Displaceability in Symplectic Geometry

A Dissertation Presented to the Faculty of the Graduate School

of Cornell University in Partial Fulﬁllment of the Requirements for the Degree of Doctor of Philosophy

We use a topological condition by Albers that is sufﬁcient for nondisplaceability to describe a large class of nondisplaceable Lagrangians, namely the anti- diagonals of closed monotone symplectic manifolds. We provide some examples of closed monotone symplectic manifolds of Euler characteristic zero.

Furthermore, we study the displaceability of fibers of the bending flow system on equilateral pentagon space. Besides torus fibers over points in the interior of moment map image, this completely integrable system has two La- grangian sphere fibers over boundary points of the moment polytope. They are nondisplaceable for topological reasons. Most of the regular torus fibers are displaceable using McDuff’s probes technique. We prove that the central torus fiber is nondisplaceable by showing that we can lift a Hamiltonian diffeomorphism displacing it to a Hamiltonian diffeomorphism displacing the central torus fiber of the Gelfand-Cetlin system on the complex Grassmannian of

2-planes. This ﬁber is known to be nondisplaceable. BIOGRAPHICAL SKETCH

Frederik De Keersmaeker was born in Belgium in 1989. He started studying at

Ghent University in 2007, where he got an M.S. in Engineering Physics in 2012 and an M.S. in Mathematics in 2013. After working as a teaching assistant for a year at KULeuven KULAK, he enrolled in the Ph.D. program in Mathematics at

Cornell University.

iii This document is dedicated to all Cornell graduate students.

iv ACKNOWLEDGEMENTS

I cannot imagine what my experience in graduate school would have been like without my adviser, Tara Holm. She is the most supportive and caring per- son. Thank you for guiding me through my research over the past six years, for encouraging me to be a better teacher and to do outreach, for suggesting I go to as many interesting workshops and conferences as possible, and for providing a safe space to talk about the ups and downs. The Cornell Math Department is a wonderful place, and I have a lot of peo- ple to thank there. Let me start with my minor committee members: Reyer Sjamaar and Allen Knutson. Thank you for reading this dissertation and for the many useful comments. I have loved teaching at Cornell; Kelly Delp and Ja- son Manning played a big role is making it a positive experience. Crossing the border to get into the U.S. would have been much harder without those letter Melissa Totman wrote for me, saying I was a valued member of the department.

The heart of the department is its graduate students. There are a lot of names to list here, and I hope I have conveyed my appreciation for you all during our time at Cornell. Thank you Ian, My, Thomas, Balazs, Daoji, Emily, David,

Lila, Smaranda, Trevor, Ryan, Ellie, Nicki, Fiona, Portia, Joe, Amin, Sasha, Jeff, Kelsey, Valente, and Voula. My symplectic buddy, Benjamin Hoffman, deserves a special shout-out. So does Hannah Keese for teaching in prison with me for two semesters. I learned a lot during the research visits I went on. I want to thank Nguyen Tien Zung at Universite´ Paul Sabatier in Toulouse and Katrin Wehrheim at UC

Berkeley for talking to me about my research. Ithaca has been a wonderful place to live with my friends. Thank you Lau- ren, Michelle, Naomi, Drew, Gennie, Jaron, Emily, Smaranda, Patrick, Margaux,

v Trevor, Benjamin, Phoebe, Jan, Patsy, and Valeria! Whenever I was back in Belgium, Emilie, Annabelle, Margo, Emma, and Margot made me feel like I had never left! Thank you for all the support!

Finally, I want to thank my parents and my family for cheering me on and believing in me.

vi TABLE OF CONTENTS

Biographical Sketch ...... iii Dedication ...... iv Acknowledgements ...... v Table of Contents ...... vii List of Tables ...... ix List of Figures ...... x

1 Introduction 1

2 Preliminary Material 3 2.1 A Brief Overview of Symplectic Geometry ...... 3 2.1.1 Symplectic Manifolds: Deﬁnition and Examples ...... 3 2.1.2 Local Normal Forms ...... 6 2.1.3 Hamiltonian Vector Fields ...... 7 2.1.4 Hamiltonian Diffeomorphisms ...... 9 2.1.5 Hamiltonian Group Actions and Symplectic Reduction . . 13 2.1.6 Compatible Almost Complex Structures ...... 15 2.2 Nondisplaceable Lagrangians ...... 16 2.2.1 Deﬁnition and Properties ...... 16 2.2.2 Examples ...... 17 2.3 Monotone and Semipositive Symplectic Manifolds ...... 18 2.4 The Maslov Class and Monotone Lagrangians ...... 21

3 Floer Homology and the PSS Isomorphism 29 3.1 Morse Theory ...... 29 3.1.1 Morse Functions ...... 29 3.1.2 The Morse Complex ...... 30 3.1.3 Continuation Maps ...... 33 3.1.4 Functoriality and Induced Maps ...... 36 3.1.5 Poincare´ Duality ...... 37 3.2 Periodic Orbits of Hamiltonian Vector Fields ...... 39 3.2.1 Deﬁnition and Properties ...... 40 3.2.2 The Arnold Conjecture ...... 41 3.3 Floer Homology ...... 42 3.3.1 The Novikov Ring ...... 43 3.3.2 The Loop Space and the Action Functional ...... 45 3.3.3 The Conley-Zehnder Index ...... 47 3.3.4 The Floer Chain Groups ...... 51 3.3.5 The Boundary Operator ...... 52 3.4 Moduli Spaces of Floer Cylinders ...... 59 3.4.1 Fredholm Operators and the Implicit Function Theorem . 59 3.4.2 Smoothness, Linearization, and Transversality ...... 61

vii 3.4.3 Compactness ...... 62 3.5 The Piunikhin-Salamon-Schwarz (PSS) Isomorphism ...... 69 3.6 Representing Submanifolds Using PSS ...... 75 3.7 An Application to Displaceability ...... 82 3.8 Nondisplaceable Anti-Diagonals ...... 84

4 Polygon Spaces 89 4.1 Deﬁnition and Elementary Properties ...... 89 4.2 The Bending Flow System ...... 90 4.3 McDuff’s Probes Technique ...... 94 4.4 Displacing Fibers and Symplectic Reduction ...... 98 4.5 Polygon Spaces and Grassmannians ...... 100 4.6 The Gelfand-Cetlin System ...... 104 4.7 Displacing the Central Fiber ...... 107

Bibliography 110

viii LIST OF TABLES

2.1 Semipositivity of products of two complex projective spaces in low dimension...... 21 2.2 Semipositivity of products of three complex projective spaces in low dimension...... 21

3.1 Different moduli spaces of Floer cylinders ...... 62

ix LIST OF FIGURES

3.1 Floer cylinder with disk cappings connecting two contractible periodic solutions (in blue)...... 55 3.2 Broken configuration with two Floer cylinders, with periodic orbits in blue...... 57 3.3 A plumber’s helper solution...... 70 3.4 The two possible cases for a broken plumber’s helper solution with one break: either the Morse flow half-line breaks (top) or the cylindrical part breaks (bottom)...... 73 3.5 A broken configuration: periodic orbits in red connected by Floer cylinders, and Lagrangian boundary condition for the rightmost cylinder, with sphere bubbles in green and disk bubbles in blue...... 78

4.1 A pentagon with its two diagonals as red dashed line segments. 91 4.2 The image of the bending flow system for r = (2, 3, 3, 2, 3)..... 92 4.3 The image of the bending flow system for r = (1, 1, 1, 1, 1). The polytope is not smooth at the two blue vertices...... 93 4.4 A equilateral pentagon with a vanishing diagonal consists of an equilateral triangle and a line segment...... 93 4.5 Bending flow fibers displaced by vertical (left) and horizontal (right) probes in equilateral pentagon space...... 96 4.6 Bending flow fibers displaced by probes in equilateral pentagon space...... 97

x CHAPTER 1 INTRODUCTION

The main subject of this dissertation is nondispaceable Lagrangians. They are symplectic analogues of n-dimensional submanifolds of an 2n-dimensional smooth manifold with nonzero self-intersection number.

We start Chapter 2 with a brief overview of symplectic geometry. We deﬁne displaceable and nondisplaceable Lagrangians and introduce monotonicity of symplectic manifolds and their Lagrangians.

In Chapter 3 we present a topological condition (Theorem 15) by Peter Al- bers that guarantees nondisplaceability. We describe a large class of nondisplaceable Lagrangians, namely the anti-diagonals of closed monotone symplectic manifolds (Theorem 16). If their Euler characteristic is nonzero, they are not displaceable by maps isotopic to the identity (a class that contains Hamiltonian diffeomorphisms). We provide some examples of closed monotone symplectic manifolds of Euler characteristic zero (Examples 9 and 10).

In Chapter 4 we study the displaceability of fibers of the bending flow system on equilateral pentagon space. Besides torus fibers over points in the interior of moment map image, this completely integrable system has two La- grangian sphere fibers over boundary points of the moment polytope. They are nondisplaceable for topological reasons (Proposition 25). Most of the regular torus fibers are displaceable using McDuff’s probes technique (Theorem 17). We prove that the central torus fiber is nondisplaceable (Theorem 23) by showing that we can lift a Hamiltonian diffeomorphism displacing it to a Hamiltonian diffeomorphism displacing the central torus fiber of the Gelfand-Cetlin system

1 on the complex Grassmannian of 2-planes. This ﬁber is known to be nondisplaceable.

2 CHAPTER 2 PRELIMINARY MATERIAL

After a brief overview of symplectic geometry, we deﬁne what it means for a Lagrangian submanifold in a symplectic manifold to be displaceable (by Hamil- tonian diffeomorphisms). We introduce monotonicity for symplectic manifolds and Lagrangians.

2.1 A Brief Overview of Symplectic Geometry

We review the basic concepts from symplectic geometry that we will need this dissertation. We use this opportunity to introduce notation and conventions.

Unless stated otherwise, the material presented in this section can be found in [MS17] or [CdS01].

2.1.1 Symplectic Manifolds: Deﬁnition and Examples

A symplectic form on a smooth manifold M is a closed nondegenerate 2-form,

2 i.e. an element ω of Ω (M) satisfying dω = 0, such that ωp : TpM × TpM R is a nondegenerate skew-symmetric bilinear form for all p ∈ M. Every nonzero → tangent vector v ∈ TpM has a nonzero symplectic buddy w ∈ TpM, unique up to multiplication by a scalar, such that ωp(v, w) 6= 0. We call the pair (M, ω) a symplectic manifold.

The existence of a symplectic form severely restricts the topology of M:

• Since nondegenerate skew-symmetric bilinear forms only exist on even-

3 dimensional vector spaces, the dimension dim(M) = 2n must be even. We will often write M2n to mean that M is a manifold of dimension 2n.

• Nondegeneracy is equivalent to the top form ωn = ω ∧ ··· ∧ ω ∈ Ω2n(M) being a volume form. Therefore M must be orientable; it carries an orientation induced by the symplectic form.

2 • Since ω is closed, it deﬁnes a de Rham cohomology class [ω] ∈ HdR(M). If M is closed, then [ω] cannot be zero: if ω = dα for some α ∈ Ω1(M), then

ωn = d(α∧ωn−1). The integral of a closed top form is zero by Stokes’ The- orem, contradicting the fact that the integral of a volume form is nonzero.

2i In fact, this argument implies that HdR(M) 6= 0 for all i ∈ {0, 1, . . . , n}.

Example 1. We give some elementary examples of symplectic manifolds.

1. Any area form on a closed orientable surface Σg is symplectic. The

closedness condition is automatically satisﬁed. In particular, the two- dimensional sphere S2 is symplectic.

2. If the k-dimensional sphere Sk admits a symplectic form for k ≥ 4, then

2 k 0 2 HdR(S ) 6= 0 since it is closed. Therefore S and S are the only spheres that are symplectic.

3. The complex projective spaces CPn admit a symplectic form, called the

Fubini-Study form ωFS. Its de Rham cohomology class is determined by the fact that it assign symplectic area π to any complex line. More gener- ally, any complex submanifold of CPn equipped with the restriction of the

Fubini-Study form is symplectic.

4. As follows from the previous example, all complex Grassmannians and

ﬂag manifolds admit symplectic forms.

4 5. Consider Cn =∼ R2n with coordinates zj = xj + iyj for j ∈ {1, . . . , n}. In light of the Darboux Theorem (see Theorem 1), we call

n i n ω := dxj ∧ dyj = dzj ∧ dzj st 2 j=1 j=1 X X the standard symplectic form.

6. Let Xn be a smooth manifold and T ∗X its cotangent bundle. Let (x1, . . . , xn) be a coordinate system deﬁned on an open subset U ⊂ X and (ξ1, . . . , ξn) the associated ﬁber coordinates on T ∗U. It is straightforward to check that

the forms

n n i i i i αU := ξ dx and ωU := dx ∧ dξ = −dαU i=1 i=1 X X on T ∗U do not depend on the chosen coordinates. They deﬁne global

∗ 1 ∗ forms on T X, called tautological form αtaut ∈ Ω (T X) and the canoni-

2 ∗ cal symplectic form ωcan ∈ Ω (T X).

A diffeomorphism f : M1 M2 between two symplectic manifolds (M1, ω1)

∗ and (M2, ω2) is a symplectomorphism if f ω2 = ω1. The symplectomorphisms → of (M, ω) form a subgroup of the diffeomorphism group Diff(M), which we denote by Sympl(M, ω).

Let (M2n, ω) be a symplectic manifold and i : S M the inclusion of a submanifold. We say S is isotropic if the restriction i∗ω of the symplectic form to → S vanishes. We say S is Lagrangian if it is maximally isotropic, i.e. isotropic and not strictly contained another isotropic submanifold. Symplectic linear algebra

1 guarantees that dim S ≤ n = 2 dim M if S is isotropic. Lagrangian submanifolds are exactly the n-dimensional isotropic submanifolds.

5 2.1.2 Local Normal Forms

We consider two local normal forms in this subsection, the ﬁrst describing what a neighborhood of a point in a symplectic manifold looks like, and the second describing what a neighborhood of a Lagrangian looks like.

Theorem 1 (Darboux, [Dar82]). Let (M2n, ω) be a symplectic manifold. Then every point in M is contained in a coordinate chart ϕ : U ⊂ M R2n such that

n ! ∗ ∗ j j ω|U = ϕ ωst = ϕ dx ∧ dy . → j=1 X

The Darboux Theorem states that any point in a symplectic manifold of dimension 2n has an open neighborhood that is symplectomorphic to an open

2n ball in the standard symplectic manifold (R , ωst). We use this opportunity to list some key differences between symplectic and Riemannian geometry.

• Since all symplectic manifolds of the same dimension are locally symplectomorphic, symplectic geometry has no local invariants other than dimen-

sion, in contrast to Riemannian geometry, which has local invariants like curvature. An example of a global symplectic invariant is the symplectic volume 1 vol(M2n, ω) := ωn. n! ZM • While any smooth manifold admits a Riemannian metric, we have already discussed that the existence of a symplectic form restricts the topology of

the underlying manifold.

• The isometry group of a compact Riemannian manifold is a compact Lie group [MS39]. Symplectomorphism groups tend to be inﬁnite-

dimensional. The natural map on the cotangent bundle T ∗X induced by a

6 ∗ diffeomorphism f : X X is a symplectomorphism, so Sympl(T X, ωcan) contains a subgroup isomorphic to the diffeomorphism group of X. Cotan- → gent bundles are not compact, but even in the compact case, the symplec-

tomorphism group contains the inﬁnite-dimensional subgroup of Hamil- tonian diffeomorphisms, which we introduce in the next subsection.

Weinstein proved in [Wei71] that the way a Lagrangian is embedded in a symplectic manifold only depends on the diffeomorphism type of the La- grangian.

Theorem 2 (Weinstein’s Lagrangian Neighborhood Theorem, [CdS01, Thm. 9.3]). Let (M, ω) be a symplectic manifold and L ⊂ M a Lagrangian. Con- sider the cotangent bundle T ∗L equipped with its canonical symplectic form.

Then the zero section L T ∗L can be extended to a symplectomorphism from a tubular neighborhood of L in M to a tubular neighborhood of the zero section → in T ∗L.

2.1.3 Hamiltonian Vector Fields

The proof of the Darboux Theorem relies on the fact that dω = 0. We discuss another consequence of closedness here: the Jacobi identity for the Poisson bracket.

A symplectic vector ﬁeld is a vector ﬁeld X on M for which LXω = 0. Equiv-

t t ∗ alently, its time-t ﬂow φX : M M satisﬁes (φX) ω = ω for all t ∈ R. Using Cartan’s magic formula and the closedness of the symplectic form, we write →

LXω = dιXω + ιXdω = dιXω.

7 A vector ﬁeld X is therefore symplectic if and only if ιXω is closed.

t Remark 1. To ensure that the time-t flow φX of a vector field X is well-defined for all t ∈ R, we assume that X is compactly supported or that the manifold M is closed.

A Hamiltonian vector ﬁeld is a vector ﬁeld X on M for which ιXω is exact.

All Hamiltonian vector ﬁelds are symplectic. The Hamiltonian vector ﬁeld Xf

1 corresponding to f ∈ C (M) is defined by the condition ω(Xf, ·) = −df. It ∞ uniquely defines Xf because ω is nondegenerate. We say that f is a Hamiltonian for Xf. Two functions define the same Hamiltonian vector field if their difference is locally constant. Since

LXf (f) = df(Xf) = ω(Xf,Xf) = 0,

the ﬂow of Xf preserves the level sets of its Hamiltonian f.

If X and Y are two symplectic vector ﬁelds, then

ι[X,Y]ω = LXιYω − ιYLXω

= dιXιYω + ιXdιYω − ιYdιXω − ιYιXdω (2.1)

= −dω(X, Y),

so their Lie bracket [X, Y] is the Hamiltonian vector ﬁeld Xω(X,Y). In the Lie algebra X(M) of vector ﬁelds on M, the space Xsympl(M, ω) of symplectic vector

ﬁelds and the space XHam(M, ω) of Hamiltonian vector ﬁelds are Lie subalge- bras. In fact, XHam(M, ω) is an ideal of Xsympl(M, ω). The Poisson bracket of two functions f, g ∈ C (M) is the function {f, g} := ω(Xf,Xg) = dg(Xf). It satis-

1 ∞ Some authors [CdS01, MS17] use the convention ω(Xf, ·) = df. In that case the map in (2.2) is a Lie algebra anti-homomorphism.

8 ﬁes the Jacobi identity since

1 {{f, g}, h} + {{g, h}, f} + {{h, f}, g} = (dω)(X ,X ,X ) = 0, 2 f g h so it is a Lie bracket on C (M). The deﬁnition of the Poisson bracket only relied ∞ on the nondegeneracy of the symplectic form. We observe that the closedness of ω is equivalent to the requirement that the Poisson bracket satisfy the Jacobi identity. The computation in (2.1) shows that the map

C (M) X(M): f 7 Xf (2.2) ∞ is a Lie algebra homomorphism. → →

We say that two functions on M are in involution if their Poisson bracket vanishes. An integrable system is a collection (f1, . . . , fk) of functions on M that are pairwise in involution and linearly independent, i.e. there is an open dense subset Ω ⊆ M such that their differentials dpf1, . . . , dpfk are linearly in-

dependent in TpM for all p ∈ Ω. Then the tangent vectors Xf1 (p),...,Xfk (p) are linearly independent and span an isotropic subspace of (TpM, ωp) for all p ∈ Ω, so k ≤ n. An integrable system is complete if k = n.

Remark 2. Some authors require the functions in an integrable system to be smooth. We will adopt a more generous requirement: we only require that the functions are continuous everywhere, and smooth on an open dense subset.

2.1.4 Hamiltonian Diffeomorphisms

We introduce Hamiltonian diffeomorphisms in this subsection as the flows of Hamiltonian vector fields. To get a sufficiently rich class of diffeomorphisms, we

9 allow the Hamiltonians to be time-dependent. The material in this subsection is from [HZ11].

Let (M, ω) be a closed symplectic manifold and H ∈ C (R × M). We can ∞ interpret H as a collection (Ht)t∈R of smooth functions on M that is smoothly indexed by time. Let (Xt)t∈R be the corresponding collection of Hamiltonian vector fields on M. It is a time-dependent vector field XH on M, which we call the time-dependent Hamiltonian vector field associated to H.

Remark 3. With a slight abuse of notation, we will often attribute objects and concepts to a Hamiltonian H that are traditionally attributed to its Hamiltonian X (φt ) vector field H. As an example, we will sometimes refer to the flow XH t∈R of t XH as the flow of H and denote it by (φH)t∈R. More abuse of notation in this vein is to follow.

t Recall that the ﬂow of XH is a collection (φH)t∈R of diffeomorphisms of M

0 determined by φH = idM and

dφt H = X ◦ φt dt H H for all t ∈ R. In the autonomous (i.e. time-independent) case, the ﬂows are

s t s+t symplectomorphisms that satisfy the group law φH ◦ φH = φH for all s, t ∈ R. While the group law no longer holds in the time-dependent case, the ﬂows are still symplectomorphisms:

d (φt )∗ω = (φt )∗L ω dt H H Xt t ∗ = (φH) (dιXt ω + ιXt dω)

= 0,

t ∗ 0 ∗ and therefore (φH) ω = (φH) ω = ω.

10 We say a diffeomorphism on M is a Hamiltonian diffeomorphism if it is the

1 time-1 ﬂow φH : M M of a time-dependent Hamiltonian vector ﬁeld. We denote the set of Hamiltonian diffeomorphisms by Ham(M, ω). Since they play → an important role in this dissertation, we explore their properties in the rest of this section.

The next result shows that we can restrict our attention to 1-periodic Hamil- tonians H : S1 × M R, where we identiﬁed the circle S1 with R/Z. In the rest of this dissertation, we will use the term periodic to mean 1-periodic. →

Proposition 1. Let H : R×M R be a time-dependent Hamiltonian. Then there exists a periodic Hamiltonian K : S1 × M R such that φ1 = φ1 . → H K

→ Proof. Let α :[0, 1] [0, 1] be a smooth function with the property that there exists > 0 such that α(t) = 0 for all t < and α(t) = 1 for all t > 1 − . Deﬁne → K : R × M R by K(t, p) = α0(t)H(α(t), p) for t ∈ [0, 1] and then periodically

α(0) 0 extending it. Note that K is smooth. Since φ = φ idM and → H H d φα(t)(p) = α0(t)X φα(t)(p) dt H Ht H α(t) 0 = Xα (t)Ht φH (p)

α(t) = XKt φH (p)

α(t) t for all t ∈ [0, 1] and p ∈ M, we conclude that φH = φK for all t ∈ [0, 1]. The result follows since α(1) = 1.

To show that compositions and inverses of Hamiltonian diffeomorphisms are Hamiltonian diffeomorphisms, we need the following operations: given two periodic Hamiltonians H and K, let H # K : S1 × M R and H : S1 × M R be

→ →

11 the functions deﬁned by

t −1 (H # K)(t, p) := H(t, p) + K t, (φH) (p) , (2.3)

t H(t, p) := −H t, φH(p) for all t ∈ R and p ∈ M.

Lemma 1. Let f : M M be a symplectomorphism and H ∈ C (M). Then ∞ ∗ −1 → XH◦f = f XH = (df) ◦ XH ◦ f.

Proof. The result follows easily from the properties of differential forms under pullbacks, interior products with vector ﬁelds, and the exterior derivative:

∗ ∗ ιXH ω = −dH f (ιXH ω) = −f (dH)

∗ ∗ ι ∗ (f ω) = −d(f H) ⇒ f XH

ι ∗ ω = −d(H ◦ f). ⇒ f XH

⇒

φt = φt ◦ φt φt = (φt )−1 t ∈ R Proposition 2. The formulas H#K H K and H H hold for all and for all periodic Hamiltonians.

t t Proof. We know that the ﬂows (φH)t∈R and (φK)t∈R are determined by dφt dφt H = X ◦ φt , K = X ◦ φt , dt H H dt K K  and   0  0  φH = idM,  φK = idM.    

12 t t −1 X t −1 = dφ ◦ XK ◦ (φ ) Using the fact that K◦(φH) H H from Lemma 1, we see that d dφt dφt φt ◦ φt = H ◦ φt + dφt ◦ K dt H K dt K H dt t t t t = (XH ◦ φH) ◦ φK + dφH ◦ (XK ◦ φK)

t t t t −1 t t = XH ◦ φH ◦ φK + dφH ◦ XK ◦ (φH) ◦ φH ◦ φK

t t t t = XH ◦ φ ◦ φ + X t −1 ◦ φ ◦ φ H K K◦(φH) H K

t t = (XH + X t −1 ) ◦ φ ◦ φ K◦(φH) H K

t t = X t −1 ◦ φ ◦ φ H+K◦(φH) H K

t t = XH#K ◦ φH ◦ φK

0 0 for all (t, p) ∈ R × M. Since in addition φH ◦ φK = idM, the ﬁrst formula follows. To prove the second formula, note that

t −1 H # H (t, p) = H(t, p) + H t, (φH) (p)

t t −1 = H(t, p) − H t, φH ◦ (φH) (p)

= H(t, p) − H (t, p) = 0.

(t, p) ∈ R × M φt ◦ φt = for all . The ﬁrst formula then guarantees that H H idM.

Corollary 1. The set Ham(M, ω) of Hamiltonian diffeomorphisms is a subgroup of Sympl(M, ω).

2.1.5 Hamiltonian Group Actions and Symplectic Reduction

Let (M, ω) be a symplectic manifold and G a Lie group. A symplectic action of G on M is a smooth map ψ : M Sympl(M, ω). The action is Hamiltonian if it additionally there exists a map µ : M g∗ (called a moment map for the → action) that satisﬁes the two follow conditions. →

13 1. For all X ∈ g, let µX : M R be the component of µ along X, i.e.

→ µX(p) = hµ(p),Xi

for all p ∈ M. Let X# be the fundamental vector ﬁeld on M generated by X, i.e. the vector ﬁeld determined by

# d X (p) = ψexp(tX)(p) dt t=0 for all p ∈ M. Then X# is a Hamiltonian vector ﬁeld with Hamiltonian µX.

X In other words, we require ιX# ω = −dµ .

2. The moment map µ is equivariant w.r.t. the given action on M and the coadjoint action on g∗:

∗ µ ◦ ψg = Adg ◦ µ

for all g ∈ G.

Let (M, ω) be a symplectic manifold and G a compact Lie group. Assume G acts Hamiltonianly on M with moment map µ : M g∗. Let a ∈ g∗ and i : µ−1(0) M the inclusion of the moment map ﬁber. If a is a regular value of → µ and G acts freely on µ−1(a), then →

−1 1. the quotient Mred = µ (a) is a smooth manifold of dimension dim M − 2 dim G;

−1 2. the quotient map π : µ (0) Mred is a principal G-bundle; and

3. there exists a symplectic form→ ωred on Mred uniquely determined by the ∗ ∗ condition π ωred = i ω.

We say that (Mred, ωred) is the symplectic reduction of M at the moment map value a.

14 2.1.6 Compatible Almost Complex Structures

An almost complex structure on a smooth manifold M is vector bundle homo-

2 morphism J : TM TM satisfying J = −idTM.

Let (M, ω) be a→ symplectic manifold. An almost complex structure J is compatible with the symplectic form if

g(v, w) = ω(v, Jw) deﬁnes a Riemannian metric on M. In that case, we also say (ω, J, g) is a compatible triple.

The first Chern class c1(M, ω) of (M, ω) is the first Chern class of the complex vector bundle (TM, J), where J is a compatible almost complex structure. The space of almost complex structures compatible with ω is nonempty and path-connected (contractible even), so the first Chern class does not depend on the choice of J.

Fix a compatible almost complex structure J.A J-holomorphic sphere is a smooth map u : S2 =∼ CP1 M that is (j, J)-complex linear, where j is the standard complex structure on CP1. In other words, its derivative satisﬁes →

J ◦ du = du ◦ j.

15 2.2 Nondisplaceable Lagrangians

2.2.1 Deﬁnition and Properties

Deﬁnition 1. Let (M, ω) be a symplectic manifold and L ⊂ M a Lagrangian. We say that L is displaceable (by Hamiltonian diffeomorphisms) if there exists a Hamiltonian diffeomorphism ϕ ∈ Ham(M, ω) such that L ∩ ϕ(L) = ∅. Otherwise we say L is nondisplaceable.

Lemma 2. Let (M, d) be a compact metric space and K ⊂ M a compact subspace. If a continuous map f : M M satisﬁes K ∩ f(K) = ∅, then there exists an open neighborhood U of K such that U ∩ f(U) = ∅. →

Proof. The distance d0 = inf d(x, y) x ∈ K, y ∈ f(K) between the disjoint compact sets K and f(K) is positive. Fix > 0 such that 2 < d0. Since M is compact, the map f is uniformly continuous; there exists δ > 0 such that

d(x, y) < δ d(f(x), f(y)) < for all x, y ∈ M. We may assume without⇒ loss of generality that δ < . The sets

[ [ U := B(x, δ) and V := B(f(y), ). x∈K y∈K are open neighborhoods of K and f(K), respectively, satisfying f(U) ⊆ V. Sup- pose that there is a point z ∈ U ∩ V. Then there exist points x, y ∈ K such that d(x, z) < δ and d(z, f(y)) < . Then

d(x, f(y)) ≤ d(x, z) + d(z, f(y)) < δ + < 2 < d0

provides a contradiction, since d0 is the minimal distance between points of K and f(K).

16 Corollary 2. If a Hamiltonian diffeomorphism displaces a compact Lagrangian in a compact symplectic manifold, it displaces an open neighborhood of it.

2.2.2 Examples

Hamiltonian diffeomorphisms are isotopic to the identity, and therefore

Ham(M, ω) is contained in the identity component Sympl0(M, ω) of the symplectomorphism group. Banyaga showed in [Ban78] that there is an exact sequence

1 0 Ham(M, ω) Sympl0(M, ω) H (M; R)/Γ 0, where Γ is a countable group. Recall that the symplectomorphisms of a two- dimensional symplectic manifold are its area-preserving diffeomorphisms. We conclude Hamiltonian diffeomorphisms of S2 or D2 are the area-preserving diffeomorphisms isotopic to the identity.

2 1 2 Example 2. Let Da be the disk of area a and Sb ⊂ Da an embedded circle that

1 encloses a region with area b. The area enclosed by Sb is preserved under sym-

b plectomorphism. If a < 2 , it is straightforward to see that there is an isotopy of 1 area-preserving diffeomorphisms that displaces Sb. Any two circles enclosing a

b region of area a, where a ≥ 2 , have to intersect.

Example 3. Similarly, the nondisplaceable Lagrangians of the two-dimensional sphere are the embedded circles that divide the sphere into two regions of equal area. Any two such circles must intersect.

17 2.3 Monotone and Semipositive Symplectic Manifolds

We introduce two classes of symplectic manifolds on which the symplectic area and first Chern number of spheres are strongly related. Having control over the interplay between those quantities will prove to be crucial in defining Floer homology. We want to avoid the existence of pseudoholomorphic spheres (which must have nonnegative symplectic area) with negative first Chern numbers. The definitions in this section are standard and can be found in [MS12].

Let (M, ω) be a symplectic manifold with ﬁrst Chern class c1 = c1(M, ω).

Denote by π2(M) the subgroup of spherical classes of H2(M), i.e. the image of the Hurewicz map π2(M) H2(M). The Hurewicz map is an isomorphism if

π1(M) = 1. Consider the homomorphisms →

Iω : π2(M) R,Iω(A) := h[ω],Ai, (2.4)

I : π (M) Z,I (A) := hc ,Ai. c1 2 → c1 1

The minimal Chern number of (→M, ω) is

∗ 2 NM := inf u c1 > 0 u : S M 2 ZS

= inf hc1,Ai > 0 A ∈ π2(M→) , with the convention that the inﬁmum of the empty set is .

Deﬁnition 2. A symplectic manifold (M, ω) is monotone∞if there exists2 λ > 0 such that Iω = 2λIc1 .

Example 4. The complex projective space CPn equipped with the Fubini-Study

n form ωFS is monotone since c1(CP ) = (n + 1)α and [ωFS] = πα, where α is the canonical generator of H2(CPn; Z). 2The reason for the factor 2 in the deﬁnition will become clear in the next section about monotone Lagrangians.

18 Example 5 ([MS12]). Consider M = CPn1 × CPn2 × · · · × CPnk equipped with the

k ∗ ni symplectic form ω = i=1 λipi ωi, where ωi is the Fubini-Study form on CP ,

ni pi : M CP is the projectionP onto the i-th factor, and λi > 0. The ﬁrst Chern class of M is → k ! k k M ∗ ni ∗ ni ∗ c1(TM) = c1 pi (TCP ) = pi c1(TCP ) = (ni + 1) pi αi, i=1 i=1 i=1 X X 2 ni where αi is the canonical generator of H (CP ; Z).

ni Let Li ∈ H2(M) denote the class of a line in the i-th factor CP . The group ∼ π2(M) = H2(M) is freely generated by the classes L1,...,Lk. Then (M, ω) is

monotone if and only if there exists λ > 0 such that Iω(Li) = 2λIc1 (Li) for all i ∈ {1, 2, . . . , k}. Since Iω(Li) = πλi and Ic1 (Li) = ni + 1, that is exactly when λ λ λ 1 = 2 = ··· = k . n1 + 1 n2 + 1 nk + 1

Example 6 ([MS17]). A compact complex manifold (X, J) with dimR X = 2n

−1 n,0 is Fano if its anticanonical bundle KX = Λ TX is ample, or, equivalently, if there exists an embedding f : X CPN (respecting the complex structures)

∗ −1 ⊗k ∗ such that f O(1) = (K ) . We can equip X with the symplectic form f ωFS, X → i.e. the pullback of the Fubini-Study form on CPN. Starting from the fact that

N [ωFS] = c1(O(1)) in CP and that the ﬁrst Chern class of a complex vector bundle and its determinant line bundle are the same, we get

∗ −1 ⊗k −1 n,0 [f ωFS] = c1 (KX ) = kc1(KX ) = kc1(Λ TX) = kc1(TX).

∗ Therefore (X, f ωFS) is monotone. Conversely, any closed projective monotone Kahler¨ manifold is Fano.

Deﬁnition 3. A symplectic manifold (M2n, ω) is semipositive if the implication

ω(A) > 0  c1(A) ≥ 0 c1(A) ≥ 3 − n  ⇒  19 holds for all A ∈ π2(M).

It is straightforward to show that (M2n, ω) is semipositive if and only if it satisﬁes at least one of the following conditions:

1. (M, ω) is monotone;

2. c1(A) = 0 for all A ∈ π2(M); or

3. the minimal Chern number satisﬁes NM ≥ n − 2.

The third condition guarantees that all symplectic manifolds up to dimension 6 are semipositive. In a closed semipositive symplectic manifold, there are no pseudoholomorphic spheres with negative ﬁrst Chern number for generic compatible almost complex structures [MS12, Lemma 6.4.7].

Example 7. We return to Example 5 and examine under what conditions the product of complex projective spaces is semipositive. We have already explored when it is monotone. Its ﬁrst Chern class does not vanish on spherical classes,

k so we are left with the case where NM ≥ n − 2. Since n = i=1 ni and NM = gcd(n1 + 1, n2 + 1, . . . , nk + 1), the condition NM ≥ n − 2 becomesP k !

gcd(n1 + 1, n2 + 1, . . . , nk + 1) ≥ ni − 2. (2.5) i=1 X If it is satisﬁed, then for any j ∈ {1, 2, . . . , k}, we have k !

nj + 1 ≥ gcd(n1 + 1, n2 + 1, . . . , nk + 1) ≥ ni − 2, i=1 X which in turn implies that k

k − 1 ≤ ni ≤ 3. i=1 Xi6=j It is now straightforward to check all the possible cases with k ≤ 4 by hand. We have already determined that all examples with k = 1 are monotone.

20 • If k = 2 then all cases with n1 ≤ n2 ≤ 3 are listed below.

n1 n2 gcd(n1 + 1, n2 + 1) n1 + n2 − 2 semipositive? 1 1 2 0 yes 1 2 1 1 yes 1 3 2 2 yes 2 2 3 2 yes 2 3 1 3 only if monotone 3 3 4 4 yes

Table 2.1: Semipositivity of products of two complex projective spaces in low dimension.

• If k = 3 then all cases with n1 ≤ n2 ≤ n3 are listed below.

n1 n2 n3 gcd(n1 + 1, n2 + 1, n3 + 1) n1 + n2 + n3 − 2 semipositive? 1 1 1 2 1 yes 1 1 2 1 2 only if monotone

Table 2.2: Semipositivity of products of three complex projective spaces in low dimension.

• If k = 4 then we only need to check the case where n1 = n2 = n3 = n4 = 1.

It is semipositive.

2.4 The Maslov Class and Monotone Lagrangians

There are several ways to introduce the Maslov class of a Lagrangian. We follow the presentation using characteristic classes from [KS18]. See [MS17] for

21 an alternative approach. The Maslov class is a cohomology class that we can evaluate on disks with boundary on the Lagrangian to get their Maslov index. We want to avoid the existence of holomorphic disks (which must have nonnegative symplectic area) with boundary on the given Lagrangian and negative Maslov index. This leads us to the deﬁnition of monotone Lagrangians.

Let (M2n,J) be an almost complex manifold and L ⊂ M a totally real submanifold, i.e. an n-dimensional submanifold satisfying TpL ∩ J(TpL) = {0} for all p ∈ L. In a symplectic manifold equipped with a compatible almost complex structure, Lagrangian submanifolds are totally real.

Consider bundle pairs (EC,ER), where EC M is a complex line bundle and

ER L is a real subbundle of rank 1 of the restriction EC|L L. Such bundle → pairs are determined up to isomorphism by the homotopy class of a classifying → → map ϕ(EC,ER) :(M, L) (BU(1), BO(1)) = (CP , RP ). ∞ ∞ Recall that the cohomology→ rings of those classifying spaces are

∗ ∼ ∗ ∼ H (CP ; Z) = Z[α], |α| = 2, H (CP ; Z2) = Z2[γ], |γ| = 2, ∞ ∞ ∗ ∼ ∗ ∼ H (RP ; Z) = Z[β]/(2β), |β| = 2, H (RP ; Z2) = Z2[ζ], |ζ| = 1. ∞ ∞ 2 We can interpret the generator α of H (CP ; Z) as the ﬁrst Chern class c1 and

1 ∞ the generator ζ of H (RP ; Z2) as the first Stiefel-Whitney class w1. The classi- ∞ fying space CP is quotient of the unit sphere S in a complex Hilbert space ∞ ∞ of countably-infinite dimension by the action of the unit circle S1 ⊂ C by scalar multiplication. The quotient of S by the subgroup {1, −1} ⊂ S1 is the classify- ∞ ing space RP . We define an embedding i : RP CP by sending orbits to ∞ ∞ ∞ orbits. The induced map i∗ : H2(CP ; Z) H2(RP ; Z) is onto [Hat02, p. 229]. → ∞ ∞ In the long exact sequence of the pair→(CP , RP ) in cohomology with in- ∞ ∞ 22 teger coefficients (suppressed from notation) below, the Maslov class µMaslov is

2 ∗ the generator of H (CP , RP ; Z) satisfying j µMaslov = 2c1.

∞ ∞ j∗ ∗ H1(RP ) δ H2(CP , RP ) H2(CP ) i H2(RP ) k ∞ ∞k ∞ k ∞ k ∞ ×2 0 Z Z Z2 Deﬁnition 4. The Maslov class of a totally real submanifold L ⊂ M2n is

∗ 2 µL := ϕL(µMaslov) ∈ H (M, L; Z),

n n where ϕL :(M, L) (CP , RP ) is a classifying map of the pair (ΛCTM, ΛRTL). ∞ ∞ → Note that the map ϕL : M CP is a classifying map for the complex line ∞ bundle ΛnTM. A complex vector bundle and its determinant line bundle have C → ∗ the same ﬁrst Chern class, so c1(TM, J) = ϕL(c1). Since the diagram j∗ H2(CP , RP ; Z) H2(CP ; Z)

∞ ∗ ∞ ∞∗ ϕL ϕL j∗ H2(M, L; Z) H2(M; Z)

∗ 2 commutes, the relation j µMaslov = 2c1 in H (CP ; Z) implies that the Maslov

∗ 2 ∞ class satisﬁes j µL = 2c1(TM, J) in H (M; Z).

We now focus on the case where (M, ω) is a symplectic manifold equipped with a compatible almost complex structure and L ⊂ M is a Lagrangian. The Maslov class of L does not depend on the chosen almost complex structure because the ﬁrst Chern class c1(M, ω) does not depend on it. Let π2(M, L) be the subgroup of H2(M, L) generated by disks with boundary on L, or, in other words, the image of the Hurewicz map π2(M, L) H2(M, L). Consider the homomorphisms → L Iω : π2(M, L) R,Iω(A) := h[ω],Ai,

I : π (M, L) Z,I (A) := hµ ,Ai. µL 2 → µL L

→ 23 The symplectic areas of disk classes and spherical classes are related: indeed, the diagram j∗ π2(M, L) π2(M)

L Iω Iω

R commutes. The minimal Maslov number of L is

∗ 2 2 NL := inf u µL > 0 u :(D , ∂D ) (M, L) 2 ZD

= inf {hµL,Ai > 0 | A ∈ π2(M, L)} , → with the convention that the inﬁmum of the empty set is .

L Deﬁnition 5. A Lagrangian L is monotone if Iω = λIµL for∞ some λ > 0.

Proposition 3. A simply-connected Lagrangian L in a monotone symplectic manifold (M, ω) is monotone itself. Its minimal Maslov number NL cannot be odd and is at least 2.

Proof. If L is simply-connected, then the restriction maps in the long exact sequences below are onto. The geometric intuition behind this fact is that, given a disk in M with boundary on L, its boundary circle is homotopic to a point on L.

j∗ π2(M) π2(M, L) π1(L) = 1

j∗ H2(M) H2(M, L) H1(L) = 0

Since (M, ω) is monotone, there exists λ > 0 such that Iω = 2λIc1 . Let A ∈

24 π2(M, L). Then there exists B ∈ π2(M) such that j∗B = A. Therefore

IµL (A) = hµL,Ai = hµL, j∗Bi

∗ = hj µL,Bi

= h2c1,Bi = 2Ic1 (B) 1 = I (B) λ ω 1 1 = IL (j B) = IL (A). λ ω ∗ λ ω

L We conclude that Iω = λIµL and that L is monotone. Moreover, since Ic1 is integer-valued and IµL (A) = 2Ic1 (B), the image of IµL is a subgroup of 2Z. The minimal Maslov number can therefore not be odd. Since it is at least 1 by deﬁ- nition, it is at least 2 in the simply-connected case.

The proof of the previous proposition suggests that the coefﬁcients λ in the deﬁnitions of monotonicity of symplectic manifolds and of Lagrangians are related.

Proposition 4. Let (M, ω) be a symplectic manifold and L ⊂ M a Lagrangian.

L Suppose that L is monotone and that Iω = λIµL for some λ > 0. Then the ambient manifold (M, ω) is monotone and Iω = 2λIc1 .

Proof. Let A ∈ π2(M) be a spherical class in M. Then

L Iω(A) = Iω(j∗A) = λIµL (j∗A) = λhµL, j∗Ai

∗ = λhj µL,Ai = λh2c1,Ai = 2λIc1 (A),

so that Iω = 2λIc1 and (M, ω) is monotone.

Proposition 5. The image of the Maslov class of a Lagrangian L under the

2 2 change of coefﬁcient homomorphism H (M, L; Z) H (M, L; Z2) is equal to

25 → the image of the ﬁrst Stiefel-Whitney class of the tangent bundle TL under the

1 2 connecting homomorphism δ : H (L; Z2) H (M, L; Z2).

→ Proof. We start by showing that the relationship holds in the cohomology of the classifying space. The rows in the commutative diagram below are parts of the long exact sequence of the pair (CP , RP ), with Z coefficients on top and ∞ ∞ Z2 coefficients on the bottom. The vertical maps are the change of coefficient homomorphisms.

∗ j∗ ∗ H1(CP ; Z) i H1(RP ; Z) δ H2(CP , RP ; Z) H2(CP ; Z) i H2(RP ; Z)

∞ ∞ ∞ ∞ ∞ ∞

∗ 1 i∗ 1 δ 2 j 2 i∗ 2 H (CP ; Z2) H (RP ; Z2) H (CP , RP ; Z2) H (CP ; Z2) H (RP ; Z2)

∞ ∞ ∞ ∞ ∞ ∞ Using the information we have already presented in this section, we get the following commutative diagram:

×2 0 0 Z Z Z2

} x . | z y 0 Z2 { Z2 Z2

2 2 The map x is the change of coefﬁcient map H (RP ; Z) H (RP ; Z2), which is ∞ ∞ an isomorphism (see [Hat02, p. 214]). If x is an isomorphism, then y must also → be an isomorphism if the rightmost square is commutative. Then z is zero by exactness, so { must be isomorphic to Z2 and | is an isomorphism by exactness.

Next we show that the change of coefficient homomorphism } is the nontrivial homomorphism Z Z2. The change of coefficient homomorphisms commute with the maps in the Universal Coefficient Theorem: →

26 ∼ 2 Z = H (CP , RP ; Z) Hom (H2(CP , RP ), Z) 0

∞} ∞ ~∞ ∞ .

∼ 2 Z2 = H (CP , RP ; Z2) Hom (H2(CP , RP ), Z2) 0 ∞ ∞ ∞ ∞ ∼ It is easy to see that H2(CP , RP ) = Z from the long exact sequence in homol- ∞ ∞ α ogy of this pair. The map ~ takes a homomorphism Z − Z to the composition α 6=0 Z − Z − Z2 and is clearly onto. Therefore } must be onto as well. →

1 →The morphisms→ in (2.6) below are both onto. The generators of H (RP ; Z2)

2 2 ∞ and H (CP , RP ; Z) are both mapped to the generator of H (CP , RP ; Z2). ∞ ∞ ∞ ∞ These generators are exactly the ﬁrst Stiefel-Whitney class w1 and the Maslov class µMaslov, so δw1 = µMaslov in the cohomology of the classifying space.

H2(CP , RP ; Z) Z

∞ ∞ } (2.6)

1 δ 2 | H (RP ; Z2) H (CP , RP ; Z2) Z2 Z2 ∞ ∞ ∞ n n Let ϕL :(M, L) (CP , RP ) be a classifying map for (ΛCTM, ΛRTL). The

∞ ∞ n restriction ϕL : L RP is a classifying map for the real vector bundle Λ TL. → R ∞ We ﬁnally conclude that →

n ∗ ∗ ∗ w1(TL) = w1(ΛRTL) = ϕL(w1) = ϕL (µMaslov) = ϕL(µMaslov) = µL.

Here we used the fact that the change of coefﬁcient homomorphisms commute with induced maps.

Corollary 3. The minimal Maslov number NL of an orientable Lagrangian L cannot be odd and is at least 2.

Proof. The following three conditions are equivalent:

1. the Lagrangian L is orientable;

27 2. the tangent bundle TL is an orientable vector bundle; and

3. the ﬁrst Stiefel-Whitney class w1(TL) vanishes.

If L is orientable, Proposition 5 guarantees that the reduction to Z2 coefﬁcients of

its Maslov index µL vanishes. The image of IµL is a subgroup of 2Z and The minimal Maslov number therefore cannot be odd. Since it is at least 1 by deﬁnition, it is at least 2 in the orientable case.

Remark 4. Since all simply-connected manifolds are orientable, Corollary 3 is a generalization of the last part of Proposition 3. The ﬁrst part of the proposition does not generalize, however: an orientable Lagrangian in a monotone symplectic manifold is not necessarily monotone itself.

28 CHAPTER 3 FLOER HOMOLOGY AND THE PSS ISOMORPHISM

The goal of this chapter is to use a topological result guaranteeing displaceability by Albers [Alb05, Alb10] to show that the anti-diagonal of a monotone symplectic manifold is nondisplaceable (see section 3.8).

We start this chapter with an introduction to Morse Theory and Floer The- ory. The two homology theories are linked by the PSS isomorphism, which is introduced next. Under certain conditions, we can represent the homology class of a Lagrangian as the image of a cycle in Floer homology under the PSS isomorphism. This representation can be exploited to study the topological im- plications of displaceability.

3.1 Morse Theory

Floer theory is a generalization of Morse theory. Morse theory is conceptually easier to understand. Its development will serve as a roadmap for the deﬁnition of Floer homology. Moreover, Morse theory will return to the forefront when we discuss the PSS isomorphism, which relates Morse and Floer theory. We give a brief overview of Morse homology based on [AD14] and [Sal99].

3.1.1 Morse Functions

Let Mn be a closed smooth manifold and f ∈ C (M). A point p ∈ M is a ∞ critical point of f if dpf : TpM TpM is zero. We denote the set of critical points of f by Crit(f). →

29 2 The Hessian dpf : TpM × TpM R of f at p ∈ Crit(f) is deﬁned by

2 (d f)p(v,→ w) := v(W(f))(p) (3.1)

for all v, w ∈ TpM, where W is a vector ﬁeld on M such that Wp = w. If V is a vector ﬁeld such that Vp = v, then

v(W(f))(p) − w(V(f))(p) = [V, W]p(f) = (dpf) ([V, W]) .

We conclude that the expression in (3.1) is independent of the chosen extension exactly when p is a critical point of f. Furthermore, it is a symmetric bilinear form. Its matrix representation in local coordinates is the classical Hessian of f, i.e. the matrix whose entries are the partial derivatives of order 2.

2 A critical point p of f is nondegenerate if the Hessian dpf is a nondegenerate bilinear form. In that case the (Morse) index µMorse(p; f) of p is equal to the

1 2 number of negative eigenvalues of dpf. We denote the set of nondegenerate critical points of f of Morse index k by Critk(f).

A function f ∈ C (M) is a Morse function if all its critical points are nonde- ∞ generate. In that case, there exists a Morse chart around every p ∈ Critk(f), i.e. a chart ϕ : U Rn such that

k n −1 2 2 → f ◦ ϕ (x1, . . . , xn) = f(p) − xi + xi . i=1 i=k+1 X X

3.1.2 The Morse Complex

Let g be a Riemannian metric on M. The negative gradient of f is the vector

g g s ﬁeld −∇ f on M determined by the condition g(∇ f, ·) = −df. Let (ϕ )s∈R be

1The symmetric matrices that represent the Hessian in different coordinate systems all have the same number of negative eigenvalues. This is exactly Sylvester’s Law of Intertia.

30 the negative gradient flow of f. For a fixed p ∈ M, the integral curve or flow line

g s γp : R M of −∇ f satisfying γp(0) = p is given by γp(s) = ϕ (p). Negative gradient ﬂow lines always connect two critical points of f. →

Proposition 6 ([AD14, Prop. 2.1.6]). Let f ∈ C (M) be a Morse function on a ∞ closed Riemannian manifold (M, g). For any ﬂow line γ : R M of −∇gf, there exist p, q ∈ Crit(f) such that lims − γ(s) = p and lims γ(s) = q. → → ∞ →∞ The stable and unstable manifold of a critical point p of a Morse function f,

Ws(p; f) := q ∈ M lim ϕs(q) = p , s

u s W (p; f) := q ∈ M →∞lim ϕ (q) = p , s −

→ ∞ are embedded submanifolds of M of respective dimensions

s dim W (p; f) = dim M − µMorse(p; f),

u dim W (p; f) = µMorse(p; f).

The stable manifold of p consists of all points on negative gradient ﬂow lines that end at p. Similarly the unstable manifold of p contains all points on negative gradient ﬂow lines that start at p.

If f ∈ C (M) is a Morse function and g a Riemannian metric, we say the ∞ pair (f, g) is Morse-Smale if Wu(p; f) and Ws(q; f) intersect transversally for all p, q ∈ Crit(f). The set of Morse functions is open and dense in C (M). ∞ Moreover, generic pairs of smooth functions and Riemannian metrics satisfy the Morse-Smale property. The usefulness of this property is that it implies the union of all negative gradient ﬂow lines from p to q, namely

MMorse(p, q; f) := Wu(p; f) ∩ Ws(q; f),

31 is a smooth manifold of dimension µMorse(p; f)−µMorse(q; f). This further implies that the Morse index decreases along ﬂow lines.

The space MMorse(p, q; f) comes with an R-action given by s · p = ϕs(p), which is free if p 6= q. The action moves points along the negative ﬂow lines. The moduli space of negative gradient ﬂow lines from p to q,

Morse Morse M (p, q; f) Mc (p, q; f) := R, is a smooth manifold of dimension µMorse(p; f)−µMorse(q; f)−1 (if it is nonempty).

Theorem 3. The moduli spaces McMorse(p, q; f) are compact in dimension zero. In dimension one, they can be compactified by adding broken flow lines: the boundary of the compactification is

[ ∂McMorse(p, q; f) = McMorse(p, r; f) × McMorse(r, q; f),

r∈Critk−1(f) if µMorse(p; f) = µMorse(q; f) + 2 = k.

Deﬁnition 6. The Morse chain groups of a Morse function f ∈ C (M) are the ∞ vector spaces over Z2 generated by the critical points of f, graded by the Morse CM (f) := L Z · p k ∈ Z index: k p∈Critk(f) 2 , for all . The Morse boundary operator of a Morse-Smale pair (f, g) is the Z2 linear map ∂M : CMk(f) CMk−1(f) deﬁned on generators by →

Morse ∂Mp := #2Mc (p, q; f) · q. q∈CritXk−1(f) We use #2 to denote the number of elements modulo 2 in the ﬁnite set.

The boundary ∂Mp encodes the number of negative gradient flow lines from p to critical points of f of index k−1. The fact that the moduli spaces are compact in dimension zero guarantees that the boundary operator is well-defined. We need the compactification result in dimension one to prove the following.

32 2 Theorem 4. The boundary operator satisﬁes ∂M = 0.

Proof. Let p ∈ Critk(f). Then   2 Morse Morse ∂Mp =  #2Mc (p, q; f) · #2Mc (q, r; f) · r. r∈CritXk−2(f) q∈CritXk−1(f)

The coefﬁcient of r ∈ Critk−2(f) is exactly number of elements mod 2 in the boundary of the one-dimensional moduli space McMorse(p, r; f). Since it is compact by Theorem 3 and the boundary of a compact one-dimensional manifold has an even number of points, the coefﬁcients are all zero.

Deﬁnition 7. The Morse homology of a Morse-Smale pair (f, g) is the homology of the Morse chain complex:

ker ∂M : CM∗(f) CM∗−1(f) HM∗(f) := . im ∂M : CM∗+1(f) CM∗(f) → Theorem 5 ([AD14, Section 4.9]). The Morse homology→ of a Morse-Smale pair on M is isomorphic to the singular homology of M with Z2 coefﬁcients.

Remark 5. We can deﬁne a Morse chain complex with integer coefﬁcients associated to a Morse-Smale pair if we choose an orientation on the stable manifold of every critical point. The resulting Morse homology groups are isomorphic to the singular homology groups of M [AD14, Section 3.3].

3.1.3 Continuation Maps

Since Morse homology is isomorphic to singular homology, it does not depend on the chosen Morse-Smale pair. It is natural to ask whether there is a canonical isomorphism between the Morse homology groups of two given Morse-Smale

33 pairs. The answer is yes: one can construct a chain map between the Morse chain complexes starting from a regular homotopy between the Morse function. The induced map on Morse homology, called a continuation map, is an isomorphism that does not depend on the chosen regular homotopy. The construc- tion is discussed in detail in [AD14] and [Sch93]. We present the material from [AD14] here.

Let (f0, g0) and (f1, g1) be Morse-Smale pairs on M.A regular homotopy

1 4 from f0 to f1 is a smooth function F : − 3 , 3 × M R with the property that 1 1 2 4 Ft = f0 for − ≤ t ≤ and Ft = f1 for ≤ t ≤ . Choose a Morse function 3 3 3 3→ α : R R that is strictly increasing on (− , 0] and [1, ) and that decreases sufﬁciently fast on [0, 1] such that → ∞ ∞ ∂F (t, p) + α0(t) < 0 ∂t

1 4 for all (t, p) ∈ [0, 1] × M. We deﬁne the function eF : − 3 , 3 × M R by

eF(t, p) = F(t, p) + α(t). →

1 4 1 1 Equip − 3 , 3 × M with a metric as follows: on − 3 , 3 × M, use the product of the standard metric and g0. Similarly, use the product of the standard metric and

2 4 1 2 g1 on 3 , 3 × M. Extend it arbitrarily on 3 , 3 × M and perturb it if necessary so we obtain a Morse-Smale pair (eF, Ge). The critical points of eF and their Morse indices are given by

Crit(eF) = Crit(f0) × {0} ∪ Crit(f1) × {1} µMorse (p, 0); eF = µMorse(p; f0) + 1 µMorse (q, 1); eF = µMorse(q; f1).

Because of the way we chose α, the negative gradient trajectories of eF either lie in {0} × M or {1} × M and are negative gradient trajectories of f0 or f1, or they

34 ﬂow from critical points in {0} × M to critical points in {1} × M. They cannot ﬂow in the opposite direction. Using the decomposition

=∼ CMk+1(eF) CMk(f0) ⊕ CMk+1(f1)

∂eF eF , M ∂M =∼ CMk(eF) CMk−1(f0) ⊕ CMk(f1) the boundary operator decomposes as   ∂f0 0 ∂eF =  M  . M   eF f1 Φ ∂M

eF The map Φ : CMk(f0) CMk(f1) is deﬁned on generators by

eF Morse Φ (p) =→ #2Mc (0, p), (1, q); eF · q. q∈CritXk(f1) By combining the facts that the boundary operators all square to zero, we learn

eF f0 f1 eF eF that Φ ◦ ∂M = ∂M ◦ Φ . In other words, Φ is a chain map and therefore deﬁnes

eF a homomorphism Φ : HM∗(f0, g0) HM∗(f1, g1), which we call the continuation map between the given Morse-Smale pairs. →

The continuation maps satisfy the following properties:

• If (f0, g0) = (f1, g1) and we choose the homotopy F = I to be trivial, then

the continuation map ΦeI is the identity on Morse homology.

• Let (f0, g0), (f1, g1), and (f2, g2) be Morse-Smale pairs on M. Suppose Fij

is a regular homotopy from (fi, gi), (fj, gj) for 0 ≤ i < j ≤ 2. Then the diagram ΦeF01 HM∗(f0, g0) HM∗(f1, g1)

ΦeF02 ΦeF12

HM∗(f2, g2) commutes.

35 Not only are the continuation maps isomorphisms, these properties show that they are independent of the chosen regular homotopy and its extension. The Morse homology groups are all canonically isomorphic via the continuation maps. We can therefore safely deﬁne the Morse homology of a smooth manifold as the Morse homology of a Morse-Smale pair on it.

3.1.4 Functoriality and Induced Maps

Let ϕ : M1 M2 be a smooth map between closed manifolds. The continuation maps allow us to deﬁne an induced map on Morse homology by using generic → Morse-Smale pairs. We follow the deﬁnition outlined in [AS10].

Let (f1, g1) and (f2, g2) be Morse-Smale pairs on M1 and M2, respectively,

s u such that ϕ|W (p;f1) is transverse to W (q; f2) for all p ∈ Crit(f1) and q ∈ Crit(f2). Note that this is a generic requirement. The set

u −1 s M(p, q; f1, f2, ϕ) := W (p; f1) ∩ ϕ W (q; f2)

is a compact submanifold of M1 of dimension µMorse(p; f1) − µMorse(q; f2). The linear map ϕ : CM∗(f1) CM∗(f2) deﬁned on generators by

ϕ∗(p→) := #2M(p, q; f1, f2, ϕ) · q q∈CritX∗(f2) commutes with the Morse boundary operator, so it induces a map

ϕ∗ : HM∗(f1, g1) HM∗(f2, g2).

It is shown in [AS10] that this map coincides→ with the map induced by ϕ on singular homology (under the standard identiﬁcation between Morse and singular homology) and that the induced maps commute with the continuation maps.

36 Example 8. Let i : Sk Mn be the inclusion of a closed submanifold. Fix a

Morse-Smale pair (fS, gS) on S. Every p ∈ Crit(f) has a slice neighborhood in → ∼ k n−k ∼ k M, i.e. an open neighborhood Up = D × D such that S ∩ Up = D × {0}.

We extend fS quadratically in the normal direction on Up and get fUp : Up R

1 2 deﬁned by fU (x, y) = fS(x) + ||y|| . We extend the metric by choosing the p 2 → standard metric on Dn−p in the normal direction. Finally we arbitrarily extend the function and the metric on the complement of these open neighborhoods to get a Morse-Smale pair (fM, gM) on M.

Note that Crit(fM) ∩ S = Crit(fS). The quadratic perturbation ensures that negative gradient ﬂow lines of fM can ﬂow into S, but not out from it, so that

µMorse(p, fM) = µMorse(p, fS) for all p ∈ Crit(fS). Furthermore, the negative gradient ﬂow lines of fS are negative gradient ﬂow lines of fM.

∼ Let [p] be the generator of HMk(fS, gS) = Z2. If q ∈ Critk(fM), then

u −1 s M(p, q; fS, fM, i) = W (p; fS) ∩ i (W (q; fM))

Morse is empty if q∈ / Critk(fS) and it is M (p, q; fs) is q ∈ Critk(fS). Transversality

Morse is guaranteed by the Morse-Smale property. The space M (p, q; fs) is empty if p 6= q. Therefore the map i∗ : HMk(fS, gS) HMk(fM, gM) induced by the inclusion sends [p] to [p]. →

3.1.5 Poincar´eDuality

Poincare´ duality is very straightforward to deﬁne using the continuation maps. The material in this section is from [Sch93] and [AB95].

Let (f, g) be a Morse-Smale pair on Mn. Let (CM∗(f), δM) be the cochain

37 complex we obtain by dualizing the Morse chain complex (CM∗(f), ∂M). In other words, the cochain groups are

k CM (f) := Hom (CMk(f), Z2) , and the coboundary δM : CMk(f) CMk+1(f) is the linear map deﬁned on

M ∗ M generators by (δ α)(p) := α(∂Mp). The homology of the complex (CM (f), δ ) → is by deﬁnition the Morse cohomology of f. It is isomorphic to the singular cohomology of M with Z2 coefﬁcients.

k Let {αp | p ∈ Crit(f)} be the basis of CM (f) dual to the basis Critk(f). In other

k 0 words, the linear map αp : CM (f) Z2 is determined by αp(p ) = δpp 0 for all

0 p, p ∈ Critk(f). Then →

M hδ αp, qi = αp(∂Mq)   Morse 0 0 = αp  #2Mc (q, p ; f) · p  0 p ∈XCritk(f) Morse 0 0 = #2Mc (q, p ; f) · αp(p ) 0 p ∈XCritk(f) Morse = #2Mc (q, p; f).

for all p ∈ Critk(f) and q ∈ Critk+1(f). We conclude that

M Morse δ αp = Mc (q, p; f) · αq. q∈CritXk+1(f) The only meaningful difference between Morse homology and cohomology is that we use negative gradient ﬂow lines in the former and gradient ﬂow lines in the latter.

Another way to work with gradient ﬂow lines is to consider the Morse homology of −f. Note that (−f, g) is a Morse-Smale pair and‘ Crit(f) = Crit(−f).

38 The Morse index satisﬁes µMorse(p; −f) = n − µMorse(p; f) for all p ∈ Crit(f). We

n−k can interpret the natural pairing CM (f) ⊗ CMn−k(f) Z2 as a nondegener-

n−k ate pairing CM (f) ⊗ CMk(−f) Z2. It is compatible with the boundary and → coboundary operator and descends to a nondegenerate pairing →

n−k HM (f, g) ⊗ HMk(−f, g) Z2.

This is the Poincare´ duality pairing. It determines an→ isomorphism

f n−k Ψ : HM (f) HMk(−f).

We deﬁne the Poincare´ duality map as the→ composition

n−k Ψf Φ−f,f HM (f, g) HMk(−f, g) HMk(f, g), where the second map is the continuation map. Poincare´ duality agrees with the continuation map on Morse homology and its transpose on Morse cohomology. Its deﬁnition is therefore independent of the chosen Morse function. It corre- sponds to Poincare´ duality under the identiﬁcation of Morse (co)homology and singular (co)homology.

3.2 Periodic Orbits of Hamiltonian Vector Fields

The periodic orbits of a nondegenerate periodic Hamiltonian H ∈ C (S1 × M) ∞ will play the role that critical points did in Morse theory. The material in this section is from [AD14].

39 3.2.1 Deﬁnition and Properties

A periodic solution or periodic orbit of a periodic time-dependent Hamiltonian

1 H : S × M R is a 1-periodic orbit of the Hamiltonian vector field XH, i.e. a solution x : R M ofx ˙ (t) = XH(t, x(t)) satisfying x(t) = x(t + 1) for all t ∈ R. → 1 1 Lemma 3. The→ periodic solutions x : S M of a Hamiltonian H : S × M R are in one-to-one correspondence with the fixed points of the time-1 flow φ1 . → →H

Proof. The correspondence sends the ﬁxed point p to the orbit x : R M de- ﬁned by x(t) = φt (p). Its inverse sends a periodic orbit x : S1 M to x(0). H → → The preceding lemma is not valid for general time-dependent Hamiltonians

1 H : R × M R: even if φH(p) = p, there is no reason to assume that XH(0, p) the tangent vectors and XH(1, p) are the same. If we want closed periodic orbits, we → need to only consider periodic Hamiltonians.

For an autonomous Hamiltonian H : M R, every point on a periodic orbit is ﬁxed by φ1 . This is no longer true in the time-dependent case. Let p be a H → 1 ﬁxed point of φH and x : R M the corresponding periodic orbit. The point

t0 1 q = x(t0) = φ (p) is a ﬁxed point of φ exactly when XH(t0, q) = XH(0, q). → H

A periodic solution x of H is nondegenerate if 1 is not an eigenvalue of

1 dx(0)φH : Tx(0)M Tx(0)M. We say that the Hamiltonian H is nondegenerate if all its periodic solutions are nondegenerate. Equivalently, H is nondegenerate → 1 if the graph of its ﬂow φH is transverse to the diagonal in M × M. Nondegener- acy is a generic condition in C (S1 × M). ∞ Proposition 7. A nondegenerate Hamiltonian H : S1 × M R on a compact symplectic manifold M has ﬁnitely many periodic solutions. →

40 1 Proof. The ﬁxed points of φH : M M correspond to intersection points of its graph and the diagonal in M × M. Due to transversality, this intersection is a → zero-dimensional manifold in M × M. Since the ﬁxed point set is closed and

1 M × M is compact, we conclude φH only has ﬁnitely many ﬁxed points.

The following results from [AD14] reinforce the notion that Floer homology is a generalization of Morse theory.

Proposition 8. Let H ∈ C (M) be an autonomous Hamiltonian. ∞

• A critical point of H is a constant periodic orbit of its Hamiltonian vector ﬁeld. It is nondegenerate as a periodic orbit if and only if it is nondegen-

erate in the sense of Morse theory.

2 • If H is sufﬁciently small in the C sense, then all the periodic orbits of XH

are constant.

3.2.2 The Arnold Conjecture

The Arnold conjecture states that the number of contractible periodic solutions of a nondegenerate periodic Hamiltonian H : S1 × M R is bounded below by

2n the sum bi(M; Z2) of the Z2 Betti numbers of M. Since nondegeneracy is a i=0 → generic condition,P it is equivalent to the claim that the number of ﬁxed points of a generic Hamiltonian diffeomorphism of M is bounded below by the sum of the Betti numbers. It was stated in this last form by Arnold in [Arn65]. The conjecture has now been proved for general closed symplectic manifolds. We give an overview of its history found in [Sal99, FW18]:

41 • It was ﬁrst proved by Eliashberg [Eli79] for Riemann surfaces.

• Conley and Zehnder [CZ83] proved it for tori T2n = R2n/Z2n.

• Gromov [Gro85] showed that generic Hamiltonian diffeomorphisms have at least one ﬁxed point on aspherical symplectic manifolds (i.e. when

ω(π2(M)) = 0).

• Floer [Flo88a, Flo88b, Flo88c, Flo89b] combined the approaches of Conley-

Zehnder and Gromov, and proved the conjecture in the aspherical case. He

deﬁned a chain complex (CF∗(H), ∂F) whose chain groups are the Z2 vector spaces generated by the contractible periodic solutions. The homology

HF∗(H) (called Floer homology) of this complex is isomorphic to the singular homology of M. The number of contractible periodic orbits is equal to the dimension of the chain complex, which satisﬁes

dim CFi(H) ≥ dim HFi(H) = bi(M; Z2). i∈Z i∈Z i=0 X X X • Floer [Flo89a] generalized his approach and proved the conjecture for

monotone symplectic manifolds.

• Hofer and Salamon [HS95], and independently Ono [Ono95], extended

Floer’s approach to semipositive symplectic manifolds.

• The general case was proved by Fukaya-Ono [FO99] and Liu-Tian [LT98].

3.3 Floer Homology

We present an introduction to Floer homology based on the very brief outline in [Alb05] and supplemented with material from [AD14, MS12, Sal99].

42 In this entire section, (M2n, ω) will always be a closed semipositive symplectic manifold, unless speciﬁcally stated otherwise.

3.3.1 The Novikov Ring

In a monotone symplectic manifold the symplectic area ω(A) and the first Chern number c1(A) of a spherical class A ∈ π2(M) are proportional. In the semipositive case, we can encounter sequences of spherical classes for which the first Chern number is constant, but the symplectic area tends to infinity. We introduce a coefficient ring tailored to this situation to do some bookkeeping.

We choose to identify two spherical homology classes when they have the same symplectic area and ﬁrst Chern number: deﬁne the group

π (M) Γ := 2 , ker Iω ∩ ker Ic1 and equip it with the grading deg(A) := 2c1(A). Let Γk be the set of elements of degree k in Γ.

L The Novikov ring is Λ := k∈Z Λk, where

#{A ∈ Γk | λA 6= 0, ω(A) ≤ c} < , ∀c ∈ R A Λk :=  λAe  . (3.2) A∈Γk λA ∈ Z2, ∀A ∈ Γk  X ∞  Elements ofΛ satisfy the Novikov condition: there are only ﬁnitely many terms whose symplectic area is smaller than any given upper bound. The multiplication is given by ! ! ! A B A+B C λAe µBe := λAµBe = λAµC−A e .

A∈Γk B∈Γl A∈Γk C∈Γk+l A∈Γk X X XB∈Γl X X

43 Proposition 9. The multiplication in Λ is well-deﬁned.

λ eA µ eB = Proof. We can rewrite the expression above as A∈Γk A B∈Γl B ν eC ν := λ µ P C ∈ Γ P c , c ∈ R C∈Γk+l C , where C A∈Γk A C−A. First ﬁx k+l and 1 2 .

SinceP both factors satisfy theP Novikov condition, all but ﬁnitely many A ∈ Γk for which λA 6= 0 and µC−A 6= 0 satisfy c1 < ω(A) < c2 − ω(C). By choosing c1 and c2 such that this range is empty, we see that νC is well-deﬁned.

ν eC Next suppose that C∈Γk+l C does not satisﬁes the Novikov condition.

Then there exist a sequenceP (Ci)i=1 in Γk+l such that limi ω(Ci) = − and ∞ νC 6= 0 for all i ∈ N. Since the factors satisfy the Novikov→∞ condition, the sym- i ∞ plectic area of the occurring terms A ∈ Γk and B ∈ Γl is bounded below. From the deﬁnition of the multiplication, however, it is clear that the symplectic area of the term occurring in the product is also bounded below.

Denote by SM the minimal positive symplectic area of a spherical homology class in M. More speciﬁcally,

min {ω(A) > 0 | A ∈ π2(M)} , if ω(π2(M)) 6= {0}, SM :=   0, otherwise.  ∼ Since Γ0 = π2(M) ker Iω = Iω(π2(M)) = Z · SM, we distinguish two cases:

∼ 1. If SM = 0, then Γ0 = {0} and Λ0 = Z2. This is the case when (M, ω) is monotone or aspherical.

∼ 2. If SM 6= 0, then Γ0 = Z. Let A0 be a class in Γ0 such that ω(A0) = SM. We can

−1 identify Λ0 with a subring of the power series ring Z2[[t, t ]] by sending

eA0 to t. The Novikov condition in this context demands that only ﬁnitely

44 many terms have negative exponents. Therefore Λ0 is isomorphic to the

−1 Laurent series ring Z2[[t]][t ].

Note that Λ0 is a ﬁeld in both cases. The Novikov ring is a vector space over Λ0.

We choose to work with Λ0 as coefﬁcient ring from now on.

3.3.2 The Loop Space and the Action Functional

Let D2 = z ∈ C |z| ≤ 1 be the unit disk and S1 = e2πit t ∈ R the unit circle in the complex plane. Let LM ⊂ C (S1,M) be the space of contractible smooth ∞ loops in M and LgM the space of equivalence classes x = [x, dx], where x ∈ LM

2 and dx : D M is a smooth extension of x to the disk. We will often call x a (disk) capping of x. Two such pairs (x, dx) and (y, dy) are equivalent if → x = y, ω(dx # dy) = 0, and c1(dx # dy) = 0. Here dx # dy denotes the sphere

2 S M we obtain by gluing the disks dx and dy along their common boundary.

Equivalently, the pairs (x, dx) and (y, dy) are equivalent if x = y, →

∗ ∗ ∗ ∗ dxω = dyω, and dxc1 = dyc1. 2 2 2 2 ZD ZD ZD ZD

We denote this equivalence by (x, dx) ∼ (y, dy).

Then LgM is a covering space2 of LM. We deﬁne the action of A ∈ Γ on

[x, dx] ∈ LgM by concatenating an A-sphere to the disk dx. The group of deck transformations of LgM can be identiﬁed with Γ.

Deﬁnition 8. Given a periodic Hamiltonian H : S1 × M R, the action func-

2 It is not the universal covering space of LM, however, which we get if we deﬁne (x, dx) ∼ (y, dy) exactly when x = y and dx # dy is a torsion class. →

45 tional AH : LgM R is deﬁned by

1 → ∗ AH([x, dx]) := dxω − H(t, x(t))dt. 2 ZD Z0

Proposition 10. The action functional AH : LgM R is well-deﬁned. Further- more, it satisﬁes AH(A · x) = AH(x) + ω(A) for all x ∈ LgM and A ∈ Γ. →

Proof. Let x = [x, dx] and y = [y, dy] be two elements of LgM. If x = y, then

1 ∗ ∗ AH(x) − AH(y) = (dxω − dyω) − H(t, x(t)) − H(t, y(t) dt 2 ZD Z0 ∗ ∗ = (dxω − dyω) = ω(dx#dy) = 0. 2 ZD

The action AH is therefore well-deﬁned. Concatenating with an A-sphere does not change the H-dependent part of the action. It increases the integral over the disk by ω(A).

Let P(H) be the set of contractible periodic orbits of H and Pe(H) ⊂ LgM the set of all cappings of elements of P(H). We will call Pe(H) the set of contractible periodic orbits of H, if there is no confusion.

Theorem 6. The set of critical points of the action functional AH is exactly the set Pe(H) of capped periodic orbits of H.

Proof. We follow the proofs given in [AD14, MS12]. Let x = [x, dx] ∈ LgM and

1 ∗ ξ ∈ C (S , x TM) = TxLM = Tx LgM. For > 0 sufﬁciently small, we can extend ∞ these maps to

x : (−, ) × S1 M, (s, t) 7 xs(t) = x(s, t),

d : (−, ) × D2 M, (s, z) 7 ds(z) = d(s, z), → →

→ →

46 such that

0 0 x (t) = x(t), d (z) = dx(z),

s 2πit s s d (e ) = x (t), ∂sx (t)|s=0 = ξ(t).

2 ∗ Furthermore we extend ξ to eξ ∈ C (D , dxTM) by setting ∞ ∂d eξ(z) := (s, z) . ∂s s=0 For any s ∈ (−, ), we have

1 s s s ∗ s AH([x , d ]) = (d ) ω − H(t, x (t))dt. 2 ZD Z0 The derivative of the ﬁrst term in the action is d d (ds)∗ω = (ds)∗ω = d∗(L ω) x eξ ds s=0 D2 D2 ds s=0 D2 Z Z Z 1 = d(d∗ι ω) = x∗(ι ω) = ω(ξ(t), x(t))dt. x eξ ξ ˙ 2 1 ZD ZS Z0 The derivative of the second term is

1 1 d s s − H(t, x (t))dt = − (dHt) (∂sx (t)|s=0) dt ds s=0 0 0 Z Z 1 = − ω ξ(t),XH(t, x(t)) dt. Z0 We conclude that

1 d s s (dAH)x(ξ) = AH([x , d ]) = ω ξ(t), x˙ (t) − XH(t, x(t)) dt (3.3) ds s=0 Z0 for all ξ ∈ C (S1, x∗TM). The result follows because ω is nondegenerate. ∞

3.3.3 The Conley-Zehnder Index

The Morse chain complex is graded by the Morse index, which is straightforward to deﬁne and very intuitive. The dimension of the space of negative gra-

47 dient trajectories connecting two given critical points is the difference of their Morse indices.

The grading in the Floer chain complex is given by the Conley-Zehnder index of a capped periodic orbit. Unlike for the Morse index, the intuition behind it lies hidden in the dimension analysis of the moduli spaces that deﬁne the Floer boundary operator. Nevertheless, we present the deﬁnition at this point.

Our treatment is based on [AD14] and [Sal99].

∗ Step one. Let x = [x, dx] ∈ Pe(H). The symplectic vector bundle dxTM is (symplectically) trivial since D2 is contractible. Fix such a trivialization. We can

t interpret dx(0)φH as a matrix A(t) ∈ Sp(2n),

d φt x(0) H Tx(0)M, ωx(0) (Tx(t)M, ωx(t))

=∼ =∼

2n A(t) 2n R ,Ω0 R ,Ω0 and obtain a path A :[0, 1] Sp(2n) satisfying A(0) = I2n. The fact that H is nondegenerate guarantees that det(A(1)−I2n) 6= 0. All symplectic trivializations → ∗ of dxTM are homotopic [MS17, Section 2.6], and therefore the homotopy class of A does not depend on the choice of trivialization.

Step two. We now associate an integer to any path A :[0, 1] Sp(2n) starting at A(0) = I2n and ending in the space →

? Sp (2n) := S ∈ Sp(2n) det(S − I2n) 6= 0 . of symplectic matrices that do not have eigenvalue 1. The integer will not depend on the homotopy class of the path. The space Sp?(2n) has two connected

48 components,

+ Sp (2n) := S ∈ Sp(2n) det(S − I2n) > 0 ,

− Sp (2n) := S ∈ Sp(2n) det(S − I2n) < 0 . + + Fix a matrix in each component: we choose W = −I2n in Sp (2n) and the diagonal matrix   2 0 0 0 ··· 0     0 1 0 0 ··· 0   2    0 0 −1 0 ··· 0  −   W =     0 0 0 −1 ··· 0    ......  ......      0 0 0 0 ··· −1 in Sp−(2n). We can extend A :[0, 1] Sp(2n) to a path Ae :[0, 2] Sp(2n) by connecting A(1) ∈ Sp±(2n) to W±. The extension is unique up to homotopy → → relative to its endpoints because the inclusion Sp±(2n) Sp(2n) induces the zero map on fundamental groups. →

Recall that symplectic matrices admit a polar decomposition: every symplectic matrix S can be uniquely written as the product of a unitary matrix S(StS)−1/2 and a symmetric, positive-deﬁnite, symplectic matrix (StS)−1/2. The map

F :[0, 1] × Sp(2n) Sp(2n) (t, S) 7 S(StS)−t/2 → is a deformation retraction of Sp(2n) onto→ U(n) = Sp(2n) ∩ O(2n), that is, a homotopy from the identity on Sp(2n) to a retraction r : Sp(2n) U(n). The induced map r∗ : π1(Sp(2n)) π1(U(n)) is an isomorphism. The complex → 1 determinant detC : U(n) S also induces an isomorphism on fundamental → → 49 groups. The rotation map is the composition

det ρ : Sp(2n) r U(n) C S1.

1 ∼ It induces an isomorphism ρ∗ : π1(Sp(2n)) π1(S ) = Z.

The matrices W± satisfy ρ(W±) = ±1.→ The composition ρ2 ◦ Ae :[0, 2] S1 is therefore a loop. The Conley-Zehnder index µ (A) of the loop A :[0, 1] CZ → Sp(2n) is the equal to the degree of the map ρ2 ◦ Ae : S1 S1. Since the degree → of a map only depends on its homotopy class, the Conley-Zehnder index only → depends on the homotopy class of the path A. It does not depend on the chosen extension.

The difference in Conley-Zehnder index of two capped periodic solutions will be the dimension of the moduli space of Floer trajectories connecting them. Since only differences in the index matter, we can choose a convenient normal- ization of µCZ, i.e. add a constant to the index. We choose not change the index as deﬁned here; this is the convention in [Alb05, Alb10].

Proposition 11 ([AD14, Cor. 7.2.2]). Let H ∈ C (M2n) be a C2-small Morse ∞ function. We can interpret H as an autonomous nondegenerate Hamiltonian. If x ∈ Crit(H) and we denote the constant periodic solution by x = [x, dx ≡ x], then µCZ(x) = n − µMorse(x; H).

We denote the set of all capped periodic orbits of a nondegenerate Hamilto-

1 nian H : S × M R of Conley-Zehnder index k by Pek(H). For an equivalent deﬁnition and properties of the Conley-Zehnder index, see [AD14, Sal99]. For → our purposes, we only need to know how the action of Γ on Pe(H) changes the index.

50 Proposition 12 ([Sal99, p. 24]). Let H : S1 × M R be a nondegenerate Hamil- tonian. Then µCZ(A · x) = µCZ(x) + 2c1(A) for any x ∈ Pe(H) and A ∈ Γ. → Remark 6. We can deﬁne a Conley-Zehnder index for uncapped contractible periodic orbits x ∈ P(H). It depends on a choice of symplectic trivialization of

∗ x TM, which is equivalent to a choice of capping [x, dx]. The previous proposition makes it clear that this version of the Conley-Zehnder index is only well- deﬁned modulo 2NM, where NM is the minimal Chern number.

3.3.4 The Floer Chain Groups

In Morse theory, we constructed chain groups using the critical points of a

Morse function. We do the same here using the critical points of the action functional, i.e. the contractible periodic orbits of the Hamiltonian.

Deﬁnition 9. For a nondegenerate Hamiltonian H : S1 × M R, we deﬁne the Floer chain groups →

# {x | ax 6= 0, AH(x) ≥ c} < , ∀c ∈ R CFk(H) := axx (3.4)   x∈P (H) ax ∈ Z2, ∀x   Xek ∞  for all k ∈ Z. Note that the elements of CFk(H) satisfy a Novikov condition . We can let Λ0 act on CFk(H) by setting !   A λAe ·  axx := λAax (−A) · x. A∈Γ A∈Γ X0 x∈XPek(H) X0 x∈XPek(H)

Proposition 13. CFk(H) is a ﬁnite-dimensional vector space over Λ0.

Proof. First note that c1(A) = 0 for all A ∈ Γ0, and therefore

µCZ(A · x) = µCZ(x) + 2c1(A) = µCZ(x) = k

51 for all x ∈ Pek(H). The action preserves the grading.

Next we need to show that the sum on the right hand side satisﬁes the Novikov condition in (3.4). Fix c ∈ R and consider all pairs (x, A) for which x ∈ Pek(H) and A ∈ Γ0 such that ax 6= 0, λA 6= 0 and

AH(x) − ω(A) = AH(x#(−A)) ≥ c.

0 The Novikov condition (3.2) guarantees that ω(A) ≥ c for all A ∈ Γ0 for which

0 λA 6= 0. Since AH(x) ≥ c + ω(A) ≥ c + c , there can only be finitely many such x because of (3.4). For a given such x, the Novikov condition (3.2) states that there can only be finitely many such A because ω(A) ≤ AH(x) − c. We conclude that there are only finitely many such pairs (x, A).

Recall that H only has ﬁnitely many (uncapped) periodic orbits, say x1, . . . , xN. Fix a capping of index k on each of them. Writing ax = ai,A when x = (−A) · xi, we can express any element of CFk(H) as

N N ! A axx = ai,Axi#(−A) = ai,Ae · xi. i=1 A∈Γ i=1 A∈Γ x∈XPek(H) X X0 X X0 a eA A ((−A) · The coefﬁcients A∈Γ0 i,A satisﬁes the Novikov condition: since H xi) = AH(xi)−ω(PA), the occurring symplectic areas are bounded below because the actions are bounded above. Therefore {x1,..., xN} forms a basis for CFk(H) as a vector space over Λ0.

3.3.5 The Boundary Operator

Let J be an almost complex structure on M compatible with the symplectic form and H : S1 ×M R a nondegenerate Hamiltonian. The compatible Riemannian

→ 52 metric on M is given by gp(v, w) = ωp(v, Jpw) for all v, w ∈ TpM. It in turn deﬁnes a metric on LgM given by

1 hξ, ρix := gx(t) ξ(t), ρ(t) dt 0 Z1 = ωx(t) ξ(t),Jx(t)ρ(t) dt, Z0 1 ∗ for all ξ, ρ ∈ C (S , x TM) = Tx LgM. The gradient of the action functional is by ∞ definition the unique vector field ∇AH on LgM that satisfies

∇xAH, ξ = (dxAH)(ξ) (3.5) for all ξ ∈ C (S1, x∗TM). Using the expression for the differential of the action ∞ in (3.3), we see that the gradient is characterized by

1 1 ωx(t) ∇x(t)AH,Jx(t)ξ(t) dt = ωx(t) ξ(t), x˙ (t) − XH(t, x(t)) dt 0 0 Z 1 Z 1 ωx(t) Jx(t)∇x(t)AH, ξ(t) dt = ωx(t) x˙ (t) − XH(t, x(t)), ξ(t) dt 0 0 Z 1 Z ⇐⇒ ωx(t) Jx(t)∇x(t)AH − x˙ (t) + XH(t, x(t)), ξ(t) dt = 0 Z0 1 ∗ for all⇐⇒ξ ∈ C (S , x TM). We conclude that the gradient of AH is given by ∞ ∇x(t)AH = −Jx(t) x˙ (t) − XH(t, x(t)) for all x ∈ LM.

Consider a gradient ﬂow line of AH under the projection LgM LM: it is a smooth map u : R × S1 M with the property that u(s, ·) is a contractible loop → for every s ∈ R, and it satisﬁes the equation → h i ∂su(s, t) + J u(s, t) ∂tu(s, t) − XH t, u(s, t) = 0.

For cylinders u : R × S1 M, we will always use s for the coordinate in the linear direction and t for the one in the circular direction. We introduce the →

53 1 1 Floer operator ∂J,H : C (R × S ,M) C (R × S , TM) deﬁned by ∞ ∞

∂J,Hu := ∂su +→J(u) (∂tu − XH(t, u)) . (3.6)

The equation ∂J,Hu = 0 is called the Floer equation. It is a Hamiltonian perturbation of the Cauchy-Riemann equation ∂su + J(u)∂tu = 0. Solutions of the Floer equation are called Floer cylinders. Since the Floer operator is elliptically regular, Floer cylinders in the class C1 are automatically in the class C . See ∞ [AD14, Chapter 12] or [MS12, Appendix A] for more information.

Just as negative gradient ﬂow lines of a Morse function connect critical points, we want to show that Floer cylinders connect contractible periodic orbits of the Hamiltonian.

Deﬁnition 10. The energy of a smooth map u : R × S1 M is

1 2 E(u) := |∂su| dt ds. → ∞ Z− Z0 Proposition 14 ([Sal99, Prop. 1.21])∞. Let u : R × S1 M be a Floer cylinder. Then the following are equivalent: → 1. Its energy is ﬁnite: E(u) < .

± 2. It connects contractible periodic∞ orbits: there exist x ∈ P(H) such that ± 1 lims ± u(s, t) = x (t) and lims ± ∂su(s, t) = 0 uniformly in t ∈ S .

3. It converges→ ∞ exponentially: there→ exist∞ constants δ > 0 and c > 0 such that

−δ|s| 1 |∂su(s, t)| ≤ ce for all (s, t) ∈ R × S .

So far we have neglected to discuss how the disc cappings are affected under the gradient flow of the action functional. The gradient flow is actually a map u : R × D2 M u = u : R × S1 M e that restricts to a Floer cylinder e R×∂D2 . If the ± ± energy of u is finite, it connects two capped periodic orbits x = [x , dx± ]. In that → →

54 2 case lims ± ue(s, z) = dx± (z) for all z ∈ D . The cylinder u is contractible, and + dx+ is equivalent→ ∞ to the capping of x we get by gluing dx− onto the cylinder u.

In the notation we used to deﬁne LgM, the cappings are related by dx− # u ∼ dx+ .

Figure 3.1: Floer cylinder with disk cappings connecting two contractible periodic solutions (in blue).

To deﬁne the boundary operator of the Floer chain complex, we need the moduli spaces

∂J,H(u) = 0, dx # u ∼ dy,

Floer :  1  M (x, y; J, H) = u : R × S M lims − u(s, ·) = x,      lim→s ∞u(s, ·) = y →     of Floer cylinders connecting the capped period orbits→∞x and y. 

Floer Proposition 15. The energy of u ∈ M (x, y; J, H) is E(u) = AH(y) − AH(x). In particular, the energy is constant, and the action increases along ﬁnite energy

Floer cylinders.

55 Proof. We can rewrite the equation ∂J,Hu = 0 as J(u)∂su = ∂tu − XH(t, u). The energy of u is

1 1 2 E(u) = |∂su| dt ds = ω ∂su, J(u)∂su dt ds ∞ ∞ Z− Z0 Z− Z0 1 = ∞ ω ∂su, ∂tu − XH(∞t, u) dt ds ∞ Z− Z0 1 1 = ∞ ω(∂su, ∂tu)dt ds − ω ∂su, XH(t, u) dt ds ∞ ∞ Z− Z0 Z− Z0 1 ∗ = ∞ u ω − ω ∂su, XH(∞t, u) dt ds. 1 ∞ ZR×S Z− Z0 ∞ For the ﬁrst term, the condition dx # u ∼ dy implies that

∗ ∗ ∗ dxω + u ω = dyω. 2 1 2 ZD ZR×S ZD If we switch the order of integration in the second term and use the fact that

ω(XHt , ·) = −dHt, we see that

1 1 ω ∂su, XH(t, u) dt ds = dHt(∂su)ds dt −∞ 0 0 −∞ Z Z Z 1 Z 1 ∞ = H(t,∞ y(t))dt − H(t, x(t))dt. Z0 Z0 We conclude that

1 1 ∗ ∗ E(u) = dyω − dxω − H(t, y(t))dt + H(t, x(t))dt 2 2 ZD ZD Z0 Z0 = AH(y) − AH(x).

Remark 7. The advantage of the approach used in the proof of Proposition 15 is that it can easily used to obtain energy bounds for other moduli spaces of

56 cylinders. A different approach is to use (3.3) to get

1 E(u) = ω ∂su, ∂tu − XH(t, u) dt ds ∞ Z− Z0 = ∞(dAH)u(s,·) ∂su(s, ·) ds ∞ Z− d = ∞ AH(u(s, ·)) ds ∞ ds Z−

= AH∞(y) − AH(x).

Theorem 7 (Floer/Salamon-Hofer). For generic choices of J and H, the moduli

Floer space M (x, y; J, H) is a smooth manifold of dimension µCZ(y) − µCZ(x). It carries a natural action of R given by (σ · u)(s, t) = u(s + σ, t), which is free if x 6= y. The quotient

Floer Floer M (x, y; J, H) Mc (x, y; J, H) := R

is a ﬁnite set if µCZ(y)−µCZ(x) = 1, and it can be compactiﬁed by adding broken cylinders if µCZ(y) − µCZ(x) = 2.

Figure 3.2: Broken conﬁguration with two Floer cylinders, with periodic orbits in blue.

57 We introduce some of the techniques needed to prove this theorem in Section 3.4. Note that, because we used the gradient ﬂow instead of negative gradient ﬂow of the action functional, the Conley-Zehnder index increases along Floer cylinders.

Deﬁnition 11. The Floer boundary operator is the Z2 linear map ∂F : CFk(H)

CFk−1(H) deﬁned on generators by → Floer ∂Fy := #2Mc (x, y; J, H) · x. (3.7) x∈PeXk−1(H)

The boundary ∂Fy encodes the numbers of Floer cylinders connecting y to periodic orbits of Conley-Zehnder index k − 1.

Lemma 4. The Floer boundary operator is a well-deﬁned linear map of vector spaces over Λ0.

2 Proposition 16. The boundary operator satisﬁes ∂F = 0.

Proof. Let y ∈ CFk(H). Then   2 Floer Floer ∂Fy =  #2Mc (x2, x1; J, H) · #2Mc (x1, y; J, H) · x2. x2∈PXek−2(H) x1∈PXek−1(H) The coefﬁcient of x2 ∈ Pek−2(H) is exactly number of elements mod 2 in the

Floer boundary of the one-dimensional moduli space Mc (x2, y; J, H). Since it is compact by Theorem 7 and the boundary of a compact one-dimensional manifold has an even number of points, those coefﬁcients are all zero.

Deﬁnition 12. The Floer homology groups of the nondegenerate Hamiltonian H : S1×M R and the compatible almost complex structure J are the homology groups of the complex (CF∗(H), ∂F): → ker (∂F : CFk(H) CFk−1(H)) HFk(H) := . im (∂F : CFk+1(H) CFk(H)) They are well-deﬁned for generic choices of H→and J. →

58 3.4 Moduli Spaces of Floer Cylinders

We discuss the tools used to study moduli spaces of various kinds of cylinders. We use the Floer cylinders (i.e. the moduli spaces appearing in the Floer boundary operator) as our leading example. The analysis always follows the same pattern: start by showing the the moduli spaces are smooth manifolds. Next study their compactness: show that they are either compact, or find an elegant way of compactifying them. Finally show that the compactified moduli spaces are still smooth manifolds. We focus on the first two steps, smoothness and compactification.

3.4.1 Fredholm Operators and the Implicit Function Theorem

The main tool to show that moduli spaces in Floer theory are smooth manifolds is an inﬁnite-dimensional version of the Implicit Function Theorem. We state some results from [AD14, Appendix C.2].

Let X and Y be Banach spaces. Recall that a linear operator D : X Y is bounded if there exists C > 0 such that ||D(x)||Y ≤ C||x||X for all x ∈ X. A linear → operator is bounded if and only if it is continuous. Its operator norm is

||D||op := inf C > 0 ||D(x)||Y ≤ C||x||X for all x ∈ X

||D(x)||Y = sup = sup ||D(x)||Y. x∈X ||x||X x∈X x6=0 ||x||X=1

A compact operator is a bounded linear operator D : X Y for which the image of the unit ball in X has compact closure in Y. They are exactly the bounded → linear operators of ﬁnite rank.

59 Deﬁnition 13. A bounded linear operator D : X Y is Fredholm if it satisﬁes the following equivalent conditions (the Fredholm property): →

1. Its image is closed in Y, and both its kernel and cokernel are ﬁnite- dimensional.

2. There exists a bounded linear operator S : Y X such that both idX −S◦D

and idY − D ◦ S are compact operators. →

Its (Fredholm) index is ind(D) := dim(ker D) − dim(coker D).

Fredholm operators are exactly the bounded linear operators that are invert- ible when ﬁnite-dimensional effects are ignored.

Proposition 17. The Fredholm property and the Fredholm index are stable under perturbation: in the space of bounded linear operators X Y equipped with the operator norm, the set of Fredholm operators is open and the Fred- → holm index is locally constant. In other words, if D : X Y is a Fredholm operator, there exists > 0 such that all bounded linear operators L : X Y → satisfying ||D − L|| < are also Fredholm and satisfy ind(L) = ind(D). op →

Deﬁnition 14. A smooth map f : X Y is Fredholm if dxf : X Y is a Fredholm operator for all x ∈ X. Its Fredholm index is the Fredholm index of its differen- → → tials. (These indices are equal because of the stability property). A value y ∈ Y

−1 is regular if dxf : X Y is onto for all x ∈ f (y).

Theorem 8 (Implicit→ Function Theorem, [AD14, Theorem C.2.13]). Let X and Y be Banach spaces and f : X Y a Fredholm map. Let M be the level set f−1(y) at a regular value y ∈ Y. Then M is a ﬁnite-dimensional smooth manifold. For → any x ∈ M we can identify the tangent space TxM with ker(dxf). Therefore

dim M = dim TxM = dim ker(dxf) = ind(dxf) = ind(f).

60 3.4.2 Smoothness, Linearization, and Transversality

1 1 Let E C (R × S ,M) be the vector bundle whose ﬁber Eu over u : R × S M

∞ 1 ∗ is the space Eu = C (R × S , u TM) of vector ﬁelds on M along u. We can → → ∞ interpret the Floer operator ∂J,H deﬁned in (3.6) as a section of this bundle, which we will denote by F : C (R × S1,M) E. We identify the space of Floer ∞ cylinders with the intersection of the image of F and the zero section of E. If this → intersection is transverse, then this space is a smooth manifold. Transversality is

1 ∗ 1 ∗ equivalent to the operator DuF : C (R × S , u TM) C (R × S , u TM) being ∞ ∞ onto for every Floer cylinder u, where →

1 ∗ 1 duF C (R × S , u TM) TuC (R × S ,M) TuE ∞ ∞ prEu DuF 1 ∗ Eu C (R × S , u TM) ∞ is the projection of the differential duF onto the ﬁber Eu.

Let x and y be two capped periodic solutions of H. The Fredholm property of DuF and the Implicit Function Theorem play a crucial role in proving that the space MFloer(x, y; J, H) of connecting Floer cylinders is a smooth manifold. Unfortunately C (R×S1, u∗TM) is not a Banach space. We therefore ﬁrst extend ∞ DuF to an operator

1,p 1 ∗ p 1 ∗ DuF : W (R × S , u TM) L (R × S , u TM) (3.8) between a Sobolev space and an Lp space→ (for 1 < p < ), which are Banach spaces. ∞

Theorem 9 ([AD14, Theorem 8.1.5]). The operator DuF in (3.8) is Fredholm for

Floer every u ∈ M (x, y; J, H), and its Fredholm index is µCZ(y) − µCZ(x).

61 While the preceding theorem guarantees that DuF is Fredholm for all Hamil- tonians and all compatible almost complex structures J, it will not always be onto. Denote the space of almost complex structures compatible with the symplectic form by J and the space C (S1 × M) of periodic Hamiltonians by H. ∞ We say a pair (J, H) ∈ J × H is regular if the operator DuF in (3.8) is onto for all Floer cylinders. The following theorem states that regularity is a generic property in J × H.

Theorem 10 ([AD14, Theorem 8.1.1]). The set (J × H)reg of regular pairs is a comeager subset of J × H, i.e. it contains the intersection of a countable collection of open dense subsets.

3.4.3 Compactness

Before we dive into the analysis of compactness, we review some notation for the relevant moduli spaces in the table below.

Moduli space Elements MFloer(J, H) Finite energy Floer cylinders McFloer(J, H) Unparametrized ﬁnite energy Floer cylinders MFloer(x, y; J, H) Floer cylinders connecting capped periodic orbits x to y McFloer(x, y; J, H) Unparametrized Floer cylinders connecting capped periodic orbits x to y

Table 3.1: Different moduli spaces of Floer cylinders

The aspherical case [AD14]. First assume the space of all ﬁnite energy Floer cylinders MFloer(J, H) is compact in the C topology. This is the case if (M, ω) ∞ 62 is aspherical [AD14, Theorem 6.5.4]. Since every ﬁnite energy Floer cylinder connects two contractible periodic orbits, this space decomposes as

G MFloer(J, H) = MFloer(x, y; J, H).

x,y∈Pe(H)

A sequence (un)n=1 of unparametrized cylinders converges to u in the quo-

Floer ∞ tient Mc (J, H) if there exists a sequence (σn)n=1 of real numbers such that the

∞ Floer sequence of rescaled cylinders (σn · un)n=1 converges to u in M (J, H). ∞ Fix two contractible periodic orbits x 6= y ∈ Pe(H). We now focus our attention on the moduli space McFloer(x, y; J, H) of unparametrized Floer cylinders connecting x to y. It is not necessarily compact in the quotient topology; our goal is to compactify it. Let d be a metric on LgM that induces the C topology. ∞ Since the periodic orbits of a nondegenerate Hamiltonian are isolated, we can choose > 0 such that the -neighborhoods B(x, ) of all x ∈ Pe(H) are pairwise disjoint.

Floer Let (un)n=1 be a sequence in M (x, y; J, H). For every n ∈ N, consider the

∞ 1 loops un(s, ·): S M for varying s ∈ R. Since un connects x to y, they cannot

1 all lie in B(x, ). Let un(σ , ·) be the ﬁrst loop to exit B(x, ), or, equivalently, → n

1 σn = inf σ ∈ R d (x, un(σ, ·)) ≥ . 1 1 The rescaled cylinder σn · un satisﬁes the property that the loop (σn · un)(s, ·) ex- its B(x, ) for the ﬁrst time at s = 0. Since MFloer(J, H) is compact, the sequence

1 (σn · un)n=1 of ﬁnite energy Floer cylinders converges (after passing to a subse-

∞ 1 quence) to a Floer cylinder u connecting x and x1 6= x. If x1 = y then (un)n=1

1 Floer ∞ converges to u in the quotient space Mc (x, y; J, H). If x1 6= y, then we repeat the process:

63 1 ∗ 1 • Since u tends to x1 as s , there exists s ∈ R such that the loop u (s, ·)

∗ lies in B(x1, ) for all s ≥ s . → ∞ 1 1 1 ∗ 1 ∗ • Since (σn · un)n=1 converges to u , the loop (σn · un)(s , ·) = un(σn + s , ·) ∞ lies in B(x1, ) for n sufﬁciently large.

1 • Since every un approaches y as s , the loops un(σn + s, ·) must exit

∗ 1 2 B(x1, ) at some s > s . Let un(σ + σ , ·) be the ﬁrst loop to exit B(x1, ), n → ∞n 2 1 ∗ or, equivalently, σn = inf σ ≥ σn + s d (x1, un(σ, ·)) ≥ .

Floer 2 • Since M (J, H) is compact, the sequence (σn · un)n=1 of ﬁnite energy ∞ Floer cylinders converges (after passing to a subsequence) to a Floer cylin-

2 der u connecting x1 and x2 6= x1.

We keep repeating this process until we reach y. Since the Conley-Zehnder index strictly increases along the cylinders we constructed, the process must terminate in a ﬁnite number of steps.

Floer Theorem 11. Let (un)n=1 be a sequence in M (x, y; J, H). Then there exist ∞ contractible periodic orbits x = x0, x1,..., xm = y, together with Floer cylin-

i Floer i ders u ∈ M (xi−1, xi; J, H) and sequences (σn)n=1 of real numbers, for all ∞ i ∈ {1, . . . , m}, such that

i i lim σn · un = u n with respect to the C topology,→∞ for all i ∈ {1, . . . , m}. ∞

If we extend the space McFloer(x, y; J, H) of unparametrized Floer cylinders by adding these broken configurations, then the convergence of sequences defined above defines the Gromov-Floer topology on this space.

Because the Conley-Zehnder index is positive along Floer cylinders, the difference µCZ(y) − µCZ(x) limits how often breaking can happen. If

64 Floer Mc (x, y; J, H) has dimension zero, i.e. if µCZ(y) − µCZ(x) = 1, there cannot be any breaking. The moduli space is already compact. In dimension one, there can be at most one break.

2 To finish the argument that ∂F = 0, we need to make sure that compactified one-dimensional moduli spaces McFloer(x, y; J, H) are manifolds when equipped with the Gromov-Floer topology. To study what they look like in the neighborhood of broken configurations, we need a gluing theorem that allows us to construct a single Floer cylinder out of a broken configuration. We refer to

[AD14, MS12, Sal99] for details.

The monotone case. It is possible that the space of all ﬁnite energy Floer cylinders MFloer(J, H) is not compact in the C topology. In the aspherical case,

∞2 one can show that the energy density |∂su| of Floer cylinders is uniformly bounded. The Arzela-Ascoli Theorem [AD14, Thm. C.1.1] guarantees that any sequence of Floer cylinders has a subsequence that converges to some map u ∈ C0(R × S1,M) in the C0 topology. Elliptic regularity [MS12, Appendix B] guarantees that u is smooth and satisﬁes the Floer equation.

In general the energy density of a sequence (un)n=1 can accumulate around a ∞ ﬁnite number of points z1, . . . , zl in the domain. The point z∗ = (s∗, t∗) ∈ R × S1 is such an accumulation point if it admits a sequence (Vn)n=1 of open neigh- T V = {z∗} |∂ u |2 =∞ borhoods such that n=1 n and supn∈N Un s n . The sequence ∞ 1 (un) converges to a Floer cylinder u : R × SR M in the C topology on n=1 ∞ ∞ ∞ compact subsets of (R × S1)\{z1, . . . , zl}. →

To capture the energy accumulation, we construct a J-holomorphic sphere at every accumulation point. Fix i ∈ {1, . . . , l}. We identify a neighborhood of zi

65 i i with C by setting z = s + it. There exists a sequence (ζn)n=1 that converges to z

i ∞ and a sequence (rn)n=1 that converges to zero, such that ∞ 2 lim |∂sun| = . i i i ZB(ζn,rn) i →∞ i ∞i i Deﬁne the map vn : Ωn ⊆ C M by vn(z) = un(ζn + rnz) for all n ∈ N, where i S i Ω is a neighborhood of 0 in C such that Ωn = C. A subsequence of (v ) n → n=1 n n=1 ∞ ∞ converges to a nonconstant J-holomorphic map vi : C M. It has ﬁnite energy and can therefore (by removal of singularities, see [MS12, Theorem 4.1.2]) be → extended to a J-holomorphic sphere vi : C ∪ { } =∼ (S2, j) (M, J), which we call a sphere bubble. Since vi( ) = u(zi), the bubble connects to the Floer ∞ → cylinder. ∞

If we add conﬁgurations with sphere bubbles to MFloer(J, H), the energy density of a sequence of Floer cylinders with sphere bubbles can accumulate at a point of the sphere bubble. As a result, we need to include conﬁgurations where sphere bubbles form on sphere bubbles. In general, we can get trees of sphere bubbles, see [MS12].

When we compactify the moduli space McFloer(x, y; J, H), we need to add conﬁgurations of broken Floer cylinders that include trees of sphere bubbles. We remember that moduli spaces of Floer cylinder whose energy is uniformly bounded, are compact up to breaking and sphere bubbling.

We can use the following results to limit the number of broken cylinders and sphere bubbles:

Floer 1. If a sequence (un)n=1 in Mc (J, H) converges to a broken conﬁguration ∞

66 with Floer cylinder (u1, . . . , uk) and sphere bubbles (v1, . . . , vl), then

k l i i E(u ) + E(v ) = lim sup E(un). i=1 i=1 n X X →∞ 2. There exists a number ~ > 0 such that any nonconstant J-holomorphic

sphere v : S2 M satisﬁes E(v) ≥ ~. If the energy of the Floer cylinders is uniformly bounded by E , then at most L spheres can bubble off, where → max E L = max . ~

3. For a sequence in McFloer(x, y; J, H), the Fredholm index is additive with respect to convergence to conﬁgurations with broken cylinders and bubbles in the sense that

k l

µCZ(y) − µCZ(x) + 1 = ind Fun = ind Fui + ind Fvi i=1 i=1 Xk Xl i = ind Fui + 2c1(v ). i=1 i=1 X X

We conclude that, if the space McFloer(x, y; J, H) has dimension zero, it is already compact. In other words, there cannot be any breaking or bubbling. In dimension one, either the Floer cylinder can break once, or a holomorphic sphere of Chern number 1 can bubble off. If sphere bubbling occurs, the limit cylinder does not depend on the s-coordinate; it is a periodic orbit. One then shows that for generic choices of J, holomorphic spheres of Chern number 1 do not intersect any periodic orbits. Sphere bubbling therefore does not occur generically in the one-dimensional case.

Remark 8. Sphere bubbling is characterized by the accumulation of energy in a shrinking sequence of open balls in the domain whose centers converge to

67 a point. We can similarly characterize breaking as the accumulation of energy, but the centers of the balls run off to ± . We encounter a third vari- ant of energy accumulation when we enforce Lagrangian boundary conditions, ∞ e.g. for holomorphic strips in Lagrangian Floer theory, or for Floer cylinders u : (− , 0] × S1 M such that u({0} × S1) ⊂ L, where L is a ﬁxed Lagrangian. If the points in the domain at which energy accumulation occurs converge to ∞ → a point on the boundary {0} × S1, we need to allow for the formation of a disk bubble. The Fredholm index remains additive: if a sequence (un)n=1 converges ∞ to a conﬁguration with Floer cylinders (u1, . . . , uk), sphere bubbles (v1, . . . , vl), and disk bubbles (w1, . . . , wm), then

k l m i j ind Fun = ind Fui + 2c1(v ) + µMaslov(w ). i=1 i=1 i=1 X X X See [Flo89a, Oh93, Aur14] for a more detailed analysis.

The semipositive case. Hofer and Salamon give a good summary in [HS95] of the challenges in extending Floer theory from the monotone to the semipositive setting:

1. The existence of nonconstant holomorphic spheres of nonpositive ﬁrst Chern number is an obstruction to the compactness of the moduli spaces.

Luckily, there are no such spheres with negative ﬁrst Chern number for generic choices of J. Just like in the monotone case, the holomorphic spheres of ﬁrst Chern number one do not generically intersect periodic

solutions. A sophisticated argument shows that the holomorphic spheres of Chern number zero cannot connect to the Floer cylinders either.

2. Unlike in the monotone case, there can be inﬁnitely many unparametrized

Floer cylinders connecting two periodic orbits, and their energies can be

68 unbounded. We introduced disk cappings and the Novikov ring to help with the bookkeeping.

3.5 The Piunikhin-Salamon-Schwarz (PSS) Isomorphism

The PSS isomorphism is one approach to study the relationship between Floer homology and Morse/singular homology. We give a very limited introduction here, with a focus on what we will need in the rest of this chapter.

Let H : S1 ×M R be a nondegenerate periodic Hamiltonian, J a compatible almost complex structure on M, g be a Riemannian metric on M, and f ∈ C (M) → ∞ a Morse function. Let q ∈ M be a critical point of f and x = [x, dx] a contractible periodic orbit of H. Fix a smooth cutoff function β : R [0, 1] satisfying β(s) = 0 if β ≤ 1 and β(s) = 1 if s ≥ 1. 2 →

Deﬁnition 15. The moduli space MPSS(q, x; J, H, f, g) of plumber’s helper solutions consists of all pairs (γ, u) such that:

1. The half-line γ : (− , 0] M satisﬁesγ ˙ + ∇gf ◦ γ = 0.

1 2. The cylinder u : R ×∞S →M satisﬁes E(u) < and

∂su +→J(u) ∂tu − β(s)XH(t,∞ u) = 0. (3.9)

3. The following boundary conditions are satisﬁed:

• The negative gradient ﬂow half-line has to converge to a critical point

of f as s − . We require that lims − γ(s) = q.

→ ∞ → ∞

69 1 • Since β(s) = 0 for s ≤ 2 , the cylinder u satisﬁes the Cauchy-Riemann 1 1 equation for such values of s. Its restriction to − , 2 × S is a holomorphic punctured disk. It has ﬁnite energy since E(u) < , so it ∞ can be continuously extended over the puncture (by removal of sin- ∞ gularities, see [MS12, Theorem 4.1.2]). We denote the value of the extension at the puncture by u(− ) and require that γ(0) = u(− ).

• The cylinder u satisﬁes the Floer∞ equation for s ≥ 1 and has ﬁnite∞

energy. It therefore converges to a contractible periodic orbit of XH as

s . We require that lims u(s, ·) = x.

→∞ 4. Recall→ that ∞u is a disk after continuously extending it by u(− ). We en-

force the homotopy conditions ω(u # dx) = 0 and c1(u # dx) = 0. ∞

Figure 3.3: A plumber’s helper solution.

Proposition 18 ([PSS96]). For generic choices of J, H, f, and g, the moduli space

PSS M (q, x; J, H, f, g) is a smooth manifold of dimension µCZ(x) + µMorse(q) − n.

For the moduli spaces of plumber’s helper solutions to be helpful, we need them to be compact manifolds in dimension zero, and we need to know how

70 to compactify them in dimension one. We start our analysis by showing their energy is uniformly bounded. In that case, we can apply familiar results from Morse (Section 3.1) and Floer theory (Subsection 3.4.3) to compactify the moduli spaces.

Proposition 19 ([Alb05]). The energy of a plumber’s helper solution (γ, u) ∈ MPSS(q, x) satisﬁes the universal energy bounds

1 0 ≤ E(u) ≤ AH(x) + sup H(t, ·)dt. Z0 M

Proof. The proof is similar to the one of Proposition 15. Following [MS12, Equa- tion (12.1.11)], we deﬁne the function Hs,t := β(s)Ht and the Hamiltonian vector

1 ﬁeld Xs,t := XHs,t for all (s, t) ∈ R × S . Then

1 1 2 E(u) = |∂su| dt ds = ω ∂su, ∂tu − Xs,t(u) dt ds ∞ ∞ Z− Z0 Z− Z0 1 ∗ = ∞ u ω − ω ∂∞su, Xs,t(u) dt ds. 1 ∞ ZR×S Z− Z0

As before, we can rewrite the∞ ﬁrst term using the condition ω(u # dx) = 0 and

∗ ∗ get R×S1 u ω = D2 dxω. Using R R ω ∂su, Xs,t(u) = dHs,t(∂su) = ∂s(Hs,t ◦ u) − (∂sHs,t)(u),

lim(Hs,t ◦ u)(s, t) = lim(β(s)Ht ◦ u)(s, t) = Ht(x(t)), s s

→∞lim (Hs,t ◦ u)(s, t) =→∞lim (β(s)Ht ◦ u)(s, t) = 0, s − s − we rewrite the→ second∞ term as → ∞

1 − ω(∂su, Xs,t(u))dt ds ∞ Z− Z0 1 1 =∞ (∂sHs,t)(u)ds dt − ∂s(Hs,t ◦ u)ds dt ∞ ∞ Z0 Z− Z0 Z− 1 1 0 = ∞ β (s)Ht(u)ds dt − H(t,∞ x(t))dt. ∞ Z0 Z− Z0 ∞ 71 1 0 We conclude that E(u) = AH(x) + β (s)H(t, u)ds dt. Since β is posi- ∞ Z0 Z− tive, the result now follows because ∞

0 0 β (s)Ht(u)ds ≤ sup H(t, ·) β (s)ds = sup H(t, ·). ∞ ∞ Z− M Z− M ∞ ∞

Corollary 4. Sequences of plumber’s helper solutions converge up to bubbling and breaking (after passing to a subsequence).

Proposition 20 ([PSS96]). The zero-dimensional moduli spaces MPSS(q, x) are compact. In dimension one, they can be compactified by adding broken configurations where either the Morse flow half-line or the cylinder has broken exactly once.

Definition 16. The Piunikhin-Salamon-Schwarz (PSS) map is the linear map from Floer homology to Morse cohomology (both with Novikov coefficients) defined on generators by

n−∗ PSS : CF∗(H) (CM(f) ⊗ Λ) ,

PSS A x 7 #2M (q, A · x) · q ⊗ e . → q∈Crit(f) XA∈Γ → A The grading on the right is given by deg(q ⊗ e ) := µMorse(q) − 2c1(A).

The PSS map is an isomorphism on (co)homology on semipositive symplectic manifolds and is therefore usually referred to as the PSS isomorphism. It was introduced in [PSS96] give a proof the Arnold conjecture for semipositive symplectic manifolds. Since Morse cohomology with Novikov coefﬁcients is isomorphic to quantum cohomology,

n−∗ ∼ n−∗ PSS : HF∗(H) (HM(f) ⊗ Λ) = QH (M, ω)

→ 72 Figure 3.4: The two possible cases for a broken plumber’s helper solution with one break: either the Morse ﬂow half-line breaks (top) or the cylindrical part breaks (bottom). is an isomorphism between Floer homology and quantum cohomology. The advantage of this approach is that PSS intertwines the pair-of-pants product on Floer homology and the quantum cup product. See [MS12, Chapter 12] for an elementary introduction.

Let Π : CM∗(f) ⊗ Λ CM∗(f) be the projection deﬁned by Π(q ⊗ eA) := q. In what follows, we will investigate the properties of the composition →

n−∗ PSS0 := Π ◦ PSS : CF∗(H) CM (f),

PSS PSS0(x) = #2M →(q, x) · q. q∈CritXn−∗(f)

73 Corollary 5. The map PSS0 commutes with the Floer boundary and Morse

n−∗ coboundary, and therefore it descends to a map PSS0 : HF∗(H) HM (f).

→ Proof. We need to show that the diagram

PSS0 n−∗ CF∗(H) CM (f)

M ∂F δ

PSS0 n−∗+1 CF∗−1(H) CM (f) commutes. Fix x ∈ Pek(H). On the one hand,   Floer (PSS0 ◦ ∂F)(x) = PSS0  #2Mc (y, x) · y y∈PeXk−1(H)   Floer PSS = #2Mc (y, x) ·  #2M (q, y) · q q∈Crit (f) y∈PeXk−1(H) Xn−k+1   PSS Floer =  #2M (q, y) · #2Mc (y, x) · q q∈Crit (f) Xn−k+1 y∈PeXk−1(H)   [ PSS Floer = #2  M (q, y) × Mc (y, x) · q. q∈Crit (f) Xn−k+1 y∈Pek−1(H)

Note that the coefﬁcient of q ∈ Critn−k+1(f) is equal to the number (modulo 2) of broken plumber’s helper conﬁgurations connecting q and x, where the cylinder part has broken once.

74 On the other hand,   M M PSS (δ ◦ PSS0)(x) = δ  #2M (p, x) · p p∈CritXn−k(f)   PSS Morse = #2M (p, x) ·  #2Mc (q, p) · q p∈CritXn−k(f) q∈CritXn−k+1(f)   Morse PSS =  #2Mc (q, p) · #2M (p, x) · q q∈CritXn−k+1(f) p∈CritXn−k(f)   [ Morse PSS = #2  Mc (q, p) × M (p, x) · q, q∈CritXn−k+1(f) p∈Critn−k(f) so the coefﬁcient of q ∈ Critn−k+1(f) is the number (modulo 2) of broken plumber’s helper conﬁgurations connecting q and x, where the negative gradient half-line part has broken once. Theorem 20 guarantees that these expres- sions are equal.

3.6 Representing Submanifolds Using PSS

Let L ⊂ M be a Lagrangian. Under certain circumstances, we can express the Poincare´ dual of L in Morse cohomology as the image of a cycle in Floer homology under the PSS0 map. Consider the following moduli space, consisting of Floer half-cylinders that approach a periodic orbit as s − and have their boundary on L: → ∞

1 ∂J,Hu = 0, u({0} × S ) ⊆ L,

− :  1  ML (x; J, H) = u : (− , 0] × S M lims − u(s, ·) = x,  .     ω(dx # u)→ = ∞µ (dx # u) = 0 ∞ → Maslov     Theorem 12 ([Alb05, Alb10]). Let (M2n, ω) be a monotone closed symplectic

75 manifold and L ⊂ M a monotone closed Lagrangian with minimal Maslov number NL ≥ 2. Then   − n PSS0  #2ML (x; J, H) · x = PD[L] ∈ HM (f) (3.10) x∈XPe0(H) for generic choices of J and H.

We push the proof to the end of this section. First we show that the moduli

− spaces ML (x; J, H) are compact in dimension zero and can be compactiﬁed by adding broken conﬁgurations in dimension one. Next we prove that the argument of PSS0 in (3.10) is a cycle in HF0(H). In the proof of Theorem 12, we then only need to show that (3.10) holds.

Theorem 13 ([FHS95]). Let (M2n, ω) be a closed symplectic manifold and L ⊂ M a closed Lagrangian. For generic choices of J and H, the moduli spaces

− ML (x; J, H) are smooth manifolds of dimension −µCZ(x).

− Lemma 5. The energy of an element u ∈ ML (x; J, H) satisﬁes the universal energy bound 1 0 ≤ E(u) ≤ −AH(x) − inf H(t, ·)dt. L Z0

Proof. The proof will be similar to the proofs of Propositions 15 and 19. We already know that

0 1 2 E(u) = |∂su| dt ds Z− Z0 0 1 ∗ = ∞ u ω − ω ∂su, XH(t, u) dt ds. 1 Z(− ,0)×S Z− Z0 ∞ ∞ For the ﬁrst term, note that ω(dx # u) = 0 implies that

∗ ∗ u ω = − dxω. 1 2 Z(− ,0)×S ZD ∞ 76 If we switch the order of integration in the second term and use the fact that

ω(XH, ·) = −dHt, we see that

0 1 1 0 ω ∂su, XH(t, u) dt ds = dHt(∂su)ds dt − 0 0 − Z Z Z 1 Z 1 ∞ = Ht,∞ u(0, t)dt − Ht, x(t)dt. Z0 Z0 We conclude that

1 1 ∗ E(u) = − dxω + H t, x(t) dt − H t, u(0, t) dt D2 0 0 Z 1Z Z = −AH(x) − H t, u(0, t) dt 0 Z 1 ≤ −AH(x) − inf H(t, ·)dt. L Z0 In the last step we used the fact that u(0, ·) ⊆ L.

Theorem 14 ([Alb05]). Let (M2n, ω) be a monotone closed symplectic manifold and L ⊂ M a monotone closed Lagrangian with minimal Maslov number

− NL ≥ 2. Then the moduli spaces ML (x; J, H) are compact in dimension zero, and they can be compactiﬁed in dimension one by adding solutions where one Floer cylinder splits off.

− Proof. Let (un)n=1 be a sequence in ML (x; J, H). The uniform energy bound ∞ in Lemma 5 allows us to apply the Gromov-Floer convergence theorem: there exists a subsequence of (un)n=1 that converges to a conﬁguration ∞ (u ; v1, . . . , vk; s1, . . . , sΣ; d1, . . . , d∆) in the Gromov-Floer topology, where

∞ − • u ∈ ML (x0; J, H) is a half-cylinder with Lagrangian boundary condition

converging∞ to a periodic orbit x0;

• vγ ∈ M(xγ, xγ−1; J, H) is a Floer cylinder connecting periodic orbits xγ and

xγ−1 for all γ ∈ {1, . . . , k}, where xΓ = x;

77 • sσ is a holomorphic sphere bubble for all σ ∈ {1, . . . , Σ}; and

• dδ is a holomrphic disk bubble for all δ ∈ {1, . . . , ∆}.

Figure 3.5: A broken conﬁguration: periodic orbits in red connected by Floer cylinders, and Lagrangian boundary condition for the rightmost cylinder, with sphere bubbles in green and disk bubbles in blue.

We denote the Fredholm operator in the proof of Theorem 13 by F. Then, for any n ≥ 1, we get

− ind(Fun ) = dim ML (x; J, H) Γ Σ ∆

= ind(Fu ) + ind(Fvγ ) + 2c1(sσ) + µMaslov(dδ) γ=1 σ=1 δ=1 ∞ X X X by Remark 8. We note the following about the quantities that appear in the formula above:

• The Fredholm index ind(Fv) is at least one if the Floer cylinder v is not independent of the s-coordinate.

• Since (M, ω) is monotone, the fact that the holomorphic sphere sσ has non-

negative symplectic area implies that it has nonnegative ﬁrst Chern number. Moreover, the ﬁrst Chern number is only zero for constant spheres.

• Similarly, since L is monotone, the Maslov number µMaslov(dδ) has to be

nonnegative. It is only zero for constant disks. Moreover, µMaslov(dδ) is at

78 least 2 for nonconstant disks since the minimal Maslov number NL is at least 2.

− We conclude that, if the dimension of ML (x; J, H) is zero, there cannot be any nontrivial breaking, sphere bubbling, or disk bubbling. The moduli space

− is compact. If the dimension of ML (x; J, H) is one, breaking can occur, but at most once.

Corollary 6. The argument of PSS0 in formula (3.10) is a cycle in Floer homology.

− Proof. 2M (x; J, H) · x We first show that x∈Pe0(H) # L is a chain by showing that is satisfies the Novikov conditionP in (3.4). Recall that H only has finitely many

(uncapped) periodic orbits, but potentially inﬁnitely many cappings of Conley- Zehnder index zero. The change of capping preserving the Conley-Zehnder index is described by the action of Γ0. Fix c ∈ R and x ∈ Pe0(H). The condition

AH(A · x) = AH(x) + ω(A) ≥ c bounds the allowed symplectic area of A below.

The uniform energy bound (Lemma 5) bounds it above. Since elements of Γ0 are determined by their symplectic area, the result follows. Furthermore, using the characterization of the boundary of the moduli space in dimension one in Theorem 14, the chain is a cycle because   − ∂F  #2ML (x; J, H) · x x∈XPe0(H)   − Floer = #2ML (x; J, H) ·  #2Mc (y, x; J, H) · y x∈XPe0(H) y∈PXe−1(H)   − Floer =  #2ML (x; J, H) · #2Mc (y, x; J, H) · y y∈PXe−1(H) x∈XPe0(H) = 0.

79 PSS,− Proof of Theorem 12. The moduli space ML (q; J, H, f, g) consists of all triples (R, γ, u), where R ≥ 0, γ : (− , 0] M is a negative gradient ﬂow half-

1 line of f such that lims − γ(s) = q, and u : (− ,R] × S M a ﬁnite en- ∞ → 3 ergy half-cylinder satisfying→ ∞ ∂su + J(u)(∂tu − β(s)XH(t, u)) = 0, the bound- ∞ → ary conditions u(R, ·) ⊆ L and u(− ) = γ(0), and the homotopy condition ω(u) = µ (u) = 0. Maslov ∞

For generic choices, this moduli space is a smooth manifold of dimension

µMorse(q) − n + 1. The energy of its elements is uniformly bounded by

1 1 0 ≤ E(u) ≤ β(R) sup H(t, ·)dt − inf H(t, ·)dt . L Z0 M Z0 A Fredholm index argument (similar to the one in the proof of Theorem 14) shows that the moduli spaces are compact in dimension zero and that they can be compactiﬁed in dimension one by adding solutions with either a broken Morse ﬂow line or a broken Floer cylinder. In the one-dimensional case, for q ∈ Critn(f), the boundary decomposes as

PSS,− [ Morse PSS,− ∂ML (q) = Mc (q, p) × ML (p)

p∈Critn−1(f)

[ PSS − ∪ M (q, x) × ML (x) (3.11)

x∈Pe0(H)

PSS,− ∪ (R, γ, u) ∈ ML (q) R = 0 . Since the moduli spaces are ﬁnite in dimension zero, we can deﬁne the chain

PSS,− n−1 θL := #2ML (p) · p ∈ CM (f). p∈CritXn−1(f) 3 1 Recall that the cutoff function β : R [0, 1] satisﬁes β(s) = 0 if s ≤ 2 and β(s) = 1 if s ≥ 1.

→ 80 and the cycles

PSS,− n ρL := #2 (R, γ, u) ∈ ML (q) R = 0 · q ∈ CM (f), q∈Critn(f) X − ΦL := #2ML (x) · x ∈ CF0(H). x∈XPe0(H) They satisfy   M PSS,− Morse δ (θL) = #2ML (p) ·  #2Mc (q, p) · q p∈CritXn−1(f) q∈CritXn(f)   Morse PSS,− =  #2Mc (q, p) · #2ML (p) · q q∈CritXn(f) p∈CritXn−1(f)   − PSS PSS0(ΦL) = #2ML (x) ·  #2M (q, x) · q q∈Critn(f) x∈XPe0(H) X   PSS − =  #2M (q, x) · #2ML (x) · q q∈Critn(f) X x∈XPe0(H) The computation implies that

M PSS,− δ (θL) + PSS0(ΦL) + ρL = #2∂ML (q) · q = 0, q (f) ∈CritXn using the boundary decomposition (3.11) of the one-dimensional moduli spaces.

Note that, if R = 0, then u is a holomorphic disk with vanishing symplectic area, so it must be constant. The coefﬁcients in ρL are counting ﬂow half-lines

γ : (− , 0] M such that lims − γ(s) = q ∈ Critn(f) to and γ(0) ∈ L. If we choose f to be a Morse function→ ∞ on L that we extended quadratically in the ∞ → normal direction (see Example 8), then it is clear that PD[L] = [ρL].

81 3.7 An Application to Displaceability

Theorem 15 ([Alb05, Alb10]). Let (M2n, ω) be a monotone closed symplectic manifold and L ⊂ M a monotone closed Lagrangian with minimal Maslov number NL ≥ 2. If L is displaceable, then [L] = 0 in Hn(M; Z2).

Proof. Since L is displaceable, there exists a Hamiltonian K : S1 × M R such that the time-1 ﬂow φ1 : M M displaces L. Lemma 2 ensures that φ1 dis- K → K places an open neighborhood UL of L. Choose an autonomous Hamiltonian → H : M [0, ) supported in UL such that H|L ≥ > 0.

→ ∞ 4 1 1 1 Fix ρ > 0. Suppose that p ∈ M satisﬁes φK#ρH(p) = (φK ◦ φρH)(p) = p. Then 1 1 φρH(p) cannot be in UL because φK displaces UL. Since H is supported in UL,

1 1 it follows that φρH(p) = p and therefore that φK(p) = p. We conclude that the

1 1 ﬁxed point sets of φK and φK#ρH are the same.

1 Let x = [x, dx] ∈ Pe(K#ρH). Then there exists a ﬁxed point p of φK such that

t t t x(t) = φK#ρH(p) = (φK ◦ φρH)(p)

1 for all t ∈ S . Adding the fact that the ﬂow of XH preserves the level sets of H,

4See (2.3) for the deﬁnition of the # operation on Hamiltonians.

82 t so ρH φρH(p) = ρH(p) = 0, we get 1 ∗ AK#ρH(x) = dxω − (K#ρH)(t, x(t))dt D2 0 Z Z 1 1 ∗ t −1 = dxω − K(t, x(t))dt − ρ H (φK) (x(t)) dt D2 0 0 Z Z 1 Z 1 ∗ t = dxω − K(t, x(t))dt − ρ H φρH(p) dt D2 0 0 Z Z 1 Z ∗ = dxω − K(t, x(t))dt 2 ZD Z0 = AK(x).

The energy bound in Lemma 5, namely

1 0 ≤ E(u) ≤ −AK#ρH(x) − inf(K#ρH)(p, t)dt 0 p∈L 1 Z = −AK(x) − inf(K#ρH)(p, t)dt, p∈L Z0 − holds for any u ∈ ML (x; J, K#ρH). The integral satisﬁes 1 1 t −1 inf(K#ρH)(p, t)dt = inf K(t, p) + ρH (φK) (p) dt 0 p∈L 0 p∈L Z Z 1 1 t −1 ≥ inf K(t, p)dt + ρ inf H (φK) (p) dt p∈L p∈L Z0 Z0 t −1 The function [0, 1] R : t 7 infp∈L H (φK) (p) is continuous and nonnegative. Since H is bounded away from zero on L, it is positive at t = 0, so its → → integral is positive. We conclude that

1 1 t −1 0 ≤ E(u) ≤ −AK(x) − inf K(t, p)dt − ρ inf H (φK) (p) dt p∈L p∈L Z0 Z0 − for all u ∈ ML (x; J, K#ρH). The upper bound is negative if ρ > 0 is sufﬁciently − large. The moduli spaces ML (x; J, K#ρH) are therefore empty for large ρ. Using the characterization of PSS0 in Theorem 12, we conclude that   − n PSS0  #2ML (x; J, K#ρH) · x = PD[L] ∈ H (M; Z2) x∈PeX0(K#ρH) and therefore that [L] = 0 in Hn(M; Z2).

83 3.8 Nondisplaceable Anti-Diagonals

We use Theorem 15 to construct a class of nondisplaceable Lagrangians.

Proposition 21. Let (M2n, ω) be a closed monotone symplectic manifold. The

∗ ∗ product M × M equipped with the symplectic form Ω = pr1ω − pr2ω is also monotone.

Proof. Let J be an almost complex structure on M compatible with ω. The product almost complex structure J × (−J) on M × M is compatible with Ω. The ﬁrst Chern class satisﬁes

∗ ∗ c1(M × M, Ω) = c1 pr1(TM, J) ⊕ pr2(TM, −J)

∗ ∗ = pr1 c1(TM, J) + pr2 c1(TM, J)

∗ ∗ = pr1 c1(M, ω) − pr2 c1(M, ω).

There exists λ > 0 such that Iω = λIc1 on π2(M). For A ∈ π2(M × M),

∗ ∗ λ hc1(M × M, Ω),Ai = λ pr1 c1(M, ω),A − λ pr2 c1(M, ω),A

= λ c1(M, ω), (pr1)∗A − λ c1(M, ω), (pr2)∗A

= [ω], (pr1)∗A − [ω], (pr2)∗A

∗ ∗ = [pr1ω − pr2ω],A = h[Ω],Ai .

We conclude that (M×M, Ω) is monotone with the same monotonicity constant as (M, ω).

Lemma 6. A diffeomorphism f : M M is a symplectomorphism if and only if its graph Γf is a Lagrangian in (M × M, Ω). →

84 Proof. Deﬁne the map F : M M×M by F(p) = (p, f(p)). Then Γf is Lagrangian if and only if F∗Ω = 0. Since →

∗ ∗ ∗ ∗ ∗ F Ω = F pr1ω − F pr2ω

∗ ∗ = (pr1 ◦ F) ω − (pr2 ◦ F) ω = ω − f∗ω, this is exactly the case when f∗ω = ω.

Proposition 22 ([Oh93]). Let (M2n, ω) be a closed monotone symplectic manifold. Suppose that the Lagrangian L ⊂ M is the ﬁxed point set of an anti-

2 symplectic involution, i.e. a smooth map σ : M M satisfying σ = idM and

σ∗ω = −ω. Then L is a monotone Lagrangian. →

Proof. Given a disk d :(D2, ∂D2) (M, L) with boundary on L, consider the sphere → d(z) z ∈ D2, 2 if u : S M, u(z) =  2  (σ ◦ d)(z) if z ∈ D .

→2 2 2 Here we decomposed S = D ∪ D into its northern and southern hemisphere. We identiﬁed both hemispheres with the unit disk D2 ⊂ C, with an opposite orientation for the southern one. Note that u is well-deﬁned on D2 ∩ D2 since σ

ﬁxes L. Then 2c1(u) = µ(d) − µ(σ ◦ d) = 2µ(d) (see [Vit87]) and

u∗ω = d∗ω + (σ ◦ d)∗ω = d∗ω + d∗σ∗ω 2 2 2 2 2 ZS ZD ZD ZD ZD = d∗ω − d∗ω = 2 d∗ω. 2 2 2 ZD ZD ZD L Suppose that Iω = 2λIc1 on π2(M). Then Iω(d) = λIc1 (u) = λIµL (d).

Corollary 7. Let (M2n, ω) be a closed monotone symplectic manifold. Then the

∗ ∗ diagonal ∆ is a monotone Lagrangian in (M × M, pr1ω − pr2ω).

85 Proof. The diagonal is Lagrangian because it is the graph of the identity on M. Let σ : M × M M × M be deﬁned by σ(p, q) = (q, p). It is an involution whose ﬁxed point set is ∆. Since → ∗ ∗ ∗ ∗ ∗ σ (pr1ω − pr2ω) = (pr1 ◦ σ) ω − (pr2 ◦ σ) ω

∗ ∗ = pr2ω − pr1ω, it is anti-symplectic, so its ﬁxed point set is monotone.

Proposition 23. The homology class [∆] ∈ H2n(M × M; Z2) is nonzero.

N i N Proof. We follow [MS74]. Let (αi)i=1 be an additive basis and (α )i=1 its Poincare´ dual basis for H∗(M; Z2). The class of the diagonal is N i [∆] = αi × α . i=1 X i N It is nonzero since (αi × α )i=1 is an additive basis for H2n(M × M; Z2).

Theorem 16. Let (M2n, ω) be a closed monotone symplectic manifold. Then the

∗ ∗ diagonal is a nondisplaceable Lagrangian in (M × M, pr1ω − pr2ω).

Proof. In addition to what we have already proved in this section, note that the minimal Maslov number of the diagonal is at least 2 since M is orientable. Apply Theorem 15.

The result is interesting when χ(M) = 0. Otherwise nondisplaceability follows from topological considerations. We give some examples of closed monotone symplectic manifolds of Euler characteristic zero.

Example 9. The torus T2n = R2n/Z2n is aspherical since

2n ∼ 1 1 π2(T ) = π2(S ) × · · · × π2(S ) = 0, and therefore it is monotone. Its Euler characteristic is zero.

86 k Example 10. Fix d = (d1, . . . , dk) ∈ Z>0. Let Xn(d) be the smooth complete intersection of k smooth hypersurfaces in CPn+k whose degrees are given by d. If n = 2m − 1 is odd and d = (2, 2), its Hodge numbers are

1 if 0 ≤ p = q ≤ n, p,q  h =  m if (p, q) = (m − 1, m) or (m, m − 1),   0 otherwise.   The Hodge diamond of X3(2, 2) is displayed below.

h3,3 1

h3,2 h2,3 0 0 h3,1 h2,2 h1,3 0 1 0 h3,0 h2,1 h1,2 h0,3 0 2 2 0

h2,0 h1,1 h0,2 0 1 0 h1,0 h0,1 0 0 h0,0 1

The Betti numbers are the sum of the Hodge numbers in each row, i.e.

1 if 0 ≤ k ≤ 2n is even, k p,k−p  bk = h =  0 if 0 ≤ k ≤ 2n is odd and k 6= n,  p=0  X n + 1 if k = n.   Its Euler characteristic (for nodd) is therefore

2n k χ(Xn(2, 2)) = (−1) bk = 0. k=0 X 2 If α ∈ H (Xn(d); Z) is the restriction of ﬁrst Chern class of the canonical bundle

n+k on CP to Xn(d), then the Chern polynomial of Xn(d) is

k n+k+1 1 c(Xn(d)) = (1 + α) . 1 + diα i=1 Y

87 See [EH16] for a detailed computation. For d = (2, 2) we get

(1 + α)n+3 1 c(X (2, 2)) = = 1 + (n − 1)α + (n2 − 3n + 6)α2 + .... n (1 + 2α)2 2

Therefore [ω] = πα = π(n − 1)c1(Xn(2, 2)), so Xn(2, 2) is monotone for n > 1. We conclude that the smooth complete intersection of two smooth hyperplanes of degree 2 in CPn+2 is a closed monotone symplectic manifold of Euler characteristic zero for n ≥ 3 odd.

88 CHAPTER 4 POLYGON SPACES

4.1 Deﬁnition and Elementary Properties

n Fix a tuple r = (r1, . . . , rn) ∈ R>0. The polygon space Pol(r) is the space of closed

3 paths in R consisting of n line segments of lengths r1, ... , rn, up to rigid motion.

We can eliminate the translational degree of freedom by having the paths start at the origin. Since the lengths of the line segments are ﬁxed, choosing an element

3 2 of Pol(r) boils down to choosing n directions in R . If we denote by Sr the sphere of radius r centered at the origin in R3, then

n n , 2 Pol(r) = (x1, . . . , xn) ∈ S xj = 0 SO(3), rj j=1 j=1 Y X where SO(3) acts diagonally.

p : n S2 S2 i Let i j=1 rj ri be the projection onto the -th factor. We equip n 2 n 1 ∗ Sr withQ the symplectic form ω = pj ωj, where ωj is the standard j=1 j → j=1 rj S2 (3) areaQ form on rj . The action of SO on thisP product of spheres is Hamiltonian µ : n S2 so(3)∗ ∼ R3 w.r.t. this symplectic form and its moment map j=1 rj = is n given by µ(x1, . . . , xn) := xi. The polygon spaceQ Pol(r) is the symplectic i=1 → n S2 (3) reduction of j=1 rj by SOP at the zero level of the moment map: Q n ,, ( ) = S2 (3). Pol r rj SO (4.1) j=1 Y 0 If Pol(r) is nonempty, 0 is a regular value of the moment map. Therefore Pol(r) is a compact symplectic orbifold of dimension

n ! ( ) = S2 − 2 (3) = 2n − 6. dim Pol r dim rj dim SO j=1 Y

89 The following properties of polygon spaces are from [KM96, Kly94].

n Proposition 24. Let r = (r1, . . . , rn) ∈ R>0.

n 1. The polygon space Pol(r) is nonempty if and only if ri ≤ j=1 rj for all j6=i i ∈ {1, . . . , n}. P

2. The polygon space Pol(r) is a smooth manifold if and only if it does not contain any degenerate polygons, i.e. polygons that lie on a line. In other

words, Pol(r) is smooth if and only if i∈I ri 6= j∈J rj for any partition I t J = {1, . . . , n}. We say that length vectorP r is genericP if it satisﬁes this condition. It ensures that the SO(3) action in (4.1) is free.

4.2 The Bending Flow System

d : ( ) R3 d ( ) := i x i = 1, . . . , n We deﬁne i Pol r by i x j=1 j for . It is the length of the diagonal connecting the origin toP the i-th vertex. The ﬁrst one and and → the last two are determined by r:

d1(x) = |x1| = r1,

n−1

dn−1(x) = xj = | − xn| = rn, j=1 X n

dn(x) = xj = 0. j=1 X We will use the term diagonals to refer to the remaining n − 3 diagonals or to their lengths, if there is no confusion. They are continuous everywhere, and they are smooth wherever they are nonzero. Kapovich and Millson showed in [KM96] that they are pairwise in involution and functionally independent on

90 the open dense subset of prodigal polygons, i.e. polygons for which none of the diagonals vanish. Therefore

n−3 Φ = (d2, . . . , dn−2): Pol(r) R is a completely integrable system on Pol(r). The→ Hamiltonian circle action induced by di rotates the polygon about the i-th diagonal. The induced torus action is not deﬁned at points where one of the diagonals vanishes.

The image of the moment map Φ : Pol(r) Rn−3 is a polytope. It consists of all possible combinations of lengths for the diagonals. We can describe it as → follows: the n − 3 diagonals divide the polygons into n − 2 triangles. The side lengths of those triangles have to satisfy the triangle inequalities. Conversely, any set of diagonal lengths for which the triangle inequalities hold is included in the image of the moment map [HK97].

Example 11. Consider pentagons with side lengths r = (r1, r2, r3, r4, r5).

d2 r1

r5 r4

Figure 4.1: A pentagon with its two diagonals as red dashed line segments.

Then the triangle inequalities are

|r1 − r2| ≤ d2 ≤ r1 + r2,   |d2 − d3| ≤ r3 ≤ d2 + d3,   |r4 − r5| ≤ d3 ≤ r4 + r5.    91 The moment map image for r = (2, 3, 3, 2, 3) is depicted below.

d3 5

d2 1 2 3 4 5

Figure 4.2: The image of the bending ﬂow system for r = (2, 3, 3, 2, 3).

Example 12. For equilateral pentagon space, for which r = (1, 1, 1, 1, 1), the moment map image is depicted in Figure 12.

Note that the moment polytope is not smooth at the two highlighted vertices, namely at (1, 0) and (0, 1). The induced torus action fails to be toric. Let us look at the fiber Φ−1(1, 0). If the action were toric, the fiber would be a point. The fiber contains exactly those equilateral pentagons (up to rigid motion) whose first diagonal is 1 and whose second diagonal is 0.

These polygons consists of an equilateral triangle with a line segment at- tached to it. We can use rigid motion to put the triangle in a standard position in R3. An element of Φ−1(1, 0) is then completely determined by the direction in which the line segment is pointing, so the ﬁber is homeomorphic to S2. It is

92 d3

d2 1 2

Figure 4.3: The image of the bending ﬂow system for r = (1, 1, 1, 1, 1). The polytope is not smooth at the two blue vertices.

Figure 4.4: A equilateral pentagon with a vanishing diagonal consists of an equilateral triangle and a line segment. proved in [Bou18] that these sphere ﬁbers are embedded Lagrangian submanifolds.

Proposition 25. The Langrangian sphere ﬁbers of the bending ﬂow system on equilateral pentagon space are non-displaceable.

Proof. Denote the Lagrangian sphere ﬁber by L. By Weinstein’s Lagrangian Neighborhood Theorem, we can identify a tubular neighborhood of L with the cotangent bundle of S2. The self-intersection number L · L is therefore (up to sign) equal to the self-intersection number of the zero section of T ∗S2, which is

93 the Euler characteristic χ(S2) = 2. Since L · L 6= 0, we cannot (topologically) displace L using a map that is isotopic to the identity. In particular, we cannot displace L by a Hamiltonian diffeomorphism.

4.3 McDuff’s Probes Technique

McDuff [McD11] developed a straightforward technique to displace ﬁbers of the Hamiltonian torus action for a compact symplectic toric manifold. We present her argument here to convince the reader that, under certain conditions, we can apply the technique to ﬁbers of completely integrable systems.

We equip Rn with the standard inner product and identify it with its dual. We do the same for the lattice Zn and its dual. An affine line L in Rn is rational if its direction vector is rational. The affine distance daff(x, y) between two points x and y on an affine line L is the ratio of the Euclidean distance between them and the minimal distance between two integral points on the line parallel to L through the origin.

A hyperplane H in Rn is rational if its normal vector is rational. Let η be a primitive integral normal vector to H. We say an integral vector λ is integrally transverse to H if it can be extended to a basis of Zn using vectors that are parallel to H. Equivalently, λ is integrally transverse to H if hλ, ηi = ±1.

Let (M2n, ω) be a compact symplectic toric manifold with moment map µ : M Lie(T)∗ =∼ Rn and moment polytope ∆ = µ(M). Let F be a facet of ∆,

n y ∈ F and λ ∈ Z integrally transverse to F. The probe pF,λ(y) is the half-open → line segment consisting of y and all points in the interior of ∆ that lie on the ray

94 starting at y in the direction λ.

Proposition 26 ([McD11]). Suppose that x ∈ int ∆ lies on the probe p = pF,λ(y). If y lies in the relative interior of F and x is less than halfway along the probe, then the ﬁber µ−1(x) is displaceable.

Proof. The inverse image µ−1(p) ⊂ M of the probe lies entirely in a Darboux chart. If we change coordinates on Rn (preserving the lattice Zn) such that F is supported on the x1-axis and λ = (1, 0, . . . , 0), then there is a Darboux chart on

M with coordinates z1 = x1 + iy1, ... , zn = xn + iyn such that

−1 n |z1| < a, µ (p) =∼ z ∈ C , |zi| = constant, if i 6= 1 where a is the affine length of the probe. We see that there exists a diffeomorphism µ−1(p) B2(a) × T n−1, where B2(a) ⊂ R2 is the open disk with area a centered at the origin. The diffeomorphism identifies the restriction of the sym- → plectic form on M with pr∗(dx ∧ dy), where pr : R2 × T n−1 R2 is the projection onto the first factor. The fiber µ−1(x) is identified with ∂B2(b) × T n−1, where b is → the affine distance between x and y.

If x is less than halfway along p, then b < a/2. We can displace the circle ∂B2(b) inside B2(a) using a Hamiltonian diffeomorphism. We can extend this to a Hamiltonian diffeomorphism on M that displaces µ−1(x) inside µ−1(p).

Remark 9. We did not use the fact that the moment polytope ∆ was smooth. We only used the fact that the inverse image of the entire probe lies in a Darboux chart. We can use McDuff’s probes technique on integrable systems as long as we verify this condition. Milena Pabiniak used this observation in [Pab15] to study displaceability of ﬁbers of the Gelfand-Cetlin system on complete ﬂag manifold F(C3).

95 Below we used probes to displace toric fibers of the bending flow system on equilateral pentagon space Pol(1, 1, 1, 1, 1). On the left we used vertical probes; on the right horizontal ones. Note that these directions are integrally transverse to all five facets. Fibers over points in shaded regions are displaceable. Each shaded region consists of probes starting on the same facet. Fibers over points on the red lines (separating the different shaded regions) are not displaceable by the chosen type of probes.

d3 d3 2 2

1 1

d2 d2 1 2 1 2

Figure 4.5: Bending ﬂow ﬁbers displaced by vertical (left) and horizontal (right) probes in equilateral pentagon space.

By combining the results from vertical and horizontal probes, we see that we can displace all the torus ﬁbers by probes, except for potentially the ones over two line segments.

Theorem 17. The bending ﬂow ﬁbers over points on the line segments

{(1, t) | 0 < t ≤ 1} and {(t, 1) | 0 < t ≤ 1} are not displaceable by probes.

Proof. Because of the symmetry of the polytope, it sufﬁces to prove the state-

96 d3 F5 2

F1 F4

F2 F3

d2 1 2

Figure 4.6: Bending ﬂow ﬁbers displaced by probes in equilateral pentagon space. ment for the segment {(1, t) | 0 < t ≤ 1}. Label the facets as indicated in Figure

4.3. The segment is too far away from F4 and F5 to displace with a probe starting at those facets.

• The rays starting at F1 that cross the segment must have direction vector (a, b) with a > 0 and b < 0. They are integrally transverse to the normal

(1, −1) iff |a − b| = 1. Two successive integers cannot have different signs.

• The rays starting at F2 that cross the segment must have direction vector (a, b) with a > 0. They are integrally transverse to the normal (1, 1) iff |a + b| = 1. Then b = −a ± 1 ≤ 0, but these probes intersect the line

segment farther than halfway.

• Similarly, the rays starting at F3 that cross the segment must have direction vector (a, b) with a < 0. They are integrally transverse to the normal (−1, 1) iff |a − b| = 1. Then b = a ± 1 ≤ 0, but these probes intersect the

line segment farther than halfway.

97 In the rest of this chapter, we show that the torus ﬁber over the center (1, 1) is nondisplaceable.

4.4 Displacing Fibers and Symplectic Reduction

We present a result by Abreu and Macarini [AM13] in this section. They work in the context of compact toric symplectic manifolds, where they consider the symplectic reduction with respect to a subtorus. The symplectic reduction is equipped with a residual Hamiltonian torus action, and its moment polytope can be embedded into the the moment polytope of the original toric symplectic manifold. They show that, if a torus ﬁber of the moment map on the symplectic reduction is displaceable, then the torus ﬁber of the moment map of the original manifold over the same point is displaceable as well. We present their argument here to convince the reader that we can apply it in the context of a completely integrable system.

Let (M2n, ω) be a (not necessarily closed) symplectic manifold equipped with a Hamiltonian action of the n-torus T with a proper moment map µ : M Lie(T)∗ =∼ Rn. Let K be a subtorus of T of dimension n − k determined by the → inclusion i : Lie(K) Lie(T) of Lie algebras. The moment map for the restricted action of K on M is → ∗ µ ∗ i ∗ µK : M − Lie(T) − Lie(K) .

∗ −1 If c ∈ Lie(K) is a regular value→ of µK such→ that K acts freely on Z := µK (c), then the quotient Mred := Z/K is a smooth manifold of dimension 2k equipped

98 ∗ ∗ with a symplectic form ωred determined by the condition π ωred = iZω. Here

π : Z Mred is the quotient map and iZ : Z M is the inclusion. We call (M , ω ) the symplectic reduction of M by K. red →red →

The symplectic reduction carries a Hamiltonian action of Tred := T/K with

∗ moment map µred : Mred Lie(Tred) determined by the commutative diagram

µ → M Lie(T)∗

π j∗ .

µred ∗ Mred Lie(Tred)

The map j : Lie(T) Lie(Tred) is the quotient map. The following two propositions are adapted from [AM13]. →

Proposition 27. For any f ∈ Ham(Mred, ωred), there exist F ∈ Ham(M, ω) such that F(Z) ⊆ Z and π(F(p)) = f(π(p)) for all p ∈ Z.

Proof. Let (ht)t∈R be a Hamiltonian on Mred whose time-1 flow is f. Lift it to a Hamiltonian (Ht)t∈R on M by arbitrarily extending ht ◦ π : Z R to a K- invariant function supported in a neighborhood of Z. Its time-1 flow satisfies → the required properties.

∗ −1 Proposition 28. Let x ∈ Lie(Tred) be a regular value of µred. If µred(x) is displaceable, then µ−1(x) is displaceable.

−1 Proof. Suppose that f ∈ Ham(Mred, ωred) displaces µred(x) and F : M M a lift of f as in Proposition 27. We will show that F displaces µ−1(x). Assume that it → does not, i.e. that there exists q ∈ µ−1(x) ∩ F(µ−1(x)). Then q = F(p) for some

−1 −1 p ∈ µ (x). Since π(p) and π(q) both lie in µred(x) and f(π(p)) = π(F(p)) = π(q), −1 this contradicts the fact that f displaces µred(x).

99 4.5 Polygon Spaces and Grassmannians

Haussmann and Knutson [HK97] discovered a connection between complex Grassmannians of 2-planes and polygon spaces. This relationship links the

Gelfand-Cetlin integrable system on the Grassmannian and the bending ﬂow system. We closely follow their paper here to explain this connection.

Consider the skew field H =∼ C ⊕ Cj of quaternions. We can identify the imaginary quaternions IH = ai + bj + ck (a, b, c) ∈ R3 with R3. This correspondence preserves length. The map H C2×2 defined by   →u v u + vj   7   −v u → for all u, v ∈ C, identifies H with a real subalgebra of C2×2. Using this identifica- tion, we can have U(2) act on the left or on the right of H.

The Hopf map φ : H IH is deﬁned by φ(q) = qiq. Alternatively, for u, v ∈ C, we ﬁnd → φ(u + vj) = (u + vj)i(u + vj) = (u − jv)i(u + vj) h i = i(u + jv)(u + vj) = i |u|2 − |v|2 + 2uvj .

It satisﬁes the property that φ(p) = φ(q) if and only if p = eiθq. For P ∈ U(2), we have φ(q · P) = P−1 · φ(q) · P. Conjugation by an element of U(2) on IH is an element of SO(IH).

n n Consider the Stiefel manifold V2(C ) of orthonormal pairs in C , i.e.   2 n 2 a1 b1 |a| = |ai| = 1   i=1 n  . .  n×2 2 n 2  V2(C ) =  . .  ∈ C |b| = P |bi| = 1 ,   i=1      n  an bn ha, bi =P i=1 aibi = 0      P  100 n n ∼ 3 n together with the map Φ : V2(C ) (IH) = (R ) deﬁned by

Φ(a b) = φ(a→1 + b1j), . . . , φ(an + bnj) . (4.2)

Note that Φ(a b) is a polygon since

n n h 2 2 i φ(ar + brj) = i |ar| − |br| + 2arbrj r=1 r=1 X hX i = i |a|2 − |b|2 + 2ha, bij = 0.

It satisﬁes the property that Φ(a b) = Φ(a0 b0) if and only if       iθ1 0 0 a1 b1 e a b      1 1   . .   .   . .   . .  =  ..   . .  ,             iθn 0 0 an bn e an bn or, in other words, exactly when (a b) and (a0 b0) lie in the same orbit of the

n n n maximal torus U(1) of U(n). The action of U(1) on V2(C ) is free on the subset of all (a b) that do not have any zero rows.

n ∼ 3 n n Let ` = (`1, . . . , `n):(IH) = (R ) R≥0 be function that assign to each polygon the lengths of its sides. Then → q 2 2 2 2 2 `r (Φ(a b)) = (|ar| − |br| ) + (Re 2arbr) + (Im 2arbr)

p 4 4 2 2 2 2 = |ar| + |br| − 2|ar| |br| + 4|ar| |br| (4.3)

2 2 = |ar| + |br| for all r ∈ {1, . . . , n}. The total perimeter of Φ(a b) is

n n 2 2 2 2 `r (Φ(a b)) = |ar| + |br| = |a| + |b| = 2. (4.4) r=1 r=1 X X Let Pol(n, R3) be the space of all n-gons in R3. It is a vector space equipped with a norm given by the total perimeter. Let Pol(n, R3)+ be the space of nonzero

101 polygons in R3 up to scaling by positive numbers. We can identify Pol(n, R3)+ with the subspace of Pol(n, R3) of all polygons with perimeter 2. We say that a polygon is proper if none of its sides are zero.

n Proposition 29 ([HK97]). We can interpret (4.2) as a map Φ : V2(C )

3 + n n 3 + Pol(n, R ) . It induces a homeomorphism Φb : V2(C )/U(1) Pol(n, R ) that → is a smooth principal U(1)n-bundle when restricted to proper polygons. →

n n n The Grassmannian Gr2(C ) of 2-planes in C is the quotient V2(C )/U(2). The action of U(1)n drops down to the quotient, but it is not faithful: the diag-

3 + 3 onal U(1) acts trivially. Let Pol(n, R )SO be the space of n-gons in R with total perimeter 2, up to rigid motion, i.e. the quotient of Pol(n, R3)+ with respect to

n the natural action of SO(3). If P ∈ U(2) and (a b) ∈ V2(C ), then

Φ(a b) · P = P−1 · Φ(a b) · P.

Recall that conjugation by U(2) on IH =∼ R3 is an element of SO(3). The action of

3 + SO(3) on Pol(n, R )SO is free on the subspaces of nondegenerate polygons (i.e. polygons that do not entirely lie on a line). The map in Proposition 29 therefore drops down to the quotient, as indicated in the diagram below.

n Φ 3 + V2(C ) Pol(n, R )

n Φ 3 + Gr2(C ) Pol(n, R )SO

Proposition 30 ([HK97]). We can interpret the bottom map in the diagram

n 3 + as a map Φ : Gr2(C ) Pol(n, R )SO. It induces a homeomorphism Φb : n n 3 + n Gr2(C )/U(1) Pol(n, R ) that is a smooth principal U(1) /U(1)-bundle → SO when restricted to proper nondegenerate polygons. →

∗ Let Hn be the space of Hermitian n × n matrices. We identify it with u(n)

102 using the nondegenerate bilinear pairing

∗ i Hn × u(n) R, (H, X) 7 tr(HX). 2

∗ The coadjoint action on u(n) is→ identiﬁed with→ the action of U(n) on Hn by conjugation.

n×2 † The map Φe : C H2 given by Φe (a b) = (a b) (a b) − I2 is the moment map of the natural action of U(2). The Grassmannian is the symplectic reduction → n n×2 n Gr2(C ) = C U(2) = V2(C ) U(2). 0

n The symplectic form on Gr2(C ) is the Kostant-Kirillov form on the coadjoint orbit of U(n) through diag(1, 1, 0, . . . , 0).

n×2 † The map Ψe : C Hn given by Ψe(a b) = (a b)(a b) is the moment map of the natural U(n) action. For any P ∈ U(2) and Q ∈ U(n) we have →

† Ψe Q(a b)P = QΨe(a b)Q , so the moment map is invariant under the action of U(2). It therefore induces a

n map Ψe : Gr2(C ) Hn that is the moment map of the induced action of U(n)

n on Gr2(C ). →

n n Theorem 18 ([HK97]). The moment map µ : Gr2(C ) R of the action of the

n n n n diagonal U(1) ⊂ U(n) on the Grassmannian Gr2(C ) is ` ◦ Φb : Gr2(C ) R . → → n Proof. We know that the moment map of the action of U(n) on Gr2(C ) is in-

n×2 duced by Φe : C Hn. We get the moment map for the action of the diagonal

n torus by composing it with the projection Hn R that maps a matrix to its → diagonal elements. We get →

2 2 2 2 µ(a b) = |a1| + |b1| ,..., |an| + |bn| ,

103 which is exactly the length vector of Φ(a b) from (4.3).

Theorem 19 ([HK97]). We obtain polygon space by performing the symplectic reduction n n Pol(r) = Gr2(C ) U(1) . r Remark 10. The symplectic form on Pol(r) is a positive multiple of the one ob- tained in [KM96]. Similarly, the effect on the symplectic form of rescaling the

n length vector such that the total perimeter satisﬁes i=1 ri = 2 is multiplication by a positive scalar. P

4.6 The Gelfand-Cetlin System

Now that we have established polygon spaces as symplectic reductions of

Grassmannians, we continue to follow [HK97] and explain the connection between the Gelfand-Cetlin system and the bending ﬂow system.

n Given M = (a b) ∈ V2(C ), consider the truncated matrices   a1 b1     a2 b2   Mi :=    . .   . .    ai bi for i ∈ {1, . . . , n}. The matrices   i |a |2 a b †  r r r Mi Mi =  2  r=1 a b |b | X r r r

104 have trace and determinant given by

i † 2 2 tr Mi Mi = |ar| + |br| , r=1 X i ! i ! i 2 M†M = |a |2 |b |2 − a b . det i i r r r r r=1 r=1 r=1 X X X The trace, the sum of the eigenvalues, is clearly nonnegative. The determinant, the product of the eigenvalues, is also nonnegative by the Cauchy-Schwarz in-

i i equality. We conclude that the eigenvalues λ1 ≥ λ2 must be nonnegative. They are given by

r 2 i 1 † 1 † † λ = tr M Mi ± tr M Mi − det M Mi 1,2 2 i 4 i i  v  i u i !2 i 2 1 u =  |a |2 + |b |2 ± t (|a |2 − |b |2) + 4 a b  . 2  r r r r r r  r=1 r=1 r=1 X X X

2 2 Since `i(Φ(a b)) = |ai| + |bi| and

di(Φ(a b)) = φ(ar + brj) r=1 vX u i !2 i 2 u t 2 2 = (|ar| − |br| ) + 4 arbr , r=1 r=1 X X we conclude that these eigenvalues only depend on the lengths of the sides and diagonals of the polygon:

i i i i i `r = λ1 + λ2 and di = λ1 − λ2. (4.5) r=1 X

† † i The matrices Mi Mi and MiMi share their non-zero eigenvalues. Let λ1 ≥

i i † i i λ2 ≥ . . . λi be the ordered eigenvalues of MiMi . They satisfy λ3 = ··· = λi =

i i+1 i 0 and λj+1 ≤ λj+1 ≤ λj. These inequalities are often displayed in a so-called

105 Gelfand-Cetlin pattern. An example is provided below in the case where n = 5.

0 0 0 1 1 = = ≤ = = = ≤ = 4 0 0 λ2 1 = ≤ ≤ = ≤ ≤ 3 3 0 λ2 λ1 ≤ ≤ ≤ ≤ 2 2 λ2 λ1 ≤ ≤ 1 λ1 (4.6)

† This collection of ordered eigenvalues of the matrices MiMi (where i ∈

1 n {1, . . . , n}) contains 2(n − 2) = 2 dim Gr2(C ) nonconstant functions. They are continuous everywhere, and smooth on the open dense subset where none of

n them are equal. They form a completely integrable system on Gr2(C ), called the Gelfand-Cetlin system [GS83]. They induce a Hamiltonian torus action on the open dense subset where they are smooth. The standard torus U(1)n is a subtorus of the Gelfand-Cetlin torus by Theorem 18 and (4.5). We have proved the following.

Theorem 20 ([HK97]). The bending ﬂow action is the residual torus action on

n polygon space from the Gelfand-Cetlin action on Gr2(C ).

By combining this result with Proposition 28, we get the following relationship between displaceability of regular ﬁbers of the bending ﬂow system and the Gelfand-Cetlin system.

Theorem 21. If a regular torus fiber of the bending flow system on Pol(r) is displaceable, then the corresponding torus fiber of the Gelfand-Cetlin system

106 n on Gr2(C ) is displaceable.

4.7 Displacing the Central Fiber

We apply this to the case of equilateral pentagon space Pol(1, 1, 1, 1, 1) to study whether or not the torus ﬁber over (d2, d3) = (1, 1) is displaceable. To respect the convention we have introduced in this chapter, we renormalize the lengths

2 2 2 2 2 2 2 such that the total perimeter is 2. We set r = 5 , 5 , 5 , 5 , 5 and (d2, d3) = 5 , 5 . The rescaling has no effect on the issue of displaceability.

By Theorem 21, it sufﬁces to show that the corresponding ﬁber of the Gelfand-Cetlin system is nondisplaceable.

0 0 0 1 1 = = ≤ = = = ≤ = 4 0 0 λ2 1 = ≤ ≤ = ≤ ≤ 3 3 0 λ2 λ1 ≤ ≤ ≤ ≤ 2 2 λ2 λ1 ≤ ≤ 1 λ1 (4.7)

i i i i i Recall that j=1 rj = λ1 +λ2 for i ∈ {1, . . . , 5} and di = λ1 −λ2 for i ∈ {2, 3}. For P

107 2 2 2 2 2 2 2 r = 5 , 5 , 5 , 5 , 5 and (d2, d3) = 5 , 5 , we get the Gelfand-Cetlin pattern below. 0 0 0 1 1

= = ≤ = = = ≤ = 3 0 0 1 5 = ≤ ≤ = ≤ ≤ 2 4 0 5 5 ≤ ≤ ≤ ≤ 1 3 5 5 ≤ ≤ 2 5 (4.8)

We claim that this fiber the center of the Gelfand-Cetlin polytope in the sense of [FOOO10], which we define now. Let ∆ ⊂ Rn be a rational polytope with facets Fi given by the equations ì(x) = 0 for i ∈ {1, . . . , M}, where

`i(x) = τi − hui, xi

n for some τi ∈ R and an outward pointing primitive normal vector ui ∈ Z . The facet presentation of the polytope is

n ∆ = x ∈ R hui, xi ≤ τi, ∀i ∈ {1, . . . , M} . The center is defined using an iterative process. In the first step, define the function s1 : ∆ R by s1(x) := min{`1(x), . . . , `M(x)}. Let S1 := supx∈∆ s1(x) and

P1 := {x ∈ ∆ | s1(x) = S1}. If P1 is a singleton, its unique element is the center → of ∆. If not, we move on to the second step of the process. We do not explain the full process here because we will only need the ﬁrst step. See [FOOO10] or

[McD11] for details.

108 Proposition 31. The point x0 in (4.8) is the center of the polytope ∆ deﬁned by the inequalities in (4.7).

Proof. The polytope ∆ has 12 facets. For a facet Fk given by an equation of the

j j+1 j+1 j form λi = λi , we have `k(x) = xi − xi. Similarly for a facet Fk given by an j+1 j j j+1 equation of the form λi+1 = λi, we have `k(x) = xi − xi+1. We easily read off that

1 2 3 1 s1(x0) = min , , = . 5 5 5 5

1 There is no point x in ∆ for which s1(x) > 5 because the inequalities

2 1 2 3 0 ≤ λ2 ≤ λ1 ≤ λ1 ≤ λ1 ≤ 1

limit the minimum successive difference. Furthermore, it is easy to see that x0 is

1 the only point in ∆ for which s1(x) = 5 .

5 Theorem 22 ([NNU10]). The torus ﬁber of the Gelfand-Celtin system on Gr2(C ) over the center of the polytope deﬁned by the inequalities in (4.7) is nondisplaceable.

Theorem 23. The torus ﬁber Φ−1(1, 1) of the bending ﬂow system on equilateral polygon space Pol(1, 1, 1, 1, 1) is nondisplaceable.

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