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QUANTIZATION OF THE SERRE SPECTRAL SEQUENCE 7 0 JEAN-FRANCOISBARRAUDAND OCTAV CORNEA 0 2 Abstract. The present paper is a continuation of [1] and [2]. It ex- n plores how the spectral sequence introduced in [1] interacts with the a presence of bubbling. As consequences are obtained some relations be- J tween binary Gromov-Witten invariants and relative Ganea-Hopf in- 2 variants,acriterionfordetectingthemonodromyofbubblingaswellas 2 algebraic criteria for the detection of periodic orbits. ] G S . h t a Contents m 1. Introduction 1 [ 1.1. Acknowledgment. 3 1 2. Notation and statement of results 3 v 5 2.1. Setting and recalls 3 2 2.2. Main statement. 6 6 3. Proof of the main theorem. 7 1 3.1. Construction of the truncated spectral sequence. 7 0 7 3.2. Invariance of the truncated spectral sequence. 11 0 3.3. Detection of monodromy. 13 / h 3.4. Quantum perturbation of the Serre spectral sequence. 15 at 4. Examples, applications and further comments 19 m 4.1. Extensions 19 : 4.2. Examples 20 v 4.3. Fibrations over S2 22 i X 4.4. Non trivial periodic orbits for Morse functions 24 r References 27 a 1. 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 lx,y M(x,y) −→ G 1 2 J.-F.BARRAUDANDO.CORNEA into some sufficiently large topological monoid G, one can define a “rich” Morse type chain complex whose differential is of the usual form dx = a y x,y X y but a , the coefficient “measuring” the moduli space M(x,y), belongs to x,y 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 notonly that themapsl are continuous butalso that they are compatible x,y in the obvious way with compactification and with the crucial boundary formula: (1) ∂M(x,y) = M(x,z)×M(z,y) . [ z 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 Hamil- tonian version of Floer homology and this is indeed the setting of this paper. In particular, the modulispaces M(x,y) consist of Floer tubes and the monoid G is the space of pointed Moore loops on M, ΩM, with (M2n,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 , and that, min 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 QUANTIZATION OF THE SERRE SPECTRAL SEQUENCE 3 M, we see that this construction provides a new symplectic invariant which consists of the first c pages (together with their differential) of a spectral min sequence which is a quantum deformation of the Serre spectral sequence. One additional important point is that, on the last defined page, the pre- sumptive differential, dr, is still defined and invariant but might not verify (dr)2 = 0. Ofcourse,thenextstageistounderstand-atleastinpart-thisquantum deformation in terms of classical Gromov-Witten invariants. In this respect we obtain that the first interesting differential dr, 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 (dr)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 en- coded in the ring structure of H (ΩM). ∗ The next interesting point is to understand what happens for the first r when (dr)2 6= 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 . The fact that dr is invariant but, simultaneously, x,y (dr)2 might not vanish is quite remarkable and, indeed, this morphism dr 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. Inthe last section we firstshortly 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 appro- priateas,earlyinthisproject,webelieved,aposteriori withoutjustification, thatthemonodromyof bubblingismuch lessrelevant anditisoneofDusa’s questions which made us reconsider the issue and appreciate the full impor- tance of this phenomenon. 2. Notation and statement of results 2.1. Setting and recalls. Fix the symplectic manifold (M2n,ω) and we suppose for now that M is closed. We assume that M is monotone in 4 J.-F.BARRAUDANDO.CORNEA the sense that the two morphisms ω : π (M) → R and c : π (M) → Z 2 1 2 are proportional with a positive constant of proportionality ρ. We denote by c the minimal Chern class and by ω the corresponding minimal min min symplectic area (so that we have ω = ρc ). min 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 (x) for each x ∈ f Crit(f). For two critical points x and y and a class α ∈ π (M) such that 2 i(x) − i(y) + 2c (α) − 2 = 0, we define GW (x,y) as the number of ele- 1 α ments 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, mod- ulo reparametrization. As such GW (x,y) is not an invariant (because x,y α might not be Morse cycles). However, if for two classes [x] = [ λ x ] i i and [y] = [ µ y ], we define GW ([x],[y]) = λ µ GW (x ,y ) tPhen we i i α i j α i j obtain an iPnvariant. For α ∈ π , let [α] be iPts image by the morphism 2 π (M) → H (ΩM). 2 1 2.1.2. The Novikov ring. Let L(M) bethe space of contractible loops in M. Let Γ be the image of the Hurevicz morphism π (M) → H (M,Z/2). 2 2 The two forms ω and c define morphisms Γ :−ω−,c→1 R,Z which under our 1 monotonicity assumption are proportional. Let Γ = Γ/ker(ω). We let Λ 0 be the associated Novikov ring which is defined as follows Λ =  λ eα . α X  α∈Γ0 where the coefficients λ belongto Z/2 suchthat α ∀c> 0, ♯{α,λ 6= 0,ω(α) ≤ c} < +∞ . α The grading of the elements in Λ is given by |eλ| = −2c (λ). 1 We also denote by L˜(M) the covering of L(M) associated to Γ : it is the 0 quotient of the space of couples (γ,∆), where γ ∈ L(M) and ∆ is a disk bounded by γ, under the equivalence relation (γ,∆) ∼ (γ′,∆′) if γ = γ′ and ω([∆−∆′]) = c ([∆−∆′]) = 0. 1 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. 2.1.3. Moduli spaces of Floer tubes. Let H :M ×S1 → R be a Hamiltonian function. The Hamiltonian flow associated to H is the flow of the (time dependent) vector field X defined by : H ω(X ,·) = −dH . Ht t QUANTIZATION OF THE SERRE SPECTRAL SEQUENCE 5 All along this paper, the periodic orbits of X will be supposed to be non H degenerate. We denote by P ⊂ L(M) the set of all contractible periodic H orbitsof thehamiltonianflowassociated toH andweletP˜ bethecovering H of P which is induced from L˜(M). H For each periodic orbit x ∈ P we fix a lift (x,∆ ) ∈ P˜ . For a generic H x H pair (H,J) and x,y ∈ P , λ ∈Γ we now consider the moduli spaces: H 0 M′(x,y;λ) = {u :R×S1 :u verifies (2)} so that the pasted sphere ∆ ∪u∪(−∆ ) is of class λ and x y (2) ∂ u+J(u)∂ u−J(u)X (u) = 0 , lim u(s,t) = x(t), lim u(s,t) =y(t) . s t H s→−∞ s→+∞ 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,∆ ) 6= (y,∆ ) they admit a free R action. We denote the x y quotient by this action by M(x,y;λ) and we have dimM(x,y;λ) = µ((x,∆ ))−µ((y,∆ ))+2c (λ)−1 x y 1 where µ((x,∆ ) is the Conley-Zehnder index of the orbit x computed with x respect to the capping disk ∆ . x 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: (3) ∂M(x,y;λ) = M(x,z;λ′)×M(z,y;λ′′) ∪ Σ x,y,λ z,λ′+[λ′′=λ Here Σ is a set of codimension 2 which consists of Floer tubes with x,y,λ at least one attached bubble. We will say that (H,J) has bubbling monodromy if there exist x,y ∈ P H and λ ∈ Γ so that: 0 H1(Σ ;Z) 6= 0 x,y,λ Thismeans,inparticular,thatπ (Σ ) 6= 0sothattherearenon-contractible 1 x,y,λ 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 (Er,dr), 0 ≤ r ≤ k is a truncated spectral sequence of order k if (Er,dr) is a chain complex for each r ≤ k − 1 which verifies H (Er,dr) = Er+1 and dk is a linear map ∗ of degree −1. A truncated spectral sequence of ∞-order is a usual spectral sequence. A morphismof order k truncated spectral sequences is a sequence of chain maps φ : (Er,dr) → (Fr,dr), 0 ≤ r ≤ k, so that H (φ ) = φ for r ∗ r r+1 0≤ r ≤ k−1. We say that two truncated spectral sequences are isomorphic 6 J.-F.BARRAUDANDO.CORNEA 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. AssumethatC isagradedrationalvector spaceandthatFiC isanincreas- ∗ ing filtration of C . We say that a linear map d : C → C is a truncated ∗ ∗ ∗−1 differential of order k compatible with the given filtration if d(FiC)⊂ FiC ∀i and (d◦d)(FrC)⊂ Fr−2kC for all r ∈ Z. 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 Zr = {v ∈ FpC : dv ∈ Fp−rC}+Fp−1C p and r-boundaries Br = {dFp+r−1C ∩FpC}+Fp−1C p it is immediate to see that Br ֒→ Zr for 0 ≤ r ≤ k which allows us to define p p the pages of the truncated spectral sequence by Er = Zr/Br. Obviously, d p p p induces differentials dr on Er when r < k as well as a degree −1 linear map dk on Ek. 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. There- fore, we assume here that (M,ω) is closed, simply-connected and monotone with c ≥ 2. min Theorem 2.1. There exists a truncated spectral sequence of order c , min (Er(M),dr), 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: E2 ∼= H (M)⊗H (ΩM)⊗Λ . ∗ ∗ ii. The differential d2 has the decomposition d2 = d2+d2 0 Q where d2 is the differential appearing in the classical Serre spectral 0 sequence of the path loop fibration ΩM → PM → M and d2x= GW (x,y)y[α]eα . Q α X y,α iii. If (dcmin)2 6= 0, then any regular pair (H,J) has bubbling monodromy. Remark 2. Clearly, if d2◦d2 = 0 - for example if c ≥ 3 - the vanishing of min the square of d2◦d2 translates into some relations between binary Gromov- Witten invariants and the classical Serre spectral sequence differential d2. 0 In turn, this differential is quite well known and rather easy to compute and QUANTIZATION OF THE SERRE SPECTRAL SEQUENCE 7 itcan beexpressedin 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 (ΩM), eα ∈ Λ and x,y ∈ H (M) so that in the square of the differential 1 ∗ appears the Pontryaguin product H (ΩM)⊗H (ΩM) → H (ΩM). 1 1 2 The relation with the Seidel homomorphism is seen by considering the spectral sequence in the case of a symplectic fibration over CP1. 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. Letx,y ∈ H (M)andλ ∈ Λ. Wewillsay thatxandyeλ (which existonthe ∗ E2 page of thespectralsequencein Theorem2.1) are dr-related if xsurvives to the r-th level of the spectral sequence and there is some γ ∈ C (ΩM) ∗ so that the product γ ⊗yeλ also survives to the r-th page of the spectral sequence and we have dr([x]) = [γ ⊗yeλ]+... Corollary 2.2. Assume that there are homology classes x,z ∈ H (M), ∗ |x| < |z|, so that x is dr-related to zeλ and H (M) ⊗ Λ = 0 for |x| > k q k + q > |zeλ|). 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 ofthesameindexhavethesamecriticalvalueandind (x) > ind (y)implies f f 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 pseudo- complex 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 ∗ betheMooreloopspaceoverX (thespaceofloopsparametrizedbyintervals 8 J.-F.BARRAUDANDO.CORNEA 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. NoticethatC (ΩM′)isadifferentialringwheretheproductisinduced ∗ by the concatenation of loops. Finally, our coefficient ring is: R = C (ΩM′)⊗Λ . ∗ ∗ 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: C(H,J) = ⊕ R x˜/ ∼ x˜∈P˜H ∗ with the identification x˜eλ ∼ x˜♯λ, where x˜♯λ stands for the capping of x obtained by gluing a sphere in the class λ to x˜. The grading of an element in x˜ ∈ P˜ is given by the respective Conley-Zehnder index. There is a H natural filtration of this complex which is given by FrC(H,J) = R < x˜ ∈P˜ : µ(x˜) ≤ r > . ∗ H 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 lift x˜ ∈ P˜ for each x ∈ P . With these conventions H H and-as assumedbefore-foragenericchoice ofJ andH -themodulispaces are smooth manifolds of dimension |x˜|−|y˜|−1 when |x˜| =6 |y˜|, 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˜,y˜) for the moduli space of Floer tubes which lift to paths inside L˜(M) joining x˜ ∈ P˜ to y˜ ∈ P˜ and we let M(x˜,y˜) be the respective compactification. H H 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˜,y˜). Let u : R×S1 → M be a parametrization of v. Since the value of the action functional a : L˜(M) → R, a ((γ,∆)) = − ∆∗ω+ H(t,γ(t))dt H H Z Z D2 S1 QUANTIZATION OF THE SERRE SPECTRAL SEQUENCE 9 is strictly decreasing along the R direction, it can be used to reparametrize u by the domain [−a(x˜),−a(y˜)]×S1, and the restriction of u to the interval [−a(x˜),−a(y˜)]×{0} defines a Moore loop in M′. This defines a map (4) σ : M(x˜,y˜) → Ω(M′) x˜,y˜ which is continuous. We will call it the “spine” map. This map should then be extended to the compactification M(x˜,y˜) of M(x˜,y˜). It is well-known that the objects in M(x˜,y˜) 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 σ extends continuously over the part of x˜,y˜ 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 ∈Γ be the class so that c (α ) = c (by our monotonicity min 0 1 min min 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 can occur 0 in a moduli space M(x˜,y˜) with y˜6= x˜♯α only if |x˜|−|y˜| ≥ 2c (α)+1 . 1 It is also important to note that bubblingof a spherein 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˜|−|y˜| ≤ 2c (α)−1. 1 We summarize this discussion: Lemma 3.1. The spine map σ extends continuously to M(x˜,y˜) if x˜,y˜ |x˜|−|y˜| ≤ 2c −1 . min In case |x˜|−|y˜|= 2c and if σ does not have such a continuous extension min to M(x˜,y˜), then y˜= x˜♯α . min Thespinemapobtainedinthiswaysatisfiesalsoacompatibility condition which we now make explicit. If M(x˜,z˜) × M(z˜,y˜) ⊂ M(x˜,y˜), then the restrictionofσ onthesetontheleftoftheinclusionequalsm◦(σ ×σ ) x˜,y˜ x˜,z˜ z˜,y˜ where m : ΩM′×ΩM′ → ΩM′ is loop concatenation. For pairs (x˜,y˜) with |x˜|−|y˜| ≤ 2c −1, we use the map σ to rep- min x˜,y˜ resent the moduli spaces M(x˜,y˜) inside the loop space Ω(M′). We then choose “chain representatives” m(x˜,y˜) ∈ C (ΩM′), i.e. chains generating ∗ the fundamental class of σ(M(x˜,y˜)) relative to its boundary, in such a way 10 J.-F.BARRAUDANDO.CORNEA that: (5) ∂m(x˜,y˜) = m(x˜,z˜)∗m(z˜,y˜) X |y|<|z|<|x| where∗is theoperation induced on C (ΩM′) by the concatenation of loops. ∗ The key point regarding this formula is that, under our assumption |x˜|− |y˜| ≤ 2c −1, the compactified moduli space M(x˜,y˜) is a manifold with min boundary. Moreover, its boundary verifies the usual formula valid in the absence of bubblingso 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: (6) dx˜ = m(x˜,y˜) y˜ X 1≤|x˜|−|y˜|≤2cmin−1 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 γ ⊗x˜ ∈ C (ΩM)⊗P˜ we have d◦d(γ ⊗x˜) = (γ ⊗(d◦d)(x˜)). We now ∗ H compute: d◦d(x˜)= d(m(x˜,y˜) y˜) X |x˜|−|y˜|≤2cmin−1 = ∂m(x˜,y˜) y˜+m(x˜,y˜) dy˜ X 1≤|x˜|−|y˜|≤2cmin−1 = m(x˜,z˜)m(z˜,y˜) y˜+ m(x˜,y˜)m(y˜,z˜) z˜ X X 1≤|x˜|−|y˜|≤2cmin−1 1≤|x˜|−|y˜|≤2cmin−1 |y˜|+1≤|z˜|≤|x˜|−1 1≤|y˜|−|z˜|≤2cmin−1 = m(x˜,y˜)m(y˜,z˜) z˜ X 1≤|x˜|−|y˜|≤2cmin−1 |y˜|−2cmin+1≤|z˜|≤|x˜|−2cmin and we see that d2 drops the filtration index by at least 2c . In the min algebraic terms of §2.1.5 we obtain: Lemma 3.2. With the definition above, d is a truncated differential of order c with respect to the canonical filtration on C(H,J) and thus it induces min a truncated spectral sequence Er(H,J) of the same order so that E2(H,J) ∼= H (M)⊗H (ΩM)⊗Λ ∗ ∗ TheisomorphisminthelemmaisobviousbecauseE1(H,J) ∼= CF (H,J)⊗ ∗ H (ΩM) and as d1 only involves the 0 dimensional moduli spaces of Floer ∗ tubes we obtain that d1 is just: d ⊗id where (CF (H,J),d ) is the usual F ∗ F 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