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Quantization of the Serre spectral sequence

The present paper explores how the spectral sequence introduced in a previous work (and obtained by taking moduli spaces of any dimension into account in the Floer construction), interacts with the presence of bubbling. As consequences are obtained some relations between binary Gromov-Witten invariants and relative Ganea-Hopf invariants, a criterion for detecting the monodromy of bubbling as well as algebraic criteria for the detection of periodic orbits.


Introduction
In [1] has been introduced an algebraic way to encode the properties of high dimensional moduli spaces of trajectories in Morse-Floer type theories. The basic idea is that, by making use of a "representation" theory of the relevant moduli spaces M(x, y) lx,y −→ G into some sufficiently large topological monoid G, one can define a "rich" Morse type chain complex whose differential is of the usual form dx = y a x,y y but a x,y , the coefficient "measuring" the moduli space M(x, y), belongs to a graded ring (for example, the ring of cubical chains of G) and is in general not zero when dim(M(x, y)) > 0. By representation theory it is meant here not only that the maps l x,y are continuous but also that they are compatible in the obvious way with compactification and with the crucial boundary formula: (1) ∂M(x, y) = z M(x, z) × M(z, y) .
The complex constructed this way comes with a natural filtration induced by the grading of the generators x, y, .... The pages of order greater than 1 of the associated spectral sequence are invariant with respect to the various choices made in the construction and their differentials encode algebraically the properties of the M(x, y)'s. This construction is described in the absence of bubbling in [1] and, in [2], it is shown to be easily extendable to cases when pseudo-holomorphic spheres and disks exist as long as we work under the threshold of bubbling.
The present paper explores what happens when bubbling does occur.
It is obvious that to study this case it is natural to start with the Hamiltonian version of Floer homology and this is indeed the setting of this paper. In particular, the moduli spaces M(x, y) consist of Floer tubes and the monoid G is the space of pointed Moore loops on M , ΩM , with (M 2n , ω) our underlying symplectic manifold. We will also restrict to the monotone case even if the machinery described here appears to extend to the general case. The reason for this is that the main phenomena we have identified are already present in this case and, at the same time, in this way we avoid to deal with the well-know transversality issues which are present in full generality.
Here is a short summary of our findings. Firstly, it is not surprising that when bubbling is possible, only some of the pages of the spectral sequence mentioned before exist. It is also expectable that the number of pages that are defined should roughly be the minimal Chern class, c min , and that, moreover, some of these pages should again be independent of the choices made in the construction.
What is remarkable is that, in general, these pages do not coincide with those associated to a Morse function: a quantum deformation is generally present. Given that in the Morse case the resulting spectral sequence is, as shown in [1], the Serre spectral sequence of the path-loop fibration over M , we see that this construction provides a new symplectic invariant which consists of the first c min pages (together with their differential) of a spectral sequence which is a quantum deformation of the Serre spectral sequence. One additional important point is that, on the last defined page, the presumptive differential, d r , is still defined and invariant but might not verify (d r ) 2 = 0.
Of course, the next stage is to understand -at least in part -this quantum deformation in terms of classical Gromov-Witten invariants. In this respect we obtain that the first interesting differential d r , can be expressed in terms of binary Gromov-Witten invariants (these are those associated to spheres with two marked points) and Ganea-Hopf invariants (these control the classical part of the differential). Moreover, in this case, the relation (d r ) 2 = 0 becomes a relation between these two types of invariants which takes place in the Pontryaguin ring H * (ΩM ). Undoubtedly, this is just a first step towards understanding the deeper relationships between the combinatorics of Gromov-Witten invariants and classical algebraic topology invariants encoded in the ring structure of H * (ΩM ).
The next interesting point is to understand what happens for the first r when (d r ) 2 = 0. Clearly, the culprit is bubbling but interestingly enough what this non-vanishing relation detects is monodromy -the fact that in the appropriate moduli space the attachment point of the bubbles turns non-trivially around Floer cylinders -which turns out to interfere with the representation maps l x,y . The fact that d r is invariant but, simultaneously, (d r ) 2 might not vanish is quite remarkable and, indeed, this morphism d r has interest in itself and is seen to be, in fact, a generalization of the Seidel homomorphism [10]. Finally, we also discuss an application of this structure to the detection of periodic orbits. This provides a sort of algebraic counterpart to the result of Hofer-Viterbo [5].
The paper is structured as follows. In the second section we introduce the main notation and give the precise statements of our results. The third section contains the proofs. In the last section we first shortly mention some possible extensions of the construction we then provide some examples and, finally, we discuss the aplication to periodic orbits.
1.1. Acknowledgment. It is our great pleasure to dedicate this paper to Dusa McDuff on the occasion of her 60st birthday. This is even more appropriate as, early in this project, we believed, a posteriori without justification, that the monodromy of bubbling is much less relevant and it is one of Dusa's questions which made us reconsider the issue and appreciate the full importance of this phenomenon.

Notation and statement of results
2.1. Setting and recalls. Fix the symplectic manifold (M 2n , ω) and we suppose for now that M is closed. We assume that M is monotone in the sense that the two morphisms ω : π 2 (M ) → R and c 1 : π 2 (M ) → Z are proportional with a positive constant of proportionality ρ. We denote by c min the minimal Chern class and by ω min the corresponding minimal symplectic area (so that we have ω min = ρc min ).
2.1.1. Binary Gromov-Witten invariants. Fix on M a generic almost complex structure J which tames ω. The binary Gromov Witten invariants we are interested in can be described as follows: pick a generic Morse function f and metric on M . Denote by i(x) = ind f (x) for each x ∈ Crit(f ). For two critical points x and y and a class α ∈ π 2 (M ) such that i(x) − i(y) + 2c 1 (α) − 2 = 0, we define GW α (x, y) as the number of elements in the moduli space M(J, α; x, y) which consists of J-holomorphic spheres in the homology class α with two marked points, one lying on the unstable manifold of x and the other on the stable manifold of y, modulo reparametrization. As such GW α (x, y) is not an invariant (because x, y might not be Morse cycles). However, if for two classes then we obtain an invariant. For α ∈ π 2 , let [α] be its image by the morphism π 2 (M ) → H 1 (ΩM ). Let Γ be the image of the Hurevicz morphism π 2 (M ) → H 2 (M, Z/2). The two forms ω and c 1 define morphisms Γ : ω,c 1 −−→ R, Z which under our monotonicity assumption are proportional. Let Γ 0 = Γ/ ker(ω). We let Λ be the associated Novikov ring which is defined as follows where the coefficients λ α belong to Z/2 such that ∀c > 0, ♯{α, λ α = 0, ω(α) ≤ c} < +∞ .
The grading of the elements in Λ is given by |e λ | = −2c 1 (λ). We also denote byL(M ) the covering of L(M ) associated to Γ 0 : it is the quotient of the space of couples (γ, ∆), where γ ∈ L(M ) and ∆ is a disk bounded by γ, under the equivalence relation (γ, Remark 1. Here and later in the paper we could also use, alternatively, rational coefficients as all the moduli spaces involved are orientable and the orientations are compatible with our formulae. ω(X Ht , ·) = −dH t .
All along this paper, the periodic orbits of X H will be supposed to be non degenerate. We denote by P H ⊂ L(M ) the set of all contractible periodic orbits of the hamiltonian flow associated to H and we letP H be the covering of P H which is induced fromL(M ).
For each periodic orbit x ∈ P H we fix a lift (x, ∆ x ) ∈P H . For a generic pair (H, J) and x, y ∈ P H , λ ∈ Γ 0 we now consider the moduli spaces: Of course, these moduli spaces are quite well-known in the subject and we refer to [9] for their properties. In particular, they have natural orientations and, when (x, ∆ x ) = (y, ∆ y ) they admit a free R action. We denote the quotient by this action by M(x, y; λ) and we have 2.1.4. Monodromy of bubbling. Among the standard properties of the moduli spaces above we recall that they admit a natural topology as well as natural compactifications, M(x, y; λ), so that the following formula is valid: Here Σ x,y,λ is a set of codimension 2 which consists of Floer tubes with at least one attached bubble.
We will say that (H, J) has bubbling monodromy if there exist x, y ∈ P H and λ ∈ Γ 0 so that: H 1 (Σ x,y,λ ; Z) = 0 This means, in particular, that π 1 (Σ x,y,λ ) = 0 so that there are non-contractible loops in the space of Floer tubes with bubbles.
2.1.5. Truncated differentials and spectral sequences. The following algebraic notions will be useful in the formulation of our results.
We say that the sequence of graded vector spaces (E r , d r ), 0 ≤ r ≤ k is a truncated spectral sequence of order k if (E r , d r ) is a chain complex for each r ≤ k − 1 which verifies H * (E r , d r ) = E r+1 and d k is a linear map of degree −1. A truncated spectral sequence of ∞-order is a usual spectral sequence. A morphism of order k truncated spectral sequences is a sequence of chain maps φ r : (E r , d r ) → (F r , d r ), 0 ≤ r ≤ k, so that H * (φ r ) = φ r+1 for 0 ≤ r ≤ k − 1. We say that two truncated spectral sequences are isomorphic starting from page s is they admit a morphism which is an isomorphism on page s (and, hence, on each later page).
The typical example of a truncated spectral sequence appears as follows. Assume that C * is a graded rational vector space and that F i C is an increasing filtration of C * . We say that a linear map d : C * → C * −1 is a truncated differential of order k compatible with the given filtration if d( It is easy to see that a truncated differential of order k induces a truncated spectral sequence of the same order. Indeed, by using the standard descriptions of the r-cycles p ֒→ Z r p for 0 ≤ r ≤ k which allows us to define the pages of the truncated spectral sequence by E r p = Z r p /B r p . Obviously, d induces differentials d r on E r when r < k as well as a degree −1 linear map d k on E k .

2.2.
Main statement. We will formulate our main statement in a simple case and we will discuss various extensions at the end of the paper. Therefore, we assume here that (M, ω) is closed, simply-connected and monotone with c min ≥ 2.
Theorem 2.1. There exists a truncated spectral sequence of order c min , (E r (M ), d r ), whose isomorphism type starting from page 2 is a symplectic invariant of (M, ω) and which has the following additional properties: i. As a bi-graded vector space we have an isomorphism: ii. The differential d 2 has the decomposition where d 2 0 is the differential appearing in the classical Serre spectral sequence of the path loop fibration ΩM → P M → M and iii. If (d c min ) 2 = 0, then any regular pair (H, J) has bubbling monodromy.
Remark 2. Clearly, if d 2 • d 2 = 0 -for example if c min ≥ 3 -the vanishing of the square of d 2 • d 2 translates into some relations between binary Gromov-Witten invariants and the classical Serre spectral sequence differential d 2 0 . In turn, this differential is quite well known and rather easy to compute and it can be expressed in many cases in terms of relative Ganea-Hopf invariants (see [4]). The interesting part about these relations is that they take place in the Pontryaguin algebra H * (ΩM ). Indeed, in the formula at ii. α ∈ H 1 (ΩM ), e α ∈ Λ and x, y ∈ H * (M ) so that in the square of the differential appears the Pontryaguin product H 1 (ΩM ) ⊗ H 1 (ΩM ) → H 2 (ΩM ).
The relation with the Seidel homomorphism is seen by considering the spectral sequence in the case of a symplectic fibration over CP 1 .
We also formulate here a very simple version of our application to the detection of periodic orbits. We specialize to the case when the manifold M admits a perfect Morse function (that is a Morse function whose associated Morse complex has trivial differential). We also need the following notion. Let x, y ∈ H * (M ) and λ ∈ Λ. We will say that x and ye λ (which exist on the E 2 page of the spectral sequence in Theorem 2.1) are d r -related if x survives to the r-th level of the spectral sequence and there is some γ ∈ C * (ΩM ) so that the product γ ⊗ ye λ also survives to the r-th page of the spectral sequence and we have d r ([x]) = [γ ⊗ ye λ ] + . . .

Corollary 2.2.
Assume that there are homology classes x, z ∈ H * (M ), |x| < |z|, so that x is d r -related to ze λ and H k (M ) ⊗ Λ q = 0 for |x| > k + q > |ze λ |). Then any self-indexed perfect Morse function on M has some non-trivial closed characteristic.
By a self-indexed Morse function f we mean here that the critical points of the same index have the same critical value and ind f (x) > ind f (y) implies f (x) > f (y).
There are many ways in which this statement can be extended and some will be discussed at the end of the paper.
3. Proof of the main theorem.
3.1. Construction of the truncated spectral sequence. In this section we fix the 1-periodic hamiltonian H and almost complex structure J compatible with ω so that the pair (H, J) is generic (of course, both are in general time-dependent). For simplicity, we will also assume to start that the manifold M is simply-connected but we will see later on that this condition can be dropped with the price that the construction becomes more complicated.
As in [1] the truncated spectral sequence we intend to discuss is induced by a natural filtration of an enriched Floer type pseudo-complex. We use the term pseudo-complex here to mean that we will not have here a true differential but rather a truncated one. The construction of this pseudocomplex is a refinement of the classical Floer construction in which the coefficient ring is replaced with the ring of cubical chains over the Moore loops on M . Here is this construction in more detail.
3.1.1. Coefficient rings. Let C * denote the "cubical" chain complex, let ΩX be the Moore loop space over X (the space of loops parametrized by intervals of arbitrary length). Consider the space M ′ obtained from M by collapsing to a point a simple path γ going through the starting point of each periodic orbit. Notice that C * (ΩM ′ ) is a differential ring where the product is induced by the concatenation of loops. Finally, our coefficient ring is: This is a (non abelian) differential ring, and its differential will be denoted by ∂.
The (pseudo)-complex we are interested in is a (left) differential module generated by the contractible periodic orbits of H over this ring: with the identificationxe λ ∼x♯λ, wherex♯λ stands for the capping of x obtained by gluing a sphere in the class λ tox. The grading of an element inx ∈P H is given by the respective Conley-Zehnder index. There is a natural filtration of this complex which is given by We will call this the canonical filtration of C(H, J).

3.1.2.
Truncated boundary operator. The next step is to introduce a truncated differential on C(H, J). We recall from §2.1.3 the definition of the moduli spaces M(x, y; λ) of Floer tubes. We recall also that this definition requires a choice of liftx ∈P H for each x ∈ P H . With these conventions and -as assumed before -for a generic choice of J and H -the moduli spaces are smooth manifolds of dimension |x| − |ỹ| − 1 when |x| = |ỹ|, and they have a natural compactification involving "breaks" of the tubes on intermediate orbits, or bubbling off of holomorphic spheres. We will write M(x,ỹ) for the moduli space of Floer tubes which lift to paths insideL(M ) joining x ∈P H toỹ ∈P H and we let M(x,ỹ) be the respective compactification. In our monotone situation these compactifications are pseudo-cycles with boundary.
To define the truncated boundary operator we proceed as in the usual Floer complex, but we intend to take into consideration the moduli spaces of arbitrary dimensions instead of restricting to the 0 dimensional ones. To associate to the (compactification of the) moduli spaces coefficients in our ring R, we first need to represent them into the loop space Ω(M ′ ), and then pick chains representing them (i.e. defining their fundamental classes relative to their boundary).
Let us start with "interior" trajectories, i.e. elements v ∈ M(x,ỹ). Let u : R × S 1 → M be a parametrization of v. Since the value of the action functional is strictly decreasing along the R direction, it can be used to reparametrize u by the domain [−a(x), −a(ỹ)] × S 1 , and the restriction of u to the interval which is continuous. We will call it the "spine" map. This map should then be extended to the compactification M(x,ỹ) of M(x,ỹ).
It is well-known that the objects in M(x,ỹ) are constituted by Floer trajectories possibly broken on some intermediate periodic orbits to which might be attached some J-holomorphic spheres that have bubbled off.
It is easy to see that the map σx ,ỹ extends continuously over the part of this set where no spheres are attached to some tube in a point belonging to the line R × {0}. Indeed, as in [1], except for these types of elements, the spine map is compatible with the breaking of Floer tubes in the sense that the loop associated to a broken trajectory is the product of the loops associated to each "tube" component.
Let α min ∈ Γ 0 be the class so that c 1 (α min ) = c min (by our monotonicity assumption there is a single such class). By using again the monotonicity assumption we see that bubbling off of a sphere in class α ∈ Γ 0 can occur in a moduli space M(x,ỹ) withỹ =x♯α only if |x| − |ỹ| ≥ 2c 1 (α) + 1 .
It is also important to note that bubbling of a sphere in the class α is also possible inside the space M(x,x♯α). In all cases, bubbling of an α sphere is never possible if |x| − |ỹ| ≤ 2c 1 (α) − 1.
We summarize this discussion: In case |x| − |ỹ| = 2c min and if σ does not have such a continuous extension to M(x,ỹ), thenỹ =x♯α min .
The spine map obtained in this way satisfies also a compatibility condition which we now make explicit. If M(x,z) × M(z,ỹ) ⊂ M(x,ỹ), then the restriction of σx ,ỹ on the set on the left of the inclusion equals m•(σx ,z ×σz ,ỹ ) where For pairs (x,ỹ) with |x| − |ỹ| ≤ 2c min − 1, we use the map σx ,ỹ to represent the moduli spaces M(x,ỹ) inside the loop space Ω(M ′ ). We then choose "chain representatives" m(x,ỹ) ∈ C * (ΩM ′ ), i.e. chains generating the fundamental class of σ(M(x,ỹ)) relative to its boundary, in such a way that: where * is the operation induced on C * (ΩM ′ ) by the concatenation of loops.
The key point regarding this formula is that, under our assumption |x| − |ỹ| ≤ 2c min − 1, the compactified moduli space M(x,ỹ) is a manifold with boundary. Moreover, its boundary verifies the usual formula valid in the absence of bubbling so that the construction of the m(−, −)'s is the same as that in the non-bubbling setting. We refer to [1] for a complete discussion of this construction.
We now define the boundary operator d by: and extend it to the full complex using the Leibnitz rule.
It is easy to check that d has degree −1 with respect to the total grading and that it is compatible with the canonical filtration. Notice first that if and we see that d 2 drops the filtration index by at least 2c min . In the algebraic terms of §2.1.5 we obtain: With the definition above, d is a truncated differential of order c min with respect to the canonical filtration on C(H, J) and thus it induces a truncated spectral sequence E r (H, J) of the same order so that The isomorphism in the lemma is obvious because E 1 (H, J) ∼ = CF * (H, J)⊗ H * (ΩM ) and as d 1 only involves the 0 dimensional moduli spaces of Floer tubes we obtain that d 1 is just: d F ⊗ id where (CF * (H, J), d F ) is the usual Floer complex (with coefficients in the Novikov ring Λ). Thus we have constructed our truncated spectral sequence and have proved property i. in Theorem 2.1 Remark 3. Without the monotonicity assumption, but still assuming that the moduli spaces in question are regular, there is no way to avoid the bubbling phenomenon, even on low-dimensional moduli spaces. However, on 2-dimensional moduli spaces, the bubbling component is 0 dimensional, and hence consists in isolated points: for each of them, the real line R × {0} can actually be deformed to avoid the point where the bubble is attached. Interpolating between these perturbed real lines with the standard one in small neighborhood of the "bubbled" trajectories defines a spine map for 2 dimensional moduli spaces, with the desired continuity and compatibility conditions.

3.2.
Invariance of the truncated spectral sequence. To show invariance we will proceed along Floer's original proof by first constructing a comparison morphism between the spectral sequences associated to two different sets of generic data (H i , J i ) i=0,1 . We will describe the construction of this morphism in more detail below but we only mention here one remarkable fact: despite the fact that in our spectral sequences we might have The construction uses a homotopy between (H 0 , J 0 ) and (H 1 , J 1 ). As in the usual Floer case, we consider a generic homotopy between them, (G,J ), and, forx ∈P H 0 andx ′ ∈P H 1 , we consider the moduli spaces N (x,x ′ ) of tubes v : R × S 1 → M which lift inL(M ) to a path joiningx tox ′ and verify the equation: The moduli space N (x,x ′ ) has properties similar to those of M ′ (−, −) except that it has no R-invariance. Its dimension is |x| − |x ′ |. Clearly, bubbling of an α-sphere inside such a moduli space is not possible if |x| − |x ′ | ≤ 2c 1 (α)− 1. As in §3.1 , sphere bubbling is the only obstruction to extend the spine map. Assuming that |x| − |x ′ | ≤ 2c min − 1 the spine map can therefore be extended over these spaces in a way compatible with the spine maps of (H 0 , J 0 ) and (H 1 , J 1 ) (as in [1]). The chain morphism between the two (truncated)-complexes is defined by a formula similar to (6): [1]). This morphism clearly respects the canonical filtrations.
We also have: Computing dΘ and Θd we get: which is not 0, but has degree at least −2c min with respect to the Maslov index.
It is easy to see that this implies that Θ induces a morphism of truncated spectral sequences:Θ Similarly to the isomorphism in Lemma 3.2 it is easy to see that E 1 (Θ) is identified with: where θ F is the Floer comparison morphism. As this morphism induces an isomorphism in homology we deduce that E 2 (Θ) and hence all ofΘ are isomorphisms for r ≥ 2 and this shows the invariance claim in the statement of Theorem 2.1.
Remark 4. A morphism of spectral sequences preserves the bi-degree, therefore to show thatΘ is a morphism we only need that dΘ − Θd drops the filtration degree by c min . In other words, a considerable part of the geometric information carried by Θ is actually forgotten in the spectral sequence. There are some ways to recover it but as this goes beyond the purpose of the present paper we will not discuss this here.
3.3. Detection of monodromy. The purpose here is to prove Theorem 2.1 iii. thus we fix a regular pair (H, J) and we assume that d c min • d c min = 0.
We start by looking again at the calculation for d • d given before Lemma 3.2. We see from that formula that d c min • d c min is given by a linear combination of terms of the form For each fixedz with |z| = |x| − 2c min this last sum can be rewritten as Suppose thatz =x♯α min . In that case, as indicated in Lemma 3.1, the spine map is well defined and continuous on the whole space M(x,z) and no bubbling is possible inside this space. But this means that we may find a representing chain m(x,z) so that, as in formula (5), (∂m(x,z))z = S(x,z) which means that S(x,z) vanishes in E r for r ≥ 2.
Thus, the only terms which count in d c min • d c min are S(x,x♯α min ) and if d c min • d c min = 0, then at least one such term survives to E r . To simplify notation we letx♯α min =x t . Notice that the moduli space M(x,x t ) is only a pseudo-cycle with boundary in the sense that it is a stratified set with three strata: i. a co-dimension two stratum: formed by the bubbled configurations. ii. a co-dimension one stratum: ∂M = ∪zM(x,z) × M(z,x t ) iii. a co-dimension zero stratum: M(x,x t ) . Fix now somex and, to simplify notation, let Σ = Σx ,x t and notice that Σ is a compact manifold. The spine map σ is defined on M(x,x t ) with the exception of Σ. Notice also that Σ ∩ ∂M = ∅. Suppose that there exists a continuous deformation σ ′ of σ which agrees with σ with the exception of a neighborhood of Σ. Then, as M(x,x t ) is a pseudo-cycle, the same argument described above for the casez =x♯α min applies also here (the point is that as Σ is of co-dimension 2, the construction of representing cycles is still possible) and it shows that S(x,x t ) does not play any role in E r for r ≥ 2. To conclude, our assumption d c min • d c min = 0, implies that there exists at least onex so that such a deformation σ ′ of σ does not exist. We now want to deduce from this that the first cohomology group of Σ does not vanish.
Given that, by definition c 1 (α min ) = c min , it follows that each u ∈ Σ is represented by a Floer tube R × S 1 to which is attached a single sphere in a point (t u , a u ) ∈ R × S 1 so that the tube is mapped in M on the constant orbitx and the sphere is mapped to a pseudo-holomorphic sphere in the class α min . Thus, there is a continuous map To show that H 1 (Σ; Z) = 0 it is enough to show that ξ is not nullhomotopic. Assume that ξ ≃ 0. Then ξ can be lifted to an applicatioñ ξ : Σ → R. Fix χ : R → R a smooth function supported on [−1, 1] and such that χ(0) = 1. For A, s 0 in R consider the function where the second map in the composition is t → e it .
The graph of this function defines a deformed spine ∆(s 0 , A) = graph(χ s 0 ,A ) on R×S 1 with the property that, if A = 2kπ, it avoids the point (s 0 , 0) . For each bubbled curve u ∈ Σ we consider the deformed line on the tube given by ∆ u = ∆(t u ,ξ(u) + π). This line avoids the point (t u , a u ) and thus avoids the "bubble". We obtain in this way a continuous spine map: σ ′ : Σ → ΩM ′ defined by σ ′ (u) = u(∆ u ) .
To conclude our proof it is enough to show that this spine map extends continuously to M(x,x t ) without modifying the standard spine map on ∂M. Due to by-now standard gluing results [9][7], for each point x ∈ Σ there exists a small neighborhood U (x) ⊂ Σ and a homeomorphism φ : C×U (x) → M(x,x t ) so that φ({0}×U (x)) = U (x). As Σ is compact we can cover it with a finite number of such neighborhoods which we denote by U i , 1 ≤ i ≤ k with corresponding homeomorphisms φ i . Denote V i = φ i (U i ) and let p i : V i → U i be the obvious projection. For a point y ∈ V (x) let d i (y) = d(y, p i (y) where d(−, −) is (some) distance in M(x,x t ). By possibly using smaller neighborhoods V i , we may assume that d i (y) < 1, ∀i, y. Finally, let h i : U i → [0, 1], 1 ≤ i ≤ k be a partition of the unity. We put U (Σ) = ∪V i . We also consider a smooth function η : [0, 1] → R which is decreasing, supported on [0, 1/2] and so that η(0) = 1. Let d ′ i : V i → R be given by With these notations we now extend σ ′ to U (Σ): we let and put σ ′ (u) = u(∆ u ). As this map coincides with σ on ∂U (Σ) we may extend σ ′ to a continuous map on all of M(x,x t ) so that it equals σ outside U (Σ). This concludes the proof. This is also the first page of the (classical) Serre path-loop spectral sequence. However, the second differential, d 2 , on this page is in general different from the classical one. To interpret d 2 in terms of binary Gromov-Witten invariants, we will use the construction of [8]. To this end, we start with a quantized-Morse version of the spectral sequence construction described before.
3.4.1. The Quantized -Morse truncated spectral sequence. To a Morse-Smale pair (f, g) on M , together with a generic almost complex structure J we associate an extended quantized Morse complex CM * = CM * (f, M, J). This is the free module generated by the critical points Crit(f ) over the ring R together with a differential which will be described below. The degree of a critical point x ∈ Crit(f ) is given by its index. Given the almost complex structure J on M , a "quantum-Morse" trajectory from a critical point x to a critical point y in class α ∈ Γ 0 , is a finite collection ((γ 0 , . . . , γ k ), (S 1 , . . . , S k )) of paths and spheres in M such that: (1) each sphere S i is a J-holomorphic sphere with a marked real line [p i,0 , p i,∞ ] on it, and i [S i ] = α, (2) ∀i, γ i is a piece of flow line of −∇ g f , joining S i−1 (p i−1,∞ ) to S i (p i,0 ) (with the convention that S −1 (p −1,∞ ) = x and S k+1 (p k+1,0 ) = y). We denote by M α (x, y) the set of all such objects. For a generic choice of (f, g, J), it is a smooth manifold of dimension There is no difficulty to prove this as regularity comes down to the usual transversality of the appropriate evaluation maps [7] (in particular, this is much simpler than the relative case discussed for example in [3]). To verify the dimension formula notice that there are not only two marked points on the spheres, but also a real line joining them.
Such a trajectory defines a path from x to y by concatenation of the flow lines and the marked real lines on the spheres. Notice that each flow line segment can be parametrized by the value of −f , while on a holomorphic sphere u : C ∪ {∞} → CP 1 → M , the map t ∈ [0, +∞) → |z|≤t u * ω is strictly increasing and defines a parametrization of the marked real line.
These independent parametrizations of the different segments can now be shifted and aligned to produce a parametrization of the full respective path by the segment [−f (x), −f (y) + ω(α)]. We also assume that the path γ used to define M ′ and turn trajectories into loops goes through all the critical points of f . As a consequence we obtain a continuous map The space M α (x, y) of course has a natural Morse-Gromov compactification M α (x, y), and the question arises again of extending σ over it. Clearly, σ extends continuously over broken trajectories as long as no bubble components appear (as in §3.1.2). However, it might fail to extend over trajectories where new spheres bubble off. The arguments used in the discussion of this point for Floer moduli spaces still apply in this situation, and the map σ can be defined with the desired continuity and compatibility conditions whenever, as in Lemma 3.1, Picking chain representatives m α (x, y) of σ(M α (x, y)), we define a truncated boundary operator on CM * in the usual way: The complex (CM * , d) admits also a differential filtration again defined by the degree of the elements in Z/2 < Crit(f ) > ⊗ Λ and this induces a truncated spectral sequence in the same way as before. The differential d 2 of this spectral sequence has a natural interpretation in terms of Gromov-Witten invariants.
To see this first notice that if an element u ∈ M α (x, y) with α = 0 is so that it contains k spheres, then the dimension of M α (x, y) is at least equal to k. Indeed, the choice of the real line on each of the spheres in u gives rise to a full S 1 parametric family of elements in this moduli space. The first consequence of this remark is that the differential d 1 in the spectral sequence is simply d M orse ⊗id which is defined on Z/2 < Crit(f ) > ⊗H * (ΩM )⊗Λ where d M orse is the usual Morse differential. Indeed, d 1 involves 0-dimensional quantized-Morse moduli spaces and the remark above shows that when α = 0 these spaces are never 0-dimensional if non-void.
Suppose now, to shorten the discussion, that f is a perfect Morse function (if not, critical points should be replaced by a basis of the Morse homology of M ).
In this case, for a critical point x of f the second differential d 2 x is defined and is given by d 2 x = α,y [m α (x, y)]y, where the sum is taken over all (x, y, α) such that M α (x, y) is 1-dimensional. Notice that we may associate a homology class in H * (ΩM ) to each such moduli space in this case because the Morse differential vanishes. We let [m α (x, y)] be this class.
In the expression for d 2 the sum of the terms where α is trivial, d 2 0 , is given by 1-parametric families of Morse trajectories and, as shown in [1], this coincides with the second differential in the Serre spectral sequence of the fibration ΩM → P M → M . On the other hand, when α is non trivial, as a second consequence of the remark above, we see that the corresponding moduli space M α (x, y) is the set of single holomorphic spheres in class α with a marked real line [p 0 , p ∞ ] such that p 0 ∈ W u (x) and p ∞ ∈ W s (y).
The choice of the real line defines an S 1 action on M α (x, y), and the quotients are 0-dimensional:

Moreover, letting
be the Gromov-Witten invariants of holomorphic spheres in class α with two marked points, associated to the homology class of x and the dual class of y, we have: In particular, GW α (x, y) is the number of components of M α (x, y). An hybrid trajectory from x toỹ is a quantized-Morse trajectory -as defined in §3.4.1 -starting at x but now ending with a disk bounded by y.
The definition for a hybrid trajectory are as in in 3.4.1, with the following modifications : i. the last sphere S k is replaced with a disk u with one cylindrical end, so that in polar coordinates and away from 0: ii. the map u satisfies a 'cut off' Floer equation. For a fixed cut-off function χ such that χ(s) = 1 for s ≥ 1 and χ(s) = 0 for s ≤ 0 we have: iii. the negative gradient flow arc γ k ends at u(0), iv. the sum of the homotopy class of u with k−1 i=1 [S i ] defines the capping y of y. For a generic choice of the data, all the relevant sub-manifolds and evaluation maps can be made transversal, so that the moduli spaces M(x,ỹ) of hybrid trajectories are smooth manifolds of dimension dim M(x,ỹ) = |x| − |ỹ| These moduli spaces admit a natural compactification -we refer to [8] for the proof. We only recall here that the key point for showing compactness is to derive a uniform bound for the energy from the "cut off" Floer equation.
To turn a hybrid trajectory in M(x,ỹ) into a path from x toỹ(0), it is enough to choose a parametrization of the real line u(R) on the terminal disk and for that we may use the energy of the curve: This choice defines a continuous spine map σ : M(x,ỹ) → Ω(M ′ ), that can again be extended to the natural compactification M(x,ỹ) up to dimension 2c min − 1. Picking compatible chain representatives of these spaces, we obtain chains m(x,ỹ) ∈ C * (Ω(M ′ )) such that: z)). Consider now the truncated morphism φ given by As expected, the map dφ − φd fails to vanish in general, but one easily checks that (dφ − φd)(x) is supported on elementsỹ with |ỹ| ≤ |x| − 2c min . This means that φ induces a morphism Φ between the respective truncated spectral sequences of order c min .
Notice that the Φ 1 coincides with: where φ ′ is the usual PSS morphism and C M orse (f, g) is the Morse complex of (f, g). But, as φ ′ induces an isomorphism in homology, this implies that Φ 2 is an isomorphism which, in particular, proves the point ii. of Theorem 2.1 and concludes the proof of this theorem.

Examples, applications and further comments
4.1. Extensions. We recall that the setting considered till now in the paper was that of a closed, simply-connected, monotone manifold for which c min ≥ 2. All the constructions described previously in the paper extend much beyond this setting. We will discuss here a few such generalizations. The coveringP H is replaced by the coveringP ′ H which is the pull-back of M → M overP H → P H → M . In this case our truncated complex is isomorphic to: B. A second possibility is localization or change of coefficients. This is maybe even more useful in applications than A and consists in replacing in all the construction the coefficient ring C * (ΩM ) by C * (ΩX) where X is some simply-connected topological space which is endowed with a map: All our moduli spaces are represented inside Ω(M ′ ) and, by composition with the map Ωη : Ω(M ′ ) → Ω(X), they are also represented inside Ω(X). The results in Theorem 2.1 remain true after this change of coefficients except that H * (ΩM ) is replaced by H * (ΩX) and the path loop fibration over M is replaced with the fibration of base M which is obtained by pull-back over the map η from the path-loop fibration over X, ΩX → P X → X.
4.1.2. c min = 1. It is easy to see that even if c min = 1 the E 2 term of our spectral sequence is well defined together with the map d 2 (which might not be a differential though) and Theorem 2.1 remains true for the E 2 term. This happens because to prove the invariance of d 2 only moduli spaces of dimension 2 are needed. In turn, as bubbling is a codimension two phenomenon this means that the bubbling points can be avoided when defining the spine map over these moduli spaces (as also discussed in Remark 3).

Rational coefficients.
The moduli spaces we use in this paper admit coherent orientations and by taking these into account we may replace everywhere Z/2 with Q.

Lack of monotonicity.
It is expected that Theorem 2.1 remains true for the E 2 term of the spectral sequence even if (M, ω) is not monotone (see also Remark 3). Of course, in this case multi-valued perturbations are needed and, thus, the use of rational coefficients is mandatory. 4.1.5. Non-compactness. Finally, it is obviously possible to extend this theory to the case when M is not compact if it is convex at infinity. In that case the Hamiltonians used should have compact support.

Examples.
4.2.1. CP 1 . Take now M = CP 1 , and consider the Morse function having only one maximum a = ∞ and one minimum b = 0 as critical points. Let us pick a simple path from b to a to serve as the base point of M ′ . As auxiliary data, we can simply stick to the standard metric and complex structure on CP 1 , and use no perturbations at all : one easily checks that genericity is fulfilled for all the moduli spaces involved in the computations below.
The page 2 of the spectral sequence is simply H * (CP 1 ) ⊗ H * (ΩS 2 ). Let α denote the identity map S 2 → CP 1 . Seen as the S 1 family of flow lines going from a down to b, α defines a cycle [α] that generates H 1 (Ω(S 2 )).
The Novikov ring is generated by the multiples of α : and we have c 1 (α) = 2. To make the differential more explicit, we will "unfold" the spectral sequence by removing the Novikov ring from the coefficients, and thinking of {ae kα } or {be kα } as free families.
To compute the differential d 2 , we have to compute all the 1-dimensional moduli spaces. Because of the invariance of the moduli spaces under the action of π 2 (S 2 ) on both ends of the trajectories, we can restrict to spaces of the form M(x, ye kα ) with x, y ∈ {a, b}. The dimension of this space is dim M(x, ye kα ) = |x| − |y| + 4k − 1 so there are only two possibilities : -k = 0, x = a and y = b, -k = 1, x = b and y = a. The first moduli space consists in classical flow lines only : it contributes to the classical part d 2 0 of d 2 , and we have : , so that the page 2 of the "classical" spectral sequence (tensored by the Novikov ring) has the following form : The second moduli space, M(b, ae α ), involves holomorphic spheres of degree 1, and determines the quantum component d 2 Q of d 2 . Since there are no flow lines going out of b or into a, it consists in holomorphic spheres of degree 1 with a marked real line from b to a. This is the same cycle as α, but with reversed orientation. as a consequence, we have d 2 Q (a) = 0 and d 2 Q (b) = ae α , and the page 2 of the full spectral sequence has the following form : Notice that (d 2 ) 2 a = [α 2 ]ae α = 0. So some bubbling has to occur on a 3 dimensional moduli space. And in fact, the moduli space M(a, ae α ) is 3 dimensional and consists in flow lines going out of a down to some point p, and a holomorphic sphere of degree 1 with a marked real line from p to a. When the point p goes to b, the flow line brakes, and we see that this space is involved in the computation of (d 2 ) 2 (a). On the other hand, when the point p goes to a, we are left with the constant trajectory from a to itself, with an (unparametrized) holomorphic sphere attached to it. Here, the critical point a is seen as a constant tube, with a marked real line : this marked line is responsible for the bubbling monodromy.
It is interesting to note that this bubbling is in fact equivalent to the fact that in the Pontryaguin algebra H * (ΩS 2 ; Z/2) the non-vanshing class in H 1 (ΩS 2 ; Z/2) has a non-vanishing square. 4.2.2. CP n for n > 1. The computation can be achieved in the same way on CP n for n > 1. Notice that the minimal first Chern class is n + 1 ≥ 3 so that the spectral sequence still exists after page 2. It is an easy computation to see that the quantum component of the differential d 2 is given by : In particular, the pages of the spectral sequence all vanish after page 2.
The contrast between the situation n = 1 and n > 1 comes from the properties of the Pontryaguin product in H 1 (ΩCP n ). This product is involved in the computation of d 2 • d 2 , in particular : where a n−1 is a generator of H 2(n−1) (CP n ). What is truly remarkable here is that as c min = n + 1, when n > 1, our construction of the truncated spectral sequence shows that d 2 • d 2 = 0 which implies [∆] * [∆] = 0 in the Pontryaguin ring. Of course, this relation is well-known by purely topological methods but it is remarkable that it is a consequence of the existence of the quantized Serre spectral sequence. Moreover, by Theorem 2.1 ii, d 2 can be expressed in terms of Gromov-Witten invariants together with classical Hopf ones and this discussion shows that the relations among them in the Pontryaguin algebra are not trivial.

4.3.
Fibrations over S 2 . Given a loop φ in Ham(M ), one can construct a fibration E φ over S 2 , obtained by gluing two trivial fibrations over the disk via φ. P. Seidel [10] used sections of this fibration to associate an invertible endomorphism on H * (M ) to each such φ, deriving strong topological restrictions on elements in π 1 (Ham(M )). We first give an outline of the construction of this morphism in the context of Morse homology and then explain how it is related to our construction .
Let Ω be a symplectic form on E φ such that its restriction to the fibers is (cohomologuous to) ω, and let J φ be an almost complex structure Ωcompatible on E φ . If x is a critical point of f , we denote by x + and x − the corresponding critical points above ∞ and 0 respectively.
For the purpose of our discussion, notice that J φ can be chosen to be the product i 0 0 J of structures on S 2 and M in local charts U 0 and U ∞ near the fibers over 0 and ∞: the only curves contained in this region are in fact contained in a fiber, and the almost complex structure is regular for them. The almost complex structure J φ can then be made regular for all curves by perturbing it in the complement of this region.
Roughly speaking, the Seidel morphism is obtained by considering 0dimensional moduli spaces of flow lines going out of a critical point x − , hitting a J φ holomorphic section of E φ , followed by a second flow line flowing from the section down to a critical point y + .
To be able to compare homology classes of section with homology classes in M we fix a section s 0 of the fibration E φ : the homology classes having degree 1 over the base are then the classes of the form s 0 +i * α, for α ∈ H 2 (M ) where i : The Seidel morphism is induced at the homology level by the map Φ: We now discuss how to interpret this morphism as a component of the differential d 2 the truncated spectral sequence associated to E φ . The version we will use is a variant of the quantized Morse one from §3.4.1. With the notation in §3.4.1, we write the differential of the quantized Morse complex m λ (x, y)ye λ with the degree considered over the base. This decomposition induces an analogue one for the differentials of the associated truncated spectral sequence which we will denote by From the statement of the Corollary recall that there are two Morse homology classes x, z, |z| > |x|, which are d r -related. Due to the self-indexing condition each of these classes is represented by a linear combination of critical points with the same critical value, Recall also that we assume that H k (M ) ⊗ Λ q = 0 for |ze λ | < k + q < |x|.
For A ∈ R large enough, consider a smooth increasing function φ A : R → R, such that: for t ≥ f (x) + 1/A For any A, the function f A = φ A • f has the same critical points as f with the same (unparametrized) flow lines, but the critical levels above f (x) are shifted upward. The critical points of the same index continue to share the same level hypersurface. Of course, the existence of non-trivial characteristics for f and f A is equivalent.
Suppose that f A has no non trivial periodic orbit. It is then easy to see that, by possibly composing f with another diffeomorphism R → R whose effect is to diminish the size of the derivatives of f close to its critical values, and using a generic almost complex structure which is time-dependent, Floer theory may be applied to the Hamiltonian f A and the Conley-Zehnder index of the critical points agrees with their Morse index. Moreover, the (not extended) Floer and Morse complexes are then the same (indeed, as the homology of the Floer complex has to be isomorphic with Morse homology it follows in this case that the Floer differential is also trivial). Thus, x and z also give Floer homology classes. We now pick the constant A so that A ≥ ρ(2n + r) where ρ is the monotonicity constant (ω(α) = ρc 1 (α)).
Consider the truncated spectral sequence associated to the Hamiltonian X f A . By hypothesis we know that x is d r -related to ze λ . This implies that there is a critical point z j so that there are Floer trajectories from one of the x i 's to z j e λ . Indeed, as |x| > k + q > |ze λ | implies H k (M ) ⊗ Λ q = 0 the differential d r is the first one relating the vertical line through |x| to the one through |ze λ |. In other words, letting p = |ze λ |, we have that E r p, * is a subgroup of E 2 p, * and so, if there are no flow lines relating some x i to a z j e λ , then x and ze λ can not be d r -related.
The same technique applies in many other variants of the situation described above. The basic idea is to ensure the existence of a sequence of trajectories, "ending at a higher level than its starting point" (there was just one such trajectory in the case above) in such a way that the relevant intermediate points can be shifted out of the action window (as done before using φ A ). For this, besides identifying a chain of differentials which relate a succession of homology classes in the spectral sequence one also needs to be able to choose appropriate chains representing these classes (the self indexing condition and the homological "gap" condition had this purpose above).
Here is such a variant valid when (d c min ) 2 = 0. Proof. Fix a self-indexed Morse function f : M → R. We assume that it has no non-trivial close characteristics.
Let ξ ∈ E c min p,q be a class so that (d c min ) 2 (ξ) = 0. As in the previous proof we may assume that the critical points of f are non-degenerate periodic orbits of X f and their Conley-Zehnder index agrees with the Morse index. Thus we may apply our construction of the truncated quantized Serre spectral sequence to the Hamiltonian f (together with a generic time-dependent almost complex structure). We may also assume, after possibly composing f with an appropriate diffeomorphism R → R, that: * for any critical points x ∈ Crit(f ) the interval [f (x), f (x) + ρ(n) + ω min ] does not contain any critical values different from f (x). Here ρ is as before the monotonicity constant so that ω min = ρc min . From the discussion in §3.3 we see that (d c min ) 2 (ξ) = 0 implies that for some critical point x ∈ Crit(f ) we have that the moduli space M(x, x#α min ) is non void and has a non-void codimension one stratum Σ 1 consisting of broken Floer trajectories as well as a non-void codimension two stratum Σ 2 consisting of Floer trajectories with some bubble attached.
Assume that among the broken trajectories in Σ 1 there is one which joins x to ye α followed by a second trajectory from ye α to xe α min .
Notice that this implies that |y| = |x|. Indeed, if |y| = |x|, the first inequality implies that c 1 (α) > 0 and the second that c 1 (α) < c min which is not possible.
Recall that f is self-indexed, |y| > |x| as well as our assumption * on the critical values of f . This implies that f (y) ≥ f (x) + ρ(n + c min ) which contradicts the first inequality in (11). The second case is |y| < |x|. Then c min ≥ c 1 (α) > −2n. This means f (y) − ω(α) ≤ f (y) − ω min < f (x) − ω min which contradicts the second inequality in (11) and concludes the proof.