SPECTRAL INVARIANTS IN LAGRANGIAN FLOER THEORY

Let $(M,\omega)$ be a symplectic manifold compact or convex at infinity. Consider a closed Lagrangian submanifold $L$ such that $\omega |_{\pi_2(M,L)}=0$ and $\mu|_{\pi_2(M,L)}=0$, where $\mu$ is the Maslov index. Given any Lagrangian submanifold $L'$, Hamiltonian isotopic to $L$, we define Lagrangian spectral invariants associated to the non zero homology classes of $L$, depending on $L$ and $L'$. We show that they naturally generalize the Hamiltonian spectral invariants introduced by Oh and Schwarz, and that they are the homological counterparts of higher order invariants, which we also introduce here, via spectral sequence machinery introduced by Barraud and Cornea. These higher order invariants are new even in the Hamiltonian case. We provide a way to distinguish them one from another and estimate their difference in terms of a geometric quantity.


Introduction
Hamiltonian spectral invariants for Floer homology have been defined by Schwarz [19] and Oh [16], following the work of Viterbo [21]. Schwarz worked in the symplectically aspherical case and Oh has treated the general case. Oh [15] and Milinković [13] also defined a Lagrangian version of these invariants for cotangent bundles.
In this paper, we define a new type of Lagrangian spectral invariants. The novelty is twofold. Firstly, we use the PSS morphism to define them (this is the same approach as Schwarz's in the Hamiltonian case). Secondly, our invariants are of higher order, in the sense that they are associated to each non zero class α ∈ E r p,q (L; X) where E(L; X) (also denoted E X (L)) is a purely topological Serre spectral sequence associated to L (and an auxiliary space X) via a construction due to Barraud and Cornea [3]. These invariants are new even in the Hamiltonian case, where they extend the classical invariants of Oh and Schwarz which are recovered for r = 2. Moreover they carry strictly more information than the classical invariants, even in the Morse case. We study some properties of these invariants and we describe, in particular, some geometric way to distinguish them -this part is again new even when applied to the usual Hamiltonian case. Here are now our main results described in more detail.
Let (M, ω) be a symplectic manifold, compact or convex at infinity, and L be a closed Lagrangian submanifold such that ω| π2(M,L) = 0 and µ| π2(M,L) = 0 with µ the Maslov index. Let (H, J) be a regular pair, formed of a Hamiltonian function and an almost complex structure. Let φ 1 H be the time-1 diffeomorphism induced by H. We denote by L ′ := φ 1 H (L) and we assume L ∩ L ′ transverse. We will start with the definition of order 2 (or homological) spectral invariants since this construction will be more familiar to the reader. All homology theories used here are with coefficients in Z 2 . In our setting, the Lagrangian Floer complex CF * (L, L ′ ; H, J) is well-defined and admits a natural filtration by action, CF ν * (L, L ′ ; H, J). In the Hamiltonian case, Piunikhin, Salamon and Schwarz [18] introduced a particular morphism between the Morse and Floer complexes inducing an isomorphism in homology. It has been adapted to the Lagrangian setting by Katić and Milinković [10] for cotangent bundles and in more generality by Barraud and Cornea [2], and Albers [1] Along these lines, homological Lagrangian spectral numbers will refer to this absolute version. The distinction relative vs absolute reflects the fact that in the Lagrangian version of Floer theory, the action functional is defined up to translation by a constant due to the choice of η, in some fixed homotopy class of paths from L to L ′ .
The construction of these numbers requires various choices but, as we shall see, they are strongly invariant. Recall that Hofer's distance between two Hamiltonian isotopic Lagrangian submanifolds, denoted ∇(L, L ′ ), is the infimum of the energies of Hamiltonian diffeomorphisms which carry L to L ′ . The most delicate part of the proof of invariance consists in showing the commutativity of a certain diagram (20). This is shown via methods inspired by Seidel [20] and which have been used in the Hamiltonian case by McDuff and Salamon [12].
These quantities are a natural generalization of the Hamiltonian spectral invariants. Indeed, let ∆ be the diagonal in M ×M , and α ∈ H * (∆) be the homology class corresponding to α ∈ H * (M ), via the obvious isomorphism. For φ ∈ Ham(M, ω), we denote by ρ(−; φ) its associated spectral invariant and by Γ φ its graph. We have c(α; ∆, Γ φ ) = ρ(α; φ) − ρ(1; φ) (1) where 1 denotes the generator of H 0 (M ). This is shown via an isomorphism due to Biran, Polterovich and Salamon [5]. The Lagrangian spectral invariants, as defined before, are easily seen to satisfy two properties, similar to the ones valid in the Hamiltonian case, and listed below. Viewing the Hamiltonian spectral invariants as particular Lagrangian ones via (1), assertion (2.) improves the well-known upper bound given by Hofer's norm of φ. We will illustrate this fact via a family of Hamiltonian diffeomorphisms given by Ostrover [17].
The Lagrangian spectral invariants defined above are actually the simplest particular cases of higher order spectral invariants which we now discuss. These higher order invariants are denoted c r X,[l] (α; L, L ′ ) and c r X,[l] (α; L, L ′ ) where l : L → X is a map with the space X simply-connected, [l] is its homotopy class and α is any non zero element of the r-th page of a spectral sequence E X (L) defined as follows. We consider the path-loop fibration ΩX → P X → X. We pull-back this fibration over the map l and we let E X (L) be the Serre spectral sequence of this pull-back fibration.
It has been shown by Barraud and Cornea [2] that this spectral sequence can be related by a PSS type isomorphism to a spectral sequence induced by a natural filtration of an enriched Floer type complex. In view of this, Definition 1.1 can be adapted to this element α (see definitions 4.1 and 4.2).
The second page of E X (L) is purely homological, in the sense that E 2 p,q (L; X) ≃ H q (ΩX) ⊗ H p (L). Thus, any element α = 0 in H p (L) can be identified with the element 1 ⊗ α in E 2 p,0 (L; X) where 1 denotes the generator of H 0 (ΩX). For such an element, all three Lagrangian invariants coincide: The higher order invariants generalize the classical spectral invariants (in the previous sense) and carry strictly more information (even in the Morse case). We will illustrate this fact with explicit computations in a particular case. As suggested by our notation, given X, [l] and r, c r X,[l] (−; L, L ′ ) and c r X,[l] (−; L, L ′ ) have the same invariance property as their homological counterparts.
The main property that will be of interest to us here is that E X (L) provides a way to distinguish among these invariants, in terms of a quantity related to the geometry of L and L ′ . This geometric constant is defined, for every pair of transverse Lagrangian submanifolds, as follows. Let x be an element of L ∩ L ′ . There exist a real number ε > 0 and an embedding, e x ε , of the ball This gives rise to the following definition.
Definition 1.4. Given L and L ′ two compact, transverse Lagrangian submanifolds, we define This definition is inspired by the geometric distance introduced by Barraud and Cornea [3]. Notice that, since L ∩ L ′ is a finite set, r(L, L ′ ) > 0. With these conventions we have: πr(L, L ′ ) 2 2 for any Lagrangian submanifold L ′ , Hamiltonian isotopic (and transverse) to L.
This property of Lagrangian spectral invariants of higher order has the following immediate consequence for the invariants of Definition 1.1. Corollary 1.6. Let X be (r − 1)-connected and let {x i } be a basis of H r−1 (ΩX). If there exists a non trivial differential d r : This corollary has interesting consequences.
Corollary 1.7. Let · denote the intersection product and let [L] be the fundamental class of L.
(1.) For α ∈ H k (L) and β ∈ H * (L), with 1 < k < n − 1 and α · β = 0, we have As we shall see, in this purely homological case, there is another way to prove these properties by adapting to the Lagrangian setting the proofs valid in the Hamiltonian case (and used by Schwarz [19]) and combining this with the definition of r(L, L ′ ). The key point of this alternate proof is an adequate description of a certain module structure on HF * (L, L ′ ) over H * (L) endowed with its intersection product. The cases k = 1 and k = n − 1 of the previous corollary are proved here via this method, in the sense that assertions (1.) and (2.) hold as long as 0 < k < n. In particular, for all non zero α ∈ H k (L), 0 ≤ c(α; L, L ′ ) ≤ c([L]; L, L ′ ).
Finally we get, as an obvious consequence of this extension of Corollary 1.7 (1.) and Proposition 1.3 (2.), a lower bound for Hofer's distance between two transverse, Hamiltonian isotopic Lagrangian submanifolds in terms of our geometric constant and the cup-length of L.
Organization of the paper. In Section 2, we first quickly recall the construction of Morse and Floer homologies and three kinds of comparison morphisms, namely the classical comparison morphism for Lagrangian Floer homology, a Lagrangian version of the PSS morphism and the naturality morphism.
In Section 3, we construct our Lagrangian spectral numbers and we prove Theorem 1.2. The invariance part relies on two distinct steps. First we show that a certain diagram commutes. In order to do this, we prove that the Lagrangian Floer homology HF * (L, L ′ ) can be viewed as a module over the homology of L, endowed with its intersection product. This algebraic structure is an adaptation of a construction made by Floer [7] in the Hamiltonian case. Then we show that the naturality and Lagrangian PSS morphisms preserve this structure. The desired commutativity follows immediately. In a second step, we adapt methods used by Schwarz in the Hamiltonian case, to get an estimate on Lagrangian spectral numbers. This estimate ends the proof of their invariance and implies the continuity part of Theorem 1.2. Finally we show how the Hamiltonian spectral invariants can be viewed as particular Lagrangian ones, by proving (1).
In Section 4, we first sketch the construction of the Barraud-Cornea spectral sequence and its relation with the Serre spectral sequence of the path-loop fibration of L. Then we introduce the Lagrangian spectral invariants of higher order and prove Theorem 1.5. We also produce explicit computations of the higher order spectral invariants and show, via this example, that they carry more information than the classical invariants.
Finally (in Section 5), we come back to the homological Lagrangian spectral invariants. Corollary 1.6 follows immediately from the results obtained on the higher order invariants. We show how it implies Corollary 1.7. Then, we prove Proposition 1.3 and Corollary 1.8.
Acknowledgements. The author would like to deeply thank his supervisor, Octav Cornea, for hours of extremely fruitful and enlightening discussions. He would also like to thank Matthias Schwarz for valuable conversations, Peter Albers for numerous interesting comments and discussions and Leonid Polterovich for useful questions.
2.1.1. Morse homology. We briefly sketch this well-known construction. We work over Z 2 , and fix a closed manifold L n , a metric g and a Morse function f such that the pair (f, g) is Morse-Smale. We denote the gradient of f with respect to g by ∇f . The gradient flow of −f , defined by d dt γ t + ∇f (γ t ) = 0, induces stable and unstable manifolds for every critical point p: Let i f (p) be the Morse index of p. As f is Morse-Smale, the connecting manifold of two critical points p and q, M p,q (f, g) As R acts on this space, one can define M p,q (f, g) := M p,q (f, g)/R. In general, these moduli spaces are not compact but admit a compactification such that Let Crit k (f ) be the set of critical points of f with Morse index k. We define free Z 2 -modules CM k (L; f, g) := Crit k (f ) Z2 and a differential ∂ by the formula where # 2 M p,q (f, g) denotes the mod 2 cardinal of M p,q (f, g). The compactification (3) implies that ∂ 2 = 0. Thus (CM * (L; f, g), ∂) is a chain complex whose homology is the Morse homology of L. It is a well-known fact that it does not depend on the Morse-Smale pair (f, g) and is the (cellular) homology of L.

Lagrangian Floer homology.
This theory is exposed in Floer [6], Oh [14]. Let (M, ω) be a symplectic manifold, compact or convex at infinity. Let L and L ′ be two closed, Hamiltonian isotopic Lagrangian submanifolds of M such that ω| π2(M,L) = 0 and µ| π2(M,L) = 0 (4) where µ is the Maslov index (its definition is recalled below). We also pick a compactly supported time-dependent Hamiltonian function H and an almost complex structure J, ω-compatible. When M is not compact but convex at infinity, J will also assumed to be adapted to the boundary (see [12] for a detailed description of this condition). From now on, I denotes the interval [0, 1].
H induces a vector field X H : R × M → T M given by ω(X t H , −) = −dH t which gives rise to a family of symplectomorphisms, φ H : We assume that L ′ and φ 1 H (L) are transverse. First we define P(L, L ′ ) := {γ ∈ C ∞ (I, M ) | γ(0) ∈ L, γ(1) ∈ L ′ } and we fix η ∈ P(L, L ′ ). Let P η (L, L ′ ) be the connected component of P(L, L ′ ) containing it. The action functional A H : P η (L, L ′ ) → R is given by: Remark 2.1. In Hamiltonian Floer homology, it is natural to normalize the Hamiltonian. This induces a translation of the action by a constant. In the Lagrangian intersection setting, one can normalize the action functional by assuming that A H (η) = 0. In that case, another choice of the reference path η ′ in the same homotopy class also implies a translation of the action by a constant. Along these lines, we will emphasize our choices of reference as soon as they become crucial.
The set of the critical points of A H , denoted by I(L, L ′ ; η, H) ⊂ P η (L, L ′ ), consists of orbits of the Hamiltonian vector field. It is in one-to-one correspondence with a subset of φ 1 H (L) ∩ L ′ (which is finite since we assume transversality and compactness). A Floer trajectory is a smooth map u : R × I → M , satisfyinḡ H (u)) = 0. Its energy is defined by the formula A Floer trajectory has finite energy if and only if its ends are orbits of the Hamiltonian vector field. For x and y ∈ I(L, L ′ ; η, H), we form the following moduli space: All its elements have the same (finite) energy, satisfying There is a R-action on M x,y (L, L ′ ; H, J) and we define: Trivializing v * T M induces a loop γ v in L(R 2n ), the set of all Lagrangian submanifolds in R 2n . The degree of µ(γ v ) (µ : L(R 2n ) → S 1 is the Poincaré dual of the Maslov cycle) does not depend on the trivialization or on the choice of v (since µ| π2(M,L) = 0). It is denoted µ(x, y). Since µ is additive, we fix z 0 and define µ := µ(−, z 0 ).

Remark 2.2.
Choosing another reference z ′ 0 for the Maslov index implies a shift of degrees. Like for the reference of the action functional, this choice will in general be implied and will be emphasized in some key places.
Similarly to the Morse case, we define When µ(x, y) = 1, M x,y (L, L ′ ; H, J) is not compact in general but admits a compactification (Gromov's compactness theorem for the compact or convex at infinity setting). As condition (4) prohibits bubbling we get (the case µ(x, y) > 1 has been treated by Barraud and Cornea [3]) In particular, this implies that ∂ 2 = 0 and thus (7) defines a chain complex. Its homology is the Lagrangian Floer homology of M relatively to L and L ′ : It is well-known that this homology does not depend on the choice of the regular pair (H, J) (see §2.2.1 and §2.2.3 for two different proofs of this fact).

Comparison morphisms.
We present here three morphisms which all induce an isomorphism in homology. . Moreover we assume that there is a compact set, containing the support of all these Hamiltonian functions.
H 01 induces a family of vector fields X 01 and a family of symplectomorphisms φ 01 via ω(X 01 , −) = −dH 01 and ∂ t φ 01 = X 01 (φ 01 ). The equation satisfied by the Floer trajectories u : ∇ s x H 01 is the gradient of H 01 induced by the metric associated to J 01 s . The boundary conditions are u(R, 0) ⊂ L and u(R, 1) ⊂ L ′ . We also require their energy to be finite: which is anew equivalent to the existence of orbits x 0 ∈ I(L, L ′ ; η, H 0 ) and y 1 ∈ I(L, L ′ ; η, H 1 ) such that u(−∞, −) = x 0 and u(+∞, −) = y 1 . For generic choice (H 01 s , J 01 s ) is regular and for such a regular pair M x0,y1 (L, L ′ ; H 01 , J 01 ) is a smooth µ(x 0 , y 1 )-dimensional manifold. After compactification, the boundary of these moduli spaces is the disjoint union: Therefore, the formula is easily seen to define a morphism of chain complexes inducing an isomorphism
we get, via (5), E( u) = E(u). Notice that the two action functionals correspond for our choices of reference η and η if we assume that they have been normalized (see Remark 2.1). Therefore b H induces a bijection which gives rise to an action preserving identification of chain complexes: The Hamiltonian Piunikhin-Salamon-Schwarz (PSS) morphism [18] has been adapted to the Lagrangian setting, for the particular case of cotangent bundles by Katić and Milinković [10] and in more generality by Barraud and Cornea [2], and Albers [1]. This morphism compares the Morse and Floer complexes and induces an isomorphism in homology.
To p ∈ Crit(f ) and x ∈ I(L, L; η, H), we associate where β(s) is a smooth, increasing function, whose value is 0 for s ≤ 1/2 and 1 for a (i f (p) − µ(x))-dimensional manifold (notice that this requires a particular choice of the reference of the Maslov index). Its 0-dimensional component is compact and its 1-dimensional component admits a compactification such that (see Figure 1) Figure 1. Definition of the Lagrangian PSS morphism The morphism φ H f : CM * (L; f, g) → CF * (L, L; H, J), defined on generators by which induces ψ f H : HF * (L, L; H, J) → HM * (L; f, g). These two morphisms commute with the classical comparison morphisms, as stated below. Lemma 2.3. The following diagrams commute: To prove it, we study cobordisms (see Figure 2) similar to the ones giving that whereJ 0 andJ 1 are 1-parameter families of almost complex structures, such that . Up to generic choices, it is a smooth (µ(y 0 ) − µ(y 1 ) + 1)-dimensional manifold. Its 0-dimensional component is compact. Hence one can define a chain homotopy ϕ : CF * (L, L; H 0 , J 0 ) → CF * (L, L; H 1 , J 1 ), by the formula

Its 1-dimensional component admits a compactification such that
x1∈I(L,L;η,H1) (12) and (15) arise when R converges to a real number, (13), (14) when it converges to 0 and to infinity. Boundary (13), M φ•ψ 0 (y 0 , y 1 ), consists of pairs of discs which are pseudo-holomorphic near their common point. They can be glued (see [9] and [4]) to give the Floer trajectories defining the comparison morphism and then where α R is a smooth cut-off function, whose value is 1 for |s| < R and 0 for |s| > R + 1. We also assume that it is C 1 -bounded to get an estimate on discs energy -see [1]) The parameter R converges to a real number in (16), (19), to infinity in (18) and to 0 in (17). When R is infinite, the products lead to the composition φ H f0 • ψ f1 H , whereas when R = 0, one gets a pseudo-holomorphic disc with boundary in L, with null symplectic area (since ω| π2(M,L) = 0). Hence it has to be constant. Therefore, when R converges to 0, we get two critical points p 0 and p 1 , with a Morse flow line (of f 01 ) passing through them (flow lines unicity), with the same indices. Thus we recover the moduli spaces defining ψ 01 Morse , which proves that diagram (11) commutes.

Proof of Theorem 1.2
We consider the action filtration of the Floer complex. For a real number ν and an integer k, one can define the Z 2 -vector space: • φ H f is, from this point of view, a more natural candidate than the PSS morphism in the study of spectral numbers.
Organization of the next four sections -In the next two, we discuss additional algebraic structures required to prove Proposition 3.1. In Section 3.1, we show that Floer homology can be viewed as a module over Morse homology (endowed with the intersection product). In Section 3.2, we show that the naturality and the Lagrangian PSS morphisms agree with this structure which will be seen to immediately prove Proposition 3.1. In Section 3.3, we show the independence on the almost complex structure and the continuity property. Then we collect all the results such as to prove Theorem 1.2. In Section 3.4, we prove an additional property, namely that these invariants are a generalization of the Hamiltonian ones, by showing that equality (1) holds.
3.1. Lagrangian Floer homology as a module over Morse homology. In this section, we recall that HM * (L) is a unitary ring and we equip HF * (L, L ′ ) with a module structure over it.
3.1.1. Morse homology ring. We recall the Morse theoretic version of the intersection product. It is defined at the chain level: where f 1 , f 2 and f 3 are Morse functions and g a metric such that for all i and j ∈ {1, 2, 3}, W u pi (f i , g) and W s pj (f j , g) intersect transversally for any p i , p j respectively critical points of f i and f j . This implies in particular that the three pairs (f i , g) (i = 1, 3) are Morse-Smale. These choices are generic.
To p ∈ Crit(f 1 ), q ∈ Crit(f 2 ) and r ∈ Crit(f 3 ), we associate the moduli space: . Under our assumptions, these spaces are manifolds of dimension: Their 0-dimensional component is compact, then one can define the product on the generators of the chain complex (and extend it by bilinearity): One can show that this formula induces a product in homology which is independent on the choices of metric and Morse functions. We recall that the unity of (HM * (L), ·) is [L], the fundamental class of L.

3.1.2.
Floer homology as a module over Morse homology. We define, analogously, an external product on HF * (L, L ′ ): As before, it is defined at the chain level. In order to shorten notation, L and L ′ do not appear in what follows. For example, the moduli spaces M x,y (L, L ′ ; H, J) defined by (6) are denoted here and in Section 3.2 by M x,y (H, J).
We choose a Morse-Smale pair (f, g) and a regular pair (H, J). We associate, to p ∈ Crit(f ) and x, y ∈ I(L, L ′ ; η, H), the following moduli space:  Proof. The main difficulty is to prove that, for any α, β in HM * (L; f, g) and any a in HF * (L, L ′ ; H, J), the following equality holds: (α · β) * a = α * (β * a).
In the following two steps, we define moduli spaces M 0 (p,q,x);y (f 1 , f 2 , g; H, J) and a morphism ϕ 0 (p, q, x) := y # 2 M 0 (p,q,x);y (f 1 , f 2 , g; H, J) · y and we conclude the proof by showing that Notice that, when q = m, the maximum of f 2 , i f2 (m) = n and thus the moduli spaces defining m * x and p · m have such a dimension that, even on the chain level, we have m * x = x and p · m = p. Hence, it holds (step 1) First, for R > 0, we consider the moduli spaces Let M 0 (p,q,x);z (f 1 , f 2 , g; H, J) be obtained from M R (p,q,x);z (f 1 , f 2 , g; H, J) when R converges to 0. Notice that we get M + (p,q,x);z (f 1 , f 2 , g; H, J), when R converges to infinity. Hence, the boundary of the compactification of its 1-dimensional component is the disjoint union of (see Figure 4) The boundary of the (compactified) 1-dimensional part, is the disjoint union when it converges respectively to −∞ (36), (37) and 0 (38) (see Figure 5). Here, Γ denotes the unique flow line of f 2 (if it exists), element of the unstable manifold of q and reaching x(0).
However, semi-pairs of pants require distinct Hamiltonian perturbations at their ends. Thus the naturality morphism is not a priori compatible with the moduli spaces appearing in this description.

Naturality and Lagrangian PSS as morphisms of modules.
In this section we prove that these two isomorphisms preserve the algebraic structures described above. More precisely, we prove the following lemma.
For every α and β in HM * (L), it holds . Boundaries are now (see Figure 6):  When S converges to a real number, one gets (see Figure 7)   Therefore, Φ is the identity and the proposition is proved.

3.3.
End of the proof of Theorem 1.2. We first give a standard result (Lemma 3.6 below) used in the Hamiltonian case by Schwarz [19]. We recall that a timedependent Hamiltonian function H is said to be normalized if it satisfies We will need the following lemma, in order to complete the proof of Theorem 1.2.

Recovering Hamiltonian spectral invariants.
We show here that the Hamiltonian spectral invariants can be viewed as particular Lagrangian ones. In order to do so, we prove equality (1) which states that they coincide via the Biran-Polterovich-Salamon isomorphism [5]. We first recall the construction of this morphism which compares the Hamiltonian Floer homology of a symplectic manifold (M, ω) and the Lagrangian Floer homology of (M , ω) := (M × M, ω ⊕ (−ω)) with respect to its Lagrangian submanifold ∆ : Let (H, J) be a regular pair on (M, ω). We define H and J on M by . In order to shorten notation, we denote Hamiltonian vector fields and flows by X t and φ t . Those induced by H are given by Since φ 1 is a symplectomorphism, its graph, denoted by Γ φ 1 , is a Lagrangian submanifold of M × M . The generators of the Floer complex of (M, ω), P 0 (H), and of the Lagrangian Floer complex of (M , ω), with respect to ∆ and Γ φ 1 , are identified by the bijection Moreover, the transversality conditions required in the Lagrangian case (∆ ∩ Γ φ 1 is transverse) and in the Hamiltonian case (det(dφ 1 − id) = 0) are equivalent.
Notice that we have u(s, 0) ∈ ∆ and u(s, 1/2) ∈ Γ φ 1 . Projecting the relation ∂ s u + J∂ t u = 0 gives the following equivalences: Since ∂ s u(s, t) = dφ t (∂ s u(s, t), ∂ s u(s, 1 − t)), the energy is preserved, in the sense that E(u) = E(u). Via (5), the action is preserved up to an additive constant. Hence we get an identification between the two chain complexes: which induces the desired isomorphism in homology.
Since the critical points of f and its flow lines with respect to g are identified with the critical points of f and its flow lines with respect to g, the above process leads to an identification between the moduli spaces defining the PSS morphisms in both cases. Therefore the following diagram is commutative: Together with the fact that the action is preserved up to a constant, this gives the correspondence between spectral invariants in both cases. The equality (1) ρ(α; φ) − ρ(1; φ) = c(α; ∆, Γ φ ) holds as claimed in the introduction.

Lagrangian spectral invariants of higher order
In this section, we use the Barraud-Cornea spectral sequence machinery to introduce Lagrangian spectral invariants of higher order, which generalize the invariants defined before. In order to do so, we first recall this machinery. Then we define the generalized invariants and prove their main property. Details and proofs about this spectral sequence machinery can be found in [2,3].

4.1.
Recollection of the Barraud-Cornea spectral sequences. At the center of this construction, there is a chain complex whose differential takes into account higher dimensional moduli spaces of Morse (or Floer) trajectories (not only their 0-dimensional component). The relevant spectral sequence is induced by a natural filtration of this extended chain complex. 4.1.1. Morse case. Let L n be a smooth manifold and f a Morse function. Let w be a path embedded in L, which passes through all the critical points of f . We denote by ρ the projection of L onto L := L/w (the critical points of f are mapped to a distinguished point, * , of L). Let X be simply-connected and l : L → X be a continuous map. Notice that L and the quotient space L are clearly homotopy equivalent via ρ. The homotopy class of the map l (rather than the map itself) being involved in the following construction, one can actually view l as a continuous map from L to X.
Let us denote by C p,q L the subset of C 0 ([0, f (p) − f (q)], L) consisting of paths from p to q and by γ p,q the map from M p,q (f, g) to C p,q L which associates to x in the connecting manifold of p and q, the flow line representing it, parameterized by [0, f (p) − f (q)]. As this map is compatible with the compactification of moduli spaces, it induces γ p,q : M p,q (f, g) → C p,q L.
Let Ω ′ Y be the space of Moore loops of Y , that is, the set of loops in Y parameterized by any compact interval [0, a], with a > 0. We denote by S * (Ω ′ Y ) the cubical chain complex of Ω ′ Y . Concatenation in Ω ′ Y defines a product on S * (Ω ′ Y ). We define Q p,q : C p,q L → Ω ′ L as the projection by ρ of the paths in L. We denote by Φ p,q the composition Q p,q • γ p,q . The map l induces maps preserving the multiplication. Notice that R X * := S * (Ω ′ X) is a differential ring. We denote by j * the map induced at the (cubical) chain level by the inclusion of (M p,q (f, g), ∅) in (M p,q (f, g), ∂M p,q (f, g)). The last ingredient is a representing chain system of M(f, g), that is, a set represents the fundamental class of M p,q (f, g) relative to the boundary, ii. ∂s pq = r s pr × s rq .
Such a system exists for M(f, g). It can be built by induction, using ii.
Let d ′ be the unique differential on C X * , satisfying d ′ (a ⊗ p) = (da) ⊗ p + a · (∂p). Barraud and Cornea showed that d ′2 = 0. Thus (C X * , d ′ ) is a graded differential module. It is called the extended Morse complex and admits the following filtration: The spectral sequence which we are interested in is the one based on this filtration: EM (L; f, g; X) = (EM r p,q (L; f, g; X), d r ). Recall from the introduction that E X (L) denotes the Serre spectral sequence associated to the fibration ΩX ֒→ E → L (pull-back of the path-loop fibration of X over l). It is shown in [3] that there is an isomorphism of spectral sequences between EM (L; f, g; X) and E X (L) at page 2. Now C x,y P stands for the subset of C 0 ([0, A H (x) − A H (y)], P η (L, L ′ )) consisting of paths with respective endpoints x and y and a map γ x,y : M x,y (L, L ′ ; H, J) → C x,y P is defined. The evaluation map at 0 is used to define Q x,y = ev 0 • ρ : C x,y P → Ω ′ L and Φ x,y : M(x, y) → Ω ′ L denotes Q x,y •γ x,y . Critical points being replaced by orbits and Morse index by Maslov index, the end of the procedure is formally identical and leads to the extended Floer complex (C X * , d ′ ) which admits a filtration We deduce the desired spectral sequence: EF (L, L ′ ; η; H, J; X) = (EF r p,q (L, L ′ ; η; H, J; X), d r ). Two of its properties (see [3]) will be useful in the next section: (1) EF 2 p,q (L, L ′ ; η; X) ≃ H q (ΩX) ⊗ HF p (L, L ′ ; η) for any integers p and q, (2) if d r = 0, there exist x and y ∈ I(L, L ′ ; η, H) with µ(x, y) ≤ r such that M x,y (L, L ′ ; H, J) is not empty.
The comparison and naturality morphisms between (suitable) Floer complexes naturally extend to the extended complexes and induce isomorphisms, Ψ H,H ′ and B H , between the respective spectral sequences. Moreover, when L ′ = L, there also exists an extension of the Lagrangian PSS morphism to the Morse and Floer extended complexes, inducing an isomorphism Φ H f between the respective spectral sequences at page 2. It restricts itself to the PSS morphism on EM 2 * ,0 (L; f, g; X) ≃ HM * (L; f, g). For these three constructions (see [2]), one has to include the components of any dimension of the moduli spaces (not only the 0-dimensional part). Therefore, in particular, Morse and Floer spectral sequences are identified at page r (r ≥ 2) with E X (L).

4.2.
Definition of Lagrangian spectral invariants of higher order. For ν ∈ R, we define a truncated, extended Floer complex by C ν = C ν (L, L; H, J) := R X * ⊗ CF ν * (L, L; H, J). We denote by C >ν the quotient C/C ν . Since the filtration is preserved, they give rise to spectral sequences denoted by EF ν (L, L; H, J; X) and EF >ν (L, L; H, J; X). Moreover, the inclusion and quotient which form the following short exact sequence 0 / / C ν i / / C q / / / / C/C ν / / 0 are morphisms of filtered graded differential modules. Hence they induce morphisms between the respective spectral sequences with equality (at least) for r = 2. In that case, let us choose a basis, {x i }, of H q (ΩX) and let α j be non zero elements in H p (L). We put α := j x j ⊗ α j in EM 2 p,q (L; f, g; X). We have Like in the homological case, we now define an absolute version which does not depend on the choice of the reference path of the action. Since EM (L; f, g; X) is a first quadrant spectral sequence, for all integer r, we have EM r 0,0 (L; f, g; X) ≃ EM 2 0,0 (L; f, g; X) ≃ H 0 (ΩX) ⊗ H 0 (L). Hence, it contains an element, denoted 1 r , which corresponds to 1⊗1 by the previous isomorphism. We use this element in order to normalize σ r X,  Notice that this absolute version still satisfies equalities (57). Remark 4.3. As suggested by our notation, these numbers satisfy the same invariance property as their homological counterpart. Indeed, the commutative diagram of Proposition 3.1 induces a commutative diagram for the respective spectral sequences and the estimate (55) (and thus Lemma 3.6) holds for them. 4.3. Spectral invariants of higher order of (S 2 × S 4 )#(S 2 × S 4 ). We compute explicitly the higher order (Morse) spectral invariants of (S 2 ×S 4 )#(S 2 ×S 4 ). First we consider S 2 × S 4 with the Morse function f defined as the sum of the height function on each sphere: The function f has 4 critical points, p 6 , p 4 , p 2 , p 0 (namely (max(f 2 ), max(f 4 )), (min(f 2 ), max(f 4 )), (max(f 2 ), min(f 4 )) and (min(f 2 ), min(f 4 ))) where i f (p i ) = i.
Theorem implies the existence of a J δ -pseudo-holomorphic limit strip u δ whose energy satisfies (60).
By definition of the e x r δ 's, they preserve the symplectic area and thus Area ω (Im u δ ∩ Im e) = Area ω0 (e −1 (Im u δ ∩ Im e)) where e may be either e x r δ or e y r δ . Moreover, by the choice of J δ , e −1 (Im u δ ∩ Im e) is a J 0 -pseudo-holomorphic curve whose boundary lies in R n ∪ iR n ∪ ∂B(0, r δ ). This extends, by symmetries to a pseudo-holomorphic curve, containing 0 in its interior, whose boundary lies in ∂B(0, r δ ). The initial area has been multiplied by 4. The isoperimetric inequality implies that the extended curve has area at least πr 2 δ . Hence we get, Area ω (Im u δ ∩ Im e) ≥ π 4 r 2 δ . Since this is true for both ends and Im e x r δ ∩ Im e y r δ = ∅, we have Therefore, for every (small) δ > 0, This ends the proof of Theorem 1.5.

Back to homological Lagrangian spectral invariants
Theorem 1.5 has a few interesting corollaries relative to the invariants of Definition 1.1. We prove two of them in Section 5.1 (Corollaries 1.6 and 1.7) and then we show Proposition 1.3 in Section 5.2. In Section 5.3, we illustrate some other potential applications of the homological Lagrangian spectral invariants. 5.1. Proof of Corollaries 1.6 and 1.7. Corollary 1.6 is just the homological counterpart of Theorem 1.5.
Proof of Corollary 1.6. If we assume that X is (r − 1)-connected, the pages s with 2 ≤ s ≤ r are identical to page 2. Hence E r p,q (L; X) ≃ H q (ΩX) ⊗ H p (L). We assume that d r α = β = 0 where α ∈ H p (L) is identified with 1 ⊗ α ∈ E r p,0 (L; X) and β denotes i x i ⊗ β i with β i in H p−r (L). Notice that once the ordered basis of H r−1 (ΩX), {x i }, is chosen, the β i 's are uniquely determined. Hence, we get as an immediate corollary that We now give the proof of Corollary 1.7. For α ∈ H k (L), we denote by α ∈ H k (L) its Hom-dual and byα ∈ H n−k (L) the Poincaré dual of α. As before, α ∈ H k (L) is identified with 1 ⊗ α ∈ E r k,0 (L; X). Proof of Corollary 1.7. Recall that for any abelian group π and any integer n, one can construct the Eilenberg-MacLane space K(π, n), such that all its homotopy groups are trivial except the n-th which is π. They satisfy ΩK(π, n) = K(π, n − 1). Moreover, we have the isomorphism H n (X, π) ≃ [X; K(π, n)].
There is a cohomological version of the Serre spectral sequence E X (L) such that i. the differential of its r-th page has bidegree (r, 1 − r), ii. its second page comes with a product which coincides (as we use Z 2 coefficients) with the cup-product on cohomology classes. We use the same process as above, replacing α by the Hom-dual of its Poincaré dual α and K(Z 2 , k) by K(Z 2 , n − k). Hence in the cohomological version of E X (L), there is a non trivial differential d n−k ι n−k ⊗ 1 =α, as soon as k < n − 1. By ii., we know that for any cohomology class γ, d n−k (ι n−k ⊗ γ) =α ∪ γ. Putting γ = α · β gives that d n−k (ι n−k ⊗ α · β) = β.
Thus the first assertion of Corollary 1.7 is proved. Putting β = [L] in this inequality, together with (61), give the second one. Corollary 1.7 is proved.
Remark 5.1. In the next section we sketch another proof of this corollary. As mentioned in the introduction this alternate proof extends the result in the sense that it remains true for any non zero homology class α ∈ H k (L), with 0 < k < n. However, the cases k = 1 and k = n − 1 should be obtained with the previous method by using a system of local coefficients. We conclude the proof of Corollary 1.8 with the last assertion of Proposition 1.3. Remark 5.3. Another immediate consequence of the assertion (2.) of Proposition 1.7 is that as soon as L and L ′ are transverse, Hamiltonian isotopic Lagrangian submanifolds, ∇(L, L ′ ) > 0. This is easily seen to give another proof of the wellknown fact that Hofer's distance for Lagrangian submanifolds is non degenerate. Moreover, following Schwarz [19], we can use the very same arguments in order to deduce some properties of the set of Lagrangian submanifolds Hamiltonian isotopic to a fixed one, in particular cases. For example, let M be the 2-dimensional torus S 1 × S 1 and L 0 be the Lagrangian submanifold {0} × S 1 . We consider for any integer k, the autonomous Hamiltonian function H k on M defined by H k (x, y) = k sin(2πx). The upper bound given by the assertion (2.) of Proposition 1.7 allows to conclude that, when k converges to infinity, ∇(L, (φ 1 H k ) −1 (L)) converges to infinity. Therefore, the diameter of the set of Lagrangian submanifolds Hamiltonian isotopic to L in M , endowed with Hofer's distance, is infinite.