Chekanov-Eliashberg Dg-Algebras and Partially Wrapped Floer Cohomology

UPPSALA DISSERTATIONS IN MATHEMATICS

120

Chekanov-Eliashberg dg-algebras and partially wrapped Floer cohomology

Johan Asplund

Department of Mathematics Uppsala University UPPSALA 2021 Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 4 June 2021 at 14:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Dr, HDR Baptiste Chantraine (Département de Mathématiques, Universite de Nantes, France).

Abstract Asplund, J. 2021. Chekanov-Eliashberg dg-algebras and partially wrapped Floer cohomology. Uppsala Dissertations in Mathematics 120. 42 pp. Uppsala: Department of Mathematics. ISBN 978-91-506-2872-2.

This thesis consists of an introduction and two research papers in the fields of symplectic and contact geometry. The focus of the thesis is on Floer theory and symplectic field theory. In Paper I we show that the partially wrapped Floer cohomology of a cotangent fiber stopped by the unit conormal of a submanifold, is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension two knots in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of the knot. This allows us to exhibit codimension two knots with infinite cyclic knot group such that the union of the unit conormal of the knot and the boundary of a cotangent fiber is not Legendrian isotopic to the union of the unit conormal of the unknot union the boundary and the same cotangent fiber. In Paper II we study the Chekanov-Eliashberg dg-algebra which is a holomorphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov- Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non- singular, establishes a proof of a conjecture by Ekholm-Lekili and independently by Sylvan. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomology and the stop removal formulas of Ganatra-Pardon-Shende.

Keywords: Symplectic geometry, Contact geometry, Chekanov-Eliashberg dg-algebra, partially wrapped Floer cohomology, Lagrangian submanifold, Legendrian submanifold, singular Legendrians, based loop space

Johan Asplund, Department of Mathematics, Box 480, Uppsala University, SE-75106 Uppsala, Sweden.

ISSN 1401-2049 ISBN 978-91-506-2872-2 urn:nbn:se:uu:diva-439476 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-439476) List of papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I J. Asplund, Fiber Floer cohomology and conormal stops, arXiv:1912.02547 (2019). To appear in Journal of Symplectic Geometry.

II J. Asplund and T. Ekholm, Chekanov–Eliashberg dg-algebras for singular Legendrians, arXiv:2102.04858 (2021). Submitted.

Reprints were made with permission from the publishers.

Contents

1 Introduction ...... 7 1.1 Brief overview of the results of the thesis ...... 7 1.2 History and motivation ...... 8 1.3 Outline of the thesis ...... 8

2 Geometry ...... 10 2.1 Symplectic geometry ...... 10 2.1.1 Background and some linear algebra ...... 10 2.1.2 Basics on symplectic manifolds ...... 12 2.1.3 Submanifolds of symplectic manifolds ...... 13 2.1.4 Hamiltonian dynamics ...... 14 2.2 Contact geometry ...... 15 2.2.1 Basics on contact manifolds ...... 16 2.2.2 Reeb dynamics ...... 17 2.2.3 Submanifolds of contact manifolds ...... 18 2.3 Liouville and Weinstein manifolds ...... 18 2.3.1 A connection between symplectic and contact geometry ...... 18 2.3.2 Weinstein manifolds ...... 19 3 Gromov’s J-holomorphic curves, Floer theory and symplectic field theory ...... 22 3.1 J-holomorphic curves ...... 22 3.2 Floer theory and the Arnold–Givental conjecture ...... 24 3.2.1 Wrapped Floer cohomology ...... 26 3.3 Symplectic field theory ...... 27 3.3.1 The Chekanov–Eliashberg dg-algebra ...... 29

4 Summary of papers ...... 32 4.1 Paper I ...... 32 4.2 Paper II ...... 33

5 Summary in Swedish ...... 36 5.1 Artikel I ...... 36 5.2 Artikel II ...... 37

6 Acknowledgments ...... 39

References ...... 40

1. Introduction

In this chapter we first give a technical and brief overview of the results of the thesis, followed by history and motivation. For an introduction to the various mathematical concepts appearing in the thesis, we refer to Chapters 2 and 3. Chapter 5 consists of a summary in Swedish intended for non-technical readers.

1.1 Brief overview of the results of the thesis In this thesis we study partially wrapped Floer cohomology from the perspectives of contact geometry and symplectic field theory. Resulting from our work is a geometric description the endomorphism algebras of the generators of the partially wrapped Fukaya category of a stopped Weinstein manifold, in terms of the contact geometry of the attaching locus of the stop. In case the stopped Weinstein manifold is a cotangent bundle stopped by the unit conormal bundle of a submanifold, our description is in terms of contact geometry of the unit conormal bundle and chains of loops of the complement of the submanifold in the base together with the Pontryagin product. In Paper I we show that the wrapped Floer cohomology algebra of a cotangent fiber F in the stopped cotangent bundle is equivalent to chains of based loops on the complement of the submanifold in the base. For codimension 2 knots K in the n-sphere we show that there is a relationship between the wrapped Floer cohomology algebra of the fiber and the Alexander invariant of n ∼ K. This allows us to exhibit codimension 2 knots K satisfying π1(S \K) = Z and such that the Legendrian submanifold ΛK ∪∂F is not Legendrian isotopic to Λunknot ∪ ∂F . In Paper II we study the Chekanov–Eliashberg dg-algebra which is a holomorphic curve invariant associated to a smooth Legendrian submanifold. We extend this definition to singular Legendrians. Using the new definition we formulate and prove a surgery formula relating the wrapped Floer cohomology algebra of the co-core disk of a stop with the Chekanov–Eliashberg dg-algebra of its attaching locus interpreted as the Weinstein neighborhood of a singular Legendrian. A special case of this surgery formula, when the Legendrian is non-singular, establishes a proof of a conjecture by Ekholm–Lekili and independently by Sylvan [EL17, Syl19]. We furthermore provide an alternative geometric proof of the pushout diagrams for partially wrapped Floer cohomology and the stop removal formulas of Ganatra–Pardon–Shende [GPS18].

7 1.2 History and motivation In physics, classical mechanics is the study of motion of macroscopic objects, such as projectiles or planets. Newton’s laws of motion describe the relationship between the motion of an object and the forces acting on it, and they form the foundation of Newtonian mechanics. A more sophisticated and general approach to classical mechanics is known as Hamiltonian mechanics and it is governed by Hamilton’s equations. The framework of Hamiltonian mechanics can also be used to describe quantum mechanics. Symplectic geometry forms the mathematical basis used to describe Hamilton’s equations and Hamilto- nian mechanics. Related to symplectic geometry is contact geometry. Contact geometry provides a mathematical framework which can be used to describe optical phenomena and the evolution of wavefronts in physics. The dawn of the modern study of symplectic and contact geometry was in 1985 when Gromov introduced the use of J-holomorphic curves to symplectic geometry. Symplectic and contact geometry are very rich subjects with contri- butions from for example Gromov, Floer, Kontsevich, Fukaya and Eliashberg. Much of the development of modern symplectic and contact geometry builds upon and uses J-holomorphic curves in a crucial way. In Floer theory and symplectic field theory, certain configurations of J-holomorphic curves are counted, and defines algebraic invariants of the geometries. In this thesis we study the Fukaya category and various versions thereof, and it can be viewed as a categorification of Floer theory. Our main motivation for studying Fukaya categories is Kontsevich’s homological mirror symmetry, which is inspired by the phenomenon of mirror symmetry found in string theory. Homological mirror symmetry is a conjecture about an intricate relationship between the Fukaya category of a symplectic manifold M and a certain category of sheaves on a complex variety M| called the mirror of M. Additional motivation comes from smooth topology of the underlying manifold and submanifolds. The study of symplectic and contact geometry can for instance lead to applications in the study of exotic spheres and in knot theory.

1.3 Outline of the thesis The main part of this thesis consists of the two appended papers. The aim of the introductory chapters is to introduce and familiarize the reader with some of the concepts used in the thesis. Where possible we give references to the literature where more details can be found. In Section 2 we give the reader an introduction to both symplectic and contact geometry, both of which is important in the main part of the thesis. In Section 3 we introduce the core concept of J-holomorphic curves, which is the main tool used to study symplectic and contact geometry in this thesis. We continue by giving a brief overview of Floer theory and of symplectic field the-

8 ory. In Section 4 we summarize the main results of the two appended papers, and in Section 5 we give a short summary of the thesis in Swedish.

9 2. Geometry

2.1 Symplectic geometry Symplectic geometry has its roots in the 19th century and Hamilton’s formulation of classical mechanics. Hamilton formulated equations which described classical systems of particles which take advantage of symmetries between particles positions and momenta. Symplectic geometry is the natural mathematical language in which we can express Hamilton’s equations in a coordinate- free way. As a geometry, symplectic geometry is a kind of non-linear geometry which do not allow us to measure distances or angles, but it rather allows us to measure areas. In these sections we will review the most important notions and concepts from symplectic geometry which we use in the main part of this thesis. For more details and a more thorough introduction to symplectic geometry, we recommend [MS17].

2.1.1 Background and some linear algebra Let V be a vector field over a field k of characteristic equal to 0. Let k ∈ N and denote by Ak(V ) the space of alternating k-multilinear maps on V . An alternating k-multilinear map is a map

φ: V k −→ k satisfying •(k-multilinearity) φ is linear with respect to the ith input, for each i ∈ {1, . . . , k}. That is for any α, β ∈ k we have

φ(v1, . . . , αvi + βwi, . . . , vk) =

αφ(v1, . . . , vi, . . . , vk) + βφ(v1, . . . , wi, . . . , vk) • (alternating) For any i ∈ {1, . . . , k − 1} we have

φ(v1, . . . , vi, vi+1, . . . , vk) = −φ(v1, . . . , vi+1, vi, . . . , vk) Note that A0(V ) = k and A1(V ) = V ∗. Taking a direct sum, we consider ⊕∞ A(V ) := Ak(V ) . k=0

10 There is an operation defined on A(V ) called the wedge product. If φ ∈ Ak(V ) and ψ ∈ Aℓ(V ), their wedge product is φ ∧ ψ ∈ Ak+ℓ(V ) and is defined by the formula

(φ ∧ ψ)(v1, . . . , vk+ℓ) := 1 ∑ (−1)σφ(v ,. . . , v )ψ(v , . . . , v ) . k!ℓ! σ(1) σ(k) σ(k+1) σ(k+ℓ) σ∈Sk+l σ Here Sk+ℓ is the symmetric group on k + ℓ elements, and (−1) is the sign of the permutation σ. Let M be a smooth manifold. A differential form k-form η on M is a smooth function on M which assigns p ∈ M to an alternating k-multilinear map ηp ∈ k A (TpM) valued in the tangent space of M at p. We denote the space of differential k-forms on M by Ωk(M), and by taking a direct sum we define ⊕∞ Ω(M) := Ωk(M) . k=0 Note that Ω0(M) = C∞(M) is the space of smooth real-valued functions on M. The wedge product is defined on Ω(M) as (η ∧ κ)p = ηp ∧ κp. This turns Ω(M) into a graded algebra. We can furthermore equip Ω(M) with the exterior derivative which is a linear map d:Ωk(M) −→ Ωk+1(M) .

Using local coordinates x = (x1, . . . , xm) on M, then the differentials

((dx1)x,..., (dxm)x) ∗ ∈ k forms a basis of Tx M. Any differential k-form η Ω (M) can be written in the local coordinate x as ∧ · · · ∧ ηx = g(x1, . . . , xn)dxi1 dxik , ∞ m where i1, . . . , ik ∈ {1, . . . , m} and g(x1, . . . , xn) ∈ C (R ). The differential of η is then expressed in these coordinates as ∑m ∂g ∧ ∧ · · · ∧ (dη)x = (x1, . . . , xn)dxi dxi1 dxik . ∂xi i=1 The exterior derivative d satisfies the∑ following properties ∞ m ∂f 1. If f ∈ C (M), then df = dxi. i=1 ∂xi 2. d(df) = 0 for f ∈ C∞(M). 3. d(η∧κ) = dη∧κ+(−1)kη∧dκ where η ∈ Ωk(M) and κ ∈ Ωℓ(M). In fact, d is the unique linear map Ωk(M) −→ Ωk+1(M) satisfying 1–3 above, and can thus be taken as definition. In fact, it follows from these three properties that for any η ∈ Ωk(M) we have d(dκ) = 0. Therefore (Ω(M), d) is a cochain complex, and (Ω(M), d, ∧) is a differential graded algebra.

11 2.1.2 Basics on symplectic manifolds A symplectic manifold is a tuple (M, ω) consisting of a smooth manifold M equipped with a differential 2-form ω ∈ Ω2(M) satisfying • (closed) dω = 0 • (non-degeneracy) For each p ∈ M, the linear map ∗ TpM −→ (TpM)

w 7−→ ωp(−, w) is an isomorphism. The existence of a non-degenerate 2-form on a smooth manifold implies that the dimension of M is even. This follows from the statement that a vector space V equipped with a non-degenerate alternating bilinear form α must have even dimension. Namely, pick a basis such that α is represented by a skew- symmetric matrix, which means that we have αT = −α. Taking determinants gives us det α = (−1)dim V det α , which means that dim V must be even. From now on we let dim M = 2n. The two most important examples of a symplectic manifold are the following.

Example 2.1. 1. For any positive integer n, consider R2n with coordinates (x1, . . . , xn, y1, . . . , yn). Then ∑n ωstd := dxi ∧ dyi , i=1 is a symplectic form on R2n, and (R2n, ω) is a symplectic manifold. 2. Let M be any n-dimensional smooth manifold. Consider the cotan- ∗ gent bundle T M with local coordinates (q1, . . . , qn, p1, . . . , pn). Here q1, . . . , qn are position coordinates and p1, . . . , pn are momentum coordinates. The tautological one-form is given by ∑n λ = pidqi . i=1 The exterior derivative of λ is then a symplectic form. In coordinates it is given by ∑n ω = dλ = dpi ∧ dqi . i=1 Therefore (T ∗M, ω) is a symplectic manifold.

Theorem 2.2 (Darboux). Let (M, ω) be a 2n-dimensional symplectic manifold, and let p ∈ M. Then there is a neighborhood U of p and local coordinates

12 (x1,. . . , xn, y1, . . . , yn) in U, such that ∑n | ∧ ω U = dxi dyi i=1 | Note by that by abuse of notation, ω U is the differential form which assigns to each p ∈ U an alternating bilinear map | × −→ R (ω U )p : TpU TpU . This very important structural theorem says that any symplectic manifold 2n locally looks like (R , ωstd). Therefore, it is impossible to distinguish between different symplectic manifolds locally. Suppose that (M, ω) and (M ′, ω′) are two symplectic manifolds. A smooth function f : M −→ M ′ , is called a symplectomorphism if it is a diffeomorphism (meaning that f is bijective and has a smooth inverse), and additionally f ∗ω′ = ω . This relationship between symplectic manifolds is the isomorphism in the category of symplectic manifolds.

2.1.3 Submanifolds of symplectic manifolds Let (M, ω) be a 2n-dimensional symplectic manifold, and let L ⊂ M be a submanifold. We say | | • L is symplectic, if ω L is non-degenerate. That is, (L, ω L) is itself a symplectic manifold | • L is isotropic if ω L = 0. | • L is Lagrangian if ω L = 0 and dim L = n In fact, if L ⊂ M is isotropic, it follows that dim L ≤ n. This follows from the analogous linear algebra statement. Namely, the linear map ∗ φL : TpM −→ (TpL) 7−→ − | v ωp(v, ) L gives dim M = dim L + dim ker φL. Since L is isotropic, we have dim L ≤ dim ker φL, which implies 2 dim L ≤ dim M ⇔ dim L ≤ n. Therefore Lagrangian submanifolds are isotropic submanifolds of maximal dimension.

R2n Example 2.3. 1. Consider the symplectic manifold∑( , ωstd) with co- n ∧ ordinates (x1, . . . , xn, y1, . . . , yn) and ωstd = i=1 dxi dyi. The

13 closed unit ball { }

∑n 2n ∈ R2n 2 2 ≤ B = (x1, . . . , xn, y1, . . . , yn) xi + yi 1 , i=1

is a symplectic submanifold. The two half dimensional spaces { } L := (x , . . . , x , 0,..., 0) ∈ R2n ⊂ R2n x { 1 n } 2n 2n Ly := (0,..., 0, y1, . . . , yn) ∈ R ⊂ R

are examples of Lagrangian submanifolds. 2. Consider a smooth n-dimensional∑ manifold M and the symplectic ∗ n manifold (T M, dλ) where λ = i=1 pidqi is the tautological one- form expressed in terms of the local coordinates (q1, . . . , qn, p1, . . . , pn). The unit disk cotangent bundle DT ∗M, which is defined as the vector bundle over M with fiber over p ∈ M being { }

∑n ∗ ∈ Rn 2 ≤ DTp M = (p1, . . . , pn) pi 1 , i=1

is a symplectic submanifold. Two examples of Lagrangian subman- ⊂ ∗ ∗ ⊂ ∗ ifolds are the zero section M T M and the fiber Tp M T M over a point p ∈ M.

2.1.4 Hamiltonian dynamics Let (M, ω) be a symplectic manifold and let H : M −→ R be a smooth real- valued function. We call such function a Hamiltonian, and associated to a Hamiltonian is the Hamiltonian vector field XH . The Hamiltonian vector field is uniquely defined by the equation

ω(XH , −) = dH .

A function H : [0, 1] × M −→ R is called a time-dependent Hamiltonian and we write Ht = H(t, −). Denote the corresponding Hamiltonian vector field t by Xt. The time-t flow generated by Xt is denoted by φH , and is defined by d φt = X (φt ) . dt H t H ∈ t For each t [0, 1], φH is a symplectomorphism. The time-1 flow of a time- 1 dependent Hamiltonian vector field φH is called a Hamiltonian symplectomorphism.

14 Example 2.4. Consider (R2, dx ∧ dy), and let H : [0, 1] × R2 −→ R be any Hamiltonian. The Hamiltonian vector field is the vector field determined by the equation ( ) ∂H ∂H ∂H ∂H X dy − X dx = dx + dy ⇔ X = , − . x y ∂x ∂y H ∂y ∂x

Note that if i is the standard complex structure on R2, meaning that it is a bundle endomorphism i: T R2 −→ T R2 such that { i(∂ ) = ∂ x y , i(∂y) = −∂x then XH = i∇H, where ∇ is the gradient with respect to the standard metric 2 on R . In fact, trajectories of XH are contained in level sets of the Hamilto- nian H, meaning that if H is a function describing energy as in Hamiltonian mechanics, the trajectories of XH are energy preserving. t If we write φH (x, y) = (x(t), y(t)), the time-t flow of the vector field XH is given by the system of differential equations { dx = ∂H dt ∂y , dy − ∂H dt = ∂x which is recognized by physicists as Hamilton’s equations, and describes the total energy of a classical mechanical system and is equivalent to Newton’s law of motion.

One cornerstone conjecture in symplectic geometry is due to Arnold [Arn65]. Let M be a smooth compact manifold, and let crit(M) denote the minimum number of critical points of a smooth real-valued function f : M −→ R.

Conjecture 2.5 (Arnold). Let (M, ω) be a compact symplectic manifold and let φ be a Hamiltonian symplectomorphism. Then φ has at least crit(M) dis- tinct fixed points.

This conjecture was proven for a special class of symplectic manifolds that are called monotone by Floer [Flo89], using techniques which today is known as Floer theory, see Section 3. For a more detailed account on the status of the Arnold conjecture the reader is referred to [MS17, Section 11].

2.2 Contact geometry Sophus Lie is considered the originator of contact geometry after his work on contact transformations, which he used to study differential equations. In fact,

15 one can find traces of contact geometry in the 17th century and Huygens work on geometric optics. In contrast to the fact that symplectic geometry only lives in even dimensions, contact geometry only lives in odd dimensions. In the literature it is referred to as symplectic geometries younger sibling [Gei08], and as we will see in Section 2.3 and in the main part of the thesis, contact and symplectic geometries often coexist in adjacent dimensions of subspaces of a single space. In this section we introduce some essential notions from contact geometry which is used throughout the thesis. The reader is referred to [Gei08] for more details about contact geometry.

2.2.1 Basics on contact manifolds Let M be a (2n + 1)-dimensional smooth manifold. Let ξ be a codimension 1 hyperplane distribution on M. That is ξ is a co-rank 1 subbundle of TM. A contact manifold is a tuple (M, ξ), with the additional requirement that ξ is completely non-integrable. By definition this means that locally around x ∈ M we have ξx = ker αx, where α is a one-form locally defined in a neighborhood of x ∈ M and satisfies

α ∧ (dα)∧n ≠ 0 .

If ξ satisfies this, we will call it a contact structure. If the one-form α above is globally defined, we say that ξ is co-oriented. In case ξ is co-oriented, we call α a contact form and we may specify a contact manifold by specifying a tuple (M, α), whereby it is understood that the contact structure is given by the kernel of α.

Example 2.6. 2.2 Relation with symplectic structures Let n ∈ N and consider 2n+1 R equipped with coordinates (x1, . . . , xn, y1, . . . , yn, z). The globally defined one-form ∑n αstd = dz − yidxi , i=1 2n+1 gives the standard contact structure ξstd := ker αstd on R . Indeed, we have ( ) ∑n ∧n ∧n (dαstd) = dxi ∧ dyi = dx1 ∧ · · · ∧ dxn ∧ dy1 ∧ · · · ∧ dyn , i=1

∧n 2n+1 and therefore αstd ∧ (dαstd) is the standard volume form on R .

Much like symplectic geometry, contact geometry also lack local invariants by the following version of the Darboux theorem.

16 Theorem 2.7 (Darboux). Let (M, ξ) be a (2n + 1)-dimensional contact manifold, and let p ∈ M. Then there is a neighborhood U of p and local coordinates (x1, . . . , xn, y1, . . . , yn, z) in U, such that ∑n | − α U = dz yidxi i=1

From the definition of a contact manifold, we can see traces of symplectic geometry emerging. If (M, α) is a contact manifold. Then for each point x ∈ M, we have that dα| is an alternating non-degenerate bilinear form on ξx the hyperplane ξ = R2n, so that (ξ , dα| ) is a symplectic vector space. It is x x ξx from this observation that contact manifolds are necessarily of odd dimension. Let (M, α) and (M ′, α′) be two contact manifolds with contact structures ξ = ker α and ξ′ = ker α′ respectively. A smooth function f :(M, α) −→ (M ′, α′) is called a contactomorphism if it is a diffeomorphism (meaning that f is bijective and has a smooth inverse), and additionally df(ξ) = ξ′ . An equivalent condition is that there is some smooth function g : M −→ R \{0} such that f ∗α′ = gα.

2.2.2 Reeb dynamics One of the cornerstone concepts in contact geometry is that of the Reeb vector field. Let (M, α) be a contact manifold. The Reeb vector field R is the unique vector field defined by { dα(R, −) = 0 . α(R) = 1 The Reeb vector field is always transverse to the contact structure ξ = ker α.

2n+1 Example 2.8. Consider (R , αstd). By writing ∑n i i R = Rz∂z + Rx∂xi + Ry∂yi , i=1 we find the Reeb vector field by solving the defining system of equations for the Reeb vector field. { ∑ − n i − i dαstd(R, ) = ∑i=1 Rxdyi Rydxi = 0 − n i αstd(R) = Rz i=1 yiRx = 1 .

17 i i ∈ { } We find Rx = Ry = 0 for each i 1, . . . , n and Rz = 1, so the Reeb vector field is R = ∂z.

One of the most important open problems in contact geometry is the Wein- stein conjecture [Wei79].

Conjecture 2.9 (Weinstein). Let (M, α) be a compact contact manifold. Then the Reeb vector field R has at least one periodic orbit.

The Weinstein conjecture is known to be true for hypersurfaces of contact type in R2n by Viterbo [Vit87], and in general when dim M = 3 by Taubes [Tau07]. For a more detailed account on the status of the Weinstein conjecture we refer the reader to [Pas12].

2.2.3 Submanifolds of contact manifolds Let (M, α) be a (2n + 1)-dimensional contact manifold and let Λ ⊂ M be a submanifold. We say ∩ | • Λ is a contact submanifold if T λ ξ Λ is a contact structure on Λ • Λ is isotropic if TΛ ⊂ ξ • Λ is Legendrian if TΛ ⊂ ξ and dim Λ = n. If Λ ⊂ M is an isotropic submanifold, then we have dim Λ ≤ n. This follows ⊂ | | from the fact that if TΛ ξ, then α Λ = 0 and hence dα Λ = 0. This means that for each x ∈ Λ, we have TxΛ ⊂ ξx is an isotropic subspace of the 2n- dimensional symplectic vector space (ξ , dα| ). It follows that dim Λ ≤ n, x ξx see Section 2.1.3.

2.3 Liouville and Weinstein manifolds For the main part of this thesis, we study a special class of symplectic manifold called Weinstein manifolds which we will introduce in this section. They are certain non-compact symplectic manifolds with particularly nice structure at infinity. In the class we consider, there is always a compact set, outside of which the symplectic manifold is equal to the symplectization of a contact manifold. For a more comprehensive and detailed account on Liouville and Weinstein manifolds we recommend [CE12].

2.3.1 A connection between symplectic and contact geometry Definition 2.10. A symplectic manifold (M, ω) is called exact if there exists a one-form λ ∈ Ω1(M) such that dλ = ω.

18 Given a co-oriented contact manifold (M, α), there is an associated sym- t plectic manifold (Rt×M, d(e α)) called the symplectization of (M, α). Some- times we can also construct the contactization of a symplectic manifold. Namely, if (M, ω) is an exact symplectic manifold and ω = dλ, then (M × Rz, dz + λ) is a contact manifold called the contactization of the symplectic manifold (M, dλ). We now introduce a class of symplectic manifolds called Liouville manifolds. Let (M ′, dλ) be a compact exact symplectic manifold with boundary ∂M ′. Define a vector field Z by the equation

dλ(Z, −) = λ .

It is called the Liouville vector field. If Z points outwards along the boundary ′ ′ | ′ ∂M , then (∂M , λ ∂M ′ ) is a contact manifold, and the tuple (M , λ) is called a Liouville domain. The completion of the Liouville domain (M ′, λ) is obtained by attaching the half-symplectization ([0, ∞) × ∂M ′, d(etλ)) to M ′ along its boundary ∂M ′. More precisely the completion of (M ′, dλ) is defined by { ′ ′ ′ dλ, in M M = M ⊔∂M ′ (R × ∂M ), ω = . d(etλ), in R × ∂M ′

Definition 2.11. A Liouville manifold (M, dλ) is a non-compact exact symplectic manifold such that it is the completion of a Liouville domain.

Remark 2.12. Our definition of a Liouville manifold is called a Liouville manifold of finite type in the literature, and we will not consider any Liouville (or Weinstein) manifold that is not of finite type.

2.3.2 Weinstein manifolds In short, a Weinstein manifold is a Liouville manifold such that the Liouville vector field is the gradient of a Morse function. Let M is a smooth n-dimensional manifold. A Morse function is a smooth real-valued function f : M −→ R such that every critical point is non-degenerate, meaning that the Hessian is non-singular at each critical point. Each critical point x has an index which is equal to the number of negative eigenvalues of the Hessian at x. Given a Morse function, there is a handle decomposition of the manifold M. A handle of index i is the smooth manifold Di ×Dn−i, where Di is an i-dimensional disk. The manifold M can be built by successively attaching handles Hx of index i for each critical point x of index i. For more details and an excellent introduction to Morse theory, the reader is referred to [Mil63].

19 Let (M, λ) be a Liouville manifold with Liouville vector field Z. Let f : M −→ R be a smooth function. We say that Z is gradient-like for f if ( ) Z(f) ≥ δ |Z|2 + |df|2 , for some δ > 0 where |Z| is the norm with respect to some Riemannian metric on M, and |df| is the dual norm.

Definition 2.13. A Weinstein manifold is a tuple (M, λ, f) of a Liouville manifold (M, λ) and a Morse function f : M −→ R which is proper and bounded from below such that the Liouville vector field Z is gradient-like for f.

The upshot of having a Morse function defined on M is that as a smooth manifold M has a handle decomposition. The situation is in fact much better; there is a Weinstein handle decomposition, which gives that the Weinstein manifold (M, λ, f) has a decomposition into a finite number of Weinstein manifolds called Weinstein handles. We wrap this section up by giving the most important examples.

2n Example 2.14. 1. Consider (R , ωstd). First, it is exact symplectic because ωstd = dλ where 1 ∑n λ = x dy − y dx . 2 i i i i i=1 For this choice of∑ primitive one-form we compute the Liouville vector n field. Let Z = i=1 Zxi ∂xi + Zyi ∂yi . Then we determine Z by considering the equation ∑n 1 ∑n ω (Z, −) = Z dy − Z dx = x dy − y dx . std xi i yi i 2 i i i i i=1 i=1 ∑ Z = 1 n x ∂ + y ∂ r2 = ∑We read off that 2 i=1 i xi i yi .We note that if n 2 2 R2n 1 i=1 xi + yi is a radial coordinate in , then Z = 2 r∂r which is a radial vector field. Therefore (R2n, λ) is a Liouville manifold that is the completion of the Liouville domain (B2n, λ). Considering the Morse function f : R2n −→ R defined by 1 ∑n f(x , . . . , x , y , . . . , y ) = x2 + y2 . 1 n 1 n 4 i i i=1 This function is proper and bounded from below by the value 0. Using the standard Euclidean metric on R2n, we note that 1 ∑n ∇f = x ∂ + y ∂ = Z, 2 i xi i yi i=1

20 so in particular Z is gradient-like for f. Therefore (R2n, λ, f) is a Weinstein manifold. 2. Let M be a compact n-dimensional∑ manifold. The cotangent bundle ∗ n (T M, dλ) where λ = i=1 pidqi is the tautological one-form expressed in terms of the local coordinates (q1, . . . , qn, p1, . . . , pn) is a Liouville manifold. The Liouville vector field is given by ∑n

Z = pi∂pi , i=1 and Z is gradient-like for the function f : T ∗M −→ R defined by

1 ∑n f(q , . . . , q , p , . . . , p ) = p2 . 1 n 1 n 2 i i=1 This function is proper and bounded from below by the value 0. How- ever, f is not a Morse function, but a Morse–Bott function. One can perturb f a little around the zero-section to get a Morse function and therefore obtain a Weinstein manifold (T ∗M, λ, f). Another technical remark is that one can relax the definition of Weinstein manifolds to allow for Morse–Bott functions or generalized Morse functions.

21 3. Gromov’s J-holomorphic curves, Floer theory and symplectic field theory

In Gromov’s seminal paper from 1985 [Gro85] he introduces a new effective approach to fundamental problems in symplectic geometry. Gromov’s idea was to study moduli spaces of J-holomorphic maps from a Riemann surface to a symplectic manifold (M, ω), where J is an almost complex structure on M compatible with the symplectic structure. He proved a striking compactness result for such J-holomorphic curves, and among his most celebrated results is Gromov’s non-squeezing theorem which says that a 2n-dimensional ball of 2n radius r, denoted by Br can be symplectically embedded in the symplectic 2 × R2n−2 ≤ cylinder BR only if r R. The modern study of rigidity phenomena in symplectic geometry, such as the non-squeezing theorem, is dominated by the study of moduli spaces of J-holomorphic maps in various ways. In the late 80’s Andreas Floer used Gromov’s J-holomorphic curves to prove the Arnold conjecture for a wide class of compact symplectic manifolds [Flo89]. Floer’s idea was to study the Morse theory of the symplectic action functional, which is defined on an infinite dimensional space. Using the compactness results for J-holomorphic maps originating from the work of Gromov, and Floer’s line of thought, one can construct powerful invariants of symplectic manifolds and Lagrangian submanifolds therein. There are many versions of these Floer-type invariants, and the collective study of these is called Floer theory. One can also use J-holomorphic curves to study contact geometry. In 2000, Eliashberg–Givental–Hofer introduced symplectic field theory [EGH00]. Their idea is to use J-holomorphic curves in the symplectization of a contact manifold to study the underlying contact manifold, and Legendrian submanifolds therein. Floer theory and symplectic field theory constitute two of the major ingredi- ents in the main part of this thesis and they are the main tools we use to study symplectic and contact geometry.

3.1 J-holomorphic curves Let (M, ω) be a symplectic manifold. An almost complex structure on M is a −→ 2 − bundle endomorphism J : TM TM satisfying Jx = idTxM for every x ∈ M. We say that the almost complex structure J is ω-compatible if it is ω-tame and ω is J-invariant

22 •(ω-tame) ωx(v, Jxw) > 0 for every x ∈ M and v, w ∈ TxM. •(J-invariant) ωx(Jxv, Jxw) = ωx(v, w) for every x ∈ M and v, w ∈ TxM. That J is ω-compatible is equivalent to gJ (−, −) := ω(−,J−) being a Rie- mannian metric. Let (Σ, j) be a Riemann surface, meaning a complex manifold of complex dimension one. A smooth map

u: Σ −→ M, is called J-holomorphic if the differential of u is a complex linear map with respect to j and J: J ◦ du = j ◦ du .

This equation can also be expressed as ∂J (u) = 0, where 1 ∂ (u) := (du + J ◦ du ◦ j) , J 2 is the complex antilinear part of du with respect to J. It is useful to express this equation in local coordinates. For each x ∈ Σ, there is a complex coordinate chart with coordinates z = s + it, where i is the standard complex structure on C. Expressed in these coordinates, the equation ∂J (u) = 0 is (( ) ( )) 1 ∂u ∂u ∂u ∂u ∂J (u) = ds + dt + J(u) − dt + ds 2 ( ∂s ∂t ) ( ∂s ∂t) 1 ∂u ∂u 1 ∂u ∂u = + J(u) ds + − J(u) dt = 0 2 ∂s ∂t 2 ∂t ∂s Hence u is J-holomorphic if and only if it satisfies the non-linear first order partial differential equation ∂u ∂u + J(u) = 0 . (3.1) ∂s ∂t

Example 3.1. Consider the symplectic manifold (R2, dx ∧ dy) equipped with the standard complex structure i defined point wise by { i (∂ ) = ∂ (x,y) x y . i(x,y)(∂y) = −∂x

2 More concisely, we can write that for each (x, y) ∈ R , i(x,y) is given by the matrix ( ) 0 1 i = . (x,y) −1 0 Let u: C −→ R2 be a smooth map, and let z = s + it ∈ C be coordinates in the complex plane. In this case we also equip C with the standard complex

23 structure i, which. Writing u(s, t) = (f(s, t), g(s, t)), we have that u is i- holomorphic if and only if (3.1) holds, which in this case gives us ( ) ( ) { ∂f ∂g ∂f ∂g ∂f = ∂g , + i , = 0 ⇔ ∂s ∂t ∂f − ∂g ∂s ∂s ∂t ∂t ∂t = ∂s We recognize this system of partial differential equations as the Cauchy–Riemann equations.

3.2 Floer theory and the Arnold–Givental conjecture In this section we start by stating conjecture by Arnold–Givental, which is similar in nature to the Arnold conjecture (see Conjecture 2.5), but deals with Lagrangian submanifolds. In fact, it is true that the Arnold–Givental conjecture implies the Arnold conjecture.

Conjecture 3.2 (Arnold–Givental). Let (M, ω) be a compact 2n-dimensional symplectic manifold and let L ⊂ M be a compact Lagrangian submanifold. Let Ht : M −→ R be a smooth time-dependent Hamiltonian and denote its 1 1 time-one flow by φH . Assume that L and φH (L) intersect transversely. Then ∑n ∩ 1 ≥ Z #(L φH (L)) dim Hk(L; 2) . k=0

From now on let (M, ω) be a compact 2n-dimensional Lagrangian submanifold and let L ⊂ M be a compact Lagrangian submanifold. Let Ht : M −→ R 1 be a smooth time-dependent Hamiltonian. Assume that L and φH (L) intersect transversely. In this situation, we can associate to L a cochain complex which we denote by CF ∗(L), and whose cohomology is known as Lagrangian Floer cohomol- ∗ ogy. The cochain complex CF (L) is freely generated over Z2 by the set ∩ 1 L φH (L). ⟨ ⟩ ∗ Z ∩ 1 CF (L) = 2 L φH (L) . 1 By compactness of L and the assumption that L and φH (L) intersect trans- ∩ 1 ∗ versely we have that the set L φH (L) is finite and thus CF (L) is a finite dimensional space. We equip this vector space with a Z-grading given by the Maslov index (see [RS93, Aur14]). The differential is defined by counting certain J-holomorphic maps, where J is an almost complex structure on M. Consider the subset S := R × [0, 1] ⊂ C equipped with the standard com- ∈ ∩ 1 plex structure i. Let x, y L φH (L) be two intersection points, and define M(x, y) to be the space of maps u: S −→ M,

24 such that

25 3.2.1 Wrapped Floer cohomology In this thesis we are concerned with a generalization of the cochain complex (CF ∗(L), ∂), when L is allowed to be a certain non-compact Lagrangian, and when the ambient symplectic manifold (M, ω) is a Liouville manifold, see Sec- tion 2.3 for the definition of a Liouville manifold. In this section we briefly give an introduction to wrapped Floer cohomology, and highlight some of its differences compared to the Lagrangian Floer cohomology discussed in Sec- tion 3.2. For a more detailed overview, see e.g. [Aur14]. Let (M, ω) be a Liouville manifold, and let L be an exact Lagrangian submanifold such that Λ := L ∩ (∂M × {0}) is a Legendrian submanifold, and L agrees with Λ × [0, ∞) in the cylindrical end of M. We pick a smooth time- 2 dependent Hamiltonian Ht : M −→ R, which has the form Ht(x, r) = r in × ∞ ∩ 1 the cylindrical end ∂M [0, )r. Then, we consider the set L φH (L) as in Section 3.2. The idea is that the time-1 flow of the Hamiltonian vector field associated to Ht will “wrap” L around itself, see Figure 3.2.

1 φH (L) ∂M

Figure 3.2. The figure shows how the time-1 Hamiltonian push-off of L (in red) “wraps” around L in the cylindrical end of the Liouville manifold M.

We can make sense of the “wrapping” as follows. The hypersurface ∂M × { } | 0 is a contact manifold with contact form α := λ ∂M×{0}, where λ is a primitive one-form of the symplectic form ω on M. By computing the Hamiltonian 2 vector field associated to the wrapping Hamiltonian of the form Ht(x, r) = r in the cylindrical end ∂M ×[0, ∞), we see that it is of the form g(t)Rλ, where g : (0, ∞) −→ R \{0} is a smooth function, and Rα is the Reeb vector field associated to the contact form α. We define a cochain complex CW ∗(L) as the graded vector space ⟨ ⟩ ∗ Z ∩ 1 CW (L) = 2 L φH (L) , equipped with a Z-grading determined by the Maslov index. By considering moduli spaces M(x, y), we can define a differential ∂ : CW ∗(L) −→ CW ∗(L) .

26 27 tact structure of the contact manifold (M, α) by ξ = ker α, and the Reeb vector field associated to α by R. In addition to requiring that J is compatible with the symplectic form d(etα), we require it to satisfy 1. J is invariant under the natural R-translation 2. J(∂t) = R 3. J(ξ) = ξ Let (Σ, j) be a compact Riemann surface of genus g with k positive punctures, and ℓ negative punctures

+ + − − ζ1 , . . . , ζk , ζ1 , . . . , ζℓ .

Assume that for each puncture we have chosen a positive or negative cylinder- like end, depending on the sign of the puncture. That is, we have chosen bi- holomorphisms

+ ∞ × 1 −→ + ∈ { } ηi : (0, ) S N(ζi ), i 1, . . . , k − −∞ × 1 −→ − ∈ { } ηj :( , 0) S N(ζj ), j 1, . . . , ℓ , where N(ζ) is a neighborhood of the puncture ζ. Consider two sets of Reeb orbits

+ Γ := {α1, . . . , αk} − Γ := {β1, . . . , βℓ} ,

+ − and let Mg(Γ ,Γ ) denote the moduli space of J-holomorphic maps

u:(Σ, j) −→ (R × M,J) , satisfying • ∂J u = 0 + ∞ ∈ 1 ∈ { } • lims→∞ u(ηi (s, t)) = ( , αi(t)) for every t S and i 1, . . . , k . − −∞ ∈ 1 ∈ { } • lims→−∞ u(ηj (s, t)) = ( , βj(t)) for every t S and j 1, . . . , ℓ . Studying this type of moduli space with several positive, several negative punctures and all genera, is referred to as full SFT in the literature. Under additional assumptions on the symplectic manifold (R × M, d(etα)) one can ask + − whether the moduli space Mg(Γ ,Γ ) is a smooth manifold (this is called transversality). Understanding the transversality for such moduli spaces is in progress. Among the modern techniques is the polyfold theory of Hofer– Wysocki–Zehnder (see [FH18] and references therein for an overview) and Kuranishi structures (see [FOOO18] and references therein). John Pardon was able to use techniques similar to that of Kuranishi structures in [Par16, Par19] to prove transversality in the case g = 1 and |Γ +| = 1. There exist compactness results in full generality for the full SFT due to Bourgeois–Eliashberg–Hofer–Wysocki–Zehnder [BEH+03].

28 3.3.1 The Chekanov–Eliashberg dg-algebra In Section 3.3 we introduced symplectic field theory, which by studying J- holomorphic curves gives rise to full SFT, which is a powerful invariant of a contact manifold. In this section we will describe a relative version of SFT which produces an invariant of a Legendrian submanifold. The ideas of such an invariant for Legendrian submanifolds predates the formulation of symplectic field theory, and was independently computed for Legendrian submanifolds in R3 by Chekanov [Che02] and Eliashberg [Eli98]. In this thesis we refer to this invariant as the Chekanov–Eliashberg dg-algebra. In the literature it is sometimes called Legendrian contact homology. For a detailed overview about the Chekanov–Eliashberg dg-algebra, see [EN18] and references therein. Our presentation is built upon and closely follows [BEE12, EL17, Ekh19]. Let (M, α) be a Weinstein fillable contact manifold and let Λ ⊂ M be a Legendrian submanifold. Denote the Reeb vector field associated to α by R. A Reeb chord of Λ is a segment of a trajectory of R which has its start and end points in Λ. Let J be a cylindrical almost complex structure in the symplectization (R × M, d(etα)) as in Section 3.3.

The graded algebra Let K be any field. Assume Λ consists of n connected components denoted { }n by Λ1,...,Λn. Then let ei i=1 be a set of mutually orthogonal idempotents over K. That is 2 ∈ { } 1. ei = ei for every i 1, . . . , n 2. eiej = 0 for every i ≠ j ⊕ n K R Consider the semi-simple ring k = i=1 ei. Let denote the set of all Reeb chords of Λ. Let Rij denote the set of Reeb chords starting on Λi and ending on Λj. We equip R with the following k-k-bimodule structure: { { c, if c ∈ Rji for some j c, if c ∈ Rij for some j ei · c = c · ei = 0, otherwise 0, otherwise

The Chekanov–Eliashberg dg-algebra is a semi-free dg-algebra freely generated over k by R with respect to the k-k-bimodule structure above CE∗(Λ) = k ⟨R⟩ .

If c is a Reeb chord of Λ, we define the grading of c as |c| := 1 − CZ(c) , where CZ denotes the Conley–Zehnder index, see [EES05, Section 2.2.]. It is extended to every word of Reeb chords by |ab| = |a| + |b|. Note that we have to make some vanishing assumptions about the first Chern class of M and the Maslov class of Λ in order to get a well-defined Z-grading on CE∗(Λ), see [EES07, Section 2.2.]. Our grading convention follows that of [EL17] and

29 differs by a minus sign from that of [EES05, EES07]. We use this grading because it is cohomological in the sense that the differential defined below will then increase the grading by 1.

The differential To describe the differential in the simplest possible case, assume that there are no contractible Reeb orbits in the contact manifold (M, α), see Remark 3.7. Let Dm+1 ⊂ C be the unit disk in C equipped with the standard complex + − − structure and standard orientation with m+1 boundary punctures ζ0 , ζ1 , . . . , ζm removed. For each boundary puncture fix a strip-like end

+ ∞ × −→ + ε0 : (0, ) [0, 1] N(ζ0 ) − −∞ × −→ − ∈ { } εi :( , 0) [0, 1] N(ζi ), i 1, . . . , m ∗ Let a, b1, . . . , bm be generators of CE (Λ). Then denote by M(a; b1, . . . , bm) the moduli space of J-holomorphic maps

u:(Dm+1, ∂Dm+1) −→ (R × M, R × Λ) , satisfying 1. ∂J u = 0 + ∞ ∈ 2. lims→∞ u(ε0 (s, t)) = ( , a(t)) for every t [0, 1] − −∞ ∈ ∈ 3. lims→−∞ u(εi (s, t)) = ( , bi(t)) for every t [0, 1] and i {1, . . . , m}. There is a natural R-action on the moduli space M(a; b1, . . . , bm) by translation in the R-factor in the symplectization R × M. We then take the quotient of M(a; b1, . . . , bm) with this R-action, while still denoting the moduli space with the same symbol by abuse of notation. After a generic choice of almost complex structure J, the moduli space M(a; b1, . . . , bm) is a smooth manifold of dimension ∑m dim M(a; b1, . . . , bm) = −|a| − 1 + |bi| , i=1 see [EES05, EES07, CEL10, EL17]. The differential is a function ∂ : CE∗(Λ) −→ CE∗(Λ) defined on genera- ∗ tors as follows. Let c ∈ CE (Λ) be a generator, and let b = b1 ··· bm be a word of Reeb chords. Then ∑ ∂a = (#M(a; b)) b , |b|=|a|+1 where #M(a; b) is the number of elements in the moduli space M(a; b) of dimension equal to 1.

30 Proposition 3.6. The function ∂ satisfies ∂2 = 0, and the stable tame isomorphism class of the dg-algebra (CE∗(Λ), ∂) is invariant under Legendrian isotopies of Λ.

Remark 3.7. In the general case when we do not exclude the existence of contractible Reeb orbits of the Reeb vector field R, one needs to consider anchored curves. The description and precise definition of anchored curves is covered in parts of Paper I and in the papers [BEE12, EL17, Ekh19]. We emphasize that dealing with the transversality problem for anchored curves requires abstract perturbations, and an explicit perturbation scheme which solves the transversality problem via a direct approach is presented in [Ekh19].

31 4. Summary of papers

4.1 Paper I Let Q be a spin orientable and closed (compact without boundary) manifold. Consider its cotangent bundle (T ∗Q, λ) equipped with the Liouville one-form ∗ ∗ λ = pdq in local coordinates (q, p) on T Q. Let F := Tξ Q be a cotangent fiber at ξ ∈ Q. Associated to F is the wrapped Floer cohomology, which we denote by HW ∗(F ) [AS10b, FSS09]. It was first proved by Abbondandolo–Schwarz [AS06] that we have an isomorphism ∗ ∼ HW (F ) = H−∗(ΩξQ) , (4.1) where ΩξQ denotes the based loop space of Q. Furthermore, it was shown in [AS10a, Abo12] that there is an isomorphism as in (4.1) which intertwines the triangle product on HW ∗ and the Pontryagin product on the homology of the based loop space. There is a refinement of wrapped Floer cohomology which is known as partially wrapped Floer cohomology [Aur10, Syl19]. It does not only involve a Liouville manifold, but it also involves a new class of symplectic objects called stops [Syl19, GPS20, EL17]. Let K ⊂ Q be any submanifold and pick a generic metric on T ∗Q. Consider the unit conormal bundle of K ∗ ∗ ΛK = {(q, p) ∈ T S | q ∈ K, ⟨p, TqK⟩ = 0, |p| = 1} ⊂ ST Q. where ST ∗Q denotes the unit cotangent bundle. In Paper I we consider the partially wrapped Floer cochains of the fiber F , stopped by ΛK and denote it by CW ∗ (F ). Let M := Q \ K. The main result of Paper I is the following ΛK K

Theorem 4.1 (Paper I). There exists a geometrically defined isomorphism of A∞-algebras Ψ : CW ∗ (F ) −→ Ccell(BM ) . ΛK −∗ K ∗ Moreover, Ψ induces an isomorphism HW (F ) −→ H−∗(Ω M ) of Z[π (M )]- ΛK ξ K 1 K modules.

Here BMK denotes the space of based piecewise geodesic loops on MK . The definition of Ψ closely resembles Abouzaid’s A∞-quasi-isomorphism in [Abo12] defined by counting holomorphic curves. Furthermore we use techniques of Cieliebak–Latschev [CL09] to obtain proper length and action filtra- tions. This theorem generalizes (4.1) in two directions: Firstly Ψ is an isomorphism already on the chain level, and secondly it holds true for K = ∅.

32 For applications of Theorem 4.1, we consider the special case of Q = Sn for n = 5 or n ≥ 7. We then show that there is a relation between the Alexander invariant of K and HW ∗ (F ) for certain knots K ⊂ Sn. This is used to prove ΛK the following

Theorem 4.2 (Paper I). Let n = 5 or n ≥ 7. Let x ∈ MK be a point. Then n ∼ there exists a codimension 2 knot K ⊂ S with π1(MK ) = Z, such that ΛK ∪ Λx is not Legendrian isotopic to Λunknot ∪ Λx.

For knots K ⊂ R3, such a relationship is not new. In fact, the Legendrian ∗ 3 isotopy class of ΛK ⊂ ST R is a complete knot invariant [She19, ENS18]. Furthermore, there is a relation between the knot contact homology [EENS13] of ΛK and the Alexander polynomial of K [Ng08, ENS18]. Our result The- orem 4.2 shows that the link ΛK ∪ ∂F is able to detect the underlying knot K in some particular cases, which extends the scope of the results in [She19, ENS18] to higher dimensions.

4.2 Paper II In this paper, which is jointly written with Tobias Ekholm we generalize the definition of the Chekanov–Eliashberg dg-algebra to Legendrian embeddings V 2n−2 ,→ ∂X, thus answering a question by Eliashberg [Eli18]. A Legendrian embedding is an embedding V −→ ∂X which extends to a contact embedding (V ×(−ε, ε)z, λ+dz) −→ ∂X. The image of the skeleton of V is then viewed as a singular Legendrian in ∂X. Choose a handle decomposition of V and denote the union of all core disks of all the critical handles by l. The Chekanov–Eliashberg dg-algebra CE∗((V, h); X) has generators which is in one-to-one correspondence with • Reeb chords of l ⊂ ∂X • Reeb chords of ∂l ⊂ ∂V0 , where V0 is the subcritical part of V . In particular, the generators of the second ∗ ∗ kind generate a dg-subalgebra CE (∂l; V0) ⊂ CE ((V, h); X). In a way, this dg-subalgebra captures the singular behavior of Λ, and can be thought of a Reeb dynamical approach to a derived local system on Λ. The Legendrian embedding V,→ ∂X can be regarded as a stop, and we can construct the Weinstein manifold XV , by attaching a stop at V × (−ε, ε). We × ∗ − view a stop as the Weinstein cobordism V DTε [ 1, 1] with Liouville vector field Z + 2x∂x − y∂y, where Z is a Liouville vector field on V . We attach this cobordism along its negative end (V ×(−ε, ε))×{−1, 1} to ∂X ⊔R×(V ×R) to obtain XV . In the case n = 2 and when Λ is a Legendrian graph, our Chekanov– Eliashberg dg-algebra is quasi-isomorphic to the Chekanov–Eliashberg dg-

33 algebra defined by An–Bae [AB20]. The main result of the paper is a surgery formula for CE∗(V ) when V is regarded as a stop.

Theorem 4.3 (Paper II). There is a quasi-isomorphism of A∞-algebras

∗ ∗ Φ: CW (C; XV ) −→ CE ((V, h); X) .

Here C is the union of cocore disks of the top handles of XV . In the next result we interpret smooth Legendrians as a singular Legendrian.

Theorem 4.4 (Paper II). Let V be a Weinstein thickening of a smooth Legen- drian submanifold Λ ⊂ ∂X with a single top handle. Then there is a quasi- isomorphism of dg-algebras

∗ ∼ ∗ CE ((V, h); X) = CE (Λ, C−∗(ΩΛ)) .

∗ The right hand side CE (Λ, C−∗(ΩΛ)) is the Chekanov–Eliashberg dg- algebra of Λ with loop space coefficients. This result together with Theo- rem 4.3 gives a proof of [EL17, Conjecture 3]. One key ingredient in this ∗ ∼ proof is to show that there is a quasi-isomorphism CE (∂l; V0) = C−∗(ΩΛ). This quasi-isomorphism is defined by counting curves (with loop space coefficients) mapping to V0 with boundary on the Lagrangian filling of ∂l ⊂ ∂V0, which is given by Λ∩V0. By counting the same curves, but without loop space ∗ coefficients, we obtain an augmentation ε: CE (∂l; V0) −→ C. Linearizing the Chekanov–Eliashberg dg-algebra with respect to this augmentation we re- cover the regular Chekanov–Eliashberg dg-algebra.

Corollary 4.5 (Paper II). Let V be a Weinstein thickening of a smooth Legen- drian submanifold Λ ⊂ ∂X with a single top handle. Then there is a quasi- isomorphism of dg-algebras

∗ ∼ ∗ CE ((V, h); X; ε) = CE (Λ) ,

where ε is the augmentation induced by the exact Lagrangian filling Λ ∩ V0 of ∂l ⊂ ∂V0.

Furthermore we prove a Seifert–van Kampen-type result. If V,→ ∂X1 and V,→ ∂X2 are Legendrian embeddings, we can join X1 and X2 together by ∗ attaching a handle at V modeled on V × DεT [−1, 1]. The result is denoted by X1#V X2. Our result gives direct proof of the cosheaf property of partially wrapped Floer cohomology, and also stop removal formulas [GPS18].

34 Theorem 4.6 (Paper II). There is a natural pushout diagram

∗ ∗ CE (∂l; V0) CE (V ; X1) . ∗ ∗ CE (V ; X2) CW (C; X1#V X2)

From this pushout diagram in Theorem 4.6 we obtain a few corollaries. Let ∗ X2 = V × D1T [−1, 1] with the Liouville vector field Z + x∂x + y∂y, where Z is the Liouville vector field on V . After rounding corners we have that V × {(−1, 0)} ⊂ ∂X2 is loose, and in particular we obtain the stop removal formula [GPS18].

∗ Corollary 4.7 (Paper II). With X2 chosen as above, we have that CW (C; X1#V X2) has trivial cohomology. Furthermore, if C′ is a cocore disk of a critical handle of X1, then we have a quasi-isomorphism ∗ ′ ∼ ∗ ′ CW (C ∪ C; X1#V X2) = CW (C ; X1) .

35 5. Summary in Swedish

Symplektisk- och kontaktgeometri har båda rötter i fysiken. Symplektisk ge- ometri har sin i år 1833 då Sir William Rowan Hamilton formulerade sina välkända ekvationer om tidsutvecklingen hos fysikaliska system som finns i klassisk mekanik såsom himlakroppars rörelse i solsystemet. Hamiltons ekvationer har också en viktig koppling med utvecklingen av kvantmekaniken. Det är först när man försöker skriva ned Hamiltons ekvationer på ett koordinatfritt sett som den symplektiska geometrin dyker upp. Kontaktgeometri har också kopplingar till fysiken, och närmare bestämt optik och hur ljusvågor breder ut sig i rummet. Modern symplektisk- och kontakgeometri som vi studerar i denna avhandling handlar till stor del om att förstå en förmodan av Maxim Kontsevich som kallas för homologisk spegelsymmetri. Denna förmodan syf- tar till att matematiskt förklara ett fenomen som kallas spegelsymmetri, som observerats inom modern teoretisk fysik och strängteori. De bakomliggande matematiska objekten som studeras är komplicerade strukturer som i någon mening beskriver hur strängar i strängteori kan breda ut sig och interagera med sig själva, andra strängar och det geometriska rummet. I denna avhandling studerar vi en viss version av den såkallade Fukayakat- egorin i vissa fall. I de fall vi studerar beskriver vi dess generatorer i termer av kontaktgeometri och strängtopologi. Dessa beskrivningar av generatorer i Fukayakategorin kan ge tillämpningar utanför symplektisk- och kontaktgeometri, exempelvis i knutteori. En matematisk knut är en cirkel i rummet som är “knuten”, och ett fundamentalt problem inom knutteorin är att avgöra när två knutar är samma knut. I de två sektioner som följer ger vi en mer teknisk sammanfattning av de två artiklarna som ingår i denna avhandling.

5.1 Artikel I I Artikel I studerar vi kotangentknippet T ∗M till en kompakt mångfald M utan rand som antas vara spin och orienterad. Vi låter K ⊂ M vara en delmångfald och betecknar dess komplement med MK . Vi betraktar dess konormalknippe ∗ ∗ ΛK ⊂ ST M. Det är en Legendrisk delmångfald i randen till T M som är en kontaktmångfald, och vi kan betrakta det som ett stopp. Vi bevisar att Floer- algebran av en fiber F över punkten ξ ∈ MK är A∞-isomorf med en Morse- teoretisk modell av kedjor på looprummet av MK . Efter ett generiskt val av

36 en metrik på MK låter vi BMK beteckna rummet av loopar baserade i punkten ξ ∈ MK som är styckvisa geodeter. Detta rummet är homotopiekvivalent med det vanliga looprummet, men är fördelaktigt på det sättet att det det kan beskrivas med hjälp av Morseteori tillämpat på verkanfunktionalen. Vi betecknar Floeralgebran av F , stoppad av Λ , för CW ∗ (F ). Vi låter Ccell(BM ) K ΛK −∗ K beteckna det cellulära kedjekomplexet definierat på BMK . Båda dessa är A∞- algebror, och vår huvudsats i Artikel I är att det finns en geometriskt definierad A∞-isomorfi Ψ : CW ∗ (F ) −→ Ccell(BM ) . ΛK −∗ K ∗ −→ Denna isomorfi inducerar även en isomorfi HWΛ (F ) H−∗(ΩMK ) av K n Z[π1(MK )]-moduler. En tillämpning av detta är att när M = S är en sfär av dimension n = 5 eller n ≥ 7, så finns det en relation mellan Alexanderin- ∗ ⊂ n varianten av K och HWΛ (F ) för vissa knutar K S . Denna relation kan K n ∼ användas för att visa att det finns knutar K ⊂ S som uppfyller π1(MK ) = Z, men att länkarna ΛK ∪ ∂F och Λoknuten ∪ ∂F inte är isotopa via en isotopi av Legendriska delmångfalder.

5.2 Artikel II I Artikel II studerar vi en Weinsteinmångfald X av dimension 2n tillsammans med ett Weinsteindomän V av dimension 2n − 2 som är Legendriskt inbäddad i kontaktranden av X. Detta innebär att det finns en inbäddning V −→ ∂X som täcks av en kontaktinbäddning V × (−ε, ε) −→ ∂X. Här betraktar vi V × (−ε, ε) som en kontaktmångfald med kontaktform dz + λ där z är en linjär koordinat på faktorn (−ε, ε), och λ är en Liouvilleform på V . Vi har fixerat en handtagsindelning h av V . Vi definierar sedan den såkallade Chekanov–Eliashbergalgebran av en sådan Legendrisk inbäddning av (V, h), och vi betecknar den med CE∗((V, h); X). En Legendrisk inbäddning av V medför att skelettet av V är ett stratifierat delrum av ∂X så att de strata av högst dimension är Legendriska delmångfalder. Det är därför befogat att i generella termer tänka på CE∗((V, h); X) som Chekanov–Eliashbergalgebran för den “singulära Legendriska delmångfald” som utgör skelettet till V . I specialfallet då V är en liten kotangentomgivning med endast ett topphandtag av en glatt Legendrisk delmångfald Λ i ∂X, så bevisar vi i Artikel II att det finns en kvasi- isomorfi ∗ ∼ ∗ CE ((V, h); X) = CE (Λ; C−∗(ΩΛ)) , (5.1) där högerledet är Chekanov–Eliashbergalgebran av Λ med looprumskoeffi- cienter. Vår huvudsats handlar om en generalisering om de såkallade kirurgiform- lerna. I den geometriska situationen med X och ett Legendriskt inbäddad We- insteindomän V i dess rand, så kan vi utföra kirurgi och sätta på ett stopp på

37 V × (−ε, ε). Detta ger oss en ny Weinsteinmångfald XV som är Weinstein- mångfalden X stoppad av V . Vårt huvudresultat och kirurgiformeln ger oss en relation mellan unionen av kokärnediskarna C av stoppet och Chekanov– Eliashbergalgebran av V , där stoppet limmades fast. Mer precist har vi en A∞-kvasi-isomorfi

∗ ∗ Φ: CW (C; XV ) −→ CE ((V, h); X) .

Om V är Legendriskt inbäddad i randen till X1 och X2, så kan vi limma ihop X1 och X2 längs V × (−ε, ε) i dess vardera rand. Detta ger oss ett pushoutdi- agram ∗ ∗ CE (∂l; V0) CE (V ; X1) . ∗ ∗ CE (V ; X2) CW (C; X1#V X2) Här betecknar l unionen av alla kärndiskar av alla handtag av V av maximal dimension, och ∂l är dess rand i randen av den subkritiska delen av V , som vi betecknar med V0. Med hjälp av huvudsatsen och kvasi-isomorfin (5.1) bevisar vi en kirurgiformel som är förmodad av Ekholm–Lekili och oberoende av Sylvan [EL17, Syl19].

38 6. Acknowledgments

First and foremost I would like to thank my advisor Tobias Ekholm. Ever since I was his bachelor advisee he has always been generous with his time and had his door open for me. I am very grateful for all the guidance and support that Tobias has provided for me during all these years. I would also like to thank my co-advisor Thomas Kragh for also advising my master thesis, for teaching me about algebraic K-theory and for inspiring me to learn about (co)sheaves once and for all. During my graduate studies I have met many mathematicians with whom I have had conversations which have contributed to my work either directly or indirectly. I would like to thank everyone in the symplectic geometry group at Uppsala for all the seminars and the friendly atmosphere over the years. I am also particularly grateful to Mohammed Abouzaid, Youngjin Bae, Bap- tiste Chantraine, Kai Cieliebak, Côme Dattin, Georgios Dimitroglou Rizell, Luís Diogo, Agnès Gadbled, Paolo Ghiggini, Marco Golla, Pavel Hájek, Yang Huang, Janko Latschev, Noémie Legout and Maksim Maydanskiy. Having been at Uppsala University nearly a decade, there are very few faculty members at the department that I have not had the fortune of being a stu- dent of, a co-teacher with, received administrative help from or had interesting conversations with in one way or another. I thank you all! I would especially like to thank Magnus Jacobsson for encouragement during the early stages of my undergraduate studies and for opening the right doors for me. I am happy to have been surrounded by all my fellow PhD students during all these years. In particular I will miss the reading seminars with my fellow geometry students Axel, Cecilia, Dan, Emilia, Johan, Lukas, Nikos, Sebastian and William. Special thanks go to Anders, Gustav and Sasha. Finally I would like to thank my mother, Emelie and Loke for their constant love, encouragement and support.

The work in this thesis was supported by the Knut and Alice Wallenberg foundation.

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