ′ ; R ) vanishes on image i ǫ ∗ .(2) h [ θ ] , [ S ] i = R S λ ′ and h [ θ ] , [ S ] i = R S λ ′ .By the property of θ we have ( i ǫ ) ∗ [ θ ] = 0 ∈ H ( E + ǫ ; R ), hence there exists a smooth function h : E + ǫ −→ R such that θ | E + ǫ = dh . Now, cutoff h near { Im z = − ν − ǫ } and extend theresulting function to a smooth function h ′ : E ′ −→ R which vanishes on { Im z ≤ − ν − ǫ } andsuch that h ′ = h on E + = { Im z > − ν } . Replacing the form λ E ′ provided by point (3) of theproposition by the form λ ′′ := λ ′ − θ + dh ′ we still obtain a primitive of ω E ′ that coincides with λ E over E + and such that the matchingspheres S , S are λ ′′ -exact. This completes the proof of point (5) of the proposition in casethe fibers of π : E −→ C are 0-dimensional. (cid:3) Floer theory in E versus E ′ Recall that the extended Lefschetz fibration π ′ : E ′ −→ C from § π : E −→ C over { z ∈ C | − ν < Im z } .Let L , L ⊂ E ′ be two marked exact Lagrangians and assume that L , L ⊂ E ′ | {− ν< Im z } = E | {− ν< Im z } . By the arguments from [BC4] the Floer complexes of ( L , L ) coincide, whenviewed in E and in E ′ , provided we choose the right Floer data. More precisely, let H be aHamiltonian function compactly supported in E | {− ν< Im z } . Then there exist regular Floer data D = ( H, J ) in E and D ′ = ( H, J ′ ) in E ′ , with the same Hamiltonian function H such thatall the Floer trajectories for ( L , L ) with respect to D coincide with those for D ′ and theyall lie inside E | {− ν< Im z } . This easily follows from the open mapping theorem for holomorphicfunctions, by choosing appropriate compatible almost complex structures J and J ′ for whichthe projections π and π ′ are holomorphic. Consequently we have a chain isomorphism (inducedby the identity map on O ( H )) CF ( L , L ; D ; E ) −→ CF ( L , L ; D ′ ; E ′ ) (21) which preserves the action filtration. The E and E ′ in the notation of the Floer complexes inthe preceding formula indicate the ambient manifold in which the respective Floer complex isbeing considered. Consequently (21) induces an action preserving isomorphism of persistencemodules HF ≤• ( L , L ; E ) ∼ = HF ≤• ( L , L ; E ′ ) , hence the spectral invariants and boundary depths of CF ( L , L ), viewed either in E or in E ′ , coincide.The above can be generalized to the Fukaya categories of E and E ′ . More specifically,denote by F uk ( E ) and F uk ( E ′ ) the Fukaya categories of E and E ′ , whose objects are theclosed marked exact Lagrangian submanifolds in E and E ′ . Let F uk ( E ; − ν ) ⊂ F uk ( E ) bethe full subcategory whose objects are closed exact Lagrangians L ⊂ E | {− ν< Im z } . As explainedin [BC4] it is possible to choose the auxiliary data required for the definitions of F uk ( E ) and F uk ( E ′ ) in such a way that the inclusion of objects Ob( F uk ( E ; − ν )) ⊂ Ob( F uk ( E ′ )) extendsto a (homologically) full and faithful A ∞ -functor Inc : F uk ( E ; − ν ) −→ F uk ( E ′ ). Moreover,if we view F uk ( E ; − ν ) and F uk ( E ′ ) as weakly filtered A ∞ -categories, we can assume thatthe functor Inc is a weakly filtered functor (see § § § 2] for the precise definitions and more details).This has the following consequence for A ∞ -modules. Let L ⊂ E ′ be a marked exact La-grangian and assume that L ⊂ E | {− ν< Im z } . Denote by L E ′ the Yoneda module of L , viewed asan A ∞ -module over F uk ( E ′ ) and by L E, − ν the Yoneda module of L over F uk ( E ; − ν ). Bothmodules are weakly filtered in the sense of [BCS] and with the right choices of auxiliary datafor F uk ( E ; − ν ), F uk ( E ′ ) we have thatInc ∗ ( L E ′ ) = L E, − ν as weakly filtered F uk ( E ; − ν )-modules.Next, we compare the Floer theory of the matching spheres S j in E ′ with the Floer theoryof the thimbles T ↑ x j in E , defined on page 18. Fix a rectangle R ⊂ C of the type R = { x + iy ∈ C | x ∈ ( a, b ) , − ν < y < ǫ } (22)such that S j ∩ π ′− ( R ) = T ↑ x j ∩ π − ( R ). (See Figure 2.)Let L ⊂ E ′ be a marked exact Lagrangian and assume that π ′ ( L ) ⊂ R . Let H be aHamiltonian function compactly supported in π − ( R ). Then there exist almost complexstructures J on E and J ′ on E ′ , compatible with ω E and ω E ′ respectively, making the Floerdata D = ( H, J ) and D ′ = ( H, J ′ ) regular and such that the Floer trajectories for ( L, S j ; D ′ )in E ′ and the Floer trajectories of ( L, T ↑ x j ; D ) in E coincide and moreover all these trajectorieslie inside π − ( R ). This follows again from an open mapping theorem argument as in [BC4].It follows that the identity map on O ( H ) gives an action preserving chain isomorphism CF ( L, S j ; D ′ ; E ′ ) −→ CF ( L, T ↑ x j ; D ; E ) . OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 25 Figure 2. The rectangle R and the projection to C of the thimbles T ↑ x j .Here we view T ↑ x j ⊂ E as a marked exact Lagrangian with primitive function adjusted suchthat it coincides with the given primitive function of S j along S j ∩ π ′− ( R ) = T ↑ x j ∩ π − ( R ).Denote by F uk ( E ; R ) ⊂ F uk ( E ′ ) the full subcategory whose objects are marked exactLagrangians L with π ( L ) ⊂ R . Similarly to Inc we have weakly filtered inclusion A ∞ -functorsInc R , − ν : F uk ( E ; R ) −→ F uk ( E ; − ν ) and Inc R ,E ′ : F uk ( E ; R ) −→ F uk ( E ′ ) with Inc R ,E ′ =Inc ◦ Inc R , − ν .Putting all these constructions together we deduce: Lemma 4.0.1. Let S j be the Yoneda module of S j and let T ↑ x j be the Yoneda module of T ↑ x j ,the latter being viewed as a module over F uk ( E ; − ν ) . With the appropriate choice of auxiliarydata, we have Inc ∗R ,E ′ ( S j ) = Inc ∗R , − ν ( T ↑ x j ) (23) as weakly filtered F uk ( E ; R ) -modules. Cone decompositions in Lefschetz fibrations Recall from [BC4] that the Yoneda modules associated to closed Lagrangian submanifolds(or more generally Lagrangian cobordisms), satisfying appropriate exactness or monotonicityconditions, in a Lefschetz fibration E can be represented as iterated cones of modules involvingthe matching spheres S j in the extended Lefschetz fibration E ′ . We will apply these resultsbelow, to the fibrations E and E ′ constructed in § § Let π : E −→ R be a real Lefschetz fibration with critical values x , . . . , x k , z , z , . . . , z l , z l and let π ′ : E ′ −→ C be the extended Lefschetz fibration, as in § ǫ > ǫ < | Im z j | for every j . Let K ⊂ E be a closed λ E -exact Lagrangian submanifold and assumethat K ⊂ E | {| Im | z<ǫ } . Consider the matching spheres S j ⊂ E ′ and denote by τ S j : E ′ −→ E ′ the Dehn-twist around S j , supported in a small neighborhood of S j . Note that τ S j is welldefined up to Hamiltonian isotopy (supported near S j ) since the sphere S j , being a matchingsphere, has a canonical smooth identification with S n (2 n = dim R E ) up to smooth isotopy.Put K (0) := K , K ( j ) := τ S j ( K ( j − ), j = 1 , . . . , k . We view these Lagrangians as objects ofthe λ E ′ -exact Fukaya category F uk ( E ′ ) of E ′ . Denote by K ( j ) the Yoneda modules associatedto K ( j ) , j = 0 , . . . , k . Write also K := K (0) for the Yoneda module of K and denote by S j , j = 1 , . . . , k , the Yoneda modules associated to the matching spheres S j .By the results of [BC4], K is quasi-isomorphic, in the A ∞ -category of modules over F uk ( E ′ ),to the following iterated cone of F uk ( E ′ )-modules: K ∼ = [ B −→ · · · −→ B k −→ K ( k ) ] , (24)where each of the modules B j , j = 1 , . . . , k , has itself an iterated cone decomposition of thefollowing type: B j = [ S j ⊗ CF ( S j , K ) −→ B j, −→ B j, −→ · · · −→ B j,j − ] . (25)In order to describe the modules B j,d , 1 ≤ d ≤ j − 1, that appear in (25) we need a bit ofnotation. Denote by I d,j − the set of all multi-indices i = ( i , . . . , i d ) with 1 ≤ i < i < · · ·
1. We order the elements of I d,j − by the lexicographic order. For each multi-index i ∈ I d,j − put C i,j := S j ⊗ CF ( S j , S i d ) ⊗ CF ( S i d , S i d − ) ⊗ · · · ⊗ CF ( S i , S i ) ⊗ CF ( S i , K ) . (26)Let m d,j − := I d,j − and order the elements of I d,j − = { i (1) , . . . , i ( m d,j − ) } in such a waythat i (1) (cid:22) i (2) (cid:22) · · · (cid:22) i ( m d,j − ) . Then B j,d = [ C i (1) ,j −→ C i (2) ,j −→ · · · −→ C i ( md,j − ,j ] . (27)Having established a cone decomposition of the module K over the A ∞ -category F uk ( E ′ )we consider its pull-back to Fukaya categories associated to E . Recall from § F uk ( E ; R ) and F uk ( E ; − ν ). We take the rectangle R from (22) to bewide enough such that it contains π ( K ). Recall also the inclusion functorInc R ,E ′ : F uk ( E ; R ) −→ F uk ( E ′ )that factors as the composition Inc R ,E ′ = Inc ◦ Inc R , − ν of the two functorsInc R , − ν : F uk ( E ; R ) −→ F uk ( E ; − ν ) , Inc : F uk ( E ; − ν ) −→ F uk ( E ′ ) . By pulling back the cone decomposition (24) via Inc ∗R ,E ′ we obtain a similar cone decompo-sition for K (now viewed as a module over F uk ( E ; R )), where the modules S j in (25) and (26)are replaced by Inc ∗R , − ν ( T ↑ x j ), see (23). (Note that the terms involving the Floer complexes of S j and and of S i l remain unchanged.) OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 27 Finally, we claim that the pullback Inc ∗R ,E ′ K ( k ) of the the module K ( k ) which appears lastin (24) is acyclic.We will outline below in § ∗R ,E ′ K ( k ) . Then in § § K looks like in case the number of real critical values of π is k = 3: K ∼ = [ S ⊗ CF ( S , K ) −→S ⊗ CF ( S , K ) −→ S ⊗ CF ( S , S ) ⊗ CF ( S , K ) −→S ⊗ CF ( S , K ) −→ S ⊗ CF ( S , S ) ⊗ CF ( S , K ) −→ S ⊗ CF ( S , S ) ⊗ CF ( S , K ) −→S ⊗ CF ( S , S ) ⊗ CF ( S , S ) ⊗ CF ( S , K ) −→ K (3) ]5.1. Exact triangles associated to Dehn twists. Let ( X n , ω = dλ ) be a Liouville domainand S n ≈ −→ S ⊂ X a parametrized Lagrangian sphere. In case n = 1 we additionally assumethat S is λ -exact. Let τ := τ S : X −→ X be a symplectomorphism, supported in Int X , whichrepresents the symplectic mapping class of the Dehn twist around S . Note that τ is an exactsymplectomorphism, hence sends exact Lagrangians to exact Lagrangians.A well known result of Seidel [Sei1, Sei2] says that for every exact Lagrangian Q ⊂ X thereis the following distinguished triangle in the derived Fukaya category F uk ( X ): S ⊗ CF ( S, Q ) / / τ ( Q ) (cid:15) (cid:15) Q g g ◆◆◆◆◆◆◆◆◆◆◆◆◆ (28)Here S , Q and τ ( Q ) stand for the A ∞ -modules corresponding to S , Q and τ ( Q ) under theYoneda embedding.The above distinguished triangle implies that, up to a quasi-isomorphism of modules, Q can be expressed as the following mapping cone: Q ∼ = [ S ⊗ CF ( S, Q ) −→ τ ( Q )] . (29)By rotating (28) we obtain also the following quasi-isomorphism: τ ( Q ) ∼ = [ Q −→ S ⊗ CF ( S, Q )] . (30)Note that here and in what follows we work in an ungraded setting, hence no grading shiftsappear in any of (28) - (30).We now turn to the cone decomposition (24), and assume that ( X, dλ ) = ( E ′ , λ E ′ ) as in § K (1) = τ S ( K ) and obtain from (29): K ∼ = [ S ⊗ CF ( S , K ) −→ K (1) ] . (31) By the same argument we also have K (1) ∼ = [ S ⊗ CF ( S , K (1) ) −→ K (2) ], which togetherwith (31) gives: K ∼ = [ S ⊗ CF ( S , K ) −→ S ⊗ CF ( S , K (1) ) −→ K (2) ] . (32)But by (30) we have K (1) ∼ = [ K −→ S ⊗ CF ( S , K )]. Substituting this into (32) yields: K ∼ = [ S ⊗ CF ( S , K ) −→ S ⊗ CF ( S , K ) −→ S ⊗ CF ( S , S ) ⊗ CF ( S , K ) −→ K (2) ] . (33)Continuing in a similar vein, decomposing K (2) , K (3) etc. we obtain the cone decomposition (24)with items as described in (25) - (27).It remains to address the acyclicity of the module Inc ∗R ,E ′ K ( k ) . (Recall K ( k ) = τ S k · · · τ S ( K )).This follows from [BC4, § φ : E ′ −→ E ′ such that φ ( K ( k ) ) ⊂ E ′ | { Im z ≤− ν } . (See also [BC2] for more details.) In particu-lar, for every Lagrangian submanifold L ⊂ π ′− ( R ) we have CF ( L, φ ( K ( k ) )) = 0.5.2. Taking filtrations into account. We now go back to the cone decomposition (24) andreview it from the perspective of action filtrations.From now on we assume all the exact Lagrangian submanifolds to be marked, unless oth-erwise stated. By a slight abuse of notation, we now redefine the objects of the Fukayacategories F uk ( E ), F uk ( E ′ ), as well as F uk ( E ; R ), F uk ( E ; − ν ), to be marked exact La-grangians, subject to the additional constraints in each of these categories. These categoriesnow become weakly filtered A ∞ -categories, where the filtrations are induced by the actionfunctional. We refer the reader to [BCS, § 2] for the definitions and basic theory of weaklyfiltered A ∞ -categories and weakly filtered modules over such.Below we will take the exact Lagrangian K ⊂ E | {| Im | z<ǫ } to have an arbitrary marking.This marking induces a marking on K ( j ) = τ S j · · · τ S ( K ), j = 1 , . . . , k , see § S j are also assumed to be marked in advance.Note that all the items in the cone decomposition (24), as detailed in (25) - (27) are weaklyfiltered modules. This is so because the S j ’s and K ( k ) are Yoneda modules over a weaklyfiltered A ∞ -category, and the chain complexes CF ( S i l , S i l − ) and CF ( S j , K ) are filtered.Next, we claim that all the maps in the iterated cones (24), (25) and (27) are weaklyfiltered maps. This means, in particular, that when evaluating these iterated cones moduleson a given exact Lagrangian L , each of these maps specializes to a filtered chain map thatshifts filtrations by an amount bounded from above uniformly in L . More specifically: Proposition 5.2.1. In the iterated cone (27) B j,d = [ C i (1) ,j ϕ ,j −−−→ [ C i (2) ,j ϕ ,j −−−→ [ · · · −→ [ C i ( md,j − − ,j ϕ md,j − − ,j −−−−−−−→ C i ( md,j − ,j ] · · · ]]] , (34) each of the module homomorphisms ϕ l,j is weakly filtered, and shifts action by ≤ s ϕ l,j , forsome s ϕ l,j ≥ . This implies that the right-hand side of (34) is filtered using the filtrations of the factors C i ( l ) ,j and the recipe (53). OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 29 In particular, for every exact Lagrangian L , the module homomorphism ϕ l,j specializes toan s ϕ l,j -filtered chain map (still denoted by ϕ l,j ): ϕ l,j : C i ( l ) ,j ( L ) −→ [ C i ( l +1) ,j ( L ) ϕ l +1 ,j −−−−→ [ · · · −→ [ C i ( md,j − − ,j ( L ) ϕ md,j − − ,j −−−−−−−→ C i ( md,j − ,j ( L )] · · · ]] . A crucial point for us will be that the filtration-shifts s ϕ l,j are independent of L .Having filtered the modules B j,d , the preceding statements apply also to the maps in theiterated cone of (25), and finally also to the right-hand side of (24). We will prove Proposition5.2.1 in § K and the (nowweakly filtered) iterated cone on the right-hand side is filtered in the following sense. Proposition 5.2.2. There exist s K ≥ and weakly-filtered module homomorphisms ϕ : K −→ [ B −→ · · · −→ B k −→ K ( k ) ] , ψ : [ B −→ · · · −→ B k −→ K ( k ) ] −→ K that shift filtrations by ≤ s K and such that ϕ ◦ ψ = id + µ mod ( h ′ ) , ψ ◦ ϕ = id + µ ( mod )1 ( h ′′ ) for weakly filtered pre-module homomorphisms h ′ , h ′′ that shift filtrations by ≤ s K . The proof of this statement is again postponed to § s K depends on K (and its marking) as well as on the marking on the spheres S , . . . , S k .In particular, the above implies that for every exact Lagrangian L we have chain maps ϕ L : CF ( L, K ) −→ [ B ( L ) −→ · · · −→ B k ( L ) −→ CF ( L, K ( k ) )] ,ψ L : [ B ( L ) −→ · · · −→ B k ( L ) −→ CF ( L, K ( k ) )] −→ CF ( L, K ) , (35)which are s K -filtered and such that ϕ L ◦ ψ L and ψ L ◦ ϕ L are chain homotopic to the identitiesvia chain homotopies that shift filtrations by ≤ s K . Once again, it is important to stress thatthe bound on the action shift s K is independent of L .Phrased in the terminology of Definition 7.5.3, the above says that the module K (resp.filtered chain complex CF ( L, K )) and the module on the right-hand side of (24) (resp. thefiltered chain complex [ B ( L ) −→ · · · −→ B k ( L ) −→ CF ( L, K ( k ) )]) are at distance ≤ s K onefrom the other.Finally, recall that the pullback module Inc ∗R ,E ′ K ( k ) is acyclic. We claim that this acyclicityholds also in the filtered sense. Namely, there exists a constant s C = s C ( K ), which depends on K , and a weakly filtered pre-module homomorphism h : Inc ∗R ,E ′ K ( k ) −→ Inc ∗R ,E ′ K ( k ) that shiftsaction by ≤ s C such that in hom mod F uk ( E ; R ) (Inc ∗R ,E ′ K ( k ) , Inc ∗R ,E ′ K ( k ) ) we have id = µ mod1 ( h ). Inparticular, for every exact Lagrangian L ⊂ π − ( R ) we have: β ( CF ( L, K ( k ) )) ≤ s C . (36)Here, β ( CF ( L, K ( k ) )) is the boundary depth of the acyclic filtered chain complex CF ( L, K ( k ) ).The inequality (36) follows from the last paragraph of § s C = 2 ρ (id , φ ), where φ : E ′ −→ E ′ is a Hamiltonian diffeomorphism that sends K ( k ) to E ′ | { Im z ≤− ν } , and ρ stands for the Hofer metric on the group of Hamiltoniandiffeomorphisms. Remark . The constant s C appearing in (36) depends apriori on K (though not on L ).A more careful argument, based on [BC4, § φ , mentioned above, can be taken to be at a uniformly bounded (in K ) Hofer-distance fromid, as long as we restrict to Lagrangians K ⊂ E | {| Im | z<ǫ } . Consequently the constant s C canbe assumed to be independent of K .However, this additional information will not be used in the rest of the paper. The reasonis that we will use the filtered cone decomposition (24) only for one Lagrangian K , namely K = N - the zero-section of T ∗ ( N ) viewed as a Lagrangian in E .5.3. Proof of the statements from § We continue to assume here all exact Lagrangiansubmanifolds (and cobordisms) to be marked.We begin with a brief digression on inclusion and product functors. Let ( Y, dλ Y ) be aLiouville manifold as in § γ : R −→ R be a smooth proper embedding sending theends of R to horizontal rays in R . By abuse of notation we denote by γ also the image ofthis embedding. By the results of [BC3, BCS] there is a weakly filtered A ∞ -functor (calledin [BC3] “inclusion functor”) I γ : F uk ( Y ) −→ F uk cob ( R × Y ) which sends the object L ⊂ Y to I γ ( L ) = γ × L ⊂ R × Y . Here F uk ( Y ) stands for the Fukaya category of closed λ Y -exactLagrangians in Y and F uk cob ( R × Y ) for the Fukaya category of exact cobordisms in R × Y ,with respect to the 1-form xdy ⊕ λ Y .Let ( X, ω = dλ ) be a Liouville manifold as in § X − the manifold X endowed with the symplectic structure − ω . Take Y = X × X − , endowed with the symplecticstructure ω ⊕− ω and Liouville form e λ := λ ⊕− λ (playing the role of λ Y ). Fix e λ ′ := xdy ⊕ λ ⊕− λ as the primitive of ω R ⊕ ω ⊕ − ω .Fix an exact Lagrangians Q ⊂ X . A slight variation on the inclusion functor I γ is the A ∞ -functor I γ,Q : F uk ( X ) −→ F uk cob ( R × X × X − ) which sends an exact Lagrangians L ⊂ X to I γ,Q ( L ) := γ × L × Q . The construction of this functor is very similar to theconstruction of I γ (for the case Y = X × X − ), as detailed in [BC3]. In fact, I γ,Q factors as I γ,Q := I γ ◦ P Q , where P Q : F uk ( X ) −→ F uk ( X × X − ) is the obvious functor that sends L ⊂ X to L × Q ⊂ X × X − .The main ingredient to show Propositions 5.2.1 and 5.2.2 is to establish a filtered versionof the Seidel’s Dehn-twist triangle (28) (or more precisely (29)). We pursue this now. Lemma 5.3.1. The mapping cone in equation (29) admits a filtered version. In the course of the proof we will indicate more precisely the relevant shifts involved andtheir dependence on the choices involved in the construction. Proof. Let ( X n , ω = dλ ) be a Liouville manifold as in § S ⊂ X , τ = τ S : X −→ X be as at the beginning of § τ (a representative of the Dehn-twist symplectic mapping class) such that τ is supported near S and moreover such that OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 31 τ ∗ λ = λ + dh τ , where h τ : X −→ R is a smooth function compactly supported near S . (Thelatter easily follows from the fact that given any neighborhood of the zero-section in T ∗ ( S n ),there is a model Dehn-twist T ∗ ( S n ) −→ T ∗ ( S n ) supported in that neighborhood which is λ can -exact, and the fact that the sphere S is λ -exact.) Note that we have: ( τ − ) ∗ λ = λ − d ( h τ ◦ τ − ).Let Q ⊂ X be a marked exact Lagrangian with primitive h Q : Q −→ R for λ | Q . Then τ ( Q )is also a marked exact Lagrangian. Indeed, h τ ( Q ) : τ ( Q ) −→ R defined by h τ ( Q ) ( x ) := h Q ( τ − ( x )) + h τ ( τ − ( x ))is a primitive of λ | τ ( Q ) . We will use this function to mark τ ( Q ).We now get back to Dehn-twists, from the perspective of Lagrangian cobordism. By aresult of Mak-Wu [MW] there exists an exact Lagrangian cobordism W ⊂ R × X × X − withtwo negative ends and one positive end, as follows. The upper negative end is S × S and thelower negative end is the graph Γ τ − of τ − . The positive end is the graph of the identity map(i.e. the diagonal in X × X − ). See Figure 3. Figure 3. Projection to R of the Mak-Wu cobordism W ⊂ R × X × X − ,and the curves γ , γ ′ .Let γ ⊂ R be the curve depicted in Figure 3, and denote by W the Yoneda module corre-sponding to W ∈ Ob( F uk cob ( R × X × X − ). Denote also by S × S , τ ( Q ) the Yoneda modules(over F uk ( X × X − ) corresponding to the Lagrangians S × S and τ ( K ), respectively. Ignoringfiltrations for the moment, a straightforward calculation (based on the theory from [BC3]) shows that the pullback module I ∗ γ,Q W coincides with a mapping cone I ∗ γ,Q W = [ S ⊗ CF ( S, Q ) ϕ −→ τ ( Q )] (37)for some module homomorphism ϕ : S ⊗ CF ( S, Q ) −→ τ ( Q ).Consider now the curve γ ′ ⊂ R from Figure 3. Ignoring filtrations again, it is easy to seethat I ∗ γ ′ ,Q W = Q , the Yoneda module corresponding to Q ⊂ X .The curves γ and γ ′ are isotopic via a Hamiltonian isotopy which is horizontal at infinity.Therefore the modules I ∗ γ,Q W and I ∗ γ ′ ,Q W are quasi-isomorphic (in the category mod F uk ( X ) ).Thus we have a quasi-isomorphism Q ∼ = [ S ⊗ CF ( S, Q ) ϕ −→ τ ( Q )] . (38)Our goal now is to derive a coarse filtered version of (38). More specifically, we will have toaddress two thing: explain why the module homomorphism ϕ is filtered, and then show thatthe quasi-isomorphism in (38) is weighted in the sense of Definition 7.5.3.Note that e λ ′ coincides with e λ along each horizontal end of W (because xdy vanishes alonghorizontal rays). We also have e λ | Γ id = 0 , e λ | S × S = λ S ⊕ − λ S , e λ | Γ τ − = d ( h τ ◦ τ − ) , where λ S := λ | S . Let h W : W −→ R be a primitive of e λ ′ | W . By the above, h W restrictsalong each of the ends of W to a primitive function for the restriction of e λ to the Lagrangiancorresponding to that end. We will use these functions, denoted by h W, Γ id , h W,S × S and h W, Γ τ − ,for primitives of e λ | Γ id , e λ | S × S and e λ | Γ τ − respectively. Note that h W, Γ id is constant, and bysubtracting this constant from h W we may assume without loss of generality that h W, Γ id ≡ W does not come with a preferred marking, andwe are free to choose h W as we wish.)Pick any marking on S , i.e. a primitive function h S : S −→ R for λ S . We have: h W,S × S ( x, y ) = h S ( x ) − h S ( y ) + C W,S × S , ∀ ( x, y ) ∈ S × S,h W, Γ τ − ( x, τ − ( x )) = h τ ( τ − ( x )) + C W, Γ τ − , ∀ x ∈ X, (39)for some constants C W,S × S , C W, Γ τ − . Fix a primitive h γ : γ −→ R of ( xdy ) | γ . Note that h γ is constant along the positive and negative ends of γ . Given any marked exact Lagrangian L ⊂ X , with a primitive function h L : L −→ R for λ | L , we will use the function h γ × L × Q := h γ + h L − h Q as a primitive for e λ ′ | γ × L × Q .Consider the Floer complex CF ( γ × L × Q, W ) with Floer data consisting of a zero Hamil-tonian and any regular almost complex structure. (We assume here without loss of generalitythat ( L × Q ) ⋔ S × S and L × Q ⋔ Γ τ − .)Given two exact Lagrangians L ′ , L ′′ in a Liouville manifold ( Y, dλ Y ), endowed with primi-tives h L ′ : L ′ −→ R , h L ′′ : L ′′ −→ R for λ Y | L ′ and λ Y | L ′′ , and given a Floer datum for ( L ′ , L ′′ )we denote by A ( − ; ( L ′ , L ′′ )) the action functional associated to the given Floer datum andthe choices of primitives h L ′ , h L ′′ . Here − stands for a path connecting a point from L ′ to apoint in L ′′ . OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 33 We will now examine the action functional A for the pairs ( γ × L × Q, W ), ( L, S ) and ( S, Q ).As before, we use here Floer data with zero Hamiltonian terms. We begin with calculating A on the intersection points of ( γ × L × Q ) ∩ W (viewed as constant paths). These intersectionpoints fall into two types:(1) ( P ′ , x , x ), where P ′ ∈ R is as depicted in Figure 3 and x , x ∈ S .(2) ( P ′′ , x , x ), where P ′′ ∈ R is as in Figure 3 and x ∈ L ∩ τ ( Q ), x = τ − ( x ).For the points of the 1’st type we have: A ( P ′ , x , x ;( γ × L × Q, W ))= h S ( x ) − h S ( x ) + C W,S × S − h γ ( p ′ ) − ( h L ( x ) − h Q ( x ))= ( h S ( x ) − h L ( x )) + ( h Q ( x ) − h S ( x )) + ( C W,S × S − h γ ( P ′ ))= A ( x ; ( L, S )) + A ( x ; ( S, Q )) + ( C W,S × S − h γ ( P ′ )) . (40)Note that the sum of the first two terms in the last equality is precisely the action-level of thegenerator x ⊗ x ∈ CF ( L, S ) ⊗ CF ( S, Q ).Turning to the intersection points of the 2’nd type, we have: A ( P ′′ , x , x ;( γ × L × Q )= h τ ( τ − ( x )) + C W, Γ τ − − h γ ( P ′′ ) − h L ( x ) + h Q ( τ − ( x ))= h τ ( Q ) ( x ) − h L ( x ) + C W, Γ τ − − h γ ( P ′′ )= A ( x ; ( L, τ ( Q ))) + ( C W, Γ τ − − h γ ( P ′′ )) . (41)Now recall from (37) that CF ( γ × L × Q, W ) = [ CF ( L, S ) ⊗ CF ( S, Q ) ϕ −→ τ ( Q )] , and that by the results of [BC3] counts Floer strips going from the intersection points of type 1to points of type 2.From the standard action-energy identity we obtain the following: if the generator x ∈ L ∩ τ ( Q ) of CF ( L, τ ( Q )) participates in ϕ ( x ⊗ x ), then: A ( x ; ( L, S )) + A ( x ; ( S, Q )) + C W,S × S − h γ ( P ′ ) ≥ A ( x ; ( L, τ ( Q ))) + C W, Γ τ − − h γ ( P ′′ ) . (42)It follows that ϕ shifts action by s ϕ ≤ h γ ( P ′′ ) − h γ ( P ′ ) + C W,S × S − C W, Γ τ − . (43)The latter quantity is a constant which is independent of Q and L .Next, consider the curve γ ′ from Figure 3 and I γ ′ ,Q : F uk ( X ) −→ F uk cob ( R × X × X − ).Recall that up to a filtration shift we have I ∗ γ ′ ,Q W = Q , therefore CF ( L × Q, Γ id ) ∼ = CF ( L, Q ),again up to a filtration shift. We will now determine this shift. To this end, recall first that h W, Γ id ≡ 0. The intersection points of ( γ ′ × L × Q ) ∩ W are of the type ( R, x, x ), x ∈ L ∩ Q .Calculating the action on such points we get: A ( R, x, x ; ( γ ′ × L × Q, W )) = − h γ ′ ( R ) − h L ( x ) + h Q ( x ) = A ( x ; ( L, Q )) − h γ ′ ( R ) . (44) Therefore the identification I ∗ γ ′ ,Q W = Q holds up to an action shift of the constant h γ ′ ( R ).Finally, there exists a constant S ( W ) ≥ W and another constant C ( h γ , h γ ′ ) ≥ h γ , h γ ′ such that the followingholds. There exist weakly filtered module homomorphisms φ : I ∗ γ,Q W −→ I ∗ γ ′ ,Q W and φ ′ : I ∗ γ ′ ,Q W −→ I ∗ γ,Q W that shift filtrations by ≤ S ( W ) + C ( h γ ′ ) such that φ ′ ◦ φ = id + µ mod1 ( H ), φ ◦ φ ′ = id + µ mod1 ( H ′ ) for some weakly filtered pre-module homomorphisms H , H ′ that shiftfiltrations by ≤ S ( W ) + C ( h γ ′ )). We refer the reader to [BCS, § 4] for more details on this.The constant S ( W ) is the shadow of the cobordism W - namely the area of the domain in R consisting of the projection of W to R together with all the bounded connected componentsof the complement of this projection.As a result, we obtain a weakly filtered quasi-isomorphism Q ∼ = [ S ⊗ CF ( S, Q ) ( ϕ,s ϕ ) −−−−→ τ ( Q )] , (45)of weight bounded from above by a constant that depends only on W and γ , γ ′ . (See Def-inition 7.5.3.) As seen above at (43) the amount of shift of ϕ is bounded from above by aconstant s ϕ which does not depend on Q . This concludes the construction of the filteredversion of the Seidel exact triangle. (cid:3) Remark . There are a number of other ways to construct Seidel’s exact triangle associatedto a Dehn twist. Certainly, Seidel’s original construction in [Sei1] and also the method in[BC4]. These methods can also be used to deduce filtered versions of the exact traingle. Weused here the method in [MW] as it appears to provide the fastest approach in our context.Propositions 5.2.1 and 5.2.2 now follow by applying the procedure indicated at the endof § § Remark. The weight of the quasi-isomorphism at (45) as well as s ϕ do depend (also) on W (hence on the specific choice of the representative τ of the symplectic mapping class of theDehn-twist), however these choices are made in advance, once and for all. The dependenciesof this weight and of s ϕ on γ , γ ′ and h γ ′ , h γ ′ can in fact be eliminated by estimating moresharply the shifts in φ , φ ′ , H , H ′ above. However this is not needed for our purposes.6. Proof of the main theorem This section contains two parts. The first, and main part, provides the proof of TheoremA. The second is concerned with the converse of the statement, as indicated in Remark 1.0.1(1).6.1. The spectral norm bound in equation (2). For the proof of the main theorem wewill need the following Lemma. Fix a tubular neighborhood V = T ∗≤ r ( N ) of the zero-section.For q ∈ N denote by F q = T ∗ q ( N ) ∩ V the part of the cotangent fiber over q that lies inside OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 35 V . We endow the exact Lagrangians F q with the 0 function as a primitive of λ can . Note thatfor every marked exact Lagrangian L ⊂ V and every q ∈ N we have HF ( L, F q ) ∼ = Z , hence σ + ( CF ( L, F q )) = σ − ( CF ( L, F q )). We denote this number by σ ( CF ( L, F q )). Lemma 6.1.1. There exist constants C = C ( V ) > and C ′ = C ′ ( V ) > , that depend onlyon V , such that for every marked exact Lagrangian L ⊂ Int ( V ) and every q ′ , q ′′ ∈ N we have | σ ( CF ( L, F q ′ )) − σ ( CF ( L, F q ′′ )) | ≤ C, | β ( CF ( L, F q ′ )) − β ( CF ( L, F q ′′ )) | ≤ C ′ . Proof. The proof is based on standard arguments, hence we will only outline it.The statements in the Lemma follows from the following somewhat stronger statement: Allthe F uk ( V ) -modules corresponding to F q , q ∈ N , are at a bounded distance one from theother in the sense of Definition 7.5.3. Here is an outline of the proof of the stronger statement. Since N is compact, it is enoughto prove the statement locally for q ∈ N . Fix q ∈ N and let B ′ ⊂ N be a ball chart around q and B ⊂ B ′ a smaller closed ball around q .We claim that there exists r ′ > r , a compact subset K ⊂ B ′ and a family of Hamiltonianfunctions H ( q ) : [0 , × T ∗ ( N ) −→ R , parametrized by q ∈ B , such that the following holds:(1) All the functions H ( q ) , q ∈ B , are compactly supported in V ′ := T ∗ Proof of Theorem A. Fix a small r > V = T ∗≤ r ( N ) of N . Recallfrom § κ : V −→ E and its image U := κ ( V ) ⊂ E .We now appeal to the cone decomposition (24) from § F uk ( E ′ ).We apply this to the Lagrangian K = N (i.e. the zero section) and its Yoneda module N .Let L ⊂ U be any exact Lagrangian. The filtered cone decomposition of N , as describedin Propositions 5.2.1 and 5.2.2, gives a filtered cone decomposition of the chain complex CF ( L, N ; E ′ ), which by the formulas (24), (25), (27) involves the following types of filteredchain complexes as well as their tensor products:(1) CF ( L, S i ; E ′ ), CF ( S i , N ; E ′ ), i = 1 , . . . , k .(2) CF ( S j ′′ , S j ′ ; E ′ ), 1 ≤ j ′ < j ′′ ≤ k .(3) CF ( L, N ( k ) ; E ′ ).The chain complexes in (2) do not depend on L . In particular their spectral invariants andboundary depths are independent of L .Formulas (24)-(27) together with Proposition 7.6.1 and Lemma 7.4.1 imply that there areconstants A , B , C > 0, that do not depend on L , such that: ρ ( CF ( L, N ; E ′ )) ≤ A e ρ ( CF ( L, S ; E ′ ) , . . . , CF ( L, S k ; E ′ )) + B k X i =1 β ( CF ( L , S i ); E ′ ) + C . Passing from E ′ to E , as described in § 4, we have action preserving chain isomorphisms CF ( L, N ; E ) ∼ = CF ( L, N ; E ′ ) and CF ( L, T ↑ x i ; E ) ∼ = CF ( L, S i ; E ′ ) for every 1 ≤ i ≤ k . Con-sequently, the spectral invariants and boundary depths of the chain complexes in E coincidewith the corresponding ones in E ′ . OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 37 Next we appeal to Proposition 2.4.1 (with W = U , V = E and L = L , L = N ) and toProposition 3.2.1 and deduce that ρ ( CF ( L, N ; U )) ≤ A e ρ ( CF ( L, F q ; U ) , . . . , CF ( L, F q k ; U )) + B k X i =1 β ( CF ( L , F q i ); E ′ ) + C , where q i = κ − ( x i ) ∈ N .Put q := q . By Lemma 6.1.1 we have that both | σ ( CF ( L, F q )) − σ ( CF ( L, F q i )) | as well as | β ( CF ( L, F q )) − β ( CF ( L, F q i )) | are uniformly bounded (with respect to L and i ), hence thereexist constants A , B > 0, that do not depend on L , such that ρ ( CF ( L, N ; U )) ≤ A + B β ( CF ( L, F q )) . Now, γ ( L, N ) ≤ ρ ( CF ( L, N ; U )), hence γ ( L, N ) ≤ A + B β ( CF ( L, F q ))for all exact Lagrangians L ⊂ U . The last inequality together with the triangle inequality for γ imply inequality (2) and conclude the proof of Theorem A. (cid:3) Boundedness of the spectral metric implies boundedness of β ( CF ( L, F q ) . Herewe outline an argument showing the statement at point (2) of Remark 1.0.1. Namely, if thefunction L ex ,N ( U ) ∋ L γ ( N, L )is bounded, then L ex ,N ( U ) ∋ L β ( CF ( L, F q ))is bounded too. In other words the conjecture of Viterbo from page 2 implies the boundednessof the boundary depths CF ( − , F q ) over the collection of exact Lagrangians L ⊂ U that areexact isotopic to the zero-section N .Here is an outline of the proof. Let L ∈ L ex ,N ( U ) and assume without loss of generalitythat L ⋔ N , L ⋔ F q . Fix an arbitrary marking for L and mark N and F q by taking theirprimitive functions to be identically 0. Put α + = c ([ N ]; N, L ) , α − = c ([ N ]; L, N ) . We have α + + α − = γ ( N, L ). Note that α + and α − depend on the marking of L but theirsum α + + α − does not. Also note that β ( CF ( N, L )) is independent of the marking of L .We will now need to carry out a chain-level calculation with Floer complexes. To this endwe take the Floer complexes CF ( N, L ), CF ( L, N ), CF ( N, F q ) and CF ( L, F q ) with Floer datahaving 0 Hamiltonian terms. We also fix a Floer datum for ( L, L ) whose Hamiltonian termis induced from a C -small Morse function L −→ R with a unique critical point of top index,so that the unity e L ∈ HF ( L, L ) has a unique representing cycle in CF ( L, L ).Fix ǫ > 0. Choose perturbation data for each of the tuples ( N, L, N ), ( L, N, L ), ( N, L, F q )and ( L, N, F q ) which are compatible with the previous choices of Floer data and such that theassociated µ -operations shift action by ≤ ǫ . Let a ∈ CF ≤ α + ( N, L ), b ∈ CF ≤ α − ( L, N ), be cycles representing the Floer homology classes N NL,N ([ N ]) and N N,LN ([ N ]) (see § ϕ : CF ( L, F q ) −→ CF ( N, F q ) , ϕ ( x ) := µ ( a, x ) ,φ : CF ( N, F q ) −→ CF ( L, F q ) , φ ( y ) := µ ( b, y ) . (47)By our choices of data, ϕ shifts action by ≤ α + and φ by ≤ α − . Note that CF ( N, F q ) = Z q and it is easy to see that ϕ ◦ φ = id. We claim that φ ◦ ϕ is chain homotopic to the identityvia a chain homotopy H that shifts action by ≤ γ ( N, L ) + ǫ , where ǫ > CF ( L, F q )and CF ( N, F q )). However in order to employ Lemma 7.1.2 we need the shifts of each of ϕ and φ to be non-negative and we also need to relate each of these shifts to the shift of thechain homotopy H which is claimed to be γ ( N, L ) + ǫ . The “problem” is that ϕ and φ haveshifts of ≤ α + and ≤ α − respectively and we do not have information on the size of each ofthem alone - we only know that α + + α − = γ ( N, L ).To go about this technical problem we proceed as follows. We shift the marking of L bya constant such that α − = 0. Consequently α + will now become equal to γ ( N, L ). We thusassume from now on that α − = 0 and α + = γ ( N, L ). Under these circumstances we can nowapply Lemma 7.1.2 and obtain that | β ( CF ( L, F q )) − β ( CF ( N, F q )) | ≤ γ ( N, L ) + 2 ǫ . Since β ( CF ( N, F q )) = 0 and by assumption γ ( N, − ) is bounded, the main statement follows.It remains to show the existence of the required chain homotopy H between φ ◦ ϕ and theid. Consider the tuple of Lagrangians ( L, N, L, F q ). Choose Floer perturbation data for thistuple, which is compatible with the previous choices of Floer data, and such that the followingholds:(1) µ ( µ ( b, a ) , x ) = x for every x ∈ CF ( L, F q ). (Note that by our choices of Floer data, µ ( b, a ) ∈ CF ( L, L ) is the unique cycle representing the unity e L ∈ HF ( L, L ).)(2) The operation µ : CF ( L, N ) ⊗ CF ( N, L ) ⊗ CF ( L, F q ) −→ CF ( L, F q ) shifts action by ≤ ǫ .By standard A ∞ -identities (applied with Z -coefficients) we have for every x ∈ CF ( L, F q ): φ ◦ ϕ ( x ) = µ ( b, µ ( a, x )) = µ ( µ ( b, a ) , x ) + µ ( b, a, µ ( x )) + µ µ ( b, a, x ) = x + µ ( b, a, µ ( x )) + µ µ ( b, a, x ) . (48)The required homotopy H : CF ( L, F q ) −→ CF ( L, F q ) is then H ( x ) := µ ( b, a, x ). And itclearly shifts action by ≤ γ ( N, L ) + ǫ . (cid:3) Remark . A similar argument appears, for a different purpose, in [KS]. At a conceptuallevel these arguments are a reflection of a Yoneda type lemma in the filtered setting that OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 39 allows translation of relations among morphisms of Yoneda modules (over the the A ∞ Fukayacategory) in terms of µ k operations. Such a result, called there the λ -lemma, appears in [BCS].7. Filtered homological algebra The purpose of this section is to establish a number of algebraic results that allow controlof the spectral range and boundary depth of filtered complexes through cone-attachments.7.1. Background on filtered complexes. We consider here filtered modules C over a ring R . We assume the filtration to be indexed by the reals and increasing, namely for every α ∈ R we have a submodule C ≤ α ⊂ C and C ≤ α ⊂ C ≤ α ′ for α ≤ α ′ . For simplicity we will alwaysassume that the filtration is exhaustive, i.e. ∪ α ∈ R C ≤ α = C .The shift of order s ∈ R of a filtered module C is the filtered module C [ s ] defined by( C [ s ]) ≤ α = C ≤ α + s . (Despite the similarity in notation, this has nothing to do with grading-shifts. In fact in this paper we work in an ungraded setting.) An R -linear map f : C −→ C ′ between two filtered modules is called s - filtered if f ( C α ) ⊂ ( C ′ ) ≤ α + s for all α ∈ R . We willrefer to such a number s as an admissible shift for the map f . We will also say that f shiftsaction by ≤ s , or sometimes that f is filtered of shift s . Notice that if f is s -filtered thenit is also s ′ -filtered for all s ′ ≥ s . An R -linear map f : C −→ C ′ is called filtered if it is s -filtered for some s ≥ 0. For reasons of convenience we will consider only shifts s that arenon-negative. There is no loss of generality in doing that as any map that shifts action by anegative amount can be viewed as 0-filtered. A slight drawback of this convention is that someof the estimates on invariants of filtered chain complexes developed below will be less sharp.Since our applications are concerned with coarse estimates this will not play an importantrole in our considerations.Let C be a filtered chain complex or R -modules. This means that C is a filtered moduleand the differential d of C preserves the filtration, i.e. d ( C ≤ α ) ⊂ C ≤ α for every α . To such achain complex we can associate a persistence module H ≤• ( C ) consisting of the homologies ofthe subcomplexes of C prescribed by the filtration: H ≤ α ( C ) = H ( C ≤ α ) , i β,α : H ≤ α ( C ) −→ H ≤ β ( C ) , α ≤ β, where the maps i β,α are induced by the inclusions C ≤ α ⊂ C ≤ β . We also have the maps i α : H ≤ α ( C ) → H ( C ) induced by the inclusions C ≤ α ⊂ C .The boundary depth of the filtered complex C is defined as: β ( C ) = inf { b ∈ [0 , ∞ ) | ∀ α ∈ R , ker( i α ) = ker( i α + b,α ) } . For every a ∈ H ( C ) we define the spectral invariant σ ( a ) by σ ( a ) = inf { α | a ∈ image i α } . We also define σ + ( C ) = inf { r ∈ R | t ≥ r ⇒ Coker( i t ) = 0 } ,σ − ( C ) = sup { s ∈ R | t ≤ s ⇒ i t = 0 } ,ρ ( C ) = σ + ( C ) − σ − ( C ) . As the notation suggests σ + ( C ) is the top (or supremal) spectral invariant of C and σ − ( C ) isthe bottom (or infemal) one. We call ρ ( C ) the spectral range of C . Remark . The notions above can easily be reformulated in terms of the modern terminol-ogy of barcodes [PRSZ]. For instance β ( C ) is the length of the longest finite bar of C . Further,if the bar code associated to H ≤• ( C ) is the collection { [ i k , j k ) } , then σ + ( C ) is the minimal i k among all bars with j k = ∞ and σ − ( C ) is the maximal i k among the same (infinite) bars.We now describe the behavior of σ and β with respect to some operations with filtered chaincomplexes. We begin with the simple remark that if f : C −→ C ′ is a quasi-isomorphism andis s -filtered then we have: σ − ( C ) ≥ σ − ( C ′ ) − s, σ + ( C ) ≥ σ + ( C ′ ) − s . (49)In particular, if f admits an s -filtered homological inverse, we deduce | σ ± ( C ) − σ ± ( C ′ ) | ≤ s , | ρ ( C ) − ρ ( C ′ ) | ≤ s . In order to relate the boundary depth of two quasi-isomorphic chain complexes we willneed the notion of boundary depth of a map. Let f : C −→ C ′ be a filtered chain map andlet s ≥ f . The map f induces a map of persistence modules f •∗ : H ≤• ( C ) −→ H ≤• ( C ′ )[ s ], f •∗ = { f α ∗ } with f α ∗ : H ≤ α ( C ) −→ H ≤ α + s ( C ′ ) induced by f . Wedefine the boundary depth of f , viewed as an s -filtered map, by: β s ( f ) = inf { b ∈ [0 , ∞ ) | ∀ α ∈ R , Image( f α ∗ ) ∩ ker( i α + s ) ⊂ ker( i α + s + b,α + s ) } . Clearly β ( C ) = β ( id C ), β s ( f ) ≤ β ( C ′ ) and for s ≤ s ′ , β s ′ ( f ) = max { , β s ( f ) − s ′ + s } .Assume now that f : C −→ C ′ , g : C ′ −→ C are s -filtered chain maps with g ∗ ◦ f ∗ = id inhomology. We have the inequality: β ( C ) ≤ max { β ( C ′ ) + 2 s, β s ( g ◦ f − id C ) } . (50)The simplest way to control the boundary depth of maps as above is by using filteredhomotopies. Let f, f ′ : C −→ C ′ be two s -filtered maps that are homotopic with a homotopy h : f ≃ f ′ which is s ′ -filtered, then: β s ( f − f ′ ) ≤ min { , s ′ − s } . (51)Assume now that f : C −→ C ′ , g : C ′ −→ C are s -filtered chain maps such that there is an s -filtered homotopy h : g ◦ f ≃ id C . In this case, β s ( g ◦ f − id C ) = 0 and we deduce that β ( C ) ≤ β ( C ′ ) + 2 s . Summing up: OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 41 Lemma 7.1.2. If f : C −→ C ′ and g : C ′ −→ C are s -filtered and there are s -filtered chainhomotopies h : g ◦ f ≃ id C and h ′ : f ◦ g ≃ id C ′ , then we have: | β ( C ) − β ( C ′ ) | ≤ s , | σ ± ( C ) − σ ± ( C ′ ) | ≤ s , | ρ ( C ) − ρ ( C ′ ) | ≤ s . (52)7.2. Mapping cones. Let A , B be filtered chain complexes and f : A −→ B an s -filteredchain map. The filtered mapping cone [ A ( f,s ) −−−→ B ] of f is the mapping cone of f endowedwith the following filtration: [ A ( f,s ) −−−→ B ] ≤ α = A ≤ α − s ⊕ B ≤ α . (53)Of course, this choice of filtration is somewhat ad-hoc and there are other possibilities. Firstly,once can shift the above filtration by any real number. The reason for the specific choice in (53)is to make the inclusion B −→ [ A ( f,s ) −−−→ B ] action preserving. Secondly, the filtration in (53)depends on s (and therefore this parameter appears in the notation).We will now estimate the boundary depth and spectral range of the mapping cone in termsof the invariants of its factors. Lemma 7.2.1. Let C := [ A ( f,s ) −−−→ B ] . We have the following inequalities: σ − ( C ) ≥ min { σ − ( B ) − β ( A ) , σ − ( A ) + s } , (54) σ + ( C ) ≤ max { σ + ( B ) , σ + ( A ) + β ( B ) + s } (55) and β ( C ) ≤ β ( A ) + β ( B ) + max { , σ + ( A ) − σ − ( B ) + s } . (56)Note that the estimates in the lemma do not depend on the chain map f (though they dodepend on the amount of shift s of f ). Proof of Lemma 7.2.1. The basic ingredient in the proof is provided by the long exact se-quences: · · · −→ H ≤ α − s ( A ) f −→ H ≤ α ( B ) h −→ H ≤ α ( C ) p −→ H ≤ α − s ( A ) −→ · · · , where h is induced by inclusion and p by the projection. The maps i α , i β,α relate functoriallythese exact sequences.To see (54) let c ∈ H ≤ α ( C ), α < min { σ − ( B ) − β ( A ) , σ − ( A )+ s } . Then p ( c ) ∈ H ≤ α − s ( A ) andas α − s < σ − ( A ), then i α − s ( p ( c )) = 0. This implies that i α + b − s,α − s ( p ( c )) = 0 for all b > β ( A ).We take b sufficiently small such that α < σ − ( B ) − b . Thus, there is c ′ ∈ H ≤ α + b ( B ) suchthat h ( c ′ ) = i α + b,α ( c ). But we also have that α + b < σ − ( B ) so that i α + b ( c ′ ) = 0. Therefore i α ( c ) = 0 which shows the the first inequality.The proof of (55) is similar. Indeed, if α > max { σ + ( B ) , σ + ( A ) + β ( B ) + s } and c ∈ H ( C ),then fix b > β ( B ) very close to β ( B ) such that α > σ + ( A )+ b + s . There exists c ′ ∈ H ≤ α − s − b ( A )such that i α − s − b ( c ′ ) = p ( c ) and moreover f ( i α − s,α − s − b ( c ′ )) = 0. Let c ′′ = i α − s,α − s − b ( c ′ ). Thereis c ′′′ ∈ H ≤ α ( C ) such that p ( c ′′′ ) = c ′′ . Now p ( i α ( c ′′′ ) − c ) = 0 therefore there is ˜ c ∈ H ( B ) such that h (˜ c ) = i α ( c ′′′ ) − c . But α > σ + ( B ) hence there is ˜ c ′ ∈ H ≤ α ( B ) such that i α (˜ c ′ ) = ˜ c .It follows that i α ( c ′′′ − h (˜ c ′ )) = c and thus i α : H ≤ α ( C ) −→ H ( C ) is surjective.Finally, to show (56), assume r > β ( A )+ β ( B )+max { , σ + ( A ) − σ − ( B ) } and let c ∈ H ≤ α ( C )such that i α ( c ) = 0. We want to show that i α + r,α ( c ) = 0. Note that i α + b − s,α − s ( p ( c )) = 0 for b > β ( A ). Let c ′ = i α + b,α ( c ). Therefore, there is c ′′ ∈ H ≤ α + b ( B ) with h ( c ′′ ) = c ′ . In case α + b < σ − ( B ), then i α + b ( c ′′ ) = 0 and thus for b ′ > β ( B ) we have i α + b + b ′ ,α + b ( c ′′ ) = 0. Thisimplies that i α + b + b ′ ,α ( c ) = 0 and, by taking b, b ′ small enough, this shows i α + r,α ( c ) = 0. Theother possibility to consider is when α + b ≥ σ − ( B ). In this case let ˆ c = i α + b ( c ′′ ). As h (ˆ c ) = 0there is ˆ c ′ ∈ H ( A ) such that f (ˆ c ′ ) = ˆ c . Now consider k > max { , σ + ( A ) + s − σ − ( B ) } .There exists ˆ c ′′ ∈ H ≤ α + b − s + k ( A ) such that i α + b − s + k (ˆ c ′′ ) = ˆ c ′ . Now f (ˆ c ′′ ) ∈ H ≤ α + b + k ( B ) and i α + b + k ( i α + b + k,α + b ( c ′′ ) − f (ˆ c ′′ )) = 0. Thus i α + b + b ′ + k,α + b + k ( i α + b + k,α + b ( c ′′ ) − f (ˆ c ′′ )) = 0 whichcombined with h ( f (ˆ c ′′ )) = 0 implies that i α + b + b ′ + k,α ( c ) = 0 which shows our claim by taking b, b ′ , k small enough. (cid:3) From inequalities (54), (55)) we deduce a simpler (but rougher) estimate for the spectralrange of C = [ A ( f,s ) −−−→ B ]: ρ ( C ) ≤ max { σ + ( A ) , σ + ( B ) } − min { σ − ( A ) , σ − ( B ) } + β ( A ) + β ( B ) + s (57)It is important to note that one can not, in general, eliminate the boundary depth fromestimates such as (54), (55)) or (57)) nor can one eliminate the spectral values σ + , σ − froman estimate like (56). Remarks . (1) Above we have considered s -morphisms with s ≥ 0. Occasionally itmakes sense to consider also the case s < − s )). The estimates (49) - (57) can be easilyadjusted to the case s < 0. However, for the applications needed in this paper it isenough to assume s ≥ A , B be two filtered chain complexes and f : A −→ B an s -filtered chain map,where we allow here any s ∈ R (also s < s ′ ≥ s . Then f is also an s ′ -filteredchain map. We can now endow the mapping cone of f with two different filtrations,following (53), once using the shift s and once the shift s ′ . Denote the correspondingfiltered mapping cones by C := [ A ( f,s ) −−−→ B ] and C ′ := [ A ( f,s ′ ) −−−→ B ]. It easily follows(e.g. from Lemma 7.1.2) that | σ ± ( C ′ ) − σ ± ( C ) | ≤ s ′ − s, | ρ ( C ′ ) − ρ ( C ) | , | β ( C ′ ) − β ( C ) | ≤ s ′ − s ) . (58)Next we analyze equivalences of mapping cones, taking into account filtrations. Considerthe following diagram: A ′ ( f ′ ,s f ′ ) / / ( ψ ′ ,s ψ ′ ) (cid:15) (cid:15) B ′ ( φ ′ ,s φ ′ ) (cid:15) (cid:15) A ′′ ( f ′′ ,s f ′′ ) / / B ′′ (59) OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 43 where A ′ , B ′ , A ′′ , B ′′ be filtered chain complexes and the notation on the edges of the squareare pairs consisting of a filtered chain map and an admissible shift. (E.g. ( f ′ , s f ′ ) means that f ′ : A ′ −→ B ′ is an s f ′ -filtered chain map etc.)We assume that (59) commutes up to an s h ′ -filtered chain homotopy h ′ : A ′ −→ B ′ (i.e. φ ′ ◦ f ′ − f ′′ ◦ ψ ′ = dh + hd ) for some s h ′ ≥ s f ′ , s φ ′ , s ψ ′ , s f ′′ . Further, assume that ψ ′ and φ ′ havefiltered homotopy inverses, i.e. there exist an s ψ ′′ -filtered chain map ψ ′′ : A ′′ −→ A ′ and an s φ ′′ -filtered chain map φ ′′ : B ′′ −→ B ′ with ψ ′′ ◦ ψ ′ = dk ′ + k ′ d, ψ ′ ◦ ψ ′′ = dk ′′ + k ′′ d,φ ′′ ◦ φ ′ = dr ′ + r ′ d, φ ′ ◦ φ ′′ = dr ′′ + r ′′ d, (60)where k ′ : A ′ −→ A ′ , k ′′ : A ′′ −→ A ′′ , r ′ : B ′ −→ B ′ , r ′′ : B ′′ −→ B ′′ are filtered linear maps.We denote by s k ′ , s k ′′ , s r ′ , s r ′′ admissible shifts for these maps.Denote by C ( f ′ , s f ′ ) := [ A ′ ( f ′ ,s f ′ ) −−−−−→ B ′ ] and by C ( f ′′ , s f ′′ ) := [ A ′′ ( f ′′ ,s f ′′ ) −−−−−→ B ′′ ] the filteredmapping cones of ( f ′ , s f ′ ) and ( f ′′ , s f ′′ ) respectively. Proposition 7.2.3. There exist filtered chain maps ϕ ′ : C ( f ′ , s f ′ ) −→ C ( f ′′ , s f ′′ ) and ϕ ′′ : C ( f ′′ , s f ′′ ) −→ C ( f ′ , s f ′ ) that fit into the following diagrams: A ′ f ′ / / ψ ′ (cid:15) (cid:15) B ′ φ ′ (cid:15) (cid:15) / / C ( f ′ , s f ′ ) ϕ ′ (cid:15) (cid:15) / / A ′ ψ ′ (cid:15) (cid:15) A ′′ f ′′ / / B / / C ( f ′′ , s f ′′ ) / / A ′′ A ′ f ′ / / B ′ / / C ( f ′ , s f ′ ) / / A ′ A ′′ ψ ′′ O O f ′′ / / B φ ′′ O O / / C ( f ′′ , s f ′′ ) ϕ ′′ O O / / A ′′ ψ ′′ O O (61) where the unmarked horizontal maps in both diagrams are the canonical chain maps associatedto cones. These maps are filtered. The left-hand square in the 2’nd diagram commutes upto a filtered chain homotopy h ′′ . The 2’nd and 3’rd squares, in each diagram, commute.The compositions ϕ ′′ ◦ ϕ ′ and ϕ ′ ◦ ϕ ′′ are chain homotopic to the identities via filtered chainhomotopies H ′ and H ′′ . Moreover, there exist admissible shifts s ϕ ′ , s ϕ ′′ , s H ′ , s H ′′ , s h ′′ for ϕ ′ , ϕ ′′ , H ′ , H ′′ , h ′′ , and a universal constant C (that depends neither on the initial diagram noron any of the other maps mentioned above) such that s ϕ ′ , s ϕ ′′ , s H ′ , s H ′′ , s h ′′ ≤ C ( s f ′ + s f ′′ + s φ ′ + s φ ′′ + s ψ ′ + s ψ ′′ + s h ′ + s k ′ + s k ′′ + s r ′ + s r ′′ ) . (62) Proof. The existence of ϕ ′ , ϕ ′′ , H ′ , H ′′ is standard homological algebra. In fact, it is straight-forward to write down explicit formulae for these maps. For example, ϕ ′ can be taken to be ϕ ′ ( a ′ , b ′ ) = ( ψ ( a ′ ) , φ ′ ( b ′ ) + h ′ ( a ′ )). One then uses the chain homotopies k ′ , k ′′ , r ′ , r ′′ to describeexplicitly h ′′ , ϕ ′′ and H ′ , H ′′ .The only possibly non-standard ingredients are the statements concerning the actions shiftsand inequality (62). These can be easily derived from the formulae for ϕ ′ , ϕ ′′ , h ′′ , H ′ , H ′′ . (cid:3) Remark . By deriving explicit formulae for ϕ ′ , ϕ ′′ , H ′ , H ′′ , h ′′ it is possible obtain sharperestimates for each of s ϕ ′ , s ϕ ′′ , s H ′ , s H ′′ , s h ′′ than the uniform bound (62). In the following we will be interested only in coarse estimates on these shifts, hence we will not need such sharpestimates.7.3. Iterated cones. Let E , F , G be filtered chain complexes and f : F −→ G an s f -filteredchain map. Let g : E −→ [ F ( f,s f ) −−−−→ G ] be an s g -filtered chain map and define C = [ E ( g,s g ) −−−−→ [ F ( f,s f ) −−−−→ G ]] . There exist a module homomorphism g ′ : E −→ F that shifts action by ≤ s g ′ := max { , s g − s f } , and another module homomorphism f ′ : [ E ( g ′ ,s g ′ ) −−−−→ F ] −→ G that shifts action by ≤ s f ′ := s f , such that the module C ′ = [[ E ( g ′ ,s g ′ ) −−−−→ F ] ( f ′ ,s f ′ ) −−−−−→ G ]is isomorphic to C by the map C −→ C ′ induced from the underlying identity map. Moreover,if s g < s f (i.e. s g ′ = 0) then this map shifts action by ≤ ( s f − s g ) and if s g ≥ s f (i.e. s g ′ ≥ ≤ Remark . The asymmetry in the action shifts comes from our convention to consideronly non-negative action shifts, i.e. to regard a map that shifts action by a negative amountas shifting action by ≤ 0. If we would have allowed for negative action-shifts then we couldtake s g ′ = s g − s f and the identity map C −→ C ′ would become action preserving. But asremarked at the beginning of § | σ ± ( C ′ ) − σ ± ( C ) | ≤ | s f − s g | , | β ( C ′ ) − β ( C ) | ≤ | s f − s g | . (63)In the following we will be interested in coarse bounds on spectral invariants and boundarydepths of iterated cones. Therefore, by abuse of notation we will often write them as K =[ A r −→ A r − −→ · · · −→ A −→ A ], whenever the maps are clear from the context andtheir action shifts are fixed up to a bounded change. The spectral invariants and boundarydepths of K will then be determined up to a bounded error.7.4. Estimating the spectral range of iterated cones. Let A (0) , . . . , A ( k ) be a finite col-lection of filtered chain complexes of R -modules. Assume that each of the A ( i ) ’s has finitespectral range. Define the following values: e σ + ( A ( k ) , . . . , A (0) ) := max { σ + ( A ( k ) ) , . . . , σ + ( A (0) ) } , e σ − ( A ( k ) , . . . , A (0) ) := min { σ − ( A ( k ) ) , . . . , σ − ( A (0) ) } , e ρ ( A ( k ) , . . . , A (0) ) := e σ + ( A ( k ) , . . . , A (0) ) − e σ − ( A ( k ) , . . . , A (0) ) . (64) OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 45 From inequalities (54) - (57), and using the notation (64), we obtain the following inequal-ities for the mapping cone C = [ A ( f,s ) −−−→ B ] of an s -filtered chain map f : A −→ B : σ + ( C ) ≤ e σ + ( A, B ) + β ( B ) + s, − σ − ( C ) ≤ − e σ − ( A, B ) + β ( A ) ,β ( C ) ≤ β ( A ) + β ( B ) + e σ + ( A, B ) − e σ − ( A, B ) + s. (65)It follows that both ρ ( C ) as well as β ( C ) can be bounded from above by the same expression: ρ ( C ) , β ( C ) ≤ e ρ ( A, B ) + β ( A ) + β ( B ) + s. (66)Turning to the case of iterated cones, let A , . . . , A r be filtered chain complexes. Put C := A . Let ϕ : A −→ C be an s -filtered chain map for some s ≥ 0. Define C :=[ A ϕ ,s ) −−−−→ C ], filtered as described in (53). Continuing inductively, assume that we haveconstructed already the filtered chain complex C i for some 1 ≤ i ≤ r − ϕ : A i +1 −→ C i be an s i +1 -filtered chain map for some s i +1 ≥ 0. Define C i +1 = [ A i +1 ( ϕ i +1 ,s i +1 ) −−−−−−−→ C i ]. We callthe final chain complex C r an iterated cone with attachments A , . . . , A r and sometime denoteit by C r = [ A r −→ [ A r − −→ · · · −→ [ A −→ [ A −→ A ]] · · · ]] , omitting references to the chain maps ϕ i and the action-shifts s i .The following Lemma follows easily from (65). Lemma 7.4.1. There exists (universal) constants a r , b r , e r > , depending only on r , suchthat for every iterated cone C r as above we have: ρ ( C r ) ≤ a r e ρ ( A r , . . . , A ) + b r r X j =0 β ( A j ) + e r r X j =1 s j . (67)7.5. Weakly filtered A ∞ -categories and modules. Recall that a weakly filtered A ∞ -category C is an A ∞ -category such that for every two objects X, Y ∈ Ob( C ) the chain complex(hom C ( X, Y ) , µ C ) is filtered and additionally each of the higher order operations µ C d , d ≥ M over suchcategories are C -modules such that for every object X ∈ Ob( C ) the chain complex ( M ( X ) , µ M )is filtered, and the higher order operations µ M d , d ≥ 2, preserve filtrations up to (uniform)bounded errors (one for each d ). One can define weakly filtered pre-module (resp. module)homomorphisms f : M −→ N between weakly filtered modules, by analogy to filtered maps(resp. chain maps). The 1’st order component f : M ( X ) −→ N ( X ), X ∈ Ob( C ), of such amap is a filtered linear map (resp. chain map) that shifts filtrations by ≤ s f , where s f is aconstant that does not depend on X . An analogous condition is imposed on the higher order f d components of f . (Sometimes, by abuse of notation we will omit the the subscript in f anddenote the 1’st order component also by f .) Finally, there is also the notion of weakly filtered A ∞ -functors between weakly filtered A ∞ -categories (in contrast to module homomorphismswhich are allowed to shift filtrations, such functors are assumed to preserve filtrations, up to bounded errors). We refer the reader to [BCS] for the basic theory and formalism of weaklyfiltered A ∞ -categories. Remark . A word of caution about terminology differences is in order. The notion “weaklyfiltered” appears in the literature with two different meanings. In the formalism of [FOOO1,FOOO2] “weakly filtered map” stands for a map between filtered chain complexes (or A ∞ -algebras) that preserves filtrations up to a shift, whereas in our terminology such maps arecalled “filtered” or s -filtered if we specify the amount of shift s . Our notion of “weaklyfiltered” means something else. For example, in the case of weakly filtered categories, the 1’storder operations (i.e. the differentials of the hom’s) preserve filtrations, but the higher orderoperations preserve filtrations only up to uniform errors (which we call in [BCS] discrepancies),and the wording “weakly” refers to that. Thus, without these discrepancies we would havecalled such categories “filtered categories”. In a similar vein we have weakly filtered functors,modules and (pre)-module homomorphisms.The contents of the entire section above ( § § A ∞ -modules over a weakly filtered A ∞ -category C ratherthan just chain complexes. For example, if one replaces the filtered chain complexes A , B byweakly filtered C -modules A , B and f : A −→ B by a module homomorphism, then one candefine an A ∞ -mapping cone module C = [ A f −→ B ] which is weakly filtered in a similar way asin (53). (See [BCS, § C ′ and C replaced by C ′ ( X ) and C ( X ) respectively, for every object X in the underlying A ∞ -category C . Similar modifications apply to (63) as well as to (52).It is important to note that in the case of A ∞ -modules the preceding inequalities holduniformly for all objects X , since the shift parameters ( s f , s g etc.) depend only on themodules and the homomorphisms between them, and not on the choice of a particular objectin the A ∞ -category. Remark . Through this paper we appeal several times to the notions of weakly filtered A ∞ -categories, functors and modules. However, from a purely formal viewpoint this is not reallynecessary. Indeed, we will never use any of the higher operations associated to A ∞ -structuresor to special features that distinguish such structures from filtered chain complexes. Thus inprinciple one can “downgrade” the entire algebraic formalism in this paper to filtered chaincomplexes and their persistent homology. The reason we opted for using a bit of A ∞ formalismis the following. A considerable part of the algebra in this paper is devoted to establishingbounds on invariants of filtered Floer chain complexes, e.g. of the type CF ( − , − ), which areuniform in the “variables” ( − , − ), or at least one of them. These variables are Lagrangiansubmanifolds, hence are objects of a Fukaya category (which is weakly filtered). As explainedat several points above, the uniformity of various quantities related to action filtration can bemore concisely expressed using the language of A ∞ -modules.We end this section with a useful definition. OUNDS ON THE LAGRANGIAN SPECTRAL METRIC IN COTANGENT BUNDLES 47 Definition 7.5.3. Let M , N be two weakly filtered A ∞ -modules. Let f : M −→ N be aweakly filtered module homomorphism and w ≥ 0. We say that f is a quasi-isomorphism ofweight ≤ w if the following holds:(1) f shifts filtration by ≤ w .(2) There exists a weakly filtered module homomorphism g : N −→ M that shifts fil-tration by ≤ w and two weakly filtered pre-module homomorphisms h : M −→ M , k : N −→ N that shift filtrations by ≤ w , such that: g ◦ f = id + µ mod1 ( h ) , f ◦ g = id + µ mod1 ( k ) . (68)We say that two weakly filtered modules M and N are at distance w one from the other ifthere exists a quasi-isomorphism f : M −→ N of weight ≤ w . Remark . Similar notions appear in relation to the so-called bottleneck distance in per-sistance module theory, for instance in [UZ], as well as in a somewhat different context in[BCS].The same definition can be easily adapted to the case when M and N are just filteredchain complexes and f : M −→ N is a w -filtered chain map. In this case, the analogue ofcondition (68) simply means that f ◦ g and g ◦ f are chain homotopic to the respective identitiesvia w -filtered chain homotopies. Note that despite being called only a “quasi-isomorphism”, f satisfies a stronger condition - it is implicitly assumed to have a homotopy inverse.7.6. Spectral range and boundary depth of tensor products. Let A , B be finite di-mensional filtered chain complexes over a field R . The tensor product (over R ) chain complex A ⊗ B inherits a filtration from A and B , where ( A ⊗ B ) ≤ α ⊂ A ⊗ B is generated by thecollection of subspaces A ≤ α − s ⊗ B ≤ s , s ∈ R . Proposition 7.6.1. For the tensor product chain complex A ⊗ B we have: σ ± ( A ⊗ B ) = σ ± ( A ) + σ ± ( B ) , ρ ( A ⊗ B ) = ρ ( A ) + ρ ( B ) ,β ( A ⊗ B ) ≤ max { β ( A ) , β ( B ) } . Proof. This follows by direct calculation of the barcode of the persistence module H ∗ ( A ⊗ B ),using the K¨unneth formula for persistence modules from [PSS]. (cid:3) References [Alb] P. 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