Floer Homology on Symplectic Manifolds

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Floer Homology on Symplectic Manifolds

KWONG, Kwok Kun

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

Master of Philosophy in

Mathematics

The Chinese University of Hong Kongc August 2008

The Chinese University of Hong Kong holds the copyright of this thesis. Any person(s) intending to use a part or whole of the materials in the thesis in a proposed publication must seek copyright release from the Dean of the Graduate School.

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Thesis/Assessment Committee

Professor Wan Yau Heng Tom (Chair)

Professor Au Kwok Keung Thomas (Thesis Supervisor) Professor Tam Luen Fai (Committee Member)

Professor Dusa McDuff (External Examiner)

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Floer Homology on Symplectic Manifolds i

Abstract

The Floer homology was invented by A. Floer to solve the famous Arnold conjecture, which gives the lower bound of the fixed points of a Hamiltonian symplectomorphism.

Floer’s theory can be regarded as an infinite dimensional version of Morse theory. The aim of this dissertation is to give an exposition on Floer homology on symplectic manifolds. We will investigate the similarities and differences between the classical Morse theory and Floer’s theory. We will also explain the relation between the Floer homology and the topology of the underlying manifold.

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Floer Homology on Symplectic Manifolds ii

d d dd d d

dddArnoldddddddddddddddddddddd

dddddA. FloerdddFloerdddddddddddddd

FloerdddddddMorseddddddddddddddd ddddddFloerdddddddddddddddMorsedd dFloerddddddddddddddFloerddddddddd ddddd

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Floer Homology on Symplectic Manifolds iii

Acknowledgements

I would like to thank my advisor Prof. Thomas Au Kwok Keung for his encouragement in writing this thesis. I am grateful to all my teachers. They have not only taught me mathematics, but also the way to appreciate it. I also thank all my schoolfellows in the Depart- ment of Mathematics and the Institute of Mathematical Sciences for many enlightening discussions and their patience in answering my questions. I have learned so much from them. My deepest gratitude goes to my mother for her love and care throughout my life.

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Introduction

In 1965 V. I. Arnold conjectured in [2] that a symplectic diffeomorphism of a compact symplectic manifold M onto itself possesses at least as many fixed points as a smooth function on M has critical points, whenever this diffeomorphism is homologous to the identity (i.e. a Hamiltonian symplectomorphism). If we require the function to have non-degenerate critical points, i.e. a Morse function, then this number would be the sum of the Betti numbers of M, as is well- known in classical Morse theory. Therefore it is natural to look for a Morse-type theory to solve this version of Arnold’s conjecture. If the diffeomorphism is sufficiently near to the identity this was solved by Arnold [1] himself and also A. Weinstein [37]. Without this assumption, there are some scattered results, like Conley-Zehnder [6]

for the 2n-torus case using variational approach and M. Gromov’s [15] result of existence of at least one fixed point when π₂(M) = 0, using pseudo-holomorphic curves. The breakthrough came from Floer’s approach of combining these two ideas in a series of papers, notably [11], which proves the Arnold conjecture in the monotone case. His method can be understood as an infinite dimensional version of Morse theory, and is known as Floer homology now. The later development can be regarded as the extension of his work. We will outline his method under some additional assumptions on the

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Introduction 2

second homotopy group of M.

In chapter 1, we introduce some materials of the classical Morse theory, which is very similar to Floer’s theory in many ways. Of particular importance are the Morse homology theorem and the Morse inequalities, which are used in the proof of the Arnold conjecture.

Readers familiar with classical Morse theory may skip this chapter and proceed to chapter 2 directly, or they may start at section 1.2 instead to skim through Floer’s approach of Morse homology.

In chapter 2, we outline the background and the necessary tools needed in the proof of Arnold’s conjecture and also the construction of Floer homology groups. This chapter provides an overview of this dissertation and should be read prior to chapter 3 and chapter 4.

In chapter 3, we present the basic knowledge of Fredholm theory which is used to establish the manifold structure of the moduli space of trajectory. We prove the index formula for a Fredholm operator using Maslov index, which is often easier to calculate than the Fredholm index. This gives the local dimension of the Moduli space. The readers can skip the details without affecting their un- derstanding of the whole picture if they are willing to accept some of the technical results.

In chapter 4, we look at the construction of Floer homology groups in a more detailed way. Two important techniques are the gluing argument and the Gromov’s compactness theorem, which can

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Introduction 3

often be combined together perfectly to give many of the important results in this chapter. The success of Floer homology comes when we relate it to the Morse homology of a (time-independent) Morse-Smale function, which is obtained by transforming the time- dependent Hamiltonian function to a nice Morse-Smale function by a process known as Floer continuation. Of course the Morse homology groups are much well-understood, and we can extract information about the fixed points from them. In particular the Arnold conjecture follows as an easy corollary after proving that the Floer homology groups are isomorphic to the singular homology groups of M up to a shift of grading.

The remarks are usually some simple observations and notes.

They are of secondary importance.

I was invited to the wonderful world of Floer’s theory by my advisor about one year ago and I would like to write some notes about it in a way which (I hope) is elementary and down to earth.

However due to my lack of knowledge and time I find it impossible to cover everything in detail even if I aim at writing only a very tiny portion of this theory. Nevertheless I still hope these notes are accessible by a graduate or undergraduate student with a basic knowledge of differentiable manifolds and preferably some classical Morse theory.

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Chapter 1 Morse Theory

Floer homology can be understood as an infinite dimensional version of the Morse theory. In fact many ideas in Floer homology are analogous to that of the Morse homology. Especially the Morse homology theorem and the Morse inequalities are useful in proving the Arnold conjecture. It is therefore useful to look at the more classical case of Morse theory first. Two good references are [4]

and [25], see also [34] for a more analytic approach which is closely related to Floer’s theory.

1.1 Introduction

Definition 1.1. Let Mⁿ be a smooth finite dimensional manifold.

Let f : M → R be a smooth function, then p ∈ M is a critical point of f if df (p) = 0. It is a Morse function if for any critical point p, the Hessian Hf (p) of f at p is non-degenerate. Equivalently, in

4

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Chapter 1. Morse Theory 5

local coordinates, the determinant of the Hessian matrix at p given by

Hf (p) = ( ∂²f

∂xi∂xj

(p))

is nonzero, where x1,· · · , xn are the local coordinates around p.

We will denote the set of critical points of f by C(f ).

Definition 1.2. Let f be a Morse function and p is a critical point of f . Then the Morse index of f at p,

λ(p) = λ(p, f ) := number of negative eigenvalues of Hf (p).

The following Morse lemma gives a nice local representation of f near a critical point:

Lemma 1.3 (Morse lemma). Let f be a Morse function and p is a critical point of f . Then there exists an open neighborhood U of p and a chart h : U → Rⁿ such that in this coordinates,

f ◦ h⁻¹(x) = f (p)− x²1 − · · · − x²k + x²_k+1 +· · · + x²n, where k = λ(p).

Corollary 1.4. The critical points of a Morse function are isolated.

In particular, if M is compact, there are only finitely many critical points.

Basically, one wants to study the global topology of a manifold M through a Morse function by extracting the local information

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from the critical points of f . More precisely, define the sublevel set M^a := {x ∈ M : f(x) ≤ a}, we would like to see the change in topology (homotopy type) of these sublevel sets when the value of f runs across a critical point. The following two theorems are useful for us to understand these changes.

Theorem 1.5. If a < b and there is no critical value of f in [a, b], then M^a and M^b are diffeomorphic:

M^a ∼= M^b.

Moreover, M^a is a deformation retract of M^b, so the inclusion M^a ֒→ M^b is a homotopy equivalence.

Theorem 1.6. Suppose p is a critical point of f with Morse index λ, f (p) = c and exists ε > 0 such that there is no other critical point in f⁻¹[c− ε, c + ε] except p, then the homotopy type of M^c+ε is obtained by attaching a λ-cell eλ := {x ∈ R^λ :||x|| ≤ 1} to M^c^−ε:

M^c+ε ≈ M^c^−ε∪φeλ

by an attaching map φ : ∂eλ → M^c^−ε, where ∂eλ is the boundary of eλ. (Here ≈ means “homotopy equivalence”. ) In fact, there is a subset e ⊂ M^c+ε diffeomorphic to e_λ such that M^c^−ε ∪ e is a deformation retract of M^c+ε.

More generally, if there are exactly k critical points p1,· · · p^k ∈ f⁻¹(c), λ(p_i) = λ_i and ε > 0 is as above, then

M^c+ε ≈ M^c^−ε∪φ1 e_λ1 ∪ · · · ∪φk e_λ_k.

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Theorem 1.6 follows more or less from the Morse lemma, at least when there is only one critical value between c− ε and c + ε. In- tuitively, let U be a neighborhood of a critical point p as given by the Morse lemma. The homotopy type of the sublevel set {x ∈ U : f (x) ≤ a} in U changes only when the value of f runs across the critical point c and by analyzing the sublevel sets, the change is exactly given by attaching a λ-cell to M^c^−ε.

Choose a Riemannian metric g on M. The choice of the particular metric is not very important as it turns out that Morse homology is independent of the choice and “most” metric are good (satisfies the Morse-Smale condition). Consider the flow ψt : M → M generated by the negative gradient vector field of f :







d

dtψ_t(x) = −∇f(ψt(x)) ψ0 = id

(1.1)

Definition 1.7. Let p be a critical point of f , then the unstable manifold of p is defined by

W^u(p) := {x ∈ M : lim_t

→−∞ψ_t(x) = p} and the stable manifold of p is defined by

W^s(p) := {x ∈ M : lim

t→∞ψt(x) = p}.

As the names suggest, W^u(p) and W^s(p) are indeed embedded submanifolds of M ([19] corollary 6.3.1).

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Definition 1.8. A Morse function f is said to satisfy the Morse- Smale condition if for any critical points p and q of f , W^u(p) and W^s(q) intersect transversally. i.e. for every x ∈ W^u(p)∩ W^s(q),

TxW^u(p) + TxW^s(q) = TxM.

The flow in (1.1) is then called a Morse-Smale flow and (f, g) is called a Morse-Smale pair.

Example 1.9. Let f : T² → R be the height function on the torus T² (with induced metric g from R³) as shown in the figure (the arrows denote the directions of the flow):

This function is not Morse-Smale. The unstable manifold of the saddle point q coincides with the stable manifold of the saddle point r. So at any point of intersection, the two tangent spaces do not span the whole tangent space at that point. However, if we perturb f (or g) a little, this phenomenon will disappear. In fact, W^u(q) and W^s(r) will not even intersect. This can be thought of intuitively as if we tilt the torus a little bit:

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In fact the Morse-Smale condition is “generic” (in the sense of Baire).

Theorem 1.10 (Kupka-Smale). ([27], [36]) For a compact smooth Riemannian manifold M, the set of smooth Morse-Smale gradient vector fields is a generic subset of the set X of smooth gradient fields on M.

Here a subset of X is “generic” means it contains a countable intersection of open dense subset of X (in C^∞ topology). As the Riemannian metric gives a homeomorphism between the space of gradient vector fields and the space of exact one-forms {df : f ∈ C^∞(M, R)} ∼= C^∞(M, R)/R on M, it implies that the set of Morse- Smale functions is also a generic subset of C^∞(M, R).

Definition 1.11. Let p, q be critical points of a Morse function f . The space of trajectoryM(p, q) = M(p, q; f, g) connecting p and

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q is defined as

M(p, q) := {u ∈ C^∞(R, M) : du

dt = −∇f(u), lim_t

→−∞u(t) = p, lim

t→∞u(t) = q}.

M(p, q) naturally embeds into M by ι : u 7→ u(0).

Under this identification, we have the diffeomorphism ([34] proposition 2.31)

ι : M(p, q) ∼= W^u(p)∩ W^s(q).

Theorem 1.12. Let (1.1) be Morse-Smale and p, q are critical points of f . Then M(p, q) is a smooth manifold and

dimM(p, q) = λ(p) − λ(q).

There is a natural action of R on M(p, q) by (τ, u) 7→ u(τ + ·).

Suppose f (q) < a < f (p) and a is not a critical value of f , define M^a(p, q) := ι(M(p, q)) ∩ f⁻¹(a). Then by the implicit function theorem and theorem 1.12, M^a(p, q) is a smooth submanifold of M of dimension λ(p)− λ(q) − 1. It is easy to show the following proposition.

Proposition 1.13. The map

Ψ^a : R × M^a(p, q) → M(p, q) (τ, x) 7→ u(τ + ·)

is a R-equivariant diffeomorphism, where u(0) = x and R acts by translation on the first factor of R× M^a(p, q).

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Definition 1.14. Define the space of unparametrized trajectory

M(p, q) := M(p, q)/R.ˆ

We will say u ∈ M(p, q) is a parametrized trajectory and its image ˆu ∈ ˆM(p, q) an unparametrized trajectory if it is necessary to distinguish these two. By proposition 1.13, we have a diffeomorphism

Ψ^a : ˆM(p, q) = M(p, q)/R → M^∼⁼ ^a(p, q).

So in particular ˆM(p, q) is also a smooth manifold and dim ˆM(p, q) = λ(p) − λ(q) − 1.

1.2 Morse Homology

In this section we assume that M is compact and let (1.1) be a Morse-Smale flow on M. We will outline the construction of Morse homology. We do this not only because some of its results are used in Floer’s proof of Arnold conjecture, but also because it is very similar to the construction of Floer homology, only simpler. A detailed account can be found in [34] and also [30]. As we will see, many of the arguments here will be carried out again in chapter 4.

For simplicity we will work with Z2 coefficient, so we can ignore the problem of orientation.

When λ(p)− λ(q) = 1, ˆM(p, q) is zero-dimensional, we would like

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to count the number of points in it. Therefore we have to know that M(p, q) is compact. First we need the notion of a n-dimensionalˆ smooth manifold with corners, which is a second countable Haus- dorff space such that each point has a neighborhood with a homeomorphism with Rⁿ^−k × [0, ∞)^k for some 0 ≤ k ≤ n, and such that the transition maps are smooth. This generalizes the concept of a manifold with boundary and for n ≤ 1 they are the same. Now recall C(f ) is defined to be the set of critical points of f . We have the following compactness result.

Proposition 1.15. Suppose p, q ∈ C(f). Then ˆM(p, q) has a natural compactification to a smooth manifold with corners ˆM(p, q) by adjoining all the order k broken (unparametrized) trajectories:

[

p0,p1,··· ,p^k+1∈C(f)

M(pˆ ⁰, p1)× ˆM(p¹, p2)× · · · × ˆM(p^k, pk+1),

where p0 = p, pk+1 = q and all pi’s are distinct. This is called the compactification by broken trajectories.

The proof has two parts. One is a compactness result, which states that any sequence ˆun ∈ ˆM(p, q) has a convergence subse- quence that converges in an appropriate sense (see theorem 4.3) towards some broken trajectories of order k. The second part is a “gluing argument” which asserts that any order k parametrized broken trajectories can by “glued” together (with a gluing parameter in [R,∞)^k) to form a trajectory in M(p, q) (see theorem 4.9).

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For example, an order one broken trajectories (called simply broken trajectories) is a pair (u, v), where u ∈ M(p, q) and v ∈ M(q, r), they can be “glued” together at q to get u#ρv ∈ M(p, r) for sufficiently large ρ, and u#ρv will converge to (u, v) as ρ → ∞ in some appropriate sense.

Figure 1.1: Convergence and gluing of simply broken trajectories.

These two arguments can be regarded as the converse of each other.

The important consequence of proposition 1.15 is that when λ(p)− λ(q) = 1, then zero-dimensional ˆM(p, q) is compact since there is no broken trajectories to be added for compactification, i.e. it is finite. So at last we are able to make the following definition.

Definition 1.16. For p, q ∈ C(f) and λ(p) − λ(q) = 1, h∂p, qi := # ˆM(p, q) (mod 2).

Denote Ck := spanZ2{p ∈ C(f) : λ(p) = k}. Then the boundary

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operator

∂_k : C_k → Ck−1

is defined by

∂kp := X

q∈C^k−1

h∂p, qiq

where p ∈ C^k. The Morse-Smale-Witten chain complex, or just the Morse complex, is defined as (C_∗, ∂_∗).

Proposition 1.17. The boundary operators satisfy

∂k ◦ ∂^k+1 = 0.

Proof. Let p∈ Ck+1, this statement is equivalent to X

r∈C^k

X

q∈C^k−1

h∂p, rih∂r, qiq = 0 (mod 2).

So fixing p ∈ Ck+1, q ∈ Ck−1, we have to prove X

r∈C^k

h∂p, rih∂r, qi = # [

r∈C^k

M(p, r) × ˆˆ M(r, q) (1.2)

is an even number.

This is proved by the following observation. Each component of the 1-dimensional compact manifold with boundary ˆM(p, q) which is not a circle must be a closed bounded interval (having two endpoints) by the classification theorem. By proposition 1.15, each endpoint of these intervals is of the form of (ˆu, ˆv) ∈ ˆM(p, r) × ˆM(r, q) for some r ∈ C^k.

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Since there must be an even number of such endpoints in ˆM(p, q), it follows that the number in (1.2) is an even number.

Definition 1.18. Define the k-th Morse homology group of (M; f, g)

HMk(M; f, g) := ker ∂k/im ∂k+1.

Example 1.19. Let f : S¹ → R be the height function given by the following figure (g can be any metric):

There are four critical points. The critical points p, q are of index 1 and the critical points r, s are of index 0.

∂p = ∂q = r + s and ∂r = ∂s = 0.

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Therefore

HM₁(S¹; f, g) = hp + qi^Z² ∼= Z2

and

HM0(S¹; f, g) =hr, si^Z²/hr + si^Z² ∼= hri^Z² ∼= Z2.

This agrees with the singular homology H_∗(S¹) of S¹. Although this example is simple, this is not an incident.

Let (f0, g0), (f1, g1) be two Morse-Smale pairs. It is a remarkable fact that the Morse homology groups HM_∗(M; f0, g0) and HM_∗(M; f1, g1) are in fact isomorphic. One approach is to identify each of these Morse homology groups to the singular homology groups of M (see theorem 1.24). However Floer found an elegant alternative approach which establish a more natural isomorphism between HM_∗(M; f0, g0) and HM_∗(M; f1, g1), through a process he called continuation, without invoking the singular homology of M. The following explicit construction is from [18].

Let (C_∗ⁱ, ∂ⁱ) be the Morse complexes of (fi, gi), i = 0, 1. The idea is that we can continuously transform (f0, g0) to (f1, g1) by a smooth homotopy (ft, gt), t ∈ [0, 1], where g^t is a Riemannian metric on M for all t. (Note that the space of all Riemannian metrics on M is contractible. ) We then define a vector field V = V (t, x) on [0, 1]× M by

V (t, x) := (1− t)t(1 + t) ∂

∂t − gradtft

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where grad_tft is the (time-dependent) gradient vector field of ft : M → R with respect to the metric gt on M. Note that the (1 − t)t(1 + t)_∂t^∂ is the negative gradient of the function (t + 1)²(t− 1)²/4 on R. It is chosen because this function has a critical point of index 1 at t = 0 and a critical point of index 0 at t = 1 with no critical point in between. Actually this V is the negative gradient vector field of the function (t + 1)²(t− 1)²/4 + ft(x) on [0, 1]× M, where the metric at the point (t, x) is given by the first fundamental form





1 0

0 I(t, x)



, I(t, x) being the first fundamental form of gt at x . We can define its critical points, stable and unstable manifolds and flow lines just as the case of gradient flow on M before. As before we also require the stable and unstable manifold to intersect each other transversely. If (f0, g0) and (f1, g1) are Morse-Smale then a generic homotopy between them satisfies this condition. Such homotopy is called admissible. However for such homotopy, it may (and often must) happen that for some time t 6= 0, 1, the pair (ft, gt) is not Morse-Smale on M.

Observe that for t = 0, 1, the flow on [0, 1]× M is the same as the flow on M of (f0, g0) and (f1, g1) respectively. Note also that there are only two kinds of critical points of V , one is of the form (0, p) where p ∈ C_∗⁰ and the other is of the form (1, q) where q ∈ C_∗¹. Also the index of (0, p) is 1 + λ(p, f0) and the index of (1, q) is λ(q, f1).

Thus ˆM((0, p), (1, q)) is zero-dimensional if λ(p, f⁰) = λ(q, f1) = k

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and is compact. We then define φ : C_k⁰ → Ck¹ by φ(p) := X

q∈Ck¹

hφp, qiq

where hφp, qi := # ˆM((0, p), (1, q)) (mod 2), i.e. the number of unparametrized trajectories in [0, 1]×M connecting (0, p) and (1, q) modulo two.

Proposition 1.20. φ is a chain map. i.e.

∂¹ ◦ φ = φ ◦ ∂⁰.

Proof. The proof is similar to that of 1.17. Denote ˆMⁱ(p, q) :=

M(p, q; fˆ i, gi) for i = 0, 1. ˆMi(p, q) can be naturally identified with M((i, p), (i, q)). Let p ∈ Cˆ _k+1⁰ and q ∈ C_k¹, we have to show

X

r∈Ck⁰

h∂⁰p, rihφr, qi = X

s∈Ck+1¹

hφp, sih∂¹s, qi. (mod 2) (1.3) Equivalently, there is an even number of pairs of unparametrized simply broken trajectories between p and q. By proposition 1.15, there are two kinds of endpoints of the one-dimensional compact manifold ˆM((0, p), (1, q)). One kind is in the form of (ˆu, ˆv) ∈ Mˆ0(p, r) × ˆM((0, r), (1, q)), where r ∈ C_k⁰. This corresponds to the term on the left of equation (1.3), the other kind of endpoint is of the type ( ˆw, ˆr) ∈ ˆM((0, p), (1, s)) × ˆM1(s, q) with s ∈ Ck+1¹ , which corresponds to the right hand side of (1.3). Since there must be an even number of endpoints, the result follows.

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Proposition 1.21. Suppose (f_t⁰, g_t⁰) and (f_t¹, g_t¹) are two smooth homotopies between (f0, g0) and (f1, g1) as above, and φ⁰, φ¹ respectively denotes their induced chain maps. Then φ⁰ and φ¹ are chain homotopic, i.e. there exists Ψ = Ψk : C_k⁰ → Ck+1¹ such that

φ⁰ − φ¹ = ∂¹ ◦ Ψ + Ψ ◦ ∂⁰.

Sketch of proof. Again the proof is similar to that of proposition 1.17. First find a smooth λ-homotopy (f_t^λ, g_t^λ) between (f_t⁰, g_t⁰) and (f_t¹, g¹_t), λ ∈ [0, 1], such that (fi^λ, g^λ_i) = (fi, gi) for all λ and i = 0, 1. Then (f_t^λ, g_t^λ) can be regarded as a family {(fd, g_d) : d ∈ D} parametrized by the 2-gon D := [0, 1] × [0, 1]/{(0, λ1) ∼ (0, λ2) and (1, λ1) ∼ (1, λ²)}.

Again we find a function h : D → R with an index two critical point at the vertex v0 := {0} × [0, 1] and an index 0 critical point at the vertex v1 := {1}×[0, 1] with no other critical point and the negative

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gradient −∇h(d) of h at the two edges are (1 − t)t(1 + t)_∂t^∂. Define the vector field V = V (d, x) on D× M by

V = −∇h − graddfd

where grad_dfd is the gradient vector field of fd with respect to the metric gd. For a generic choice of (fd, gd), V is Morse-Smale. The only critical points of V on D× M are of the type (v0, p) with index λ(p, f0) + 2 where p ∈ C_k⁰ and (v1, q) with index λ(q, f1) where q ∈ Ck¹. Define Ψ : C_k⁰ → Ck+1¹ by

Ψp := X

q∈C^k+1

# ˆM((v0, p), (v1, q))q.

By analyzing the endpoints of the one-dimensional manifold ˆM((v0, p)(v1, r)) for p∈ Ck⁰, r ∈ Ck¹, we get the result.

Proposition 1.21 shows that there exists a homomorphism of the Morse homology groups

φ_∗ : HM_∗(M; f₀, g₀) → HM∗(M; f₁, g₁).

If γ1 is a homotopy from (f0, g0) to (f1, g1) as above, we denote the induced chain map by φγ1. Let γ2 be a homotopy from (f1, g1) to (f₂, g₂). Then we can concatenate the two paths, which by reparametrizing and perturbing it if necessary, can be assumed to be a smooth admissible homotopy from (f0, g0) to (f2, g2), call it γ2 ∗ γ1.

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Proposition 1.22. φγ2∗γ¹ and φγ2 ◦ φ^γ¹ are chain homotopic.

This is again proved by the compactness-gluing argument and is omitted here. For a constant homotopy from (f, g) to itself, the induced homomorphism of homology is obviously the identity. For two pairs (f0, g0) and (f1, g1) and any homotopy γ = (ft, gt) between them, since the inverse homotopy compose with it is homotopic is identity, therefore by the previous proposition each such φ_{γ ∗} is an isomorphism. Therefore

Theorem 1.23. For two Morse-Smale pairs (f0, g0) and (f1, g1) on M, the corresponding Morse homology groups are isomorphic

HM_∗(M; f0, g0) ∼= HM_∗(M; f1, g1).

So we can speak of “the” Morse homology of M without actually specifying a particular Morse-Smale pair. Furthermore, it is actually the same as singular homology of M.

Theorem 1.24 (Morse homology theorem). The Morse homology is isomorphic to the singular homology of M

HMk(M; f, g) ∼= Hk(M; Z2).

There are many proofs, see for example [12], [30], [38]. One idea is to relate the singular homology of M with that of a CW complex. We can build a CW complex Kwhose k-cells corresponds to the critical points of f with Morse index k as follows. Since

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M is compact there exists c0 < · · · < c^l such that ci’s are all the critical values of f . Suppose a is not a critical value with c_k₋₁ <

a < ck and that Mâ has the homotopy type of a CW complex, that is Mâ is homotopy equivalent to a CW complex K. By theorem 1.5 and 1.6, M^c^k^+ε is homotopy equivalent to Mâ ∪φ1 e_λ1 ∪ · · · ∪φj

eλj where λ1,· · · , λj are exactly the indices of the j critical points corresponding to ck. Then M^c^k is homotopy equivalent to K ∪^ψ¹ eλ1 ∪ · · · ∪ψj eλj for some gluing maps ψi : ∂eλi → K. (See [25]).

M^a is empty if a < c0 and by induction M^a has the homotopy type of a CW complex. Let a0,· · · , a^l are such that c0 < a0 < c1 < · · · <

cl < al, then there is a sequence of homotopy equivalences M^a⁰ ⊂ M^a¹ ⊂ · · · ⊂ M^l = M

↓ ↓ ↓

K0 ⊂ K¹ ⊂ · · · ⊂ K^l = K

each extending the previous one. So M is homotopy equivalent to the CW complex K. Then the singular homology of M is isomorphic to the cellular homology of K. Both the CW complex and the Morse complex are generated by the critical points of f graded by the indices, furthermore it can be proved that the boundary operator of the CW complex and that of the Morse complex are the same (after identification). Intuitively, it is because the “attaching degree” of a k-cell relative to a (k− 1)-cell is equal to the number of components of the intersection between the unstable manifold

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of the corresponding index-k critical point and the stable manifold of the corresponding critical point with index k− 1 (mod 2). The following is a corollary of the Morse homology theorem.

Theorem 1.25 (Weak Morse inequalities). Let Mⁿ be a compact manifold. Let c_k denotes the number of critical points of index k of f and bk := dim Hk(M, Q) denotes the k-th Betti number. Then

bk ≤ c^k and χ(M) =

n

X

k=0

(−1)^kbk =

n

X

k=0

(−1)^kck.

In particular the number of critical points of f is bounded below by the sum of the Betti numbers:

#C(f ) =

n

X

k=0

ck ≥

n

X

k=0

bk.

We also have the stronger inequalities.

Theorem 1.26 (Morse inequalities). For a compact manifold,

bk−b^k−1+· · ·+(−1)^kb0 ≤ c^k−c^k−1+· · ·+(−1)^kc0 for all k = 0,· · · , n For the proof, see for example [25].

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Chapter 2 Symplectic Fixed Points and Arnold Conjecture

2.1 Introduction

Let (M²ⁿ, ω) be a connected 2n-dimensional compact symplectic manifold without boundary, i.e. ω is a closed non-degenerate 2- form on M. Then ω determines an isomorphism Iω : T^∗M −→ T M,^∼⁼ namely for α ∈ Tp^∗M, α 7→ v, where v ∈ T^pM is the unique vector satisfying α = ωp(v,·). Let H = H(t, x) : R × M → R be a smooth function on M, called a Hamiltonian function, such that it is periodic in time (periodic means 1-periodic unless otherwise stated):

H(t, x) = H(t + 1, x).

Then Ht = H(t,·) can be regarded as a time dependent periodic family of function on M. The image of the one form−dHt under I_ω,

24

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Chapter 2. Symplectic Fixed Points and Arnold Conjecture 25

denoted by Xt, is called the Hamiltonian vector field generated by H_t. That is,

−dHt = ω(Xt,·).

Consider the Hamiltonian system of ordinary differential equations

˙x(t) = Xt(x(t)). (2.1)

The solutions for (2.1) generates a flow ψt : M → M:







d

dtψ_t = X_t(ψ_t) ψ0 = id

Let ψ = ψ₁ be the time 1 map. Clearly, the fixed points of ψ corresponds to the periodic solutions to (2.1).

Definition 2.1. Define

P (H) := {periodic solutions of (2.1)}.

As the solutions are periodic, we can also define it as P (H) := {x : R/Z → M | x solves (2.1)}.

By identifying x with x(0), sometimes we will use x to denote either a periodic solution of (2.1) or a fixed point of ψ. We will also identify R/Z with S¹ throughout.

Remark 2.2. For all t, ψt is a symplectomorphism, i.e. ψ_t^∗ω = ω, as ψ₀^∗ω = ω and by Cartan’s formula,

d

dtψ^∗_tω = ψ^∗_t(LXtω) = ψ^∗_t(dιXtω+ιXtdω) = ψ^∗_t(dιXtω) = ψ^∗_t(−ddHt) = 0,

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where LXt denotes the Lie derivative along Xt. A symplectomorphism generated by a Hamiltonian vector field is called a Hamilto- nian or exact symplectomorphism.

Definition 2.3. A fixed point x is called non-degenerate if det(I − dψ(x(0))) 6= 0.

i.e. 1 is not an eigenvalue of dψ(x(0)). H is said to be regular if all its corresponding fixed points are non-degenerate.

Arnold conjectured that the number of non-degenerate periodic solutions to this equation is at least the sum of the Betti numbers of M.

Conjecture 2.4 (Arnold conjecture). Suppose all the periodic solutions of (2.1) are non-degenerate. Then

#P (H) ≥

2n

X

i=0

bi

where bi = dim Hi(M, Q) is the i-th Betti number of M.

Remark 2.5. 1. A fixed point x can be identified with the point (x, x)at the intersection of the graph Γ := {(x, ψ(x)) : x ∈ M}

of ψ with the diagonal ∆ := {(x, x) : x ∈ M} in M × M.

Then x is non-degenerate if and only if Γ intersects with ∆

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transversely at (x, x):

T(x,x)(M × M) = T(x,x)Γ + T(x,x)∆

= T(x,x)Γ⊕ T(x,x)∆ (i.e. T(x,x)Γ∩ T(x,x)∆ = 0)

= {(v, dψx(v)) : v ∈ TxM} ⊕ {(v, v) : v ∈ TxM}

⇔ dψ^x(v) 6= v for non-zero v ∈ T^xM. i.e. x is non-degenerate.

2. Non-degenerate x ∈ P (H) are isolated: by choosing a suitable local coordinates, we can regard ψ : R²ⁿ → R²ⁿ such that x has local coordinates 0 and so ψ(0) = 0. Then d(ψ − id)(0) has non-zero determinant and thus ψ−id is a local diffeomorphism around 0 by inverse mapping theorem. So locally ψ(x) 6= x except x = 0. Therefore for compact M, P (H) consists of finite number of points.

Remark 2.6. 1. Since ψ is isotopic to the identity map, by the Lefschetz fixed point theorem, the number of fixed points of ψ is greater than or equal to |

2n

X

i=0

(−1)ⁱb_i|. So Arnold conjecture gives a stronger estimate in this case.

2. The comparison of Lefschetz fixed point theorem with Arnold conjecture is analogous to that of Poincare-Hopf theorem, which states that for a smooth vector field V on M with non-degenerate zeroes,

#{x ∈ M : V (x) = 0} ≥ |

2n

X

i=0

(−1)ⁱbi|,

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with Morse theory, which states that for a gradient vector field

∇f induced by f (by giving M a Riemannian metric),

#{x ∈ M : ∇f(x) = 0} = #{x ∈ M : df(x) = 0} ≥

2n

X

i=0

bi. The last inequality comes from theorem 1.25, the weak Morse inequality. Actually the original statement of Arnold is that every Hamiltonian symplectomorphism on M has at least as many fixed points as a function on M has critical points (see [1], [2]), this is clear in particular when H is a time independent Morse function:

3. For the special case where Ht ≡ H, i.e. H is independent of t.

Then

x is a critical point of H ⇔ dH(x) = 0

⇔ X^H(x) = 0

⇔ x(t) ≡ x ∈ P (H).

In particular if H is a Morse function, then #P (H)≥

2n

X

i=0

bi. The Arnold conjecture of the above form has now been proved in full generality. Floer ([7], [8], [9], [12]) invented the Floer homology for the monotone case, which is the analogue of Morse homology on finite dimensional smooth manifolds, by studying the “gradient flow” of a certain action functional on the loop space of M. There is

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also another version of Arnold conjecture for degenerate fixed points (see [16], [10]).

2.2 The Variational Approach

According to classical Morse theory, the existence problem of closed geodesics is restated by the variational approach as the existence of the critical points (which are by definition loops in M) of the energy functional E

E(x) :=

Z

S¹| ˙x|²dt

for x : S¹ → M in some appropriate loop space of M. One then is naturally led to apply the same method to study the critical points of the action functional associated to a Hamiltonian system:

A(x) = − Z

D

u^∗ω + Z

S¹

Ht(x(t))dt (2.2) where u|∂D = x : S¹ → M.

A natural inner product structure is introduced on the appropriate loop space so as to define the gradient of A. Then the zeroes of the gradient of A, i.e. its critical points can be identified to the solutions of the Hamiltonian equation (2.1).

However the classical Morse theory approach fails in this infinite dimensional setting due to several reasons.

Unlike the energy functional, the action functional is both unbounded above and below, so there is no absolute minimum or maximum

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which we can start a Morse complex for cellular decomposition.

Moreover, the “Morse index” would not be finite as in the classical cases, as the subspaces on which the Hessian is positive or negative definite are both infinite dimensional. Finally the gradient of the action functional grad A defined above does not give a well-defined flow on the loop space we considered.

However there is still hope. Floer realized that the essential conditions for Morse theory is still satisfied if we reduce it to the relative gradient flow, that is a flow between two fixed critical points x and y of A. We also use a relative Morse index which, roughly speaking, measures the codimension of the “unstable manifold” of y with respect to the “unstable manifold” of x. Floer found the right analytical setup to analyze the space M(x, y). He then used the structures of these spaces to extract an invariant which is now called Floer homology.

2.3 Action Functional and Moduli Space

Definition 2.7. The contractible loop space L of M is defined to be all the contractible loops in M. i.e. L := {x ∈ C^∞(R/Z, M) | x is contractible}.

Denote D := {z ∈ C : |z| ≤ 1} to be the closed unit disk.

So for x ∈ L, there exists an extension u : D → M such that u(e^i2πt) = x(t).

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We will assume throughout that ω vanish over the second homotopy group π₂(M) of M. i.e.

Z

S²

v^∗ω = 0 (2.3)

for any smooth v : S² → M, noting that this integral depend only on the homotopy class of v. This assumption is needed for the well- definedness of the action functional on M and is also crucial for the compactness of the so called moduli space. A symplecic manifold with this condition is called aspherical and we will denote this condition as ω(π2(M)) = 0.

Definition 2.8. The action functional A = AH : L → R is defined by

A_H(x) := − Z

D

u^∗ω + Z 1

0

H_t(x(t))dt, where u : D → M is an extension of x to the unit disk.

Remark 2.9. A is well defined by the following reason. Suppose u1, u2 both extends x, then we can “glue” the two maps along their boundary to get a map from S² to M. More precisely, let S² ⊂ R³ be the unit sphere and let π : (x, y, z) 7→ (x, y) by the projection onto the x− y plane. Define v : S² → M by

v(p) =







u1(π(p)) if p is on the upper hemisphere, u2(π(p)) if p is on the lower hemisphere.

Then v is a well defined continuous map and R

S²u^∗₁ω − u^∗2ω = R

S²v^∗ω = 0. Therefore A is a well defined function.

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It turns out that Arnold conjecture is easier to prove in the so called monotone case. Here we will make an even stronger assumption. Let J be an almost complex structure which is compatible with ω, i.e. J ∈ C^∞(End(TM)), J² = −I and

g(ξ, η) = hξ, ηi := ω(ξ, Jη), ξ, η ∈ T^xM, (2.4) defines a Riemannian metric on M. Then by the symmetry of g, both ω and g are J-invariant, i.e. hJξ, Jηi = hξ, ηi and ω(Jξ, Jη) = ω(ξ, η). Such J exists in abundance and in fact the space J of all compatible almost complex structures of M is contractible (see for example [23]). Then (T M, J) is a complex vector bundle over M with first Chern class c1 = c1(T M, J) ∈ H²(M, Z). c1 is independent of the choice of J as we can join two such complex structures J1, J2 by a path and thus induce an isomorphism between (T M, J1) and (T M, J2) as complex vector bundle.

We will assume throughout, as in (2.3), that c1 vanishes on π2(M):

Z

S²

v^∗c1 = 0 (2.5)

for any v : S² → M. This assumption, denoted by c¹(π2(M)) = 0, is needed to give a well-defined Maslov type index for the critical

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point of A and thus a grading of the Floer homology groups. Let us state again our assumptions.

Assumption 2.10. We will assume throughout that both ω and c1

vanishes on π₂(M):

Z

S²

v^∗ω = 0 and Z

S²

v^∗c1 = 0 for any v ∈ C^∞(S², M).

Remark 2.11. 1. Floer [11] actually proved the Arnold conjecture in the more general case where M is monotone. This means

Z

S²

v^∗c1 = c Z

S²

v^∗ω

for any v : S² → M, where c is a positive constant. The weakly monotone case was proved by Hofer-Salamon [17] and Ono [26]. The general case was proved by Fukaya-Ono [14], Liu-Tian [21] and Ruan [29].

2. In the monotone case, by rescaling ω if necessary, R

S²v^∗ω ∈ Z for any v : S¹ → M. The argument in remark 2.9 shows that A is a well defined circle-valued function.

L is a very large space and is not a finite dimensional manifold (except when M is a point). For x∈ L, we define a “tangent vector”

ξ to be a vector field on x, i.e. ξ(t) ∈ Tx(t)M. In other words, ξ is a section of the induced bundle x^∗T M. Fix ξ, let ys = y(s,·) be a one- parameter variation of contractible loop such that ^∂y_∂s(0, t) = ξ(t)

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and y(0, t) = x(t). Explicitly, we can choose y(s, t) = exp_x(t)(s · ξ) (M is a compact Riemannian manifold once J is chosen). Let u : D→ M be an extension of x in the sense that u(e^i2πt) = x(t). Note that we can extend y(s,·) by the “gluing” map y|^S¹×[0,s]#u for each s.

Then

dA(x)ξ = d ds

_s=0

(− Z

D

u^∗ω− Z s

0

Z 1 0

y^∗ω + Z 1

0

Ht(y(s, t))dt)

= d

ds s=0

( Z s

0

Z 1 0

ω(∂y

∂t, ∂y

∂s)dt ds + Z 1

0

Ht(y(s, t))dt)

= Z 1

0

(ω(dx

dt, ξ) + dHt(ξ))dt.

Therefore dA(x) = 0 ⇔ ω( ˙x, ·) = −dH^t. i.e. ˙x(t) = Xt(x(t)).

So critical points of the action functional correspond to the contractible periodic solutions to the Hamiltonian equation. Define

J := {ω-compatible almost complex structure on M}

and let J ∈ J . Let g be the induced Riemannian metric. Let ξ, η ∈ TxL, so ξ(t), η(t) ∈ Tx(t)M. We define a Riemannian metric

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on L by

hξ, ηi :=

Z 1 0

g(ξ(t), η(t))dt.

Then

hgrad A, ξi = dA(ξ)

= Z 1

0

(ω( ˙x, ξ) + dHt(ξ))dt

= Z 1

0

(ω(J ˙x, Jξ) +h∇Ht, ξi)dt (∇ is the gradient w.r.t. g)

= Z 1

0 hJ ˙x + ∇Ht, ξidt.

So

grad A(x)(t) = J ˙x(t) +∇Ht(x(t)). (2.6) A negative gradient flow line of A is u : R → L, s 7→ us(·) such that

du

ds = −grad A(us).

By the above calculations, regarding u = u(s, t) := us(t) : R × (R/Z) → M, u is given by the partial differential equation

∂u

∂s = −J∂u

∂t − ∇Ht(u) i.e. ∂u

∂s + J(u)∂u

∂t +∇H(t, u) = 0. (2.7) We denote the left hand side of the above equation by

∂(u) = ∂H,J(u) := ∂u

∂s + J(u)∂u

∂t +∇H(t, u).

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Remark 2.12. 1. The equation (2.7) can also be written as

∂u

∂s + J(∂u

∂t − Xt) = 0.

This is because J∇Ht = Xt as −dHt = h−∇Ht,·i = ω(−∇Ht, J·) = ω(J∇H^t,·) = ω(X^t,·).

2. If u(s, t) ≡ x(t) satisfying (2.7) is independent of s, then x(t) is a critical point of A as grad A(x) = −^∂u_∂s = 0, thus it is a periodic solution for (2.1).

If Ht ≡ constant, then (2.7) becomes

∂u

∂s + J∂u

∂t = 0,

that means u is a J-holomorphic curve. (A J-holomorphic curve is a map u from a Riemann surface (Σ, i) to an almost complex manifold (M, J) such that J ◦ du = du ◦ i, see [15]) 3. Finally if H(t, x) = H(x) is independent of t, then for those

solutions u = u(s) to (2.7) which is also independent of t satisfies

du

ds = −∇H(u). (2.8)

i.e. it satisfies the gradient flow equation for H. This observation will be useful to relate the Morse homology with the Floer homology as we will see later in this section (see also section 4.5).

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We are going to apply Morse-type theory to study the gradient flow line of A and more importantly, understand how the behavior of the critical points of A relates with the topology of M. However there are some problems preventing us from directly applying the classical Morse theory to the action functional.

In finite dimensional Morse theory, every gradient flow line of a Morse function f on a compact manifold M “begins” and “ends” at a critical point. More precisely, if γ(t) is a gradient flow line then

t→±∞lim γ(t) exists and the two limits are both critical points. However this is not true for the symplectic Floer theory. Actually this is true only when u is bounded.

Definition 2.13. Let u ∈ C^∞(R × S¹, M). The energy of u is defined by

E(u) := 1 2

Z 1 0

Z _∞

−∞

(

∂u

∂s

2

+

∂u

∂t − Xt(u)

2

)ds dt.

u is said to be bounded if E(u) < ∞.

Theorem 2.14. Suppose u = u(s, t) ∈ C^∞(R × (R/Z), M) is a contractible solution of (2.7). Then E(u) < ∞ if and only if there exists x^± ∈ P (H) such that

s→±∞lim u(s, t) = x^±(t), (2.9) the limits being uniform in t.

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Figure 2.1: Flow line of symplectic action.

Proof. We prove “⇐”. For u satisfying (2.9), E(u) = 1

2 Z 1

0

Z _∞

−∞

(

∂u

∂s

2

+

∂u

∂t − Xt(u)

2

)ds dt

= 1 2

Z _∞

−∞

Z 1 0

(

∂u

∂s

2

+

J∂u

∂t +∇Ht(u)

2

)dt ds

=

Z _∞

−∞

Z 1 0

∂u

∂s

2

dt ds

=

Z _∞

−∞

du ds

2

ds where || · || is the norm in L

=

Z _∞

−∞hdu

ds,−grad Aids

=

Z _∞

−∞

d

ds(−A(us))ds

= A(x⁻)− A(x⁺) < ∞. (2.10) We will prove the converse in theorem 4.2.

Suppose u solves (2.7) and E(u) < ∞, then u is called a bounded solution of (2.7) and denote the space of all bounded solutions of (2.7) by M. Given x^± ∈ P (H), we also define

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Definition 2.15. The moduli space of bounded solutions M(x⁻, x⁺) =M(x⁻, x⁺; H, J)

:= {u ∈ C^∞(R× S¹, M) | u solves (2.7) and lim_s

→±∞u(s, t) = x^±(t)}

= {u ∈ C^∞(R× S¹, M) | ∂(u) = 0 and lim

s→±∞u(s, t) = x^±(t)}.

In view of theorem 2.14, it is clear that

M = [

x,y∈P (H)

M(x, y).

Theorem 2.14 suggests that suggests that instead of considering the flow of A on the loop space L, one should look at the space of bounded energy solutions of its gradient flow equation.

Remark 2.16. If u ∈ M(x⁻, x⁺) with E(u) = 0, then ^∂u_∂s ≡ 0 and so u = u(t) satisfies ^du_dt = X_t(u), therefore u(t) = x⁻(t) = x⁺(t) ∈ P (H). In particular if x⁻ 6= x⁺ and M(x⁻, x⁺) 6= φ then the proof of theorem 2.14 shows that A(x⁻) > A(x⁺). This is analogous to the fact in classical Morse theory that the value of a Morse function f on M decreases strictly along a non-constant gradient flow line.

Remark 2.17. M(x⁻, x⁺) minimizes E among all curves with bound-

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ary conditions (2.9), this follows from E(u) = 1

2 Z _∞

−∞

Z 1 0

(

∂u

∂s

2

+

J∂u

∂t +∇H

2

)dt ds

= 1 2

Z _∞

−∞

Z 1 0

(

∂u

∂s + J∂u

∂t +∇H

2

− 2h∂u

∂s, J∂u

∂t +∇Hi)dt ds

= 1 2

Z _∞

−∞

Z 1 0

∂u

∂s + J∂u

∂t + ∇H

2

dt ds + Z _∞

−∞hdu

ds,−grad A ◦ ui)ds

= 1 2

Z _∞

−∞

Z 1 0

∂u

∂s + J∂u

∂t + ∇H

2

dt ds + A(x⁻)− A(x⁺).

There is a natural R-action on M(x⁻, x⁺) given by r · u(s, t) = u(r+s, t) for r ∈ R. Define the moduli space of unparametrized bounded solutions

M(xˆ ⁻, x⁺) := M(x⁻, x⁺)/R.

We study M(x, y; H, J) locally by linearizing ∂ at u ∈ M(x, y) to get a differential operator F (u). More precisely, differentiating equation (2.7) in the direction of a vector field ξ ∈ C^∞(u^∗T M) on u leads to the first order linear differential operator F (u) = D∂(u):

F (u)ξ = ∇^sξ + J(u)∇^tξ + (∇^ξJ(u))∂u

∂t +∇^ξ∇H^t(u)

where ∇ denotes the covariant derivative with respect to the Rie- mannian metric given by (2.4). It turns out that if x, y are non- degenerate then F (u) is a Fredholm operator with a finite index between two appropriate Sobolev spaces. We want to smoothly extend ∂ as a section of a Banach bundle over a Banach manifold, so thatM(x⁻, x⁺) is the zero section of ∂ and argue that 0 is a regular

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value of ∂ if we choose a suitable J. After this has been done we can apply implicit function theorem to conclude that M(x⁻, x⁺) is a smooth manifold with the dimension near u given by the index of F (u).

Definition 2.18. (H, J) is called a regular pair if 1. All contractible x∈ P (H) are non-degenerate, and

2. If x^± ∈ P (H) are contractible and u ∈ M(x⁻, x⁺), then F (u) is surjective.

Due to an infinite dimensional version of Sard’s theorem by Smale [35] (see also [13], [33] for the details), we have a transversality result:

Proposition 2.19. There is a dense subset of smooth almost complex structure J^reg ⊂ C^∞(End(TM)) such that for all J ∈ J^reg and u ∈ M, F (u) is onto, i.e. (H, J) is regular.

So by implicit function theorem and by choosing a regular pair, we have

Theorem 2.20. For a regular pair (H, J), M(x⁻, x⁺; H, J) is a finite dimensional smooth manifold for x^± ∈ P (H) and the dimension of M(x⁻, x⁺) is given by the index:

dimM(x⁻, x⁺) = index F (u).

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We will see that index F (u) can be in turn calculated by the difference of a Maslov-type index µ:

index F (u) = µ(x⁻)− µ(x⁺).

The Maslov index µ : P (H) → Z associates an integer to every contractible periodic solution x ∈ P (H) of (2.1). The Maslov index will play the role of grading the Floer homology groups just like the Morse index does in Morse homology.

2.4 Construction of Floer Homology

It follows from the manifold structure ofM(x,⁻, x⁺) and a compactness- gluing argument using the Gromov’s compactness theorem [15] that we have the following theorem.

Proposition 2.21. Suppose (H, J) is regular, x^± ∈ P (H) such that µ(x⁻) − µ(x⁺) = 1. Then ˆM(x⁻, x⁺) is a compact 0-dimensional manifold, i.e. it consists of finite number of points. In other words the set of connecting trajectories (modulo shifting) between x⁻ and x⁺ is finite.

This result is used to construct the boundary operator in Floer homology. The so called compactness-gluing argument is very useful and it is used repeatedly to prove several results as we will see in chapter 4. Let (H, J) be a fixed regular pair. For simplicity, we work with Z2 coefficient only.

Floer Homology on Symplectic Manifolds