Research statement Fran¸cois Charette November 29, 2016

1

Introduction

My main research interest is the topology of Lagrangian submanifolds, whose role in symplectic geometry is central. They are closely related to theoretical physics; for instance, the Homological mirror symmetry conjecture of Kontsevich relates the derived Fukaya category, a symplectic object, to coherent sheaves, matrix factorizations and Landau-Ginzburg potentials. I am also interested at the interaction of Lagrangians with the group of Hamiltonian diffeomorphisms. In my thesis I studied the Hofer geometry of Lagrangians and I proved a conjecture of Barraud and Cornea under a monotonicity assumption. Counterexamples were recently found by Rizell [Riz13] when the Lagrangians involved are not monotone. I recently adapted the construction of Lagrangian quantum homology of Biran and Cornea to the case of orientable Lagrangian surfaces, whether they are monotone or not, thus extending my results on Hofer’s geometry to this case as well. Rizell’s counterexamples mentionned above apply to non orientable Lagrangian surfaces. This shows that the hypothesis of orientability of my construction is not technical, the results do not hold without further restrictions. Borman and McLean [BM13] also obtained related results for non monotone Lagrangians, but with quite different methods and restrictions involving the Riemannian geometry of the Lagrangian. In a joint work with Octav Cornea, we defined actions of Hamiltonian isotopies on Lagrangian cobordisms and on the derived Fukaya category. The former action is given by Lagrangian suspension, a construction that goes back to Arnol’d. We show that a functor from cobordisms to the derived Fukaya category is equivariant with respect to these actions. From this, we conclude that the endomorphisms of one Lagrangian in the cobordism category contain at least as many elements as there are units in quantum homology of the ambient symplectic manifold. Apart from Lagrangian surgery, this is the only method available to produce non-trivial monotone Lagrangian cobordisms. Recently, these results were used by Tonkonog [Ton15] to find generators of the derived Fukaya category over fields of characteristic two, based on Abouzaid’s [Abo10] generation criterion. My recent research [Cha15b] focuses on quantum Reidemeister torsion, motivated by the following observations. Lagrangian Floer homology can detect Lagrangian intersection when it does not vanish. In [Fuk97], Fukaya made a striking conjecture, called the symplectic 1

s-cobordism conjecture, which states that if the Floer homology of a pair (L0 , L1 ) vanishes and, moreover, the Whitehead torsion of the Floer complex is trivial, then one can find a Hamiltonian isotopy displacing L0 from L1 . Related results were obtained by Hutchings and Lee [HL99], Lee [Lee05a, Lee05b] and Sullivan [Sul02]. Suarez [Sua14] also applied torsion to prove triviality of certain exact Lagrangian cobordisms, which have played an important role in the study of the Fukaya category due to work of Biran and Cornea [BC13, BC14]. Recently, I have investigated a simpler version of torsion for Lagrangian Floer homology, that I call quantum Reidemeister torsion, and proved that it can be expressed as a certain rational function of open Gromov-Witten invariants. Fukaya’s conjecture implies that this function has to be trivial, thus restricting the behaviour of Gromov-Witten invariants. Quantum torsion is quite computable when the dimension of the Lagrangian is small. This led me to new obstructions in the topology of monotone Lagrangian 3-manifolds. Most results in the litterature are concerned with Lagrangians in C3 , whereas my methods and results are valid for a much wider class of symplectic manifolds. Finally, this torsion can be used to associate invariants to so-called trivial components of the Fukaya category, where each object has vanishing Floer homology. These components are usually ignored because they do not contain homological information. However, my results show that they encode open Gromov-Witten invariants. Related research questions will be given in the last section.

2 2.1

Preliminary notions Symplectic topology

A symplectic manifold (M, ω) is a smooth manifold endowed with a bilinear antisymmetric two-form ω ∈ Ω2 (M ) which is both closed (dω = 0) and non-degenerate, that is, it induces an isomorphism ιω : T∗ M → T ∗ M , X 7→ ω(X, ·). Some linear algebra shows that dim M = 2n and ω n defines a volume form. Symplectic manifolds appear naturally in Hamiltonian mechanics, as the phase space of a system Pof n particles, the set of coordinates (qi , pi ), i = 1, ..., n, endowed with the symplectic form i dqi ∧ dpi . In this framework, qi is the position of a particle and pi its impulsion. According to Darboux’s theorem, every symplectic manifold is locally symplectomorphic to phase-space. Another natural example is the complex projective space CP n with its Fubini-Study structure ωF S ; this is obtained as a quotient of the standard symplectic structure w0 := P n+1 n+1 by the action of C∗ given by complex multiplication. i=1 dxi ∧ dyi in C The cotangent bundle of the interval T ∗ [0, 1] has a symplectic area form coming from R2 . This simple example is central in the study of Lagrangian cobordisms as defined below. More generally, the cotangent bundle of any manifold L admits a symplectic structure ωcan = dλ, where λ ∈ Ω1 (T ∗ L) is characterized by the property that σ ∗ λ = σ for every section σ : L → T ∗ L.

2

2.2

Lagrangian submanifolds

A submanifold L ⊂ (M, ω) is Lagrangian if dim L = n = 12 dim M and if ω|L = 0. n The typical example is Pthe real part of C , i.e. the subset (x1 , 0, x2 , 0..., xn , 0); a simple computation shows that dxi ∧ dyi vanishes on it. By definition of the Fubini-Study form n n on CP , RP is a Lagrangian submanifold in it. The product of Lagrangians in the product of symplectic manifolds yield Lagrangians, as well as any curve in R2 , since a symplectic form always vanishes on a one-dimensional space. Finally, the zero-section of a cotangent bundle (T ∗ L, ωcan ) is also Lagrangian. Lagrangians play an essential role in the classification of symplectic manifolds and exhibit rigidity properties. The first results in this direction were discovered by Gromov: Theorem 2.1 ([Gro85]) • Let L be a closed (i.e. connected, compact and without 2 (Cn , L). boundary) Lagrangian submanifold of (Cn , ω0 ), then 0 6= [ω0 ] ∈ HdeRham • There exists a symplectic structure ω ∈ Ω2 (Cn ) and a closed Lagrangian L such that 0 = [ω] ∈ H 2 (Cn , L). 2.2.1

Monotonicity restrictions

We mention the definition of monotonicity for completeness, but we will not try to explain its importance; let it only be said that monotone Lagrangians exhibit some rigidity properties that are not shared by all Lagrangians. For example, the only monotone circle on the 2sphere is a circle dividing it into two hemispheres of equal area, and this circle cannot be separated from itself by a Hamiltonian isotopies. Other circles however can be separated. A Lagrangian L is called monotone if the following two conditions hold: • There exists ρ > 0 such that the two morphisms ω : π2 (M, L) → R µ : π2 (M, L) → Z given respectively by integration and the Maslov index satisfy ω = ρµ. • The positive generator of the image of µ, denoted by NL , is greater or equal than two.

2.3

Hamiltonian isotopies and Hofer’s geometry

A Hamiltonian is a compactly supported smooth function H : S 1 ×M → R, (t, m) 7→ Ht (m). Its associated vector field is defined by the equation ω(XtH , ·) = −dHt (·), whose flow φH t satisfies d H ( φt )|t=0 = XtH (φH t ), φ0 = id. dt The set of all these flows at time t = 1 is denoted by Ham(M, ω) and called the group of Hamiltonian isotopies. The energy of φ ∈ Ham is Z 1 E(φ) := inf{ (max Ht − min Ht )dt | φH 1 = φ}. 0

M

M

3

Given two Lagrangians L0 , L1 that are Hamiltonian isotopic, the Hofer distance between them is defined by the quantity d(L0 , L1 ) :=

inf

{E(φ)}.

φ∈Ham|φ(L0 )=L1

One easily checks that d is symmetric and satisfies the triangle inequality, however the non-degeneracy property is highly non-trivial and requires the use of holomorphic curves technique. It was first proved by Chekanov [Che98] for symplectic manifolds tame at infinity and I obtained a different proof for monotone Lagrangians; the exact statement is in §3.2. Part of my research focuses on the interaction between the group Ham(M ) with Lagrangian submanifolds and cobordisms.

2.4

Lagrangian cobordisms

Biran and Cornea have recently introduced the Lagrangian cobordism category [BC13, BC14], which has fundamental importance in the study of Lagrangians. This category is denoted Cob(M, ω) or simply Cob(M ). Its objects are ordered tuples (L1 , L2 , ..., Lk ) of closed monotone Lagrangians. Without the monotonicity assumption, the notion of cobordism has been studied before and is quite flexible, it lies in the realm of algebraic, rather than symplectic, topology, as illustrated by the work of Audin` [Aud85]. A morphism V : L → (L1 , ..., Lk ) is a cobordism (V ; L; ki=1 Li ) endowed with a monotone Lagrangian embedding V → (T ∗ [0, 1] × M, dt ∧ dy ⊕ ω) that is cylindrical at the end: V |(1−,1]×R×M = (1 − , 1] × {1} × L V |[0,)×R×M

k a = [0, ) × {i} × Li i=1

(L01 , ..., L0s )

A general morphism V : → (L1 , ..., Lr ) is a disjoint union of morphisms having at most one positive end; see Figure 1 below. Examples. a. Let γ be a curve in T ∗ [0, 1] starting at (0, 1) and ending at (1, 1), L be a given ¯ := γ × L is the identity morphism in Cob(M ). monotone Lagrangian. Then the product L b. In section §3.4, we will state a theorem obtained jointly with Octav Cornea that yields pairwise distinct families of cobordisms V : L → L.

2.5

The Donaldson category

Lagrangian cobordisms encode all Lagrangians in a geometric way. The algebraic counterpart is given by the Donaldson category, via the Floer homology package. Define Don(M, ω) the category whose objects are closed monotone Lagrangians. A morphism α : L0 → L1 is given by a Floer homology class, which we very quickly define, omitting 4

Figure 1: A morphism V : (L01 , L02 , L03 ) −→ (L1 , . . . , L6 ), V = V1 + V2 + V3 , projected on T ∗ [0, 1]. all technical details; these are numerous but by now standard. Among many references, the reader can look at [Flo88, Oh93, Oh95, FOOO09a, FOOO09b, Sei08]. Given two Lagrangians intersecting transversaly L0 , L1 , the Floer complex CF (L0 , L1 , J) is the Z2 vector space spanned by the intersection points L0 ∩ L1 . The differential is defined by first choosing an almost complex structure compatible with the symplectic structure ω, i.e. an endomorphism J of T M satisfying J 2 = −I, ω(J·, J·) = ω(·, ·) and such that ω(·, J·) is a scalar product. The basic example is the standard complex structure on (Cn , ω0 ). Let p, q ∈ L0 ∩ L1 . The set of J-holomorphic, or Floer, strips from p to q is the set M(p, q, J) of functions u : R × [0, 1] → M such that: u(−∞, [0, 1]) = p, u(∞, [0, 1]) = q u(R, 0) ⊂ L0 , u(R, 1) ⊂ L1 ∂u ∂u +J = 0. ∂s ∂t For a generic choice of J, these sets are smooth manifolds whose zero-dimensional component, denoted by M0 (p, q, J), is compact. The Floer differential can thus be defined by: X ∂ : CF (L0 , L1 ) → CF (L0 , L1 ), p 7→ (#2 M0 (p, q, J))q q

Floer proved that ∂ 2 = 0 and that the homology of this complex is independant of generic choices of J; it is denoted by HF (L0 , L1 ). In the Donaldson category, we set Mor(L0 , L1 ) = HF (L0 , L1 ). Composition of morphisms is given by the ”triangle product” and will be omitted. Theorem 2.2 ([BC14]) There exists a functor Fe : Cob(M ) → Don(M ) which maps an object (L1 , ..., Lk ) to Lk . 5

Remark. Biran and Cornea’s result is much richer than what has been stated here. They define a functor with values in triangular decompositions of the derived Fukaya category. In order to keep things relatively simple, we will stick in this text to the Donaldson categoy; the same will hold in §3.4.

3 3.1

Past results Quantum Reidemeister torsion

Classically, the torsion of a chain complex is an invariant that is extracted from the Z[π1 (X]˜ of a finite CW -complex X. It is equivariant cellular chain complex of the universal cover X defined for representations of π1 (X) for which this homology vanishes and is a finer invariant than homology. For example, it has been used to classify 3-dimensional lens spaces up to homeomorphism and is a major ingredient in the s-cobordism theorem, see Milnor [Mil66]. I recently adapted this notion to the pearl complex of Biran and Cornea. To a generic triple D made of a Riemannian metric, a Morse-Smale function and an almost complex structure, one associates the pearl complex of a Lagrangian. Moreover, one can twist the differential of this complex by using field representations ϕ : π2 (M, L) → F× , the resulting twisted quantum homology of L is denoted by QH ϕ (L) and does not depend on D. If this homology vanishes, then one can define the torsion of this complex τϕ (L, D), as in the classical case. However, proving invariance is quite delicate and, in similar contexts, usually requires a bifurcation analysis of the chain complex. I identified a class of Lagrangians, which I call E 1 -narrow (the precise definition involves a spectral sequence of Oh and will be left out), where bifurcation analysis is not required. In this context, I proved the following results, which all appear in [Cha15b]. Theorem 3.1 Let L be a closed, orientable, monotone and spin Lagrangian submanifold of (M, ω). If L is E 1 -narrow, then the quantum Reidemeister torsion τϕ (L, D) is independent of D and can be expressed as a rational function of open Gromov-Witten invariants of L, weighted by ϕ. Theorem 3.2 Assume that L = S 2k+1 × V and that the following Gromov-Witten invariant does not vanish: X rϕ = GW (∗ × [V ], pt)ϕ(A) µ(A)=NL −χ(V )

Then L is E 1 -narrow and its torsion is given by τϕ (L) = rϕ characteristic.

, where χ denotes the Euler

This class of Lagrangian contains many known examples, indeed: Corollary 3.1 The hypothesis of the previous theorems are satisfied for the following list of Lagrangians:

6

• L = S 1 × ∗. Then τϕ (L) =

1 . rϕ

• L = S 1 × (T k ), where T k = S 1 × · · · × S 1 (k times) is the k-torus. Then τϕ (L) = 1 • L = S 1 × Σg , where Σg is a closed orientable surface of genus g ≥ 1. Then τϕ (L) = 2(g−1) rϕ • L = S 1 × V is a product Lagrangian embedding in S 2 × X, where π2 (X, V ) = 0. 3.1.1

Torsion and orientable Lagrangian 3-manifolds

When L is a monotone, closed and orientable Lagrangian 3-manifold, it might not be E 1 narrow, but it turns out that quantum torsion is always invariant of generic choices and is again expressed in terms of open Gromov-Witten invariants. Let us introduce some notions required to state the result more precisely. Denote by IC : H2 (L; C)⊗3 → C the 3-form defined by the triple intersection product on H2 (L; C). This was first studied by Sullivan in [Sul75] and Turaev [Tur84]. Open Gromov-Witten invariants with two marked points yield two linear maps: A : H1 (L; C) → H2 (L; C),

r : H2 (L; C) → H3 (L; C) ∼ =C

In the same manner, open Gromov-Witten invariants with three ordered marked points yield a bilinear map: Q : H2 (L; C) ⊗ H2 (L; C) → C Denote by b1 (L; Z) the first Betti number of L, by |Torev (L)| the order of the torsion subgroup of ⊕k even Hk (L; Z) and by |Torodd (L)| the order of the torsion subgroup of ⊕k odd Hk (L; Z). The result is now: Theorem 3.3 Let L be a monotone, closed and orientable Lagrangian 3-manifold in (M, ω), with minimal Maslov class NL = 2. Suppose that QH(L) = 0. There is the following alternative: 1. b1 (L; Z) is odd: there exists v ∈ H2 (L; C) such that IC (v, ·, ·) is symplectic on H2 (L; C)/hvi. Quantum Reidemeister torsion is given by the formula τ (L) =

r(v)b1 (L;Z)−3 |Torev (L)| · ∈ C× / ± 1 |Torodd (L)| det IC (v, ·, ·)|H2 /hvi

2. b1 (L; Z) is even: then IC = 0. Moreover, A is an isomorphism, Q is symplectic, and b1 (L;Z)

τ (L)

 =

|Torev (L)| |Torodd (L)|

b1 (L;Z)

7

det Ab1 (L;Z)−1 ∈ C× / ± 1 det Q

3.2

Lagrangian Gromov width and Hofer’s geometry

This was part of my PhD thesis and is concerned with many aspects of Lagrangian topology. My main interest was a conjecture of Barraud–Cornea [BC06] whose context we now explain. Given two Lagrangians L0 , L1 , the Gromov width of L0 relative to L1 is:    π There exists a symplectic embedding  e : B 2n (r) → M \L1 , wG (M \L1 , L0 ) := sup r2 2  e−1 (L0 ) = B 2n (r) ∩ Rn Here B 2n (r) denotes the ball of radius r in (R2n , ω0 ) and Rn is taken as the real part of Cn , so that it is Lagrangian. Barraud and Cornea related this radius to the Hofer distance between the two Lagrangians and stated the conjecture, which is the statement of Theorem 3.5 below, that I proved in the monotone setting. The general case has recently been disproved by Rizell [Riz13]. See §3.3 for a discussion and partial results in the case of orientable Lagrangian surfaces. Assume now that there is a symplectic embedding e : B 2n (r) → M \L1 such that e−1 (L0 ) = B 2n (r) ∩ Rn ; such embeddings exist for small values of r, by a local argument for Lagrangians. Theorem 3.4 [Cha12] Let L0 , L1 be two closed monotone Lagrangians in the same Hamiltonian isotopy class. Then πr2 /2 ≤ d(L0 , L1 ). In particular, 0 < wG (M \L1 , L0 ) ≤ d(L0 , L1 ). This result, albeit simple to state, has remarkable consequences. For example, it shows that the Hofer distance is indeed non-degenerate, a result first obtained by Chekanov [Che98]. The theorem is a consequence of the following stronger result: Theorem 3.5 [Cha12] Let L, L0 be two closed monotone Lagrangians in the same Hamiltonian isotopy class. Then for every generic compatible almost complex structure J, for every x ∈ L\L0 , there exists a non-constant J-holomorphic curve that is either a Floer strip with boundary on L and L0 , a sphere or a disc with boundary on L, such that x ∈ Im(u) and R ∗ u ω ≤ d(L, L0 ). If moreover L and L0 intersect transversaly, then the Maslov index of u is at most the dimension of L. Applying this to two Lagrangians that are close enough, one recovers a result of Barraud and Cornea [BC07] stating that there is always a non constant Floer strip through any generic point of L, whose energy is smaller than the Hofer distance. When the two Lagrangians are disjoint, we recover a result of Albers [Alb08], stating that there is a J-holomorphic disc with boudary on L through any generic point of L whose area is smaller than the displacement energy. These two direct applications are two extreme cases, one in which no Floer strips appear, and another where no discs appear. The strength of the result resides in the intermediate situation, where the existence of a J-holomorphic curve is a priori not clear.

8

3.3

Quantum homology of non-monotone orientable surfaces

As Theorem 3.4 illustrates, the relative Gromov radius of two Lagrangians plays an important role in understanding Hofer’s geometry. The main ingredient to obtain a lower bound is a uniruling result by holomorphic curves given in Theorem 3.5. The algebraic techniques used to proved this result are quantum homology and Lagrangian Floer homology, both of which strongly rely on monotonicity. An immediate consequence of these results is that the Gromov radius of a single monotone Lagrangian L, which can be displaced from itself by a Hamiltonian isotopy, is bounded from above. Generalizing this a bit, we have: Theorem 3.6 ([Cha15a]) Let L be a closed orientable Lagrangian surface, not necessarily monotone, and L0 Hamiltonian isotopic to L. Then the statements in Theorems 3.4 and 3.5 stll hold. In particular, if L is displaceable, then its Gromov width is finite. These hypotheses imposed on L are not purely technical. Indeed recent results of Rizell [Riz13] show that there are closed Lagrangians in Cn , for every n ≥ 3, whose Gromov width is infinite. There is also an example of a non-orientable closed surface. Hence this theorem is optimal without further restrictions on L. Borman and McLean also proved that if L is an orientable Lagrangian admitting a metric of non-positive sectional curvature in a Liouville manifold, then the width of L is bounded above by four times its displacement energy. In order to prove this result, I adapted the construction of the Lagrangian quantum homology by Biran and Cornea [BC09a, BC09b] to the non-monotone situation. The key observation is that on an orientable Lagrangian surface, the Maslov index of non-constant pseudo-holomorphic discs is at least two, hence such surfaces behave like monotone ones. The resulting homology QH(L; J) depends a priori on the connected component of regular almost structures for which it is defined. Proposition 3.1 There exists a Lagrangian two-torus L ⊂ CP 2 and J0 , J1 two almost complex structures such that QH(L; J0 ) = 0 and QH(L; J1 ) 6= 0. This raises further questions, that I am currently investigating, such as: • How do the enumerative invariants of Biran–Cornea [BC12] change with respect to J; this is of course related to wall-crossing phenomena. • If homology is not invariant, can one still extract an invariant that does not depend on J?

3.4

Lagrangian suspension and categorification of Seidel’s representation

As monotone Lagrangian cobordisms are quite rigid and new, there are few methods to generate examples. This project, done jointly with Octav Cornea, yields a systematic way to build Lagrangian cobordisms that are diffeomorphic to a cylinder [0, 1]×L, yet are not trivial 9

as elements of MorCob (L, L). The key tool involves the so-called Lagrangian suspension, a notion going back to Arnol’d [Arn80a, Arn80b]. Given a monotone Lagrangian L and a path of Hamiltonian diffeomorphisms {φt }t∈[0,1] , φt ∈ Ham(M ), generated by a function Ht , the suspension of L by φt is a morphism Σ(φt )(L) : φ1 (L) → φ0 (L) given by the Lagrangian embedding L × [0, 1] → T ∗ [0, 1] × M (p, t) 7→ (t, Ht (φt (p)), φt (p)). Homotopy classes of path in Ham(M ) make up the morphisms in the fundamental groupoid Π(Ham(M )), seen as a category whose objects are elements of Ham(M ); composition of morhisms is given by concatenation. In this context we show: Proposition 3.2 Lagrangian suspension extends to an action of Π(Ham(M )) on Cob(M ). In other words, there is a monoidal functor Σ : Π(Ham(M )) → End(Cob(M )) such that Σ(φt )(L) is the suspension defined above. This action is used primarily to study the monoid MorCob (L, L) and we give examples containing many non-trivial elements. Theorem 3.7 [CC14] Let M be a monotone toric symplectic manifold whose minimal Chern class is at least two. • If L is the fixed part of the canonical anti-symplectic involution on M , then MorCob (L, L) contains a subgroup isomorphic to QH(M )× , the multiplicative units of the quantum homology of M . • For any monotone Lagrangian of M whose Lagrangian quantum homology does not vanish and whose minimal Maslov number is at least 3, MorCob (L, L) is not trivial. A more concrete example is given by RP n ⊂ CP n , for which we deduce that MorCob (RP n , RP n ) contains a subgroup isomorphic to Z/(n + 1)Z. The non-trivial element in the particular case n = 1 is obtained by a half-rotation of S 2 sending the equator back to itself, with the reversed orientation. We also show that the classical Seidel representation of the fundamental group π1 (Ham(M )), which naturally sits inside Π(Ham(M )), extends to a monoidal action of Π(Ham(M )) on the Donaldson category Don(M ). As mentionned before, we actually show more, our original statement involves the derived Fukaya category and cone decompositions thereof. Recall from Theorem 2.2 the existence of a functor Fe : Cob(M ) → Don(M ). Theorem 3.8 [CC14] The functor Fe is equivariant with respect to the actions of Π(Ham(M )). This new perspective sheds some geometric light on the Seidel representation. Indeed, the main ingredient in the proof shows that Lagrangian suspension is mapped by Fe to Lagrangian Seidel elements as defined by Hu–Lalonde [HL10] and Hu–Lalonde–Leclercq [HLL11]. These 10

elements can be computed in many favorable cases, such as toric manifolds, thanks to recent results of Haug [Hau13] and Hyvrier [Hyv13]. A natural question, which I plan on investigating, is then: are there suspensions which are not trivial but whose Seidel elements (in the Fukaya category sense) are trivial? What tools are there to study them? Partial answers are given by recent results of Tonkonog [Ton15], who shows that higher order terms of the Seidel representation in Hochschild cohomology are not always trivial.

4

Future plans and ongoing research

I will collect here some questions that I am currently thinking about, some of which were already mentionned in the previous sections.

4.1

Quantum Reidemeister torsion, Fukaya categories and Lagrangian cobordisms

In light of Fukaya’s symplectic s-cobordism conjecture [Fuk97], it is important to study torsion of Lagrangians which are displaceable by a Hamiltonian isotopy. Presumably this quantum torsion is trivial, since it is trivial for the Floer complex. The subtle point here is that the pearl complex and the Floer complex may not be simple homotopy equivalent, even though they are quasi-isomorphic. A longer term project I am working on is to do a bifurcation analysis of the pearl complex, in order to prove invariance of torsion under different choices of generic data. Note that so far invariance is proven only for E 1 -narrow Lagrangians as well as orientable Lagrangian 3-folds. In a joint project with Lara Suarez, we are trying to prove that quantum Reidemeister torsion is invariant under monotone Lagrangian cobordism, which would imply that many open Gromov-Witten invariants are preserved by cobordisms, given my results in [Cha15b]. Complementary results have been obtained by Biran and Membrez [BM15], who prove that a certain quadratic form involving open Gromov-Witten invariants is preserved by cobordism. As pointed out above, torsion is also related to this quadratic form. Reidemeister torsion provides a way to attach certain invariants to trivial objects of the Fukaya category, i.e. Lagrangians L with a local system ϕ such that the quantum homology QH ϕ (L) vanishes. Such Lagrangians are usually discarded because they do not provide intersection theoretic invariants. Perhaps one can deduce non trivial information by considering their torsion. The computations done in Theorem 3.2 show that the trivial summands of the Fukaya category already contain non trivial information: they encode open Gromov-Witten invariants. Finally, there are other notions of torsion that exist, such as L2 -torsion. It would be interesting to see if they can be applied to the symplectic world.

11

4.2

Trivial Seidel elements and cobordisms

As mentionned in §3.4, it would be interesting to have an example of a suspension which is not trivial but for which the Seidel element of the loop of Lagrangian is trivial. As it turns out, Tonkonog expands in [Ton15] on my joint result with Cornea [CC14] to do precisely this. What he finds are examples where the Seidel element is trivial in Floer homology, but is not trivial in the Hochschild cohomology algebra generated by the Lagrangian. This uses the full strength of the Fukaya category, where one works on the A∞ level, rather than only on homology. There is another potentially fruitful way to recover such results by considering a spectral sequence of Barraud and Cornea [BC07] which intertwines Floer homology and the loop space of the Lagrangian. Using this, one expects that higher order Seidel elements can be detected by pages of that spectral sequence. It would be quite interesting to see if these higher Seidel elements correspond with the Hochschild cohomology ones found by Tonkonog. If so, it means that the loop space plays a role in the Fukaya category of the ambient symplectic manifold. I plan on investigating these questions jointly with Jean-Fran¸cois Barraud from Toulouse University, France.

4.3

Invariants of Lagrangian surfaces and wall-crossing

The quantum homology defined in §3.3 depends on a connected component of almost complex structures, hence it is natural to study the potential changes in homology as one varies that component, a phenomenon known as wall-crossing. For example, Auroux shows in [Aur07] that the superpotential is transformed by a holomorphic changes of coordinates whenever wall-crossing occurs. Since the superpotential controls the homology, with a suitable choice of coefficients, it seems that knowledge a priori of the homology may give information on the topology of the space of regular almost complex structures. I am investigating this line of work. There are related results of Welschinger [Wel15] in this direction, it would be interesting to see if they can be applied.

References [Abo10]

Mohammed Abouzaid. A geometric criterion for generating the Fukaya cate´ gory. Publ. Math. Inst. Hautes Etudes Sci., (112):191–240, 2010.

[Alb08]

P. Albers. A Lagrangian Piunikhin-Salamon-Schwarz morphism and two comparison homomorphisms in Floer homology. Int. Math. Res. Not. IMRN, (4):Art. ID rnm134, 56, 2008.

[Arn80a]

V. I. Arnol0 d. Lagrange and Legendre cobordisms. I. Funktsional. Anal. i Prilozhen., 14(3):1–13, 96, 1980.

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[Arn80b]

V. I. Arnol0 d. Lagrange and Legendre cobordisms. II. Funktsional. Anal. i Prilozhen., 14(4):8–17, 95, 1980.

[Aud85]

M. Audin. Quelques calculs en cobordisme lagrangien. Ann. Inst. Fourier (Grenoble), 35(3):159–194, 1985.

[Aur07]

D. Auroux. Mirror symmetry and T -duality in the complement of an anticanonical divisor. J. G¨okova Geom. Topol. GGT, 1:51–91, 2007.

[BC06]

J.-F. Barraud and O. Cornea. Homotopic dynamics in symplectic topology. In P. Biran, O. Cornea, and F. Lalonde, editors, Morse theoretic methods in nonlinear analysis and in symplectic topology, volume 217 of NATO Sci. Ser. II Math. Phys. Chem., pages 109–148, Dordrecht, 2006. Springer.

[BC07]

J.-F. Barraud and O. Cornea. Lagrangian intersections and the Serre spectral sequence. Annals of Mathematics, 166:657–722, 2007.

[BC09a]

P. Biran and O. Cornea. A Lagrangian quantum homology. In New perspectives and challenges in symplectic field theory, volume 49 of CRM Proc. Lecture Notes, pages 1–44. Amer. Math. Soc., Providence, RI, 2009.

[BC09b]

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