Exact Lagrangian cobordism and pseudo-isotopy

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Université de Montréal

Exact Lagrangian cobordism and pseudo-isotopy

par

Lara Simone Suárez López

Département de mathématiques et de statistique

Faculté des arts et des sciences

Thèse présentée à la Faculté des études supérieures en vue de l'obtention du grade de

Philosophiæ Doctor (Ph.D.) en Mathématiques

Septembre 2014

c

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Université de Montréal

Faculté des études supérieures Cette thèse intitulée

Exact Lagrangian cobordism and pseudo-isotopy

présentée par

Lara Simone Suárez López

a été évaluée par un jury composé des personnes suivantes : François Lalonde (président-rapporteur) Octav Cornea (directeur de recherche) Iosif Polterovich (membre du jury) Michael Sullivan (examinateur externe)

(représentant du doyen de la FES)

Thèse acceptée le: 30 September 2014

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RÉSUMÉ

Dans cette thèse, on étudie les propriétés des sous-variétés lagrangiennes dans une variété symplectique en utilisant la relation de cobordisme lagrangien. Plus pré-cisément, on s'intéresse à déterminer les conditions pour lesquelles les cobordismes lagrangiens élémentaires sont en fait triviaux.

En utilisant des techniques de l'homologie de Floer et le théorème du s-cobordisme on démontre que, sous certaines hypothèses topologiques, un cobordisme lagrangien exact est une pseudo-isotopie lagrangienne. Ce resultat est une forme faible d'une conjecture due à Biran et Cornea qui stipule qu'un cobordisme lagrangien exact est hamiltonien isotope à une suspension lagrangianenne.

Mots-clés: Sous-variétés Lagrangiennes, Cobordisme Lagrangien, Pseudo-isotopie Lagrangienne, Homologie de Floer, Torsion de Whitehead.

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ABSTRACT

In this thesis we study the properties of Lagrangian submanifolds of a symplectic manifold by using the relation of Lagrangian cobordism. More precisely, we are interested in determining when an elementary Lagrangian cobordism is trivial.

Using techniques coming from Floer homology and the s-cobordism theorem, we show that under some topological assumptions, an exact Lagrangian cobordism is a Lagrangian pseudo-isotopy. This is a weaker version of a conjecture proposed by Biran and Cornea, which states that any exact Lagrangian cobordism is Hamiltonian isotopic to a Lagrangian suspension.

Key-words : Lagrangian submanifold, Lagrangian cobordism, Lagrangian pseudo-isotopy, Floer homology, Whitehead torsion.

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LIST OF FIGURES

0.1 A Lagrangian cobordism projected to R2_{. . . .} ₉ 1.1 Example with d2 _{= 0. . . 21} 1.2 Perturbation of a Lagrangian cobordism. . . 31 3.1 Image of a strip in M1

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REMERCIEMENTS

This thesis would have not been possible without all the ideas and support of my great adviser, Professor Octav Cornea, who made my student life easier with his contagious happiness and generosity. Your continuous encouragement was fundamental for me and made me feel lucky to be your student.

I thank Jean-François Barraud, Baptiste Chantraine, François Lalonde and Michael Sullivan for their interest in this work and for helpful conversations.

I want to thank my Colombian professors, from the Universidad Nacional de Colombia, Stella Huérfano, Germán Fonseca and Fernando Zalamea.

Agradezco a mis amigas de Montréal Adelaida, Cata, Dianita, Florecita, Masou, Silvanita, y a mis amigos y amigas en otros lugares.

Finalmente doy gracias a mi amada familia. A mi padre, por su descomplicada manera de ver la vida, que me ha aliviado tantas veces. A mi madre, la mujer que más admiro, uno de esos raros ejemplos en donde las circunstancias no determinaron a la persona. Mis hermanos, mis mejores amigos. A nuestro sol Simón y a nuestra abuela. À mon aimé Daniel, qui a parcouru avec moi ce chemin, je te remercie de tout mon âme tout ce que tu as fait pour moi, grâce à toi et à ta merveilleuse ville j'ai trouvé un nouveau chez moi.

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INTRODUCTION

The subject of this thesis is symplectic topology, in particular we study the topology of Lagrangian submanifolds.

In a similar fashion to the study of smooth submanifolds, in order to understand the topology of Lagrangian submanifolds, one studies equivalence relations relating them. Among such a relations we nd Lagrangian cobordism, Lagrangian pseudo-isotopy, and Hamiltonian isotopy; the strongest relation being the last one. These relations can be analyzed using powerful invariants such as Floer homology, Gromov-Witten invariants and the Fukaya category.

The relation of Lagrangian cobordism was rst introduced by Arnold in [Arn80a] and [Arn80b]. Later, Eliashberg [Eli83] and Audin [Aud85] independently classied immersed Lagrangian cobordisms using algebraic topology. As for embedded La-grangian cobordisms, they were studied by Chekanov [Che97]. He showed that in the monotone case, the classication question does not reduce to purely algebraic topological methods, since the cobordisms in this class are more rigid. Recently, Bi-ran and Cornea [BC13] studied monotone LagBi-rangian cobordisms and showed that such cobordisms preserve Floer homology.

Understanding of exact Lagrangians is important, since any Lagrangian is mod-eled locally by an exact Lagrangian. The main examples of exact Lagrangians are the zero-section of the cotangent bundle and its Hamiltonian perturbations. Arnold

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conjectured that these are the only exact Lagrangians in cotangent bundles. This statement is known as the nearby Lagrangian conjecture. This conjecture implies that any Lagrangian in the cotangent bundle is Lagrangian pseudo-isotopic to the zero-section.

Despite recently mayor progress done by Abouzaid [Abo12] and Kragh [Kra13], the full nearby Lagrangian conjecture is not yet established. A more accessible question is to nd criteria determining when two exact Lagrangians are Lagrangian pseudo-isotopic. The main theorem of this thesis is such a criterion. Under some topological and dimensional restrictions, we establish that two exact Lagrangians that are cobordant via an exact Lagrangian cobordism are Lagrangian pseudo-isotopic (see Theorem 3.1.2).

Main Theorem. Let (W; L0, L1) be an exact, orientable and spin Lagrangian cobor-dism equipped with a choice of spin structure. Assume dim(W) > 5. If the map i_# : π1(Li) → π1(W) induced by the inclusion Li → W is an isomorphism for i = 0, 1, then (W; L0, L1) is a Lagrangian pseudo-isotopy.

The main tools we use are the Z(π1W)-Floer homology (studied before in [Sul02], [Dam10], [BC07]), the Whitehead torsion of the acyclic Z(π1W)-Floer complexes (fol-lowing [Sul02]), and the classical s-cobordism Theorem as well as machinery specic to Lagrangian cobordism developed in [BC13]. These four distinct components then blend together and form the proof of the main theorem. Below we describe in more detail how we address each of these constituents.

In chapter 1we study the homology of the Z(π1W)-Floer complex of a monotone Lagrangian cobordism. We show that the Floer homology of this complex is preserved by monotone Lagrangian cobordisms. The technical aspect of the chapter concerns the denition of the Floer complex, and is treated following the work of Biran-Cornea [BC13]. As a result, we obtain that a cobordism which satises the hypotheses of

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5 the Main Theorem is automatically an h-cobordism, and as such it has a well-dened Whitehead torsion.

In chapter 2 we briey recall some denitions and known results on the Whitehead torsion. In particular, we state the s-cobordism theorem ([Bar63], [Maz63], [Sta67]), which will constitute a fundamental ingredient in the proof of the Main Theorem.

In chapter 3 we study the invariance (under Hamiltonian isotopies) of the White-head torsion of the Z(π1W)-Floer complex associated to an exact Lagrangian cobor-dism (see Theorem 3.1.1). This is, perhaps, the most technical part in the proof of the Main Theorem due to the bifurcation analysis involved. The proof of Theo-rem 3.1.1 is based on the work of Sullivan [Sul02] and Lee [Lee05b]. We apply the technique of stabilization of Sullivan, and a gluing theorem for a degenerate Floer strip, proved by Lee for the Floer theory of Hamiltonian orbits. It follows that if a cobordism satises the hypotheses of the Main Theorem, then its torsion agrees with the torsion of the Floer complex. We then apply the displaceability of the cobor-dism, which yields that the Whitehead torsion of the cobordism vanishes. Hence, we conclude with an application of the s-cobordism theorem, that the cobordism in question, is a Lagrangian pseudo-isotopy. This concludes the proof of the Main Theorem, modulo orientation issues.

Finally, in chapter 4 we address the orientation issues which arose in chapters 1 and 3. This technical part is based on the work of FOOO [KF09] and Lee [Lee05b]. Our main result is a weak form of the following conjecture which motivate our work:

Conjecture (Biran and Cornea, 2012). Any exact Lagrangian cobordism is Hamil-tonian isotopic to a Lagrangian suspension.

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PRELIMINARIES

This chapter is brief recall of the basic concepts in symplectic topology. We follow the exposition in [AL94] and [DM05].

0.1. Symplectic manifolds

Symplectic topology is the study of geometric objects called symplectic manifolds. Denition 0.1.1. A Symplectic manifold (M, ω) is a 2n-dimensional smooth man-ifold M with a two-form ω : TM × TM → R, called the symplectic form such that:

• ω is non-degenerate (ω∧n _{is a volume form),}

• ω is closed (dω = 0).

Some examples of symplectic manifolds are:

• (R2n_,_dxi_{∧ dyi)}

• Every orientable surface (Σ, ω) where ω is the area form.

• The cotangent bundle of any manifold, (T∗_M,_{−dθ), where θ is the Liouville}

form.

Denition 0.1.2. Two symplectic manifolds (M1, ω1), (M2, ω2) are symplectomor-phic if there is a dieomorphism φ : M1 → M2 that preserves the symplectic form φ∗(ω2) = ω1. Such a dieomorphism is called symplectomorphism.

The study of symplectic manifolds is the study of global symplectic invariants since locally symplectic manifolds are all the same:

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Theorem 0.1.1 (Darboux). Let (M, ω) be a symplectic manifold, and let p be any point in M. Then we can nd a coordinate system (U, x1,· · · , xn, y1,· · · , yn) centered at p such that on U, the symplectic form is the standard one, ω =dxi∧ dyi.

The main objects of interest in our work are the Lagrangian submanifolds of a symplectic manifold.

Denition 0.1.3. A Lagrangian submanifold L ⊂ (M, ω) is a submanifold with dim(L) = n, such that ω |TL×TL= 0.

Some examples of Lagrangian submanifolds are:

• Rn _{→ (R}2n_,_dxi_{∧ dyi)}

• The equator in (S2_{, ω}_).

• The zero section M → (T∗_M,_−dθ).

Some of the equivalence relations used to classify Lagrangian submanifolds in a given symplectic manifold are isotopy (Hamiltonian or Lagrangian), Lagrangian cobordism and Lagrangian pseudo-isotopy. We introduce these relations here.

0.1.1. Hamiltonian isotopy, Lagrangian cobordism and Lagrangian pseudo-isotopy.

A time dependent Hamiltonian map is a smooth map H : M × [0, 1] −→ R. The vector eld XHt given by the following equation:

ωx(XHt(x), −) = −(dHt)x(−)

is called the Hamiltonian vector eld associated to Ht. The Hamiltonian ow gener-ated by Ht is dened by:

d dtψ t _{= XH} t◦ ψ t_. The isotopy {ψt_}t

∈[0,1] is called Hamiltonian isotopy.

Denition 0.1.4. Two Lagrangians L0, L1 ⊂ (M, ω) are Hamiltonian isotopic if

there is a Hamiltonian isotopy {ψt_}t

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9 Denition 0.1.5 (Denition 2.1.1 [BC13]). An (elementary) Lagrangian cobordism (W; L0, L1) is a smooth cobordism W between L0 and L1 (connected Lagrangians) such that:

• W → (M= ([0, 1] × R) × M, dx ∧ dy ⊕ ω) is an embedded Lagrangian,

• There is > 0 such that:

W∩ ([0, ) × R × M) = [0, ) × {1} × L0, W∩ ((1 − , 1] × R × M) = (1 − , 1] × {1} × L1.

Let π : ([0, 1] × R) × M −→ C. Then the image of W under the projection is represented in the following picture:

Fig. 0.1. A Lagrangian cobordism projected to R2_.

An example of a Lagrangian cobordism is the Lagrangian suspension [AL94]. Let {ψt_}t

∈[0,1] be a Hamiltonian isotopy generated by some Hamiltonian H and L ⊂

(M, ω) a Lagrangian submanifold. Then, the image of the map [0, 1] × L → [0, 1] × R × M

(t, x) → (t, −H(t, ψt_{(x)), ψ}t_(x))

isa Lagrangian cobordism (W; L, ψ1_{(L)). Notice that thiscobordism hascylindrical} endsafter re-parametrization of the Hamiltonian ow.

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Denition 0.1.6 (Denition 2.1 in [Pol93]). A Lagrangian cobordism (W; L0, L1) is called a Lagrangian pseudo-isotopy if it is dieomorphic to [0, 1] × L0.

Any two Lagrangians that are Hamiltonian isotopicare Lagrangian pseudo-isotopic. The converse is not true in general, see [Che97] for counter-examples.

In the present work we will see that for exact Lagrangians (under dimensional and topological constraints), the relation of cobordism implies the relation of Lagrangian pseudo-isotopy.

To distinguish among dierent equivalence classes of Lagrangians we use in-variants that come in big part from holomorphic techniques. The pseudo-holomorphic techniques were introduced to symplectc geometry by Gromov [Gro85] and developed since by many others, for a self-contained modern treatment see [DM10].

0.1.2. Almost-complex structures and J-holomorphic curves

Symplectic manifolds are contained in a bigger class of manifolds, the class of almost complex manifolds.

Denition 0.1.7. A smooth almost complex structure J, on a manifold M is a smooth section of the bundle of endomorphisms of the tangent space of M,

J: M → End(TM), x → Jx : TxM→ TxM

such that J2

x = −Id for every x ∈ M.

A manifold together with an almost complex structure (M, J) is called an almost complex manifold.

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11 The basic example is (R2n_{, J0), where at x ∈ R}2n_{, the automorphism (J0)x} _{: R}2n _→ R2n _{is given by the matrix}

(J0)x = ⎡ ⎣ 0 −I I 0 ⎤ ⎦ ,

This almost complex structure is called the standard almost complex structure on R2n_.

Denition 0.1.8. Let (M, ω) be a symplectic manifold. An almost complex struc-ture J on M is compatible with ω, if the bilinear form x, y = ω(x, Jy) denes a Riemannian metric on M.

We denote by Jω(M) the space of ω-compatible almost complex structures. The following proposition implies that compatible almost complex structures on a symplectic manifold always exist.

Proposition 0.1.1 (Proposition 4.1 in [DM05]). Let (M, ω) be a symplectic mani-fold. Then Jω(M) is non-empty and contractible.

An almost complex structure allows us to study maps from Riemann-surfaces to the given manifold satisfying a Cauchy-Riemann equation. These maps are called pseudo-holomorphic curves and the study of their moduli spaces is one of the most powerful technique in symplectic topology.

Denition 0.1.9. Let (Σ, j) be a Riemann surface with the standard complex struc-ture j, and let (M, J) be an almost complex manifold. A J-holomorphic curve is a map u : (Σ, j) → (M, J), such that

∂Ju= 1

2(du + J ◦ du ◦ j) = 0.

The space of J-holomorphic curves on a given symplectic manifold was studied by Gromov in [Gro85].

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0.1.3. The class of exact and monotone Lagrangians

In this thesis we restrict the study of Lagrangians submanifolds to the classes of monotone Lagrangians and the even more restricting class of exact Lagrangians. The later being more restricting due to the absence of pseudo-holomorphic disks for the Lagrangians in this class, that for example, reduces invariants as Floer homology to the classical singular homology.

Denition 0.1.10. Let (M, ω) be a symplectic manifold where the two-form ω = dλ is exact. If the one-form restricted to a Lagrangian L is also exact λ |TL= df, then we say that the Lagrangian L is an exact Lagrangian.

The main example of an exact Lagrangain submanifold is the zero-section of the contangent bundle L → (T∗_{L, ω). The class of exact Lagrangian submanifolds is a}

very important class of Lagrangian submanifolds since the exact Lagrangians give a "local" understanding of Lagrangians submanifolds, due to the following theorem of Weinstein:

Theorem 0.1.2 (Weinstein). Let L ⊂ (M, ω) be a Lagrangian submanifold. There exists a neighborhood of L in M, (U(L), ω |U(L)⊂ (M, ω) symplectomorphic to a neighborhood (U0, dλU₀) ⊂ T∗_L _{of the zero-section L ⊂ (T}∗_{L, dλ).}

The exact condition implies that there are no non-constant pseudo-holomorphic disks with boundary on an exact Lagrangian. Using this Floer [Flo88b] showed that for the Lagrangians in this class, the Floer homology and the singular homology coincide.

We now introduce the class of monotone Lagrangians. Given a Lagrangian L ⊂ M there are two canonical homomorphisms, the symplectic area

ω: π2(M, L) −→ R, [u] → ω(u) :=

D2

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13 and the Maslov index:

μ: π2(M, L) −→ Z.

Let Λ(n) ∼= U(n)/O(n) be the Lagrangian Grassmannian. Since π1(Λ(n)) Z, for a map u : (D2_{, S}1_{) → (M, L) representing an element in the Lagrangian π2(M, L),} the vector bundle u∗_{(TM) is trivial and any two trivializations are homotopic. After}

trivialization, the subbundle (u |S1)∗(TL) is a Lagrangian loop in Λ(n). The Maslov

index is dened to be the class of this loop μ(u) := [(u |S1)∗(TL)] ∈ Z. Note that the

Maslov index is invariant up to symplectic isotopy.

The minimal Maslov number of L is the integer NL dened by NL:= min{μ(A) > 0 | A ∈ π2(M, L)}.

Denition 0.1.11. A Lagrangian submanifold L ⊂ M is monotone if there exist a constant ρ > 0, such that,

ω(A) = ρμ(A),

for all A ∈ π2(M, L) and NL≥ 2. We call ρ the monotonicity constant.

An example of a monotone Lagrangian is S1 _{⊂ (R}2_{, ωst), where NS}

1 = 2 and

ρ = π₂. Taking products of S1 _{we nd monotone Lagrangian tori in symplectic} manifolds obtained as products of (R2_{, ωst}_{). Thus from the Darboux theorem, any} symplectic manifold has locally a monotone Lagrangian.

An exact Lagrangain is a spacial case of monotone Lagrangian, when ρ = 0. Remark 0.1.1. We can also restrict the study of Lagrangian cobordisms to the class of exact Lagrangian cobordisms and monotone Lagrangian cobordisms. A Lagrangian cobordism (W, L0, L1) ⊂ (M,ω) is an exact (monotone) Lagrangian cobordism if L0, L1 and W are exact (monotone) Lagrangians.

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Chapter 1

FLOER HOMOLOGY WITH

Z(π

1

W

)-COEFFICIENTS

One of the main tools used to study Lagrangian submanifolds is Floer homology. In this chapter we study Floer homology with twisted coecients, associated to a monotone Lagrangian cobordisms (W; L0, L1).

Floer homology for Lagrangian cobordisms was dened in [BC13], where it is shown that it is invariant with respect to monotone Lagrangian cobordism. In this chapter we show that the Floer homology with twisted coecients is also invariant under monotone Lagrangian cobordisms. Moreover, when the Lagrangian cobordism is exact, under an additional topological condition the cobordism itself is an h-cobordism.

The main theorem of this chapter is the following:

Theorem 1.0.3. Let (W; L0, L1) be an orientable, spin1 _{and monotone Lagrangian} cobordism, equipped with a choice of spin structure. Let HF(Li; Z(π1W)) denote the Floer homology with Z(π1W)-coecients of Li for i = 0, 1. Then,

HF(L0; Z(π1W)) ∼= HF(L1; Z(π1W)).

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As main consequences we will show the following corollaries:

Corollary 1.0.1. If (W, L0, L1) is an orientable and spin, exact Lagrangian cobordism and i# : π1(Li) → π1(W) is an isomorphism for i = 0, 1, then (W; L0, L1) is an h-cobordism.

Corollary 1.0.2. If (W; L0, L1) is an orientable and spin, exact Lagrangian cobor-dism with dim(W) > 5 and Li (i = 0, 1), W are simply connected, then (W; L0, L1) is a Lagrangian pseudo-isotopy.

Remark 1.0.2. Corollary 1.0.2has been shown in [Tan14] for the cotangent bundle using dierent techniques involving previous work of Abouzaid [Abo12] and Kragh [Kra13].

To extend the result of corollary 1.0.2to the non-simply connected setting is nec-essary the study of the Whitehead torsion, in order to apply the s-cobordism theorem.

1.1. Floer homology with twisted coefficients

1.1.1. Setting

In this section we x the properties on the symplectic manifold and Lagrangian submanifolds assumed through this chapter.

The symplectic manifolds (M, ω) are supposed to be connected and tame (bounded geometry) symplectic manifold.

Denition 1.1.1 ([AL94, Denition 4.1.1]). Let (M, J, μ) be an almost complex manifold with a Riemannian metric μ. This manifold is tame if μ is complete and there exist positive constants r0, C1, C2 with the following properties:

(T1) For all x ∈ M the map expx : B(0, r0) → B(x, r0) is a dieomorphism(Injectivity radius bounded from below).

(T2) Every loop γ in M contained in a ball B = B(x, r) with r ≤ r0 bounds a disc in B of area less than C1lenght(γ)2_.

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17 (T3) On every ball B = B(x, r0), there exist a symplectic form ωx such that

ωx ≤ 1 and |X|2 ≤ C2ωx(X, JX) (taming property).

A closed symplectic manifolds is tame. Other examples include (Cn_{, ωst) and} cotangent bundles (T∗_{M, dθ).}

The Lagrangian submanifolds are closed, orientable and spin and we x a spin structure on them.

The Lagrangian cobordisms (W, L0, L1), are orientable, monotone and spin and we assume that the choice of spin structure on W, is such that when we restricted it to Li it coincides with the chosen spin structures on Li for i = 0, 1.

A fundamental assumption to have a well dened Floer complex is that the minimal Maslov number of any Lagrangian L, NLis strictly larger than two (NL > 2). We will explain this latter.

1.1.2. The Floer complex with Z(π1L0)-coecients for Lagrangian in-tersections

Let (L0, L1) ⊂ (M, ω) be a pair of Lagrangian submanifolds. In this section we recall the construction of the Floer complex with Z(π1L0)-coecients associated to the pair (L0, L1). We also show the expected properties of the corresponding homol-ogy (invariance under Hamiltonian perturbation, relation with singular homolhomol-ogy). We follow the exposition in [BC13].

The Floer complex with twisted coecients has been studied before in [Sul02] and a similar version called lifted Floer complex is dened in [Dam10].

The ingredients to dene a Floer Complex are:

(1) A choice of a ring of coecients R, and a set of generators denoted by Oη(H). The free R-module R Oη(H) is the module used in the denition of the Floer complex.

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(2) The denition of the dierential involves moduli spaces that verify regularity and compactness conditions.

(3) Denition of a grading.

Once a Floer complex is well dened, we show that its homology does not depend on the data and on the (Hamiltonian) isotopy class of the Lagrangian pair. The Floer homology is sometimes isomorphic to a well known homology. This is the case for exact Lagrangians, where the Floer homology HF(L, L) is isomorphic to the singular homology H(L).

1.1.2.1. Ring of coecients and set of generators

The ring of coecients is composed by the following rings:

The integral group ring Z(G) of the group G (in our case G = π1L0), is the set of nite formal sums:

Z(G) := { i

aigi | ni ∈ Z, gi ∈ G}, with the natural addition and multiplication.

The universal Novikov ring is denoted by A;

A := { ∞ k=0 akTλk | ak∈ Z, lim k→∞λk= ∞} with the addition and multiplication of power series.

To dene the set of generators we need to introduce some additional notation. Let P(L0, L1) be the space of paths in M connecting L0 with L1,

P(L0, L1) = {γ ∈ C0_{([0, 1], M) | γ(0) ∈ L0, γ}_{(1) ∈ L1}.}

In particular L0∩ L1 ⊂ P(L0, L1). F or η ∈ π0(P(L0, L1)), let Pη(L0, L1) denote the path connected component of η.

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19 If L0 L1the set of generators of the complex is L0∩L1. If not, let H : M×[0, 1] → R be a Hamiltonian function with Hamiltonian ow ψH

t such that ψH1(L0) is transverse to L1.

Denote Oη(H) = {γ ∈ Pη(L0, L1) | ∃x ∈ L0, γ(t) = ψH

t(x)}, the space of paths connecting L0 with L1 that are orbits of the Hamiltonian ow ψH

t.

Fixing η, H, the Floer complex is the Z(π1L0)- complex generated bythe subset of orbits of the Hamiltonian ow ψH

t, Oη(H).

CFη((L0, L1; Z(π1L0), H, J) = (Z(π1L0) ⊗ A Oη(H), d). 1.1.2.2. The dierential

The dierential is dened as usual, bycounting elements in the moduli spaces of perturbed pseudo-holomorphic Floer strips, but we also consider elements in the integral group ring Z(π1L0) following the same construction that [Sul02],[BC07], and a similar version in [Dam10].

For the denition of the moduli space we choose a generic, compatible, time dependent almost complex structure J = {Jt}t∈[0,1] (with Jt ∈ Jω for all t).

For anypair γ−, γ+ ∈ Oη(H), denote by M(γ−, γ+; H, J) the space consisting of maps u ∈ C∞_{(R × [0, 1], M) such that:}

(1) The map u satises the equation: ∂u(s, t) ∂s + Jt(u(s, t)) ∂u(s, t) ∂t + ∇ t_{Ht(u(s, t)) = 0} _(1.1.1) (2) u(s, i) ∈ Li for i = 0, 1, (3) lim s_→−∞u(s, t) = γ−(t), lims_→∞u(s, t) = γ+(t), (4) The energyof E(u), is bounded:

E(u) =

R×[0,1]|

∂u(s, t)

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The space ^M(γ−, γ+; H, J) = M(γ−, γ+; H, J)/R denotes the quotient by the R-action on M(γ−, γ+; H, J) given by reparametrization: τ ∈ R acts by u → u(s−τ, t). For a generic choice of (H, J), the spaces M(γ−, γ+; H, J) and ^M(γ−, γ+; H, J) are manifolds [Lee03]. Denote by Mn_(γ−_{, γ}_{+; H, J) and ^}_Mn_(γ−_{, γ}_{+; H, J) their} n-dimensional component. For u ∈ ^M0_(γ

−, γ+; H, J) the space ^

M0_(γ−_{, γ}_{+; H, J, [u]) = {v ∈ ^}_M0_(γ−_{, γ}_{+; H, J) | [v] = [u] ∈ π2(M, L0}_{∪ γ}

−∪ L1∪ γ+)} is compact.

To involve the integral group ring Z(π1L0) in the construction, consider the space obtained from L0 by contracting to a point an embedded path w(t) : [0, 1] → L0, passing through each point in {γ(0) | γ ∈ Oη(H)}. Denote the resulting space by L∗0 = L0/w∼ ∗, and note that it has the same homotopy type as L0.

There is a natural map dened as follows:

Θ: M(γ₋, γ₊; H, J) −→ Ω(L∗0), u→ u(s, 0), (1.1.2) where Ω(L∗_{0) is the loop space of L}∗_0.

The dierential of the Z(π1L0)-Floer complex is given by: d(γ−) = γ+∈Oη(H) ^u∈ ^M0_(γ₋_,γ₊_;H,J) sign(u)[Θ(u)]Tω(u)_γ +. (1.1.3)

Where u ∈ M1_(γ−_{, γ}_{+; H, J) is a representative of ^u ∈ ^}_M0_(γ−_{, γ}_{+; H, J). Note that} this is well-dened since for a xed homotopy class [u], the space ^M0_(γ

−, γ+; H, J, [u]) is an oriented and compact 0-dimensional manifold so the expression in the right side of equation (1.1.3) makes sense (see chapter 4 for denition of sign(u)). Moreover, the homotopy class of the loop dened by the map Θ is invariant under the R-action. We do not discuss the way we can assign a grading to the Floer complex in this setting, but we will do it in the next section when working with exact Lagrangians.

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21 Remark 1.1.1. (1) When the pair (L0, L1) intersect transversely, we set H = 0. In this case the moduli spaces are composed by Jt-holomorphic strips connect-ing intersection points.

(2) A proof of the regularity of the moduli spaces M(γ−, γ+; H, J) in the rela-tive case (Lagrangian boundary condition), can be found in [Lee03, Section 3.2.2]. The proof follows the arguments in [AF95], adapted to the monotone Lagrangian setting.

1.1.2.3. The square of the dierential is zero

There is no additional diculty to see d2 _{= 0 in this setting. The proof follows} the same argument than in the monotone setting with coecients ring A.

The only dierence is that if the minimal Maslov number NL_i = 2 for i = 0, 1, then we can have d2 _{= 0. This can be illustrated by the following example from} [Dam10].

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22

To see d2 _{= 0 we show the following equation:}

γ+, d2(γ−) =

γ

( ^u, ^u)∈ ^M0_(γ₋_,γ_{)× ^}_M0_(γ_,γ₊₎

sign(u_)sign(u_)[Θ(u_)][Θ(u_)]Tω_(u_)+ω(u₎_{= 0.}

(1.1.4)

Since the pair (H, J) is assumed to be generic and NL_i > 2 for i = 0, 1, the Gro-mov compactness, the gluing theorem and a choice of compatible orientations (with gluing) on the 1-dimensional unparametrized moduli spaces imply the following de-composition of the moduli spaces for a xed homotopy class [u]:

∂ ^M1(γ−, γ+; H, J, [u]) = γ,_[u_]+[u_]=[u]

^

M0_(γ−_{, γ}_{; H, J, [u}_{]) × ^}_M0_(γ_{, γ}_{+; H, J, [u}_]).

In addition, [−] ◦ Θ ( the homotopy class o a the loop given by Θ), is constant on a xed class [u] and observe that [Θ(u_)][Θ(u_{)] = [Θ(u}_#u_)]:

Let γ0(0) for a xed γ0 ∈ Oη(H) be the base point of Ω(L0). The map w(t) denes a path connecting γ(0) with γ0, for any γ ∈ Oη(H). Denote by wγ(t) this path. Then we have that [Θ(u_)][Θ(u_{)] = [(wγ}

−(t))−1u(s, 0)wγ(t)][(wγ(t))−1u(s, 0)wγ+(t)] =

[(wγ−(t))−1u(s, 0)#u(s, 0)wγ+(t)] = [Θ(u#u)].

Since the signed sum of all boundary components of a 1-dimensional manifold is zero, we have (1.1.4) is zero for each γ+ and therefore d2 _{= 0.}

The homology of this complex is denoted by FHη(L0, L1; Z(πL0), (H, J)).

The complex CF(L0, L1; Z(πL0), (H; J)) is dened by considering the sum over all the components η ∈ π0(P(L0, L1),

CF(L0, L1; Z(πL0), (H, J)) = ⊕ηCFη(L0, L1; Z(πL0), (H, J)) and the corresponding homology is denoted FH(L0, L1; Z(πL0), (H, J)).

The resulting homology is independent on the choice of regular pairs (H, J). The proof of this is completely analog to the standard case.

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23 We will recall the proof of the invariance when changing the Hamiltonian H, that turns out to be equivalent (by naturality of Floer equation) to the invariance of the homology under Hamiltonian perturbations of a the Lagrangian L1, assuming L0 L1.

1.1.2.4. Invariance under Hamiltonian perturbations We follow the exposition in Section 3.2 of [BC13].

Let {ϕt}t∈[0,1] be a Hamiltonian isotopy with ϕ0= 1 associated to a Hamiltonian

function G.The isotopy ϕt induces a map:

ϕ_∗ : π0(P(L0, L1)) −→ π0(P(L0, ϕ1(L1))), η = [γ] → ϕt(γ(t)). (1.1.5) Assume that L0 L1 and L0 ϕ1(L1).Fix the data (η, J). To compare the ho-mology of the two complexes CFη(L0, L1; Z(π1L0), J) and CFφ∗η(L0, ϕ1(L1); Z(π1L0), J),

a chain map

cϕ : CFη(L0, L1; Z(π1L0), J) → CFϕ∗η(L0, ϕ1(L1); Z(π1L0), J) (1.1.6)

is dened using moving boundary conditions. The moduli spaces with moving bound-ary conditions were introduced by Oh in [Oh93].

In order to dene this map we need to introduce some notation.

Let β : R −→ [0, 1], be a smooth function such that β(s) = 0 for s ≤ 0, β(s) = 1 for s ≥ 1 and β is strictly increasing in (0, 1).

For γ− ∈ L0∩ L1 and γ₊ ∈ L0∩ ϕ1(L1), consider Mϕ(γ₋, γ₊) to be the moduli space of maps u ∈ C∞_{(R × [0, 1], M) such that:}

(1) For all s, u(s, 0) ∈ L0 and u(s, 1) ∈ ϕβ(s)(L1), (2) lim

s→−∞u(s, t) = γ−(t), lims→∞u(s, t) = γ+(t), (3) ∂Ju= 0.

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24

Let ψt = (ϕt)−1 _{and x γ0} _{a path in the connected component η. Dene the} functional: ΦG : Pϕ_∗η(L0, ϕ1(L1)) −→ R, γ → 1 0 G(ψt(γ(t)))dt − 1 0 G(γ0(t))dt. (1.1.7) For each map u ∈ Mϕ(γ−, γ+) let vu: R × [0, 1] −→ M denote the map dened by vu(s, t) = ψtβ(s)(u(s, t)).

The the chain map isdened asfollows:

cϕ(γ−) = γ₊∈L₀∩ϕ₁(L₁) u∈M0 ϕ(γ−,γ+) sign(u)[Θ(u)]Tω(v_u)−Φ_G(γ+)_γ +. (1.1.8) In order to obtain a loop Θ(u) ∈ Ω(L0), we consider the space L1/w∼ ∗ asin the previoussection, where the embedded path w(t) : [0, 1] → L0 passes through each point in (L0∩ L1) ∪ (L0∩ ϕ1(L1)).

To see that the map cϕ(γ−) isa chain map we compare the following equations:

d(cϕ(γ−)), γ+ = γ∈L₀∩ϕ₁(L₁) (u, ^w)∈M0 ϕ(γ−,γ)× ^M0(γ,γ+) sign(u)sign(w)[Θ(u)][Θ(w)]Tω(v_u)+ω(w)−Φ_G(γ) cϕ(d(γ−)), γ+ = γ∈L₀∩L₁ ( ^u,w)∈ ^M0_(γ₋_,γ_)×M0 ϕ(γ,γ+)

sign(u_)sign(w_)[Θ(u_)][Θ(w_)]Tω(v_w)+ω(u)−Φ_G(γ+)_.

For generic J and xed homotopy class A ∈ π2(M, L0∪L1), the space M1_ϕ(γ

−, γ+; J, A) admitsa compactication into a 1-dimensional manifold M1ϕ(γ−, γ+; J, [u]), with boundary given by:

γ,B+C=A M0_ϕ(γ −, γ; J, B) × ^M0(γ, γ+; J, C) ∪ γ,B+C=A ^ M0_(γ −, γ; J, B) × M0ϕ(γ, γ+; J, C)

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25 Then, the choices of orientations and spin structures on L0, L1induce canonical orien-tations on this spaces, compatible with the gluing map. 2 _{Moreover, any two} bound-ary components (u, ^w), ( ^u, w) of M1ϕ(γ₋, γ₊; J, [u]) in the same connected compo-nent satisfy [Θ(u)][Θ(w)] = [Θ(u_)][Θ(w_{)] and ω(vw}_)+ω(u_{)−ΦG(γ+) = ω(vu)+}

ω((ψt(w)) − ΦG(γ+) = ω(vu) + E((ψt(w)) − ΦG(γ) = ω(vu) + ω(w) − ΦG(γ). Therefor, the sum of the two expressions above vanishes. In a similar way we dene a chain map

cϕ‘: CFη(L0, ϕ1(L1); Z(π1L0), J), → CFϕ‘_∗η(L0, L1; Z(π1L0), J) (1.1.9) such that cϕ‘ is a quasi-inverse of cϕ, concluding that the map;

cϕ : HFη(L0, L1; Z(π1L0), J) → HFϕ_∗η(L0, ϕ1(L1); Z(π1L0), J), is an isomorphism.

We summarize this section in the following proposition:

Proposition 1.1.1. Let (L0, L1) be a pair of oriented and spin monotone Lagrangians, equipped with a choice of spin structure. Assume that the minimal Maslov num-ber satises NLi > 2 for i = 0, 1. Let (H, J) be generic, then the Floer complex

CF(L0, L1; Z(π1L0), (H, J)) is a chain complex.

Moreover, the homology H(CF(L0, L1; Z(π1L0), (H, J))) does not depend on the choice of generic pair (H, J) and is invariant under Hamiltonian isotopies of L0 or L1. Remark 1.1.2. Proposition 1.1.1 is a Z(π1L0)-version of the same result due to Oh [Oh93] in the monotone setting.

For twisted coecients, the Z2(π1L0)-version of proposition 1.1.1 appears in [Sul02] for a pair (L0, L1) of non-compact Lagrangians with L1 = φH

1(L0) and ω |π2(M,L0)= 0.

In the monotone setting, a similar result for the lifted Floer homology appears in [Dam10].

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26

1.2. Lagrangians with cylindrical ends

In this section we adapt the chain complex dened in the last section in order to associate such a complex to a Lagrangian cobordism (W; L0, L1). Instead of working with a Lagrangian cobordism we use the corresponding non-compact Lagrangian with cylindrical ends (obtained when extending the ends of the cobordism). The Floer theory for Lagrangian with cylindrical ends was developed in [BC13].

For a symplectic manifold (M, ω), the symplectic manifold (M,ω) is composed by M= R2_{× M and}_ω_{= ωst}_{⊕ ω. Denote by π the projection π : R}2_{× M → R}2_. Denition 1.2.1 ([BC13, Section 4.1]). An (elementary) Lagrangian submanifold with cylindrical ends, W ⊂ (M,ω) is a Lagrangian submanifold without boundary and with the following properties:

(1) For every a < b the subset W ∩ ([a, b] × R) × M is compact. (2) There exists R−, R+ ∈ R with R− ≤ R+ such that:

W∩ ([R₊,∞) × R) × M = ([R₊,∞) × {a₊}) × L1

W∩ ((−∞, R−] × R) × M = ((−∞, R−] × {a−}) × L0 for some pair of Lagrangian L0, L1 ⊂ M and a−, a+∈ R.

We use the word elementary since the general denition appearing in [BC13], considers Lagrangians with more than one positive or negative end.

For every R ≥ R+, let E+R(W) = W ∩ ([R, ∞) × R) × M denote the positive cylindrical end of W, and similarly for R ≤ R−, E−

R(W) will denote the negative cylindrical end of W.

Denition 1.2.2. Let W0, W1 ⊂ M be two Lagrangians with cylindrical ends. We say that they are cylindrically distinct at innity if there exists R > 0 such that π(E−_R(W0)) ∩ π(E−_R(W1)) = ∅ and π(E+_R(W0)) ∩ π(E+_R(W1)) = ∅.

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27 Any Lagrangian cobordism (W; L0, L1) extends to a Lagrangian with cylindrical ends W in the following way;

W = ((−∞, 0] × {1}) × L0) ∪ W ∪ ([1, ∞) × {1}) × L1).

In the class of Lagrangians with cylindrical ends we can consider Hamiltonian iso-topies that "x" the cylindrical ends in the following sense.

Denition 1.2.3 ([BC13, Denition 4.1.s]). An isotopy {Wt}t∈[0,1] of Lagrangian

submanifolds with cylindrical ends of M is called Horizontal isotopy if there exists a Hamiltonian isotopy {ψt}t∈[0,1] of M, with ψ0 = I and the following properties:

• Wt= ψt(W0) for all t ∈ [0, 1].

• There exist real numbers R− < R+such that for all t ∈ [0, 1] and x ∈ E±R_±(W0), we have ψt(x) ∈ E±

R_±(W0).

• There is a constant K > 0 such that for all x ∈ E±

R±(W0), |dπx(Xt(x))| < K.

Here, Xt is the time dependent vector eld of the ow {ψt}t∈[0,1].

We now proceed to dene the Z(π1W1)-FLoer complex associated to a pair of monotone Lagrangians with cylindrical ends (W0, W1).

1.2.1. The Z(π1W1)-Floer complex for Lagrangian with cylindrical ends Consider (W0, W1) a pair of monotone Lagrangians with cylindrical ends with N_W

i > 2 for i = 0, 1. In [BC13] it is shown that the Floer homology with

Z2-coecients, HF(W0, W1,[f]) is well dened and depends on an additional data [f] coming from the choice of a Hamiltonian perturbation of the ends of W1, which makes the image of W1 under this perturbation, cylindrically distinct from W0 at innity.

When working with Z(π1W0)-coecients, the construction is completely analog under additional choices of spin structures on the pair (W0, W1). The construction of the Floer complex follows the schema presented in the previous section.

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28

Given the non-compactness of the Lagrangian pair, additional choices have to be made to ensure the compactness of the moduli spaces.

The Floer complex is dened using the following data:

• A xed component η ∈ π0(P(W0, W1).

• A perturbation (H, f) where H : [0, 1] × M → R and f : R2 _{→ R are two} Hamiltonians satisfying:

(1) H has compact support.

(2) For R± big enough so that W1 is cylindrical on E±R±(W1), the function

f is such that the support of f is contained in a neighborhoods U_± of π(E±_R_±(W1)) where f(x, y) = α_±x+ β_± with α_± ∈ R small enough so that the associated Hamiltonian isotopy φf

t keeps the sets [R++2, ∞)× {a+} and (−∞, R−− 2] × {a−} inside U± respectively.

We denote the space of pairs (H, f) as above by H(W, W_).

• An almost complex structure J on (M,ω). We will restrict to a family of time dependent complex structure JB where B ⊂ R2 is a compact set, and J = {Jt}t∈[0,1] ∈ JB satises the following properties:

(1) For every t, Jt is an ω-tamed almost complex structure on M.

(2) For every t, the projection π is (Jt, i)-holomorphic on (R2_{\ B) × M. If} B= ∅, we write J .

The additional choice of perturbation (H, f) ∈ H(W0, W1) guaranties that W0 and φf◦π

1 (W1) are cylindrically distinct at innity. If the ends of W0 coincide with those of W1, the space H(W0, W1) has four connected components xed by the four possible choices of the function f (depending on the signs of α±). We denote by [f]

the path component of H(W0, W1) associated to a perturbation (H, f).

The choice of almost complex structure J ∈ JB implies the compactness of the moduli spaces of Jt-holomorphic strips (in a xed relative to the boundary homotopy

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29 class), since an application of the open mapping theorem shows that any pseudo-holomorphic curve with nite energy has its image in a xed compact set. For the space of perturbed Jt-holomorphic strips, with boundary on a Lagrangian pair (W0, W1) cylindrically distinct at innity, we apply a Hamiltonian perturbation ψH 1 to the second Lagrangian W1, such that the pair (W0, ψH

1(W1)) intersect transversely. After a naturality argument we can conclude that the moduli spaces of perturbed Jt-holomorphic strips is compact. For a detailed proof of compactness in the cobordism setting, see Lemma 4.2.1 in [BC13].

For generic data ((H, f), J), the Floer complex

CFη(W0, W1; Z(π1W0), (H, f), J) := CF(φf◦π₎_∗η(W0, φf₁◦π(W1); Z(π1W0), H, J)

is well-dened [BC13, Section 4.3]. As in previous section, CF(W0, W1; Z(π1W0), (H, f), J) denotes the sum along all the class of Hamiltonian chords η ∈ π0(P(W0, W1)). From now on, we make no distinction in the notation of a cobordism (W; L0, L1) and its associated Lagrangian with cylindrical ends W.

Proposition 1.2.1. Let (W0, W1) be a pair of oriented and spin monotone La-grangians with cylindrical ends equipped with a choice of spin structure. Assume that the minimal Maslov number satises NWi > 2 for i = 0, 1. Let ((H, f), J) be

generic. Then the Floer complex CF(W0, W1; Z(π1W0), (H, f), J) is a chain complex. Moreover, H(CF(W0, W1; Z(π1W0), (H, f), J)) does not depend on the choice of generic pair (H, J) but depends on the path connected component [f] ∈ π0(H(W0, W1)) and is invariant under horizontal isotopies of W0 or W1,up to an isomorphism that only depends on the homotopy class of the path of Hamiltonian dieomorphisms in the horizontal isotopy with xed end-points.

Once the compactness of the moduli spaces is guarantied, the proof of this propo-sition follows from last section. The only additional argument concerns the invari-ance of the homology for dierent perturbations f, f _{∈ π}_{0(H(W0, W1)) with [f] = [f}_].

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30

This issue is addressed in the proof of the analog statement for Z2-coecients, Propo-sition 4.3.1 in [BC13]. The idea is to studya chain map dened using moving bound-aryconditions induced byan homotopyfτ= β(τ)f+(1−β(τ))f_{, connecting f, with}

f.

The homologyof the complex CF(W0, W1; Z(π1W0), (H, f), J) is denoted by HF(W0, W1; Z(π1W0), [f]).

1.2.2. Proof of Theorem 1.0.3

Proof of theorem 1.0.3. This theorem is a consequence of Proposition 1.1.1 and Proposition 1.2.1. Let (W, L0, L1) be an orientable, spin, monotone, Lagrangian cobordism viewed as a Lagrangian with cylindrical ends. Consider (H, f) ∈ H(W, W) a perturbation where f is locallygiven byf(x, y) = α±x+β±with α− < 0and α+> 0. Such a perturbation makes W cylindrically distinct at innity from φf◦π

1 (W). Denote by W _{the Lagrangian φ}f◦π

1 (W).

Consider now the Floer complex CF(W, W_{; Z(π1W), (H, f), J). From Proposition}

1.2.1 follows that HF(W, W_{; Z(π1W}_{), [f]) depends onlyon [f].}

Notice that there are horizontal isotopies in the same homotopyclass {ψt(W_{)}, {ψ}_t(W _)}

(see gure), such that:

CF(W, ψ1(W_{); Z(π1W), (H, f), J) = CF(L0, ψ1(L0); Z(π1W), J),}

and in the same way

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31

Fig. 1.2. Perturbation of a Lagrangian cobordism.

From the invariance of the homology under horizontal isotopies we can conclude that

H(CF(L0, ψ1(L0); Z(π1W), J)) ∼= H(CF(L1, ψ1(L1); Z(π1W ), J)).

Moreover, the relative homology is independent of the choice of almost complex struc-ture. Denoting by HF(Li; Z(π1W)) the homology of the complex CF(Li, ψ1(Li); Z(π1W), J), for i = 0, 1, we have that

HF(L0; Z(π1W)) ∼= HF(L1; Z(π1W)).

1.2.3. Exact Lagrangians with cylindrical ends

In this section we apply the previous results to the class of exact Lagrangian cobordisms. Since for an exact Lagrangian submanifold L ⊂ (M, ω) the Floer ho-molgy HF(L), is known to be isomorphic to the singular homology, we can expect the Floer homology HF(L; Z(π1(L)) to be isomorphic to the singular homology H(L), where L denotes the universal covering of L.

Proposition 1.2.2. Let L ⊂ (M, ω) be an orientable and spin exact Lagrangian submanifold. Then HF(L; Z(π1L)) = H(L).

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32

Before we start the proof of this proposition, we will dene the version of the Floer complex for exact Lagrangians used here. For this section we follow the exposition in [Aur14].

1.2.3.1. Coecient ring

In the denition of the Floer complex we have used the universal Novikov ring

A. This ring is not anymore necessary when working with exact Lagrangians.

Consider a pair of exact Lagrangians (L0, L1) ⊂ (M, dλ). There are functions fi : C∞(Li; R) such that λ |L_i= dfi for i = 0, 1. The symplectic area ω(u) of any strip with boundary conditions on L0∪ L1 connecting two intersection points x, y, is given by u∗dλ= (f1(y) − f1(x)) − (f0(y) − f0(x)). Then, there are uniform bounds on the energy in a xed moduli space M(x, y), independently of the homotopy class of the strip [u] ∈ π2(M, L0 ∪ L1). Thus, is not necessary to consider the universal Novikov ring A in this case.

The module used in the denition of the Floer complex CF(L0; Z(π1(L0), (H, J)) is Z(π1L0)(L0 ∩ L1). Here, L1 = ψH

1(L0) is the image of L0 under the time-one Hamiltonian dieomorphism associated to H.

1.2.3.2. Grading

In the module Z(π)1(L0))(L0 ∩ L1), there is a natural graduation dened using the Maslov-Viterbo index.

Let u : R × [0, 1] → M be a strip with boundary conditions on L0, L1 joining two intersection points x, y ∈ L0∪ L1. The Maslov-Viterbo index of u, denoted by μ(u), is the Maslov index of a loop obtained from a trivialization of the bundle u∗_(TM).

Let λ : [−∞, ∞] → L0, λ _{: [−∞, ∞] → L1} _{be two paths dened by λ(s) = u(s, 0)}

and λ_{(s) = u(s, 1). Consider the path of Lagrangians Tλ}

(s)L0 concatenated with a path of Lagrangians λ_{(s) ⊂ Tλ}

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33 The Maslov-Viterbo index of u, is the Maslov index of the loop Tλ(s)L0#(−λ(s)) after trivialization.

Fix a point x0 ∈ (L0 ∩ L1). Consider the path-connected component of x0 in

P(L0, L1). For any other point y ∈ (L0∩ L1) ∩ Px0(L0, L1) we dene μ(y, x0) = μ(y)

to be the Maslov-Viterbo index of a path y in Px0(L0, L1) connecting y with x0.

This is independent on the choice of path y.Assuming that μ(x0) = k ∈ Z, we set μ(y) := μ(y, x0) − k.

Proof of Proposition 1.2.2. By the Weinstein theorem there exists a neighbor-hood of L, UL symplectomorphic to a tubular neighborhood of the zero-section. Let f: L → R be a Morse function and for small > 0 let L = graph(df). Note that L ⊂ (T∗L,dxi∧ dyi) is Hamiltonian isotopic to L. The set of intersection points L L is the set Crit(f). The relation between the Floer and Morse grading, is given by μ(x) = |x| − μ(x0), where |x| denotes the Morse index (see[Flo89]). This implies that the grading of the Floer and the Morse complexes agree up to a shift. Moreover, from [Flo89] there exist a time dependent almost complex structure J for which the moduli spaces of Floer strips correspond to moduli spaces of connecting ow lines.

On the other hand, in [Sch93, Theorem 13, Appendix B] was shown that the orientations of the Morse complex obtained from the geometrical constructions can be extended to canonical orientations induced by the determinant bundle, the last ones are the ones used to dene the Floer complex.

Then CF(L, L, ; Z(π1L), J) = (Z(π1L)(Crit(f)), d) = CM(L; Z(π1L)).

Here, CM(L; Z(π1L)) denotes the Morse complex with Z(π1L)-coecients, with the dierential dened in the same way as for the Z(π1L)-Floer complex.

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34

Now, from the invariance under Hamiltonian perturbations, and the independence of the choices dening the Floer homology, we can conclude that

H(CF(L, L; Z(π1L), J)) ∼= HF(L; Z(π1L)) ∼= H(CM(L; Z(π1L))).

To see that this last homology H(CM(L; Z(π1L))) is, as a group, the singular ho-mology of the universal covering H(L), rst notice that if p : L → L denotes the projection, for a Morse function f : L → R we have that the set of generators Z(π1L)(Crit(f)) = Z(p−1_{(Crit(f))) = Z(Crit(f)) coincide, where f is Morse} func-tion obtained by lifting f to a funcfunc-tion on L. The stable manifolds of the gradient of f denes a a CW-decomposition of L, whose lift to L computes its homology. We now consider the exact Lagrangian with cylindrical ends associated to an ex-act Lagrangian cobordism (W, L0, L1). For such a Lagrangian the Floer homology depends on the choice of path component [f], dening the perturbation on the cylin-drical ends. Dierent choices lead to dierent homologies.

In order to relate the Floer homology HF(W; Z(π1(L), [f]) with the singular ho-mology, let S be one of the ends of W. There are four options; S = ∅, L0, L1, L0∪ L1. Denote by [fS] the class of perturbation with α+ > 0if L1 ⊂ S and α+ < 0if not. As well as α− < 0if L0 ⊂ S and α0 > 0if not.

Proposition 1.2.3. Let (W, L0, L1) be an orientable and spin exact Lagrangian cobor-dism equipped with a choice of spin structure. We have the following isomorphisms:

HF(W; Z(π1W), [fS]) ∼= H(W, S; Z(π1W)).

Proof. Let J ∈ JBbe an autonomous almost complex structure and denote by gJits induced Riemannian metric. Let fS: W → R be a Morse function with the following properties:

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35 (1) The negative gradient of −∇g_JfS is transverse to ∂W and it points outside of

W along S and inside of W on the complement of S in ∂W.

(2) There is a compact set B_{, such that the Morse function fS}_{extends to a linear}

function fS : W → R in ((R2\ B) × M) ∩ W, given by fS |E±

R±+ε (x, y, p) =

α_±x+ β±, for some α_±, β±∈ R. Here x, y are the R2_{-coordinates.}

Consider a Weinstein neighborhood UW of W, where we see W as the zero-section in (T W, dλstd) with the standard exact symplectic form.

Let W = graph(dfS) be the image of W under the Hamiltonian isotopy (ψH t), generated by the Hamiltonian H = fS◦ πW. Here, πW : T∗W → W.

The pair (W, W) is cylindrically distinct at innity, since the Hamiltonian ow ψH is a vertical translation outside the compact set B, and intersects transversely.

From [Flo89], there exist a time dependent almost complex structure J, such that there is a bijection between the moduli space of Floer strips and the moduli space of ow lines connecting two critical points x, y ∈ W ∩ W.

Condition 2 on the Morse function fSensures that J ∈ JB. Since the Hamiltonian isotopy is linear outside the set (R2_{\ B) × M then (ψ}H

t)∗ = Id at each time on this set. From the identity J = (ψH

t)∗J(ψHt)−1∗ follows the claim.

Then the Floer complex CF(W, W; Z(π1(W), J) is well dened and H(CF(W, W; Z(π1(W), J)) = H(CM((fS, gJ); Z(π1W))).

The result follows from the relation between Morse and singular homologies, since every Morse function induces a cellular decomposition, using the unstable/stable manifolds associated to the critical points, then:

H(CM((fS, gJ); Z(π1(W))) ∼= H(W, S; Z(π1W)).

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Remark 1.2.1. Notice that if S = ∅ or L0∪ L1, then the connected path [fS] corre-sponds to perturbations f(x, y) = α±x+ β± with α− and α+ of opposite signs due to condition 1 on fS.

Using the invariance of the Floer homology under horizontal Hamiltonian pertur-bations (Theorem 1.0.3 and Proposition 1.2.2), we have the following isomorphisms:

HF(W; Z(π1(W)), [fS]) ∼= HM(L0; Z(π1W)) ∼= HM(L1; Z(π1W)). In the case S = L0, L1, we have

HF(W; Z(π1(W)), [fL_i]) ∼= H(W, Li; Z(π1(W))) = 0 1.2.4. Proof of Corollaries 1.0.1 and 1.0.2

Corollary. 1.0.1 If (W, L0, L1) is an orientable and spin, exact Lagrangian cobordism and i# : π1(Li) → π1(W) is an isomorphism for i = 0, 1, then (W; L0, L1) is an h-cobordism.

Corollary. 1.0.2 If (W; L0, L1) is an orientable and spin, exact Lagrangian cobordism with dim(W) > 5 and Li (i = 0, 1), W are simply connected, then (W; L0, L1) is a Lagrangian pseudo-isotopy.

To prove these Corollaries we rst recall the h-cobordism theorem.

Denition 1.2.4. An h-cobordism (W; L0, L1) is a cobordism where the inclusion maps i : Li → 1 are homotopy equivalences for i = 0, 1.

Theorem 1.2.1 (Cobordism (Smale)). If (W; L0, L1) is a simply connected h-cobordism with dim(W) > 5, then the h-cobordism (W; L0, L1) is dieomorphic to the trivial cobordism ([0, 1] × L0; L0, L0).

From the remark below, we have that if (W; L0, L1) is anexact Lagrangian cobordism, orientable and spin, equipped with a choice of spin structure, then H(W, Li; Z(π1(W))) = 0. The homology is generated by a Morse function, that

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37 induces a structure of CW-complex on W. We also observe in Proposition 1.2.1, that H(W, Li; Z(π1(W))) = H(W, L). Here, the homology H(W, L) is the relative homologyinduced from a CW-decomposition of W and the induced decomposition on L.

The hypothesis on the fundamental groups i# : π1(Li) → π1(W) combined with H(W, Li; Z(π1(W))) = 0, implythat the inclusions are homotopyequivalences. This follows from one of the Whitehead Theorems [Whi49] Theorem 3. From this we deduce Corollary1.0.1.

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Chapter 2

SIMPLE HOMOTOPY THEORY

2.1. Simple homotopy type

This chapter presents the results on simple homotopy theory used in this thesis. We follow the exposition of Cohen [Coh70].

The simple homotopy theory is the study of the relation of simple homotopy type between cellular complexes.

Denition 2.1.1. Let K and L be nite CW-complexes. There is an elementary collapse from K to L (or an elementary expansion from L to K) if K = L ∪f|_Dn−1 Dn−1∪fDn, where:

(1) f : Dn _{→ K is the attachingmap for D}n _{and f |D}

n−1: Dn−1 → K is the

attachingmap for Dn−1_.

(2) The new cells attached by f are not in L.

(3) The closure of ∂Dn_−Dn−1_{denoted by ∂D}n− Dn₋₁, satises f(∂Dn− Dn₋₁) ⊂ Ln−1.

Two CW-complexes L and K have the same simple-homotopy type if they are related by a nite sequence of collapses and expansions, denoted K ∼s _L_.

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40

A map f : L → K is a simple-homotopy equivalence if f is homotopic to a map obtained by a nite number of compositions of the maps induced by collapses and expansions.

The relation of simple homotopy type is ner than that of homotopy type. There are CW-complexes that are homotopy equivalent but not simple homotopy equiva-lent, examples can be found in [Coh70].

Given a xed complex L, consider the set of pairs (K, L), where K is a CW-complex homotopy equivalent to L, that contains L as a sub-CW-complex. The Whitehead group of L, Wh(L), is dened to be the set of equivalence classes of pairs [K, L] under the relation (K, L) ∼ (K_{, L}_{) if K ∼}s _K _{rel L. This means that K and K} _{are related by} a nite set of collapses and expansions for which no cell of L is ever removed. The group operation is

[K, L] + [K_{, L}_{] = [K ∪L}_K_{, L}_].

Where K ∪LK is the disjoint union of K and K identied by the identity map on L. The Whitehead group of L, Wh(L), is an abelian group (see 6.1 of [Coh70]).

Given a homotopy equivalence f : K → L, we dene its Whitehead torsion τ(f) := [Mf∪KMf, L], where Mf denotes the mapping cylinder of f.

The Whitehead torsion is the obstruction for a homotopy equivalence to be a simple-homotopy equivalence as stated in the following theorem.

Theorem 2.1.1 (22.2 in [Coh70]). A homotopy equivalence f : K → L, is a simple homotopy equivalence if and only if τ(f) = 0.

The geometric denitions of Whitehead group is hard to work with, instead there is an algebraic version that we introduce in the next section.

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2.2. The Whitehead torsion

2.2.1. The Whitehead torsion of a ring

Let Z(G) be the integral group ring of the group G. The group GL(n, Z(G)) denotes the group of non-singular n × n matrices over Z(G). Identifying each M ∈ GL(n, Z(G)) with the matrix

⎡

⎣ M 0

0 1

⎤ ⎦ ,

we obtain an injection GL(n, Z(G)) ⊂ GL(n + 1, Z(G)). The innite general linear group of Z(G) is the union over all n of the groups GL(n, Z(G)) and is denoted by GL(Z(G)). Amatrix in GL(Z(G)) is called elementary if it coincides with the identity matrix except for one o-diagonal element.

The set of elementary matrices is the commutator subgroup of GL(Z(G)). Let E(G) denote the group generated by all the elementary matrices and all matrices that coincide with the identity matrix except for one diagonal element g∈ ±G.

The Whitehead group of G is the group:

Wh(G) = GL(Z(G))/E(G).

The class of a matrix [A] ∈ Wh(G) is called the torsion of the matrix and is denoted by τ(A).

2.2.2. The Whitehead torsion of an acyclic complex

To dene the torsion of an acyclic complex we need to introduce some language. Denition 2.2.1. A Z(G)-module is a free Z(G)-module M with a distinguished family of bases B which satisfy: If b and b _{are two bases of M and if b ∈ B then}

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42

b ∈ B if and only if τ([b/b]) = 0 ∈ Wh(G), where [b/b] represents the matrix that changes the base b to the base b_.

An isomorphism f : M1 → M2 is a simple isomorphism of Z(G)-modules if τ(bf) = 0, where bf is the matrix of f with respect to any distinguished basis of M1. Denition 2.2.2. A Z(G)-complex is a free chain complex over Z(G), C = (C, d) such that each C∗ is a Z(G)-module. A preferred basis of C mean a basis c = ∪ici

where ci is a preferred basis of Ci.

A simple isomorphism of Z(G)-complexes, f : C → C_{, is a chain mapping such}

that f |C∗: C∗ → C∗ is a simple isomorphisms for all ∗.

Another useful notion is the following. Two Z(G)-complexes, (C, d) and (C_{, d}_),

are simple homotopy equivalent, (C, d) ∼s _(C_{, d}_{), if there exist trivial}1 _Z(G)-chain complexes (T, t) and (T_{, t}_{) such that the chain complex (C ⊕ T, d ⊕ t) is simple}

isomorphic to the chain complex (C_{⊕ T}_{, d}_{⊕ t}_).

Let (C, d) be an acyclic Z(G)-complex. Then (C, d) is contractible, so there exist a chain contraction δ : C → C[1] such that δd + dδ = 1. If Codd = C1⊕ C3⊕ · · · and Ceven = C0⊕ C2⊕ · · · , then the torsion of the complex (C, d) is dened to be the torsion of the map:

(d + δ)odd = (d + δ) |Codd: Codd→ Ceven,

τ(C) = τ((d + δ)odd) ∈ Wh(G).

The Whitehead torsion of a complex satises the following properties:

(1) If f : C → C _{is a simple isomorphism of chain Z(G)-complexes, then τ(C) =}

τ(C).

(2) The torsion of a direct sum of Z(G)-complexes C ⊕ C _{satises τ(C ⊕ C}_{) =}

τ(C) + τ(C_).

1_{A Z(G)-complex (T, t) : 0 → T}

i+1 → Ti→ 0 is elementary trivial if τ(T) = 0. A trivial chain

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43 If f : C → C _{is a homotopy equivalence of Z(G)-complexes, then the Whitehead}

torsion of f is dened by τ(f) = τ(Cone(f)).

2.2.3. The Whitehead torsion of CW-pair

Let (W, L) be a pair of nite, connected CW-complex where L is a sub-complex of W. Let C(W, L) denote the cellular complex, where C(W, L)_∗ = H_∗(W∗∪L, W∗−1∪L) is the free module generated by the ∗-cells of W − L and the dierential is dened by the boundary operator on the exact sequence for singular homology of the triple (W∗_{∪ L, W}∗−1_{∪ L, W}∗−2_{∪ L).}

If L → W is a homotopy equivalence, then the induced map π1(L) → π1(W) is an isomorphism. In this case, if p : W → W is the universal covering of W, then p−1(L) = L is the universal covering of L and L → W is also a homotopy equivalence. The projection map p induces a structure of CW-complex on W, L from the ones on W, L. With this cellular structure, the cellular complex C(W, L) is a Z(G)-complex.

Denition 2.2.3. Let (W, L) be a pair of nite, connected CW-complexes, and sup-pose that L → W is a homotopy equivalence. The Whitehead torsion of the pair is dened by

τ(W, L) = τ(C(W, L)) ∈ Wh(π1L), where (W, L) is the universal covering of (W, L).

2.2.4. The s-cobordism theorem

One application of the theory, briey recalled in this chapter, is the s-cobordism theorem.

Theorem 2.2.1 (s-Cobordism, Mazur [Maz63], Barden [Bar63], Stallings [Sta67]). Let (W; L0, L1) be an h-cobordism of dimension n > 5. Then the torsion τ(W, L0) vanishes if and only if W is dieomorphic to the product L0× [0, 1].

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Notice that if (W; L0, L1) is an h-cobordism, then we can also dene the Whitehead torsion of a pair (W, L0) using a Morse function. Recall that any nice2 _{Morse function} dened on an h-cobordism (W; L0, L1) induces a CW-structure on it, that lifts to the universal covering W. The following theorem of Milnor establishes that the torsion of the complex dened by the CW-pair induced by the lift of any nice Morse function is the torsion of the pair (W, L0).

Theorem 2.2.2 (Theorem 9.3 [Mil66]). If (W; L, L_{) is an h-cobordism, then}

τ(W, L) = τ(C(W, L; f) does not depend on the choice of nice Morse function.

Remark 2.2.1. Recall that under some conditions established in Corollary 1.0.1, if (W; L0, L1) is an exact Lagrangian cobordism, then it is an h-cobordism and the torsion of the pair (W, L0) is well dened. From Milnors's theorem follows that τ(W, L0) = τ(CM(W; Z(π1W)(fL₀, gJ)), where (fL₀, gJ) is as dened as in the proof of Proposition 1.2.3.

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Chapter 3

WHITEHEAD TORSION AND FLOER

COMPLEX

By the s-cobordism theorem, an h-cobordism of dimension higher than 5 is a pseudo-isotopy if its Whitehead torsion vanishes.

In the symplectic category, the cobordism studied has an additional structure: it is a Lagrangian submanifold with Lagrangian boundary.

To compute the Whitehead torsion of an exact Lagrangian h-cobordism we study the torsion of the Z(π1W)-Floer complexassociated to a pair of exact Lagrangian cobordisms (W, W_{), when it is dened.}

In the previous chapter we saw that any nite acyclic Z(π1W)-complexwith a preferred family of bases has a well dened Whitehead torsion. The Z(π1W)-Floer complexassociated to a pair of Lagrangian cobordisms (W, W_{) has a preferred base}

given by the set W ∩ W_{, and is an acyclic complexfor the class of perturbations}

that displace W _{away from W; therefore it has a well-dened Whitehead torsion.}

In this chapter we show that the Whitehead torsion of the Floer complexdoes not depend on the choice of horizontal Hamiltonian perturbations of the cobordism. We extend a result of M. Sullivan [Sul02] to the oriented-cobordism setting.

Exact Lagrangian cobordism and pseudo-isotopy

Université de Montréal

Exact Lagrangian cobordism and pseudo-isotopy

Lara Simone Suárez López

Université de Montréal

Exact Lagrangian cobordism and pseudo-isotopy

Lara Simone Suárez López

RÉSUMÉ

ABSTRACT

CONTENTS

LIST OF FIGURES

REMERCIEMENTS

INTRODUCTION

PRELIMINARIES

0.1. Symplectic manifolds

Chapter 1

FLOER HOMOLOGY WITH

Z(π

W

)-COEFFICIENTS

1.1. Floer homology with twisted coefficients

1.2. Lagrangians with cylindrical ends

Chapter 2

SIMPLE HOMOTOPY THEORY

2.1. Simple homotopy type

2.2. The Whitehead torsion

Chapter 3

WHITEHEAD TORSION AND FLOER

COMPLEX