The Floer Homology of Cotangent Bundles

Contents

1 Introduction 2

2 Background 62.1 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Construction of the Floer Homology of T∗M 123.1 Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Finite-Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . 173.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 The Floer Complex . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Construction of the Lagrangian Morse Homology 39

5 The Isomorphism between the Floer and Morse Homologies 425.1 Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Finite-Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . 435.3 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 The Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1

1 Introduction

The aim of this essay is to present the construction of Floer homology for thecotangent bundle T∗M of a compact connected oriented smooth manifold M,and to describe the approach of Abbondandolo and Schwarz in [2] to showingthat this is isomorphic to the singular homology of the free loop space Λ(M) =C0(S1,M) ofM.

The original inspiration for the construction of Floer homology was theArnol’d conjecture: If (X,ω) is a compact symplectic manifold and H is a 1-periodic smooth function on X (called a 1-periodic Hamiltonian), then H inducesa vector field XH viaω(XH, –) = −dH, and this in turn induces an integral flowφtH; Arnol’d conjectured that the number of fixed points of the time-1 map φ1His bounded below by the sum of the Betti numbers of X. For suitable manifoldsX, Andreas Floer constructed a chain complex whose groups are generated bythese fixed points and showed that its homology, the Floer homology of X, is iso-morphic to the singular homology of X. From this, the proof of the conjectureis immediate.

Floer homology can also be constructed on certain non-compact symplecticmanifolds, given suitable restrictions on the function H. In particular, this ispossible in the case of cotangent bundles of compact connected oriented man-ifolds, which is the situation we are concerned with here. Let us sketch howFloer homology is constructed in this case:

1. Observe that the fixed points ofφ1H correspond bijectively to the 1-periodicsolutions of the Hamiltonian equation

x(t) = XH(t, x(t)).

Let P(H) be the set of these.

2. Introduce a time-dependent 1-periodic almost complex structure J onT∗M. Then the elements of P(H) are the 1-periodic stationary solutionsof the equation

∂su− J(t, u)(∂tu− XH(t, u)) = 0.

For x± in P(H), let M(x−, x+) be the set of smooth 1-periodic solutions ofthis that tend asymptotically to x± as t→ ±∞.

3. Construct an integral index µ, the Maslov index, for the elements of P(H).

4. Show that for x± ∈ P(H) the space M(x−, x+) is a smooth manifold ofdimension µ(x−) − µ(x+).

5. Show that M(x−, x+) is precompact in the C∞loc-topology.

6. M(x−, x+) has a free R-action given by translation in the first coordi-nate; let M(x−, x+) be the quotient of M(x−, x+) by this. Then if µ(x−) −

µ(x+) = 1, the space M(x−, x+) is a precompact 0-dimensional manifold,and so it is a finite set of points.

7. Show that the manifolds M(x−, x+) are orientable.

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8. Let CFk(H) be the free abelian group generated by the elements of P(H)of Maslov index k. If µ(x−) − µ(x+) = 1, let n(x−, x+) be the numberof elements of M(x−, x+), counted with signs given by the orientation.Then define the boundary map ∂ by, for x ∈ P(H),

∂x =∑

y∈P(H)µ(x−)−µ(x+)=1

n(x, y)y.

9. Prove that ∂2 = 0. Thus (CF∗(H), ∂) is a chain complex. Let HF∗(H, J) beits homology; this is the Floer homology of H.

10. Show that HF∗(H, J) is independent of H, J, and the chosen orientationsof the manifolds M(x−, x+).

For a non-compact manifold, the Floer homology is no longer necessarilythe same as the singular homology. For the cotangent bundle T∗M, ClaudeViterbo conjectured in 1994 that its Floer homology is isomorphic to the singu-lar homology of the free loop space Λ(M). So far, three different proofs of thisresult have been found: The first was published by Viterbo in 1996, the secondby Dietmar Salamon and Joa Weber in 2003, and the third by Alberto Abbon-dandolo and Matthias Schwarz in 2004 ([10, p. 6]). It is this third proof that wewill discuss here.

The key idea of the proof of Abbondandolo and Schwarz is to constructan isomorphism between the Floer homology of T∗M and a certain Morse ho-mology on a space of loops, which we already know to be isomorphic to thesingular homology of Λ(M). Let us outline how this is done:

1. A 1-periodic Lagrangian is a smooth 1-periodic function L on the tangentbundle TM. A Lagrangian L defines an action functional E on Λ1(M) (thespace of loops of Sobolev classW1,2) by

E(x) =

∫10

L(t, x, x)dt.

Let P(L) be the set of critical points of E.

2. A procedure known as the Legendre transform produces Hamiltonians Hon T∗M from sufficiently nice Lagrangians L. Show that under the Legen-dre transform the elements of P(L) correspond bijectively to the elementsof P(H), and that the Morse index m(q) of an element q ∈ P(L) (i.e. thenumber of non-negative eigenvalues of the Hessian of E at q) equals theMaslov index µ(x) of the corresponding element x ∈ P(H).

3. Construct the Morse homology of E:

(i) Let∇E be the gradient of E with respect to a suitable metric, and letψt be the integral flow of −∇E. Set

Ws(x) = q ∈ Λ1(M) : limt→∞ψt(q) = x,

Wu(x) = q ∈ Λ1(M) : limt→−∞ψt(q) = x.

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(ii) Show that, for x, y ∈ P(L), the spaceWu(x)∩Ws(y) is an orientablemanifold of dimension m(x) −m(y), and that if m(x) −m(y) = 1

then the setN(x, y) := Wu(x) ∩Ws(y)/R

is compact, and hence a finite set of points.

(iii) Define CMk(L) to be the free abelian group on the elements of P(L)of Morse index k. Ifm(x)−m(y) = 1, let ν(x, y) be the number of el-ements of N(x, y), counted with orientation, then define the bound-ary map ∂ by, for x ∈ P(L),

∂x =∑

y∈P(L)m(x)−m(y)=1

ν(x, y)y.

(iv) Show that ∂2 = 0. Thus (CM∗(L), ∂) is a chain complex. LetHM∗(L, g)be its homology; this is the Morse homology of L.

(v) Show that HM∗(L, g) is independent of L and the chosen metric,and that it is isomorphic to the singular homology of Λ1(M). SinceΛ1(M) is homotopy equivalent to Λ(M), it follows that

HM∗(L, g) ∼= H∗(Λ(M)).

4. For q ∈ P(L), x ∈ P(H), let M+(q, x) be the set of smooth maps u :[0,∞)× T→ T∗M such that:

(i) Projecting u(0, –) toM gives an element ofWu(q).

(ii) u satisfies∂su− J(t, u)(∂tu− XH(t, u)) = 0

on (0,∞).

(iii) u converges to x as t→∞.

(The map u must also satisfy some technical conditions.) In a sense weare looking at orbits that go from q to x.

5. Show that M+(q, x) is a smooth manifold of dimensionm(q) − µ(x).

6. Show that M+(q, x) is precompact in the C∞loc-topology. Thus, if m(q) =

µ(x), M+(q, x) is a finite set of points.

7. Show that M+(q, x) is orientable.

8. Ifm(q) = µ(x), let n+(q, x) be the number of points of M+(q, x), countedwith signs from the orientation. For q ∈ P(L), define

Θ(q) =∑

x∈P(H)m(q)=µ(x)

n+(q, x) x.

9. Show that Θ is a chain complex homomorphism.

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10. Show that Θ is invertible. Thus Θ is a chain complex isomorphism from(CM∗(L), ∂) to (CF∗(H), ∂). This implies that HM∗(L, g) and HF∗(H) areisomorphic, and so HF∗(H) is isomorphic to H∗(Λ(M)).

Having sketched the main results we will consider, let us now give an out-line of the contents of this essay:

We begin by giving some relevant background material from linear algebra,geometry and analysis in section 2. In particular, we discuss basic definitionsand results relating to symplectic manifolds, Sobolev spaces, and Fredholmoperators. The algebra and geometry parts of this section are mainly based onchapters 2 and 3 of the book Introduction to Symplectic Topology [6] by McDuffand Salamon, and the analysis part on appendices A and B of J-HolomorphicCurves and Symplectic Topology [7] by the same authors.

In section 3 we describe the construction of the Floer homology of T∗M.We will focus on the results concerning the finite-dimensionality and precom-pactness of the moduli spaces, and try to give relatively complete proofs ofthese, but we will also discuss the other steps needed to construct the Floercomplex, though in far less detail. This section is mainly based on the paperOn the Floer homology of Cotangent Bundles [2] by Abbondandolo and Schwarzand Salamon’s Lectures on Floer Homology [9], though part of the proof of com-pactness is taken from the paper Morse Theory, the Conley Index and Floer Homol-ogy [8] by Salamon (namely the proofs of Corollary 3.30 and Proposition 3.31).The detailed proofs of similar results for J-holomorphic curves in J-HolomorphicCurves and Symplectic Topology [7], which often resemble the proofs for Floer ho-mology, were also helpful.

Section 4 sketches the construction of the Morse homology of a function onTM, essentially without any proofs.

In section 5, we sketch the construction of an isomorphism between thethe Morse and Floer chain complexes, without going into the details of theproofs. The material of sections 4 and 5 is essentially taken from On the Floerhomology of Cotangent Bundles [2], though the discussion of Lagrangians and theLegendre transform also draws on sections 3.5 and 3.6 of the book Foundationsof Mechanics [3] by Abraham and Marsden.

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2 Background

In this section, we will state some of the definitions and results from symplecticgeometry and analysis that are needed to understand the rest of this essay.

2.1 Linear Algebra

We begin by defining symplectic and complex structures on vector spaces, as aprelude to defining their equivalents on manifolds.

Definition 2.1. A symplectic form on a finite-dimensional real vector space Vis a non-degenerate skew-symmetric bilinear form. This means that for all v,w ∈ V ,

ω(v,w) = −ω(w, v),

and for every v ∈ V there is some w ∈ V such that ω(v,w) 6= 0. We call (V,ω)a symplectic vector space.

Example. If e1, . . . , e2n is the standard basis of R2n, set xi = ei and yi = en+i

for i = 1, . . . , n. Then the standard symplectic formω0 on R2n is given by

ω0 =

n∑i=1

dxi ∧ dyi,

where dxi, dyi are dual to xi, yi, respectively. In matrix form, this is(0 −II 0

),

which is clearly non-degenerate.

Definition 2.2. Suppose W is a linear subspace of a symplectic vector space(V,ω). We say thatW is Lagrangian ifW is equal to its symplectic complementWω, which is the space of all vectors v ∈ V such that ω(v,w) = 0 for everyw ∈W.

Example. The vertical space

λ0 = 0× Rn = span yi

is easily seen to be a Lagrangian subspace of (R2n, ω0). This space λ0 will beused later on.

Definition 2.3. A linear symplectomorphism ψ of a symplectic vector space(V,ω) is a linear automorphism ψ : V → V such that ψ∗ω = ω (i.e. for all v,w ∈ V we haveω(ψ(v), ψ(w)) = ω(v,w)). The group of such maps is denotedSp(V,ω). The symplectic group Sp(2n) is Sp(R2n,ω0).

Definition 2.4. A complex structure on a real vector space V is a linear auto-morphism J : V → V such that J2 = −I. A complex structure J is compatiblewith a symplectic formω onV ifω(–, J–) is positive-definite andω(J–, J–) ≡ ω.

Example. The standard complex structure J0 on R2n is given, with respectto the standard basis, by

(0 −II 0

). By writing ω0(–, J0–) and ω0(J0–, J0–) as

matrices, we see that J0 is compatible withω0.

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2.2 Geometry

We now give the basic definitions of symplectic geometry:

Definition 2.5. A symplectic manifold (X,ω) is a finite-dimensional smoothmanifold X equipped with a closed non-degenerate 2-form ω ∈ Ω2(X) (sodω = 0 and each fibre TpX is a symplectic vector space when equipped withωp).

Example (Cotangent bundles). Let M be a finite-dimensional smooth mani-fold; the cotangent bundle T∗M is then a symplectic manifold in a natural way.The symplectic form on T∗M is constructed as follows: Let τ∗ : T∗M → M bethe standard projection, sending every element of T∗qM to q. Then dτ∗ is a mapT T∗M → TM. An element x of T∗qM is a linear functional on TqM. We cantherefore define the Liouville form θ on T∗M by

θx = x dxτ∗ : TxT∗M→ R.

Thenω = dθ is a symplectic form on T∗M—ω is clearly closed, and to see thatit is non-degenerate we work in coordinates: If q1, . . . , qn are local coordinatesonM and p1, . . . , pn are a local coframe for T∗M, then we can write

dxτ∗ =∑i

∂

∂qi

∣∣∣∣x

dqi.

Then if x is (q1, . . . , qn;p1, . . . , pn) in coordinates, we have

x dxτ∗ =∑i

x

(∂

∂qi

∣∣∣∣x

)dqi =

∑i

pidqi.

Thusω = dθ =

∑i

dpi ∧ dqi,

which is obviously non-degenerate.Note also that the vertical space TVx T∗M = kerdxτ∗ is a Lagrangian sub-

space of (TxT∗M,ωx), since it is clearly the span of

∂∂pi

.

Definition 2.6. A symplectomorphism on a symplectic manifold (X,ω) is adiffeomorphism ψ : X→ X such that ψ∗ω = ω.

Definition 2.7. If (X,ω) is a symplectic manifold, and Y ⊂ X is an embed-ded submanifold, then Y is a Lagrangian submanifold if TpY is a Lagrangiansubspace of TpX for every p ∈ Y.

Definition 2.8. A Hamiltonian function H on a symplectic manifold (X,ω) isjust a smooth function H : X → R, possibly time-dependent (i.e. H can be afunction R× X→ R). The Hamiltonian vector field XH of H is defined by

ιXHω = ω(XH, –) = −dH,

and the Hamiltonian flow associated toH is the integral flow φtH generated byXH, i.e. the 1-parameter group of diffeomorphisms φtH satisfying

d

dtφtH = XH(t, φtH)

and φ0H = idX. The maps φtH are symplectomorphisms (see [6, p. 83]).

7

Next, we define some notions related to complex structures and symplecticand Hermitian vector bundles:

Definition 2.9. A symplectic vector bundle (E,ω) over a manifold X is a realvector bundle E → X with a symplectic form ωq on each fibre Eq, varyingsmoothly with q (i.e. ω is a smooth section of Λ2E∗). If E has rank 2n over X, asymplectic trivialization of E is a smooth bundle isomorphism

Φ : X× R2n → E

such thatΦ∗ω = ω0 (on each fibre).

Definition 2.10. A complex structure on a vector bundle E → M is an auto-morphism J of E such that J2 = −idE. When E is the tangent bundle TX of amanifold X, a complex structure on E is called an almost complex structure onX. If (E,ω) is a symplectic vector bundle, then a complex structure J on E iscompatible with ω if ω(–, J–) is positive-definite and ω(J–, J–) = ω (so if E isTX,ω(–, J–) defines a Riemannian metric on X).

Definition 2.11. A Hermitian vector bundle (E,ω, J) over a manifold X is asymplectic vector bundle (E,ω) over X equipped with an ω-compatible com-plex structure J. If E has rank 2n over X, a unitary trivialization of E is asmooth bundle isomorphism Φ : X× R2n → E such that Φ∗ω = ω0, Φ∗J = J0and Φ∗ω(–, J–) is the standard inner product on R2n.

Theorem 2.12. Let (E,ω) be a symplectic vector bundle. Then there exists acomplex structure J on E compatible withω.

See [6, p. 70] for a proof of this. The following definitions are also conve-niently given here:

Definition 2.13. The Hessian of a smooth function f : X → R on a smoothmanifold X is the map Hess(f, p) : TpX→ T∗pX,

Hess(f, p)(v) = ∇v(df),

where ∇ is some connection on TX. (If p is a critical point of f, i.e. dpf = 0,then Hess(f, p) is independent of the choice of ∇.) In local coordinates, this isthe matrix of second-order partial derivatives.

Definition 2.14. If X is a topological space, then the free loop space Λ(X) of Xis C0(S1, X), i.e. the space of all continuous loops in X.

Definition 2.15. Banach manifolds and Hilbert manifolds are defined in ex-actly the same way as real manifolds, except that charts go to open sets in Ba-nach spaces and Hilbert spaces, respectively, rather than in Rn. A Cl Banachmanifold is defined similarly, but the transition functions between charts areonly required to be of class Cl, rather than smooth.

Definition 2.16. A Banach bundle over a Banach manifold X is a Banach man-ifold E equipped with a smooth map π : E → X, such that for all p ∈ X theinverse image π−1(p) is a vector space isomorphic to some Banach space F.The space Emust also be locally trivial in the obvious sense.

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2.3 Analysis

In this subsection we give some important definitions and results from analy-sis, mainly related to Fredholm operators and Sobolev spaces. We begin withthe former:

Definition 2.17. A Fredholm operator is a bounded linear operator betweenBanach spaces that has closed range and whose kernel and cokernel are bothfinite-dimensional. If Φ : E → F is a Fredholm operator, then its Fredholmindex indΦ is

dim kerΦ− dim cokerΦ.

Proposition 2.18. Suppose that X, Y, Z are Banach spaces, Φ : X → Y is abounded linear operator, and K : X → Z is a compact linear operator. If thereexists a constant c > 0 such that for every x ∈ X,

‖x‖X ≤ c(‖Φx‖Y + ‖Kx‖Z),

then Φ has closed range and finite-dimensional kernel.

A proof can be found in [7, p. 493].

Theorem 2.19. Let Φ : E→ F be a Fredholm operator.

(i) There exists an ε > 0 such that if Ψ : E → F is a bounded linear operatorwith ‖Ψ‖ < ε, thenΦ+Ψ is a Fredholm operator and ind(Φ+Ψ) = indΦ.

(ii) If K : E→ F is a compact operator, thenΦ+K is a Fredholm operator, andind(Φ+ K) = indΦ.

For a proof, see [7, p. 494].

Definition 2.20. A differentiable map f : U → Y between open subsets of Ba-nach spaces is a Fredholm map if dxf is a Fredholm operator for every x ∈ U.The Fredholm index inddxf can be shown to be independent of x, so we candenote it by ind f.

Theorem 2.21 (Banach Space Implicit Function Theorem). Suppose that X, Yare Banach spaces, U ⊂ X is open, and f : U→ Y is of class Cl. If y is a regularvalue of f (i.e. dxf is surjective for all x ∈ f−1(y)), then f−1(y) is a Cl Banachmanifold and

Txf−1(y) = kerdxf

for all x ∈ f−1(y).

Thus, if f is a smooth Fredholm map, f−1(y) is a smooth manifold of di-mension ind f; a proof of this result is given in [7, pp. 504–505].

We now turn to Sobolev spaces and some related results:

Notation. For the following definitions, let Ω be an open subset of Rn. Wewrite C∞(Ω) for the space of restrictions of smooth functions on Rn to Ω, andC∞0 (Ω) for the space of smooth compactly supported functions onΩ.

9

Definition 2.22. Let u : Ω → R be locally integrable. A locally integrablefunction w : Ω → R is the weak derivative of u corresponding to a multi-index ν if for every φ ∈ C∞

0 (Ω) the following identity holds:∫Ω

u(x)∂νφ(x)dx = (−1)|ν|

∫Ω

w(x)φ(x)dx.

If such a w exists, it is unique up to measure 0. Weak derivatives are the sameas derivatives when these exist.

Definition 2.23. The Sobolev space Wk,p(Ω) is the space of all (equivalenceclasses of) functions u ∈ Lp(Ω) such that u has all weak derivatives of order≤ k and these all have finite Lp-norm. For 1 ≤ p < ∞, the Wk,p-norm ofu ∈Wk,p(Ω) is

‖u‖k,p =

∫Ω

∑|ν|≤k

|∂νu(x)|pdx

1/p(where ∂νu is the weak derivative corresponding to the multi-index ν). Forp =∞,

‖u‖k,∞ = max|ν|≤k

‖∂νu‖∞.

With these norms, the spaces Wk,p(Ω) are Banach spaces. We define Wk,ploc (Ω)

to be the space of all (equivalence classes of) locally Lp-integrable functions u :Ω→ R such that u|U lies inWk,p(U) for all precompact open sets U containedinΩ. Also,Wk,p

0 (Ω) is defined to be the closure of C∞0 (Ω) inWk,p(Ω).

The following definition is needed to state the Sobolev Embedding Theo-rem:

Definition 2.24. An open setΩ ⊂ Rn is a Lipschitz domain if its boundary canbe locally represented as the graph of a Lipschitz function. By this we meanthat for all x ∈ ∂Ω there is a neighbourhood U of x, a unit vector ξ, a constantδ > 0, and a Lipschitz-continuous function f : ξ⊥ → R such that f(0) = 0 and

Ω ∩U = x+ η+ tξ : η ∈ ξ⊥, ‖η‖ < δ, f(η) < t < δ.

Theorem 2.25 (Sobolev Embedding Theorem). Let Ω be a bounded Lipschitzdomain.

(i) If kp > n and 0 < k − n/p < 1, then the inclusion Wk,p(Ω) → C0(Ω) iscompact.

(ii) If kp < n and q < np/(n − kp), then the inclusion Wk,p(Ω) → Lq(Ω) iscompact.

(iii) If kp = n, thenWk,p embeds continuously into Lq for 1 ≤ q <∞.

For a proof, see [7, pp. 517-521].

Theorem 2.26. For every p > 1 and positive integer n, there is a constant c > 0such that all u ∈ C∞

0 (Rn) satisfyn∑

j,k=1

‖∂j∂ku‖Lp ≤ c‖∆u‖Lp ,

where ∆ =∑ni=1 ∂

2i is the Laplacian.

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This is a corollary of the Calderon-Zygmund inequality; see [7, p. 527] fora proof. We conclude this section with some results on the Cauchy-Riemannoperator

∂

∂z=1

2

(∂

∂x+ i

∂

∂y

),

where z = x + iy is the standard complex coordinate on C = R2; for proofs ofthese see [5, pp. 190–192].

Lemma 2.27. Let N(z) = 1πz . If f : C → C is a smooth compactly supported

function then the convolution u = N ∗ f solves the equation

∂u

∂z= f.

We say that u is a weak solution of the differential equation

∂u

∂z= f

if ∫ ⟨∂φ

∂z, u

⟩= −

∫〈φ, f〉

for every φ ∈ C∞0 (C).

Lemma 2.28. Let u, f ∈ Lp(C), both with compact support. Then u is a weaksolution of

∂u

∂z= f

if and only if u = N ∗ f.

Theorem 2.29. Let 1 < p < ∞, let k be a non-negative integer, and let Ω ⊂ Cbe an open domain. If u ∈ Lploc(Ω) is a weak solution of

∂u

∂z= f

with f ∈Wk,ploc (Ω), then u lies inWk+1,p

loc (Ω).

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3 Construction of the Floer Homology of T ∗M

In this section we will describe the construction of Floer homology on cotan-gent bundles.

3.1 Set-Up

LetM be a compact connected oriented smooth manifold of dimensionn. ThenT∗M is a symplectic manifold with its standard symplectic form ω, as definedabove. A 1-periodic Hamiltonian on T∗M is a smooth functionH : T×T∗M→R. This determines a 1-periodic vector field XH via ιXHω = −dH, with integralflow φtH. The Hamiltonian equation for H is

x(t) = XH(t, x(t)). (1)

Let P(H) be the set of 1-periodic solutions of this equation.The following simple lemma characterizes P(H), and will prove useful later

on:

Lemma 3.1. The map x 7→ x(0) gives a bijection between P(H) and the set offixed points of φ1H.

Proof. If x ∈ P(H), set γ(t) = φtH(x(0)). Then by the definition of φtH we have

d

dtγ(t) =

(d

dtφtH

)(x(0)) = XH(t, φtH(x(0))) = XH(t, γ(t)).

Thus x(t) andφtH(x(0)) are both solutions of the Hamiltonian equation (1) withthe same value at 0. By standard uniqueness results for differential equationsof this type they must therefore be equal. Hence φ1H(x(0)) = x(1) = x(0), andso x(0) is a fixed point of φ1H. This shows that the map is well-defined, and theuniqueness result also implies that it is injective.

Now suppose that x0 ∈ T∗M is a fixed point of φ1H. If we let γ(t) = φtH(x0)then as above γ(t) = XH(t, γ(t)). Since γ(0) = φ0H(x0) = x0 and γ(1) =φ1H(x0) = x0, we see that γ is 1-periodic and hence lies in P(H). Thus the mapis also surjective.

Next, we will describe P(H) as the set of critical points of a function, namelythe Hamiltonian action functional A on T∗M. For a path x : [0, 1]→ T∗M, thisis defined by

A(x) =

∫10

(θ(x(t)) −H(t, x(t)))dt.

The functional A is smooth on the space Λ1(T∗M) of all loops T → T∗M ofSobolev classW1,2, and its derivative is given by

dxA(ζ) =

∫10

(ω(ζ, x) − d(t,x)H(ζ))dt.

Using the definition of XH, we can rewrite this as

dxA(ζ) =

∫10

ω(ζ, x− XH(t, x))dt.

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Thus, the elements of P(H) are critical points of A. In fact, they can be shownto be all the critical points of A in Λ1(T∗M).

Next, we want to look at solutions of a negative gradient equation for A.To obtain a tractable equation, we choose a smooth time-dependent 1-periodicalmost complex structure J on T∗M that is compatible withω; by Theorem 2.12,such a J certainly exists. Let 〈–, –〉t be the metricω(–, J(t)–); then

dxA(ζ) =

∫10

〈ζ,−J(t, x)(x− XH(t, x))〉tdt.

Let ∇J be the gradient determined by the metric∫10〈–, –〉tdt on Λ1(T∗M).

Then the gradient of A is

∇JA(x) = −J(t, x)(x− XH(t, x)),

whose zeros are still the elements of P(H).We consider the negative gradient equation for A with respect to this met-

ric, namely∂su(s, t) = −∇JA(u(s, t)),

with u ∈ C∞(R× T, T∗M), which we can expand to

∂su− J(t, u)(∂tu− XH(t, u)) = 0. (2)

We observe that P(H) is the set of solutions of this that are stationary withrespect to s.

Definition 3.2. For x± ∈ P(H), the moduli space M(x−, x+) is the space ofsolutions u of the negative gradient equation (2) for which

lims→±∞u(s, t) = x±(t),

with the limit uniform in t. Translation in s defines a free R-action on M(x−, x+),and we define the space M(x−, x+) to be the quotient of M(x−, x+) by this.

As discussed in the introduction, we will construct our chain complex byproving that these moduli spaces have various nice properties. The first ofthese is given by the following lemma:

Lemma 3.3. The action functional A is strictly decreasing along all non-stationarysolutions of the negative gradient equation (2). Hence M(x, x) contains only thestationary solution x, and M(x−, x+) is empty if A(x−) ≤ A(x+) and x− 6= x+.

Proof. Suppose u solves the negative gradient equation (2). Then

∂tu− XH(t, u) = −J(t, u)∂su.

Hence the derivative of A at u(s, –) is given by

du(s,–)A(ζ) =

∫10

ω(ζ, ∂tu− XH(t, x))dt

= −

∫10

ω(ζ, J(t, u)∂su)dt

= −

∫10

〈ζ, ∂su〉tdt.

13

Then by applying the chain rule we get

∂sA(u(s, –)) = du(s,–)A(∂su(s, –))

= −

∫10

〈∂su, ∂su〉tdt

≤ 0.

We have equality at s = s0 if and only if ∂s|s0u = 0 for all t, i.e. if and onlyif u(s0, –) ∈ P(H). But by standard uniqueness results if two solutions of thenegative gradient equation (2) coincide at a point and have equal derivativesthere, then they must be equal everywhere. Hence u must be stationary withrespect to s if ∂sA(u(s, –)) is zero anywhere in R.

Therefore, if u is not stationary then A(u(s, –)) is strictly decreasing in s.

To construct a chain complex we of course need an integral grading forthe elements of P(H). This is provided by the so-called Maslov index, and weconclude this subsection by showing how this is constructed from the Conley-Zehnder index:

Fact. Let S be the space of smooth paths γ : [0, 1]→ Sp(2n) that satisfy γ(0) = I

and det(I − γ(1)) 6= 0. To every γ ∈ S we can assign an integer µCZ(γ), theConley-Zehnder index of γ, with the following properties:

• (Naturality) For any pathΦ : [0, 1]→ Sp(2n), µCZ(ΦγΦ−1) = µCZ(γ).

• (Homotopy) µCZ takes the same value on paths that are homotopic in S.

For the definition and further details we refer to [9, pp. 20–21]. To be ableto apply this we must from now on require that our Hamiltonian H is non-degenerate, meaning that for every x ∈ P(H) the operator dx(0)φ

1H does not

have 1 as an eigenvalue.We wish to use dx(0)φ

tH to construct a path in Sp(2n) . To do this, we must

first prove the following lemma:

Lemma 3.4. For x ∈ P(H), the symplectic vector bundle x∗(T T∗M) admits aunitary trivialization Φ : T × R2n → x∗(T T∗M) such that Φ(t)λ0 = TVx(t)T

∗M

for all t ∈ T.

Proof. First we show that we can identify the bundle TVT∗M with (τ∗)∗T∗M:Use local coordinates q1, . . . , qn on M, q1, . . . , qn, p1, . . . , pn on T∗M, andwrite elements of T∗qM as (q, p). Then τ∗(q, p) = q, and

dτ∗(∂

∂qi

)=

∂

∂qi, dτ∗

(∂

∂pi

)= 0.

Hence

kerd(q,p)τ∗ = span

∂

∂pi

.

Then if (q, r) ∈ ((τ∗)∗T∗M)(q,p) = T∗qM, the map

(q, r) 7→∑i

pi(r)∂

∂pi

∣∣∣∣(q,p)

∈ kerd(q,p)τ∗

14

gives an isomorphism from (τ∗)∗T∗M to TVT∗Mwith inverse

∑i

ai∂

∂pi

∣∣∣∣(q,p)

7→ (q, r),

where pi(r) = ai.Therefore, x∗(TVT∗M) ∼= x∗(τ∗)∗T∗M. SinceM is an oriented manifold, the

vector bundle T∗M is oriented, and hence so are the bundles E := x∗(T T∗M)and EV := x∗(TVT∗M). But it is a theorem of algebraic topology that everyorientable vector bundle over the circle T is trivial, so E and EV must thereforebe trival.

Let ψ : T × R → EV be a trivialization. The symplectic form ω induces asymplectic form on E, which we also denote by ω. Let J be an ω-compatible1-periodic almost complex structure on E. Then Et = J(t)EVt ⊕ EVt since forv ∈ EVt we haveω(v, J(t)v) 6= 0, andω(v,w) = 0 for every w ∈ EVt .

Using this splitting, we defineΦ : T× Rn ⊕ Rn → E by

Φ(t) = (−J(t)ψ(t)J0)⊕ψ(t).

This is clearly a trivialization.If e1, . . . , en is a basis for Rn, then since TVT∗M is a Lagrangian subspace

of T T∗Mwe compute that

Φ∗ω((t, ei, 0), (t, ej, 0)) = 0,

Φ∗ω((t, 0, ei), (t, 0, ej)) = 0,

Φ∗ω((t, ei, 0), (t, 0, ej)) = ω(ψ(t)(ej), J(t)ψ(t)(ei)).

Choose ei(t) to be the image underψ−1 of an orthonormal basis forω(–, J(t)–),varying smoothly with t. Then we clearly get Φ∗ω = ω0 if we identify ei(t)with the standard basis for Rn for all t by means of a bundle isomorphism.

A similar calculation shows that Φ∗J = J0, and hence Φ is a unitary trivial-ization of E. By construction it maps λ0 to EV .

For x ∈ P(H), we can use such a trivialization Φ to construct a path γΦ inSp(2n):

γΦ(t) = Φ(t)−1dx(0)φtHΦ(0).

Since Φ is a symplectic trivialization and φtH is a symplectomorphism, this isindeed a path of symplectic matrices. Moreover, γΦ(0) = I since φ0H = id, andγΦ(1) does not have eigenvalue 1 since γΦ(1)v = v if and only if

dx(0)φ1HΦ(0)(v) = Φ(1)(v) = Φ(0)(v)

(as Φ is 1-periodic), which is impossible because H is non-degenerate. Thusthe path γΦ is an element of S, and so the Conley-Zehnder index µCZ(γΦ) ofγΦ is defined.

Lemma 3.5. The Conley-Zehnder index µCZ(γΦ) is independent of the choiceof trivialization Φ.

15

Proof. For x ∈ P(H), letΦ andΨ be two symplectic trivializations of x∗(T T∗M)such that

Φ(t)λ0 = Ψ(t)λ0 = TVx(t)T∗M

for all t ∈ T. Then we can express γΨ as

γΨ(t) = Ψ(t)−1Φ(t)γΦ(t)Φ(0)−1Ψ(0)

= α(t)−1γΦ(t)α(0),

where α = Φ−1Ψ.The naturality property of µCZ implies that

µCZ(γΦ) = µCZ(α(0)−1γΦα(0)).

We wish to prove that the paths α(t)−1γΦ(t)α(0) and α(0)−1γΦ(t)α(0) arehomotopic, since by the homotopy invariance of µCZ this will imply µCZ(γΦ) =µCZ(γΨ).

The path α is obviously a loop in the space

Sp(2n, λ0) := S ∈ Sp(2n) : Sλ0 = λ0.

Claim. Any loop in Sp(2n, λ0) is null-homotopic in Sp(2n).Given this, the loop α is null-homotopic, and so there must exist a map

A : [0, 1]× T→ Sp(2n)

such that A(s, 0) = α(0) = α(1) = A(s, 1), A(0, –) = α, and A(1, t) = α(0) forall t. Then χ : [0, 1]× T→ Sp(2n) defined by

χ(s, t) = A(s, t)−1γΦ(t)α(0)

is a homotopy fromα(t)−1γΦ(t)α(0) to α(0)−1γΦ(t)α(0). Moreover χ(s, 0) = I

for all s andχ(s, 1) = α(1)−1γΦ(1)α(0) = γΨ(1),

so for all s the map χ(s, 1) does not have eigenvalue 1. Hence χ is a path in thespace S that µCZ acts on, and so by homotopy-invariance of µCZ,

µCZ(γΦ) = µCZ(γΨ).

It remains to prove the claim:

Proof of Claim. We first show that Sp(2n, λ0) retracts to Sp(2n, λ0) ∩ U(n) con-tinuously. We have (see [6, p. 20])

Sp(2n) =

(A B

C D

): ATC = CTA, BTD = DTB, ATD− CTB = I

We can thus express Sp(2n, λ0) as follows:

Sp(2n, λ0) =

(A 0

C D

): ATC = CTA, ATD = I

.

16

Let X be the space (A 0

0 D

): ATD = I

,

and consider the map φ : Sp(2n, λ0)→ X defined by

φ

(A 0

C D

)=

(A 0

0 D

).

This is a retraction: if j : X → Sp(2n, λ0) is the inclusion, then φj = idX and

φt :

(A 0

C D

)7→ (

A 0

tC D

)gives a homotopy from jφ to idSp(2n,λ0).

Now the map F : [0, 1]× Sp(2n)→ Sp(2n) given by

(t, Ψ) 7→ (ΨΨT )−t/2Ψ

is a retraction of Sp(2n) onto U(n) (see [6, p. 46]). Restricting this to X we seethat F|X is a retraction from X to X ∩ U(n). So Sp(2n, λ0) retracts to X ∩ U(n).But U(n) as a subgroup of GL(2n,R) is(

X −YY X

): XTY = YTX, XTX+ YTY = I

,

hence

Sp(2n, λ0) ∩U(n) = X ∩U(n) =

(R 0

0 R

): R ∈ O(n)

.

Since(R 00 R

)here correspoinds to R ∈ GL(n,C), the map det : U(n)→ S1 takes

values ±1 on Sp(2n, λ0) ∩U(n).But the map det : U(n) → S1 induces an isomorphism of fundamental

groups (see [6, p. 46]). Therefore, as det of a loop in Sp(2n, λ0) ∩ U(n) isconstant, such a loop is null-homotopic in U(n) and hence, as U(n) ⊂ Sp(2n),in Sp(2n).

It follows that any loop in Sp(2n, λ0) is null-homotopic in Sp(2n).

This result allows us to define our index as follows:

Definition 3.6. For x ∈ P(H), the Maslov index µ(x) of x is µCZ(γΦ), where Φis any symplectic trivialization of x∗(T T∗M) with Φ(t)λ0 = TVx(t)T

∗M for all t.

3.2 Finite-Dimensionality

In this section, we consider the following theorem:

Theorem 3.7. If M(x−, x+) is non-empty, for x± ∈ P(H), then it is a finite-dimensional smooth manifold of dimension µ(x−) − µ(x+).

We will give a fairly complete proof of the finite-dimensionality of M(x−, x+),but we will only cursorily describe the computation of the dimension. The keystep in the proof of finite-dimensionality is to apply the Banach Space Implicit

17

Function Theorem, Theorem 2.21. To do this, we first embed M(x−, x+) in aBanach manifold:

Fix some r > 2, and choose a metric on T∗M. Let B be the set of all mapsu : R×T→ T∗M of Sobolev classW1,r

loc with respect to this metric which satisfy,for some s0 (depending on u),

u(s, t) =

expx−(t)(ζ

−(s, t)), s ≤ −s0

expx+(t)(ζ+(s, t)), s ≥ s0

,

where ζ−, ζ+ are W1,r-sections of the bundles x−∗(T T∗M) → (−∞,−s0) × T,x+∗(T T∗M) → (s0,∞) × T, respectively. The space B is a smooth Banachmanifold, and for u ∈ B the space TuB is isomorphic to the space of W1,r-sections of u∗(T T∗M).

Now let W be the Banach bundle over B whose fibre Wu at u ∈ B is thespace of Lr-sections of u∗(T T∗M).

Proposition 3.8. M(x−, x+) is the set of zeros of the smooth section ∂J,H : B→W defined by

∂J,H(u) = ∂su− J(t, u)(∂tu− XH(t, u)).

We will not prove this here; roughly speaking we need to show that ev-ery element of M(x−, x+) decays exponentially, which is related to the resultproved in [9, pp. 12, 25–26], and that every zero of ∂J,H is smooth, which isdone by an elliptic bootstrapping argument.

For u ∈M(x−, x+) define the operatorDu : TuB→Wu to be the composite

TuBdu∂J,H−−−−−→ T(u,0)W ∼= TuB⊕Wu

projection−−−−−→Wu,

where the isomorphism comes from splitting T(u,0)W into vectors tangent tothe zero section and vectors orthogonal to these. Thus Du is surjective if andonly if ∂J,H is transverse to the zero section at u, which means that

im(du∂J,H) + TuB = T(u,0)W.

We will assume the following result without proof; according to [2, p. 283],the proof is analogous to that found in [4] for compact symplectic manifolds.

Theorem 3.9. For a dense set of almost complex structures J in the space J(ω)of smooth t-dependent 1-periodic almost complex structures compatible withω, the section ∂J,H is transverse to the zero section everywhere. Let Jreg(H) bethe set of such J, and call them regular.

From now on we assume that our J lies in Jreg(H). The difficult part of theproof of Theorem 3.7 is then to show:

Theorem 3.10. Du is a Fredholm operator for all u ∈M(x−, x+).

Theorem 3.11. Du has Fredholm index µ(x−) − µ(x+).

Assuming these results without proof for now, we can complete the proofas follows:

Define Fu : TuB→Wu by

Fu(ξ) = Ψu(ξ)−1∂J,H(expu(ξ)),

18

where Ψu(ξ) : TuT∗M → Texpu(ξ)T

∗M is parallel transport for the Levi-Civitaconnection along t 7→ expu(tξ). This definition makes sense since ∂J,H(expu(ξ))can be identified with a section of (expu(ξ))∗(T T∗M).) The following lemmarelates this map to Du:

Lemma 3.12. d0Fu = Du.

Proof. Define Gu : TuB→W by

Gu(ξ) = ∂J,H(expu(ξ)).

For a sufficiently small neighbourhood U of 0 = ∂J,H(u) in W we have a mapHu : U→Wu given by

η ∈Wexpu(ξ) 7→ Ψu(ξ)−1 η.

Then for ξ sufficiently close to 0, Fu = Hu Gu, so by the chain rule

d0Fu = dGu(0)Hu d0Gu = d(u,0)Hu d0Gu.

Now as expu(0) = u and d0 expu = id, we can write d0Gu as

d0Gu = du∂J,H d0 expu = du∂J,H.

We also observe that

d(u,0)Hu : T(u,0)U = T(u,0)W ∼= TuB⊕Wu → T0Wu∼= Wu.

Within a given fibre, Hu is linear, and so for tangent vectors η ∈ Wu (in thesplitting above), d0Hu(η) = Hu(η) = η. For tangent vectors ζ ∈ TuB, whendifferentiating in the ζ-direction we are just parallel-transporting 0 along somecurve, so d0Hu(ζ) = 0. Thus the operator d0Hu(ζ) is projection TuB⊕Wu →Wu. By the definition of Du, this proves that the operators d0Fu and Du areequal.

The next lemma shows that we can apply the Banach space implicit func-tion theorem to Fu:

Lemma 3.13. Any u ∈ B near u can be expressed uniquely as expu(ξ) forsome ξ ∈ TuB near 0. Under this identification the elements of M(x−, x+) nearu correspond to the zeros of Fu near 0. Moreover, dξFu is surjective and aFredholm operator when ξ is such a zero.

Proof. The first statement clearly holds since expu is a local diffeomorphism.Also, since parallel transport is a linear isomorphism, we have that Fu(ξ) = 0 ifand only if ∂J,H(expu(ξ)) = 0, which by Proposition 3.8 is true precisely whenexpu(ξ) ∈M(x−, x+). This proves the second statement.

We can write Fu as the composite

TuBexpu−−−→ B

(expu)−1

−−−−−−→ TuBFu−−→Wu →Wu.

Here expu and expu are local diffeomorphisms, and the final map is a diffeo-morphism since it is induced by a smooth family of isomorphisms

Tu(s,t)T∗M−→Tu(s,t)T

∗M.

19

Hence dξFu is the composite

TξTuB−→T0TuB ∼= TuBDu−−→Wu−→Wu.

From this we conclude that dξFu is surjective and a Fredholm map, becausethis is true of Du.

Thus 0 is a regular value of Fu|V , for some sufficiently small neighbourhoodV of 0 in TuB. By the Banach Space Implicit Function Theorem, Theorem 2.21,X = (Fu|V)−1(0) is therefore a smooth manifold. Moreover, since dξFu is Fred-holm for all ξ ∈ (Fu|V)−1(0), the manifold X is finite-dimensional of dimension

indd0Fu = indDu = µ(x−) − µ(x+).

expu is a diffeomorphism on V , so the same is true of expu(X), an open neigh-bourhood of u in M(x−, x+), and hence of M(x−, x+) (since u was arbitrary).This is precisely what we were trying to prove.

We now show that Du is a Fredholm operator for every u ∈M(x−, x+), i.e.we prove Theorem 3.10. We do this by first showing that there exists a nicetrivialization of u∗(T T∗M), with respect to which Du has a simple form:

Lemma 3.14. Suppose u ∈ M(x−, x+), and that Φ± : T × R2n → x±∗(T T∗M)are unitary trivializations such that Φ±(t)λ0 = TVx±(t)T

∗M. Then there existsa unitary trivialization Φ : R × T × R2n → u∗(T T∗M) such that Φ(±∞, t) =Φ±(t).

Proof. As in the proof of Lemma 3.4, we can show that

u∗(TVT∗M) ∼= (τ∗ u)∗T∗M,

hence as M is orientable, u∗(TVT∗M) is orientable. It is a theorem of alge-braic topology that every orientable vector bundle over R × S1 is trivial, sou∗(TVT∗M) is trivial.

Let ψ : R × T × Rn → u∗(TVT∗M) be a trivialization. Then as in the proofof Lemma 3.4 we can construct a unitary trivialization Ψ : R × T × R2n →u∗(T T∗M) such that Ψ(s, t)λ0 = TVu(s,t)T

∗M for all (s, t) ∈ R× T.Consider the loops α± in Sp(2n, λ0) ∩U(n) defined by

α±(t) = Ψ(±∞, t)−1Φ±(t).

In the proof of Lemma 3.5 we saw that any loop in Sp(2n, λ0) ∩ U(n) is con-tractible in U(n), so α± are homotopic to the constant loops at α±(0), respec-tively. Moreover, as U(n) is path-connected, these constant loops are homo-topic to each other. Therefore, there exists a homotopy α : R × T → U(n)such that α(±∞, –) = α±; we may assume without loss of generality that α issmooth.

Then the mapΦ : R× T× R2n → u∗(T T∗M) defined by

Φ(s, t) = Ψ(s, t)α(s, t)

is a unitary trivialization withΦ(±∞, –) = Φ±.

20

Lemma 3.15. With respect to a trivializationΦ as above, Du is an operator DSof the form

∂s − J0∂t − S(s, t),

where S is a smooth family of endomorphisms of R2n, 1-periodic in t, such thatS±(t) = lims→±∞ S(s, t) are symmetric matrices for all t.

Proof. We begin by deriving an expression for Du. Since Du = d0Fu we havethat for ξ ∈ TuB,

Duξ =d

dλ

∣∣∣∣λ=0

Fu(λξ).

Let γ be the path defined by γ(λ) = expu(λξ). Then

Ψu(λξ)Fu(λξ) = ∂J,H(γ(λ)).

Now in general, for∇ a connection on the tangent bundle TN of some man-ifold N, we can express∇ using parallel transport as

∇XpY = limh→0

1

h(τ−1h Yc(h) − Yp),

where c is a smooth curve such that c(0) = p and c ′(0) = Xp ∈ TpN, the mapτh is parallel transport TpN → Tc(h)N along c, and Y is a vector field on aneighbourhood of p.

Hence we can write

d

dλ

∣∣∣∣λ=0

Fu(λξ) = limλ→0

1

λ(Fu(λξ) − Fu(0))

= limλ→0

1

λ(Ψu(λξ)−1Ψu(λξ)Fu(λξ) − Ψu(0)Fu(0))

= ∇λΨ(λξ)Fu(λξ)|0

= ∇λ∂J,H(γ(λ))|0,

where ∇ is the Levi-Civita connection and∇λ = ∇d0c( ddλ ) for c(λ) = λξ.Substituting the definition of ∂J,H, this becomes

Duξ = ∇λ [∂sγ(λ) − J(t, γ(λ))(∂tγ(λ) − XH(t, γ(λ)))] .

Since ∇ is torsion-free, there is a coordinate system about any point such thatthe Christoffel symbols of ∇ for these coordinates vanish at the point. Usingsuch coordinates, we see that

∇λ∂sγ = ∇s∂λγ,

since in these coordinates we just get ∂λ∂s and ∂s∂λ at the distinguished point,which are equal. Similarly,

∇λ∂tγ = ∇t∂λγ.

Using this, the expression for Du becomes

Duξ = ∇sγ ′(0) − (∇λJ(t, γ(λ))|λ=0)(∂tγ(0) − XH(t, γ(0)))

−J(t, γ(0))(∇tγ ′(0) −∇λXH(t, γ(λ))|λ=0)

= ∇sξ− (∇ξJ(t, u))(∂tu− XH(t, u)) − J(t, u)(∇tξ−∇ξXH(t, u))

= ∇sξ− J(t, u)∇tξ− (∇ξJ(t, u))(∂tu− XH(t, u)) + J(t, u)∇ξXH(t, u).

21

Now if η ∈ W1,r(R × T,R2n) and Φ : R × T × R2n → u∗(T T∗M) is atrivialization as in Lemma 3.14 we get

∇s(Φη) = (∇sΦ)η+Φ∇sη = (∇sΦ)η+Φ∂sη,

where ∇ is the connection on the homomorphism bundle induced by∇. Simi-larly,

∇t(Φη) = (∇tΦ)η+Φ∂tη.

Therefore, with respect to the trivializationΦ, the operator Du becomes

DSη = ∂sη− J0∂tη− Sη,

where

Sη = −Φ−1[(∇sΦ)η+ J(t, u)(∇tΦ)η+∇ΦηJ(t, u)(∂tu− XH(t, u))

−J(t, u)∇ΦηXH(t, u)].

S is clearly smooth and 1-periodic in t, so it only remains to show that thelimit matrices S± are symmetric. We can express S± as

S±η = −(Φ±)−1[J(t, x±)(∇tΦ±)η− J(t, x±)∇Φ±ηXH(t, x±)],

since Φ± does not depend on s and x± satisfy the Hamiltonian equation (1).It can be shown that the formal adjointD∗u ofDu with respect to our metric

is

D∗uξ = −∇sξ− J(t, u)∇tξ−∇ξJ(t, u)(∂tu− XH(t, u)) + J(t, u)∇ξXH(t, u).

This means that DS has adjoint

D∗S = −∂s − J0∂t − S∗,

where

S∗η = −Φ−1[−(∇sΦ)η+ J(t, u)(∇tΦ)η+∇ΦηJ(t, u)(∂tu− XH(t, u))

−J(t, u)∇ΦηXH(t, u)].

Since the ±∇sΦ-term disappears at infinity, we see that S±∗ = S±, i.e. thematrices S± are symmetric.

We now wish to use Proposition 2.18 to deduce that the operator DS hasclosed range and finite-dimensional kernel. The next three lemmas derive therequired estimates:

Lemma 3.16. For every T > 0 there exists a constant c > 0 such that

‖ξ‖W1,r ≤ c(‖DSξ‖Lr + ‖ξ‖Lr)

for every ξ ∈W1,r(R× T,R2n) that is compactly supported on [−T, T ]× T.

22

Proof.Claim. There exists a constant C > 0 such that

‖ξ‖W1,r ≤ C(‖(∂s − J0∂t)ξ‖Lr + ‖ξ‖Lr)

for all ξ ∈W1,r(R× T,R2n) that are compactly supported on [−T, T ]× T.Given this, we have

‖(∂s − J0∂t)ξ‖Lr = ‖DSξ+ Sξ‖Lr≤ ‖DSξ‖Lr + ‖Sξ‖Lr .

We can bound the second term by

‖Sξ‖Lr ≤ sup(s,t)∈[−T,T ]×T

‖S(s, t)‖‖ξ‖Lr ,

since ξ vanishes outside the compact set [−T, T ]× T. Thus we get

‖ξ‖W1,r ≤ c(‖DSξ‖Lr + ‖ξ‖Lr),

for c = C+ sup(s,t)∈[−T,T ]×T ‖S(s, t)‖.

Proof of Claim. By Lemma 2.27 and Theorem 2.29 it follows that we can expressξ as (∂s + J0∂t)u for some u of classW2,r, which is also compactly supported.Then we can the express theW1,r-norm of ξ as

‖ξ‖W1,r = (‖ξ‖rLr + ‖∂sξ‖rLr + ‖∂tξ‖rLr)1/r

≤ ‖ξ‖Lr + ‖∂sξ‖Lr + ‖∂tξ‖Lr= ‖ξ‖Lr + ‖∂2su+ J0∂s∂tu‖Lr + ‖∂t∂su+ J0∂

2tu‖Lr

≤ ‖ξ‖Lr + ‖∂2su‖Lr + ‖∂s∂tu‖Lr + ‖∂t∂uu‖Lr + ‖∂2tu‖Lr .

By Theorem 2.26, if u is smooth there is a constant C > 0 such that

‖∂2su‖Lr + ‖∂s∂tu‖Lr + ‖∂t∂su‖Lr + ‖∂2tu‖Lr ≤ C‖∆u‖Lr= C‖(∂s − J0∂t)(∂s + J0∂t)u‖Lr= C‖(∂s − J0∂t)ξ‖Lr .

Hence if u is smooth we can bound ‖ξ‖W1,r by

‖ξ‖W1,r ≤ ‖ξ‖Lr + C‖(∂s − J0∂t)ξ‖Lr .

But this must be true in general since the smooth functions are dense in W2,r,and so we are done.

Lemma 3.17. The solution of

d

dtγ±(t) = J0S

±(t)γ±(t)

that satisfies γ±(0) = I is γΦ±(t) = Φ±(t)−1dx±(0)φtHΦ±(0).

23

Proof. Observe that

(∂t − J0S±)γ± = −J0(J0∂t + S±)γ± = J0DS|(±∞,t)γ±,

which is zero if and only ifDS|(±∞,t)γ± = 0. But under the trivializationΦ, theexpression DS|(±∞,t)γΦ± clearly corresponds to Du|(±∞,t)dx±(0)φtH

, which is

(projection T(u,0)W→Wu) du∂J,H|(±∞,t) dx±(0)φtH.

By Lemma 3.1,u(±∞, t) = x±(t) = φtH(x±(0)).

Therefore, du∂J,H|(±∞,t) dx±(0)φtH equals dx±(0)(∂J,Hφ

tH). But ∂J,HφtH = 0

since ∂tφtH − XH(t, φtH) = 0 and ∂sφtH = 0. Thus DS|(±∞,t)γΦ± = 0, and soγΦ± is indeed the required solution.

Lemma 3.18. There exist constants T > 0 and c > 0 such that for any ξ ∈W1,r(R× T,R2n) for which ξ(s, t) = 0when |s| ≤ T − 1, we have

‖ξ‖W1,r ≤ c‖DSξ‖Lr .

Proof. The proof is based on the following result, proved in [9, pp. 16–18]:

Suppose that S(s, t) = S(t) is independent of s, and that the mapΨ : [0, 1]→ Sp(2n) defined by

∂tΨ(t) = J0S(t)Ψ(t), ψ(0) = I

satisfies det(I− Ψ(1)) 6= 0. Then

∂s − J0∂t − S : W1,r(R× T,R2n)→ Lr(R× T,R2n)

is bijective for 1 < r <∞.

By Lemma 3.17, the solution of ∂tΨ = J0S±Ψ with Ψ(0) = I is γΦ± , and

det(I− γΦ±) 6= 0 as H is non-degenerate. Thus the operators

DS± = ∂s − J0∂t − S±

are bijective. This must still be true for sufficiently small continuous perturba-tions of S±.

It follows that there is some K > 0 such that DS is bijective on

W1,r((−∞,−K)× T,R2n)→ Lr((−∞,−K)× T,R2n)

and onW1,r((K,∞)× T,R2n)→ Lr((K,∞)× T,R2n).

Let D1, D2 denote these restrictions, respectively. D1, D2 are continuous bi-jective linear maps, hence (by the Open Mapping Theorem) D−1

1 , D−12 are also

continuous and therefore bounded. Then if ξ1 ∈W1,r((−∞,−K)×T,R2n) andξ2 ∈W1,r((K,∞)× T,R2n), we have

‖ξi‖W1,r = ‖D−1i Diξi‖W1,r ≤ ‖D−1

i ‖‖Diξ‖Lr ,

24

for i = 1, 2.Thus if ξ ∈W1,r(R×T,R2n) vanishes on [−K,K]×T, and ξ1 = ξ|(−∞,−K)×T,

ξ2 = ξ|(K,∞)×T, we get

‖ξ‖rW1,r = ‖ξ1‖rW1,r + ‖ξ2‖rW1,r

≤ ‖D−11 ‖‖D1ξ1‖

rW1,r + ‖D−1

2 ‖‖D2ξ2‖rW1,r

≤ max ‖D−11 ‖, ‖D

−12 ‖ (‖D1ξ1‖

rW1,r + ‖D2ξ2‖rW1,r)

= max ‖D−11 ‖, ‖D

−12 ‖ ‖DSξ‖

rW1,r ,

and so ‖ξ‖W1,r ≤ c‖DSξ‖Lr with c = (max ‖D−11 ‖, ‖D

−12 ‖)1/r. This com-

pletes the proof, as we can set T = K+ 1.

Corollary 3.19. DS has closed range and finite-dimensional kernel.

Proof. Choose a smooth cutoff function β : R → [0, 1] such that β(s) = 0 for|s| ≥ T and β(s) = 1 for |s| ≤ T − 1. For ξ ∈ W1,r(R × T,R2n), applyingLemma 3.16 to βξ and Lemma 3.18 to (1 − β)ξ we get (with c1 the constant inLemma 3.16 and c2 the constant in Lemma 3.18)

‖ξ‖W1,r = ‖βξ+ (1− β)ξ‖W1,r

≤ ‖βξ‖W1,r + ‖(1− β)ξ‖W1,r

≤ c1(‖DSβξ‖Lr + ‖βξ‖Lr) + c2‖DS(1− β)ξ‖Lr≤ (c1 + c2)(‖DSξ‖Lr + ‖ξ‖Lr([−T,T ])).

By Theorem 2.25, the inclusion

W1,r([−T, T ]× T) → C0([−T, T ]× T)

is compact, hence the composition of this with

C0([−T, T ]× T) → Lr([−T, T ]× T)

is also compact. It follows that the restriction

W1,r(R× T)→ Lr([−T, T ]× T)

is compact. By Proposition 2.18, this and the last inequality above togetherimply that DS has closed range and finite-dimensional kernel.

To prove that DS is a Fredholm operator, all that remains to do is to showthat cokerDS is finite-dimensional; we will only sketch the proof of this.

Lemma 3.20. DS has finite-dimensional cokernel.

Sketch Proof. The dual space of Lr(R× T,R2n) is Lq(R× T,R2n), where 1r +

1q = 1. The formal adjoint D∗S of DS is, as in the proof of Lemma 3.15,

−∂s − J0∂t − S.

The cokernel ofDS is isomorphic to the annihilator of imDS; we will showthat the latter coincides with the kernel ofD∗S. It is clear that elements of kerD∗Sannihilate imDS. Suppose η ∈ Lq(R × T,R2n) annihilates imDS; we need toshow that η ∈ kerD∗S. It can be shown (by elliptic regularity) that η ∈ W1,q

loc .

25

Moreover, we can repeat the proofs of Lemmas 3.16, 3.17 and 3.18 for D∗S toshow, as in the proof of Corollary 3.19, that there is a constant c > 0 such thatfor every k ∈ Z and ξ ∈W1,q(R× T,R2n),

‖ξ‖W1,q([k,k+1]×T) ≤ c(‖D∗Sξ‖Lq([k−1,k+2]×T) + ‖ξ‖Lq([k−1,k+2]×T)).

Since D∗Sη = 0, this implies that

‖η‖W1,q([k,k+1]×T) ≤ c‖η‖Lq([k−1,k+2]×T).

Taking the qth power of this and summing over k, we find

‖η‖qW1,q ≤ 3cq‖η‖qLq <∞,

and so η ∈W1,q(R× T,R2n).Thus the annihilator of imDS is the same as the kernel of D∗S : W1,q →

Lq. By the same argument as in the proof of Corollary 3.19, kerD∗S is finite-dimensional, and hence so is cokerDS.

This completes the proof that Du is a Fredholm operator. We now turnto Theorem 3.11, and briefly outline the proof that indDu = µ(x−) − µ(x+).Further details can be found in [9, pp. 21-23].

By a compact perturbation ofDS, which does not change the index ofDS byTheorem 2.19, we may without loss of generality assume that S is symmetriceverywhere. We write DS as ∂s +A(s), where

A(s) = −J0∂t − S(s, –) : W1,2(T,R2n)→ L2(T,R2n).

The index of DS is computed by considering the spectral flow µspec(A) of A.Roughly speaking, this is the number of eigenvalues of A(s) that cross zerofrom negative to positive as s goes from −∞ to∞. The following results thenhold:

Lemma 3.21. There is a symplectic matrix-valued function γ : R × [0, 1] →Sp(2n) associated to S by

J0∂tγ+ Sγ = 0, γ(s, 0) = I.

The spectral flow µspec(A) of A is the same as that of s 7→ γ(s, 1).

Lemma 3.22. Observe that lims→±∞ γ = γΦ± . Without loss of generalitysuppose γ(s, t) = γΦ−(t) for s ≤ T and γ(s, t) = γΦ+(t) for s ≥ T . ThenµCZ(γΦ−) − µCZ(γΦ+) = µspec(γ(–, 1)).

Theorem 3.23. µspec(A) = indDS.

Since by definition µ(x±) = µCZ(γΦ±), putting these results together weget

indDS = µspec(A) = µspec(γ(–, 1)) = µ(x−) − µ(x+),

which is Theorem 3.11.

26

3.3 Compactness

In this section we aim to prove that the moduli spaces M(x−, x+) for x± ∈P(H) are precompact in the C∞

loc-topology, provided the HamiltonianH and thealmost complex structure J satisfy certain technical conditions. It will followfrom this that M(x−, x+) is finite when µ(x−) − µ(x+) = 1.

To prove precompactness, we must fix a metric 〈–, –〉 on M. This inducesmetrics on TM and T∗Mwhich we also denote by 〈–, –〉. The metric determinesan isometry TM → T∗M and an orthogonal complement THT∗M of TVT∗M,which we call the horizontal bundle. There is a preferredω-compatible almostcomplex structure J on T∗M, which with respect to the splitting

T(q,p)T∗M = TH(q,p)T

∗M⊕ TV(q,p)T∗M

takes the form

J =

(0 I

−I 0

).

Let∇H,∇V be the horizontal and vertical components of the gradient∇ of themetric 〈–, –〉. The precise result we will prove is then as follows:

Theorem 3.24. Assume H satisfies the following two conditions:

(H1) There exist constants h0 > 0 and h1 ≥ 0 such that for all t ∈ T and(q, p) ∈ T∗M,

d(t,q,p)H(ηθ) −H(t, q, p) ≥ h0‖p‖2 − h1,

where ηθ is the Liouville vector field defined by θ = ω(ηθ, –).

(H2) There exists a constant h2 ≥ 0 such that for all t ∈ T and (q, p) ∈ T∗M,

‖(∇HH)(t, q, p)‖ ≤ h2(1+ ‖p‖2),

‖(∇VH)(t, q, p)‖ ≤ h2(1+ ‖p‖).

Then there is a constant j0 > 0 such that if ‖J − J‖ < j0, then M(x−, x+) isprecompact in C∞

loc(R× T, T∗M) for all x± ∈ P(H).

The conditions (H1), (H2) are independent of the choice of metric (up tochanging the values of the constants); see [2, pp. 273-274]. From now on weassume that our Hamiltonian H satisfies (H1) and (H2). The following simpleconsequence of these conditions will be repeatedly useful later on:

Lemma 3.25. There is a constant h3 > 0 such that for all (q, p) ∈ T∗M,

‖XH(t, q, p)‖ ≤ h3(1+ ‖p(t)‖2).

Proof. We can write

XH(t, q, p) = J∇H(t, q, p) = ∇VH(t, q, p) −∇HH(t, q, p).

Taking norms, this gives

‖XH(t, q, p)‖ = ‖J∇H(t, q, p)‖ = ‖∇VH(t, q, p) −∇HH(t, q, p)‖≤ ‖∇VH(t, q, p)‖+ ‖∇HH(t, q, p)‖.

27

Applying (H2), we get

‖XH(t, q, p)‖ ≤ h2(2+ ‖p‖+ ‖p‖2)≤ 2h2(1+ ‖p‖)2

≤ 8h2(1+ ‖p‖2)

(using (a + b)2 ≤ (2max a, b)2 ≤ 4max a, b2 ≤ 4(a2 + b2) in the last step).This proves the result with h3 = 8h2.

Now define U(a, b), for a < b real numbers, to be the set of smooth soluionsu : R× T→ T∗M of the negative gradient equation (2) with

a ≤ A(u(s, –)) ≤ b

for all s. We will prove that U(a, b) is precompact in the C∞loc-topology for all

a, b. Since M(x−, x+) is contained in U(A(x+),A(x−)) by Lemma 3.3, it willfollow that M(x−, x+) is also precompact in this topology.

To prove the precompactness of U(a, b), we will first show that for u ∈U(a, b), u and ∇u are uniformly bounded. By applying a “bootstrapping”result we get W1,r-bounds for all the derivatives of u. We can then show thata sequence in U(a, b) has a convergent subsequence by using thatW1,r(K), forK a compact subset of R2, embeds compactly in C0.

We begin by proving that U(a, b) is bounded in the L∞-norm; as a first steptowards this we show the following:

Lemma 3.26. Suppose ‖J‖∞ <∞. Then there is a constant c1 > 0 such that forall u ∈ U(a, b),

‖∂su‖L2(R×T) ≤ c1.

Proof. For real numbers s0 < s1, we have

‖∂su‖2L2((s0,s1)×T) =

∫s1s0

∫10

‖∂su‖2dtds.

We wish to rewrite this in terms of the inner products 〈–, –〉t = ω(–, J(t)–). Let‖–‖t denote the norm of 〈–, –〉t; we need to estimate the norm ‖–‖ in terms ofthis norm. Observe that

‖–‖2t = ω(J–, JJ(t)–) = 〈J–, J(t)–〉 ≤ ‖J‖‖J(t)‖‖–‖2 = ‖J(t)‖‖–‖2 ≤ ‖J‖∞‖–‖2.Similarly,

‖–‖2 ≤ ‖J‖t‖–‖2t .

Thus we find that for every vector v,

‖Jv‖2t = 〈Jv, Jv〉t ≤ ‖J(t)‖〈Jv, Jv〉 = ‖J(t)‖‖v‖2 ≤ ‖J(t)‖‖J‖t‖v‖2t .

Choose vn such that ‖vn‖t = 1 and ‖Jvn‖t → ‖J‖t. Setting v = vn in theinequality above and taking the limit n→∞, we get

‖J‖2t ≤ ‖J(t)‖‖J‖t,

so ‖J‖t ≤ ‖J(t)‖.

28

Applying this, we get

‖∂su‖2L2((s0,s1)×T) ≤ ‖J‖∞∫s1s0

∫10

‖∂su‖2tdtds.

Since u solves the negative gradient equation (2),

〈∂su, ∂su〉t = 〈−∇JA(u(s, –)), ∂su〉t.

Integrating over (0, 1), this gives∫10

〈−∇JA(u(s, –)), ∂su〉tdt = −du(s,–)A(∂su(s, –)) = −∂sA(u(s, –)),

since ∇J is the gradient of the metric∫10〈–, –〉tdt. By integrating this over

(s0, s1), we get

‖∂su‖2L2((s0,s1)×T) ≤ −‖J‖∞∫s1s0

∂sA(u(s, –))ds

= ‖J‖∞(A(u(s0, –)) − A(u(s1, –)))

≤ ‖J‖∞(b− a).

Since this inequality holds for all s0, s1, taking limits we get

‖∂su‖L2(R×T) ≤ c1,

with c1 =√‖J‖∞(b− a).

The estimates of the following Proposition are a key step of the proof thatU(a, b) is L∞-bounded:

Proposition 3.27. Suppose ‖J‖∞ <∞. Then there is a constant c > 0 such thatfor every u = (q, p) ∈ U(a, b), and every interval I ⊂ R, we have

‖p‖L2(I×(0,1)) ≤ c|I|1/2,

‖∇p‖L2(I×(0,1)) ≤ c(|I|1/2 + 1).

As the proof is long and rather technical, we omit it here. It can be foundin [2, pp. 275–279]. Given this result, we are ready to prove that U(a, b) isL∞-bounded for suitable J:

Theorem 3.28. There exists a constant j0 > 0 such that if ‖J − J‖ < j0, thenU(a, b) is bounded in L∞(R× T, T∗M) for all a, b ∈ R.

Proof. The idea of the proof is to bound u ∈ U(a, b) locally in W1,r by thesame bound everywhere and then use the Sobolev Embedding Theorem (The-orem 2.25) to get the same local L∞-bound everywhere. We will use the factthat there exists an isometric embedding of M into RN for some N, which in-duces isometric embeddings of TM and T∗M into R2N. Under this embedding,the metric-induced almost complex structure J is the restriction of J0 to T T∗M.Therefore, the negative gradient equation (2) can be rewritten as

(∂s − J0∂t)u = (J(t, u) − J0)∂tu− J(t, u)XH(t, u).

Let χ : R→ R be a smooth function that satisfies:

29

(i) χ ≡ 1 on [0, 1].

(ii) suppχ ⊂ (−1, 2).

(iii) 0 ≤ χ ≤ 1 everywhere.

(iv) |χ ′| ≤ 2 everywhere.

For k ∈ Z set χk(s) = χ(s− k).Let u = (q, p) ∈ U(a, b) and define v to be χk · u. Then

∂sv = χ ′ku+ χk∂su,

∂tv = χk∂tu.

Therefore,

(∂s − J0∂t)v = χ ′ku+ χk(∂s − J0∂tu)

= χ ′ku+ χk((J(t, u) − J0)∂tu− J(t, u)XH(t, u))

= χ ′ku+ (J(t, u) − J0)∂tv− χkJ(t, u)XH(t, u).

We now use the following result, a consequence of the Calderon-Zygmundinequality:

For every r > 0 there is a constant Cr ≥ 1 such that every w ∈W1,rλ0

(R × T,R2N) := W1,r0 (R × T,RN) ×W1,r(R × T,RN) (i.e. the

space ofW1,r-maps taking values in λ0 on the boundary) satisfies

‖∇w‖Lr(R×T) ≤ Cr‖(∂s − J0∂t)w‖Lr(R×T).

For a proof of this, see [2, p. 280].Our v clearly has compact support, and hence lies in W1,r

λ0(R × T,R2N).

Therefore, with r some number > 2,

‖∇v‖Lr(R×T) ≤ Cr‖(∂s − J0∂t)v‖Lr(R×T)

= Cr‖χ ′ku+ (J(t, u) − J0)∂tv− χkJ(t, u)XH(t, u)‖Lr(R×T)

≤ Cr(‖χ ′k‖∞‖u‖Lr((k−1,k+2)×T) + ‖J− J0‖∞‖∂tv‖Lr(R×T)

+‖χk‖∞‖J‖∞‖XH(–, u)‖Lr((k−1,k+2)×T))

≤ Cr(2(d+ ‖p‖Lr((k−1,k+2)×T)) + ‖J− J0‖∞‖∂tv‖Lr(R×T)

+‖J‖∞‖XH(–, u)‖Lr((k−1,k+2)×T)),

where d = maxq∈M ‖q‖; this is finite sinceM is compact.We now wish to bound ‖p‖Lr((k−1,k+2)×T)) and ‖XH(–, u)‖Lr((k−1,k+2)×T).

If ‖J − J0‖∞ = ‖J − J‖∞ is finite, then ‖J‖∞ is bounded, so we can applyLemma 3.26:

‖p‖W1,2((k−1,k+2)×T) = (‖p‖2L2((k−1,k+2)×T) + ‖∇p‖2L2((k−1,k+2)×T))1/2

≤ ((c√3)2 + (c(1+

√3))2)1/2

≤ 4c.

30

By Theorem 2.25,W1,2((k−1, k+2)×T) embeds continuously in Lr((k−1, k+2)× T); let Sr be the norm of this map. Then

‖p‖Lr((k−1,k+2)×T) ≤ Sr‖p‖W1,2((k−1,k+2)×T) ≤ 4Src := K.

By Lemma 3.25, ‖XH(–, u)‖ ≤ h3(1+ ‖p‖2). Hence,

‖XH(–, u)‖Lr((k−1,k+2)×T) ≤ ‖h3(1+ ‖p‖2)‖Lr((k−1,k+2)×T)

≤ h3(‖1‖Lr((k−1,k+2)×T) + ‖p‖2L2r((k−1,k+2)×T))

≤ h3(31/r + S22r‖p‖2W1,2((k−1,k+2)×T))

≤ h3(31/r + 16c2S22r) := K ′.

Putting these estimates together, we get

‖∇v‖Lr(R×T) ≤ 2dCr + 2KCr + Cr‖J− J0‖∞‖∂tv‖Lr(R×T) + ‖J‖∞CrK ′.Since ‖∂tv‖Lr(R×T) ≤ ‖∇v‖Lr(R×T), if ‖J− J0‖∞ < 1/Cr this implies the bound

‖∇v‖Lr(R×T) ≤(2d+ 2K+ ‖J‖∞K ′)Cr1− ‖J− J0‖∞Cr := K.

Thus∇v is uniformly bounded in Lr(R× T).We therefore get a localW1,r-bound for u by

‖u‖W1,r((k,k+1)×T) ≤ (‖p‖rLr((k,k+1)×T) + dr + ‖∇u‖rLr((k,k+1)×T))1/r

≤ (dr + ‖p‖rLr((k−1,k+2)×T) + ‖∇v‖rLr(R×T))1/r

≤ (dr + Kr + Kr)1/r,

so u is uniformly bounded inW1,r((k, k+1)×T). Since r > 2, by Theorem 2.25there exists a constant c such that

‖u‖L∞((k,k+1)×T) ≤ c‖u‖W1,r((k,k+1)×T),

and therefore u is uniformly bounded in L∞((k, k + 1) × T). But k is arbitraryand the bound is independent of k, hence u is uniformly bounded in L∞(R×T).Thus we have proved the result, with j0 = supr>2 1/Cr, say.

Our next step is to prove that for every u ∈ U(a, b), the gradient ∇u hasfinite L∞-norm. To do this we need the following technical lemma:

Lemma 3.29. There exists a constant ε > 0 such that if u : Br(z) ⊂ C→ T∗M isa smooth solution of the negative gradient equation (2) that satisfies∫

Br(z)

‖∂su‖2dtds ≤ ε,

then‖∂su(z)‖2 ≤ 1+

8

πr2

∫Br(z)

‖∂su‖2dtds.

For a proof of this, see [8, pp. 136–138].

31

Corollary 3.30. If u ∈ U(a, b), then ‖∇u‖L∞ <∞.

Proof.Claim. Given c > 0, there exists r0 ∈ (0, 1) such that for all z ∈ C,∫

Br0(z)

‖∂su‖2 ≤ c

(with u considered as a periodic function C→ T∗M).Given this, we can apply it with c = ε from Lemma 3.29 to get that for

z ∈ R× T,

‖∂su(z)‖2 ≤ 1+8

πr20

∫Br0(z)

‖∂su‖2

≤ 1+8

πr20‖∂su‖2L2(R×T)

≤ 1+8

πr20c21,

using Lemma 3.26. Thus ‖∂su‖L∞ ≤ Kwith K :=√1+ 8c21/πr

20.

Since u solves the negative gradient equation,

∂tu = −J(t, u)∂su+ XH(t, u).

Therefore we can bound ∂tu, if u = (q, p), by

‖∂tu‖L∞ ≤ ‖J‖∞‖∂su‖L∞ + ‖XH(t, u)‖L∞≤ ‖J‖∞K+ h3(1+ ‖p‖2L∞),

by Lemma 3.25. But by Theorem 3.28, ‖p‖L∞ ≤ ‖u‖L∞ <∞, hence

‖∇u‖L∞ ≤ ‖∂su‖L∞ + ‖∂tu‖L∞ <∞.It remains to prove the claim above:

Proof of Claim. Suppose no such r0 exists. Then for all ρ > 0, there exists azρ ∈ C such that ∫

Bρ(zρ)

‖∂su‖2 > c.

From this we see thatsupBρ(zρ)

‖∂su‖2 >c

πρ2,

which is greater than c for ρ sufficiently small.It can be shown that ∂su → 0 as s → ±∞ since u lies in U(a, b) (see [9, pp.

12, 25–26]). Therefore, there must exist some s0 ∈ R such that

‖∂su(s, –)‖2L2(T) < c

32

for all |s| > s0. This implies that if |s| > s0 + 1 and ρ < 1/2 then, for z = (s, t),

‖∂su‖2L2(Bρ(z)) ≤ ‖∂su‖2L2((s−ρ,s+ρ)×T)

=

∫s+ρs−ρ

‖∂su(σ, –)‖2dσ

<

∫s+ρs−ρ

cdσ

= 2cρ < c.

Hence for ρ < 1/2we may choose zρ to lie in the compact set [−s0−1, s0+1]×[0, 1] (by periodicity).

Let wn = z1/(n+2). Then wn is a sequence in a compact set in C, hence ithas a convergent subsequencewnk ,wnk → w, say. But this means that for anyr > 0 there is an integer K such that

‖w−wnk‖ <r

2

for every k ≥ K. Hence if k ≥ K and k is large enough that 1/(nk + 2) < r/2,then

B1/(nk+2)(wnk) ⊂ Br(w).

This gives a lower bound for the integral over Br(w):∫Br(w)

‖∂su‖2 ≥∫B1/(nk+2)(wnk)

‖∂su‖2 > c.

So we have shown that∫Br(w)

‖∂su‖2 > c for every r > 0. But this isimpossible since ∂su is smooth, and so an r0 as required must exist.

The next result will show that U(a, b) is precompact, provided that ∇u isuniformly L∞-bounded for u ∈ U(a, b).

Proposition 3.31. Suppose ‖J − J‖ < j0. Then every sequence un : C → T∗M

in U(a, b) satisfyingsupn

‖∇un‖L∞ <∞has a subsequence converging uniformly, with all its derivatives, on every com-pact subset of C.

Proof. The proof depends on the following estimate:

For every K ⊂ C compact, c > 0, p ≥ 1, and k ∈ N with kp > 2,there exists a constant C > 0 such that all smooth functions u : C→T∗Mwith ‖∂su‖+ ‖∂tu‖ ≤ c everywhere satisfy the inequality

‖u‖Wk+1,p(K) ≤ C(‖∂su−J(t, u)(∂tu−XH(t, u))‖Wk,p(K)+‖u‖Wk,p(K)).

The proof of this is rather technical and we omit it here; see [8, pp. 138–139].Choose a compact set K ⊂ C whose interior is a Lipschitz domain. By

Theorem 2.25, the inclusionW1,r(K) → C0(K) is compact, so a closed set whichis bounded in theW1,r-norm is compact under the sup-norm.

33

By assumption and Theorem 3.28 the sequence un has bounded W1,r(K)-norm, and so lies in a compact set with respect to the sup-norm. Hence we canchoose a subsequence v0n that converges under the sup-norm.

As un solves the negative gradient equation, the estimate above impliesthat

supn

‖un‖Wk,r(K) <∞for all k. Thus if we have constructed a subsequence vk−1

n that converges uni-formly with all its derivatives up to order k − 1 in K, then the sequences ofkth-order derivatives have bounded W1,r-norm. Hence they have convergentsubsequences under the sup-norm. We can therefore choose a subsequence vknof vk−1

n that converges uniformly with all derivatives up to order k. By induc-tion, we get such subsequences vkn for all k, and then by setting wn = vnn, weget a subsequencewn of un converging uniformly with all its derivatives in K.

Now we can inductively choose subsequences win of un converging withall derivatives in the open ball Bi(0), with wi+1n a subsequence of win for alli. Then wn := wnn defines a subsequence of un converging uniformly with allderivatives on every compact subset of C, which is what we wanted to find.

Theorem 3.32. If ‖J − J‖ < j0, then for all a < b there is a constant C > 0 suchthat for every u ∈ U(a, b) we have

‖∇u‖L∞ ≤ C.We will not prove this here; a proof for the case when J is time-independent

is found in [8, pp.134–135]. The idea of the proof is to show that if the bounddoes not exist then we can construct a non-constant J-holomorphic sphere inT∗M, meaning a smooth map from S2 to T∗M which intertwines J and thenatural complex structure on S2 = CP1. This then gives a contradiction be-cause non-constant J-holomorphic spheres cannot exist in T∗M as the sym-plectic formω is exact.

Corollary 3.33. U(a, b) is precompact in the C∞loc-topology.

Proof. Suppose we have a sequence un in U(a, b). Then by Theorem 3.32 thereis someC > 0 such that ‖∇un‖ ≤ C for alln. By Proposition 3.31, it follows thatui has a subsequence that converges in the C∞

loc-topology. Hence, the closure ofU(a, b) is compact in the C∞

loc-topology, i.e. U(a, b) is precompact.

By Lemma 3.3, the action functional A decreases strictly along non-stationarysolutions of the negative gradient equation (2). This means that for x± ∈ P(H)the moduli space M(x−, x+) is contained in U(A(x+),A(x−)). It follows thatM(x−, x+) is also precompact if ‖J− J‖ < j0, which proves Theorem 3.24.

Corollary 3.34. If µ(x−)−µ(x+) = 1 and ‖J− J‖ < j0, then M(x−, x+) is a finiteset of points.

Proof. By Theorem 3.7, M(x−, x+) is a 1-dimensional manifold. Therefore, byTheorem 3.24, the space M(x−, x+) is a precompact 0-dimensional manifold.But a 0-dimensional manifold is always closed, since it must have a discretetopology. So M(x−, x+) is in fact a compact 0-dimensional manifold, and there-fore it has to consist of finitely many points.

34

3.4 The Floer Complex

In this subsection, we will finally define the Floer chain complex. As in theprevious subsection, we require H to satisfy the conditions (H1) and (H2), andto be non-degenerate. We also require J to be regular and to satisfy ‖J− J‖ < j0;such a J must exist since the set Jreg(H) of regular ω-compatible almost com-plex structures is dense in the set of all ω-compatible almost complex struc-tures.

To construct the boundary map for the Floer complex, we wish to count theelements of M(x−, x+) when µ(x−) − µ(x+) = 1. However, for the boundarymap to square to zero we need to count them with sign. It is therefore necessaryto show that the manifolds M(x−, x+) are orientable. We will not go into thedetails of how this is done here; a sketch of the argument and further referencescan be found in [2, pp. 269–272].

Let CFk(H) be the free abelian group generated by those x ∈ P(H) whoseMaslov index µ(x) is k. When µ(x−)−µ(x+) = 1, orienting M(x−, x+) assigns asign ε([u]) ∈ ±1 to every [u] ∈ M(x−, x+). For x, y ∈ P(H) with µ(x)−µ(y) =1, set

n(x, y) :=∑

[u]∈cM(x,y)

ε([u]).

We then define the boundary homomorphism ∂ = ∂(H, J) : CFk(H)→ CFk−1(H)by, for x ∈ P(H),

∂x =∑

y∈P(H)µ(y)−µ(x)=1

n(x, y)y.

The following lemma implies that this is well-defined:

Lemma 3.35. For all a ∈ R, the set of x ∈ P(H) with A(x) ≤ a is finite.

Proof. Let x = (q, p) ∈ P(H) with A(x) ≤ a. We will prove that there areonly finitely many such x by showing that the set of these is bounded in theL∞-norm, which implies that the set of x(0) for x ∈ P(H) ∩ A ≤ a is compact.Finiteness then follows by showing that the set of x(0) for x ∈ P(H) is discrete.

To get the L∞-bound, we construct a sequence of other bounds, beginningby observing that

a ≥ A(x) =

∫10

(θ(x) −H(t, x))dt

=

∫10

(ω(ηθ, x) −H(t, x))dt (where ηθ is the Liouville vector field)

=

∫10

(ω(ηθ, XH(t, x)) −H(t, x))dt (since x ∈ P(H))

=

∫10

(d(t,x)H(ηθ) −H(t, x))dt (by definition of XH)

≥∫10

(h0‖p(t)‖2 − h1)dt (by (H1))

= h0‖p‖2L2(T) − h1.

35

So ‖p‖L2(T) ≤ K :=√

(a+ h1)/h0. Now by Lemma 3.25,

‖x(t)‖ = ‖XH(x(t))‖ ≤ h3(1+ ‖p(t)‖2),

so by integrating we get∫10

‖x‖dt ≤ h3(1+ ‖p‖2L2(T)) ≤ h3(1+ K2).

By Holder’s inequality we have ‖p‖L1(T) ≤ ‖p‖L2(T), and since M is com-pact we can bound q. Therefore, we have shown that P(H)∩A ≤ a is boundedunder theW1,1-norm. But this implies that the set is bounded in L∞: We provethis in the case of a smooth function y : [0, 1]→ R with

∫10

|y| ≤ c and∫10

|y| ≤ cfor some constant c > 0. We can do the following calculation:

|y(t)| =

∣∣∣∣∫t0

y(τ)dτ+ y(0)

∣∣∣∣≤∫t0

|y(τ)|dτ+ |y(0)|

≤ c+ |y(0)|.

But we also have

|y(0)| =

∣∣∣∣y(t) −

∫t0

y(τ)dτ

∣∣∣∣≤ |y(t)| +

∫10

|y(τ)|dτ

≤ |y(t)| + c.

Integrating this with respect to t, we get

|y(0)| ≤ 2c,

and so|y(t)| ≤ 3c

for all t ∈ [0, 1]. Applying this to the components of x in coordinates, we get anL∞-bound for P(H) ∩ A ≤ a. Thus the set of x(0) with x ∈ P(H), A(x) ≤ a isbounded in T∗M, and it is therefore precompact.Claim. x(0) : x ∈ P(H) is discrete.

Given this, we conclude that x(0) : x ∈ P(H),A(x) ≤ a is discrete andprecompact, and so it must be a finite set. By Lemma 3.1, this set bijects withP(H) ∩ A ≤ a and it follows that this must also be finite.

Proof of Claim. If x ∈ P(H), its initial value y = x(0) is a fixed point of φ1H (byLemma 3.1), and dyφ1H(ζ) 6= ζ for every ζ ∈ TyT∗M (by the non-degeneracy ofH). Hence there is some ε > 0 such that

‖(dyφ1H − id)(ζ)‖ ≥ ε‖ζ‖

36

for all ζ ∈ TyT∗M — otherwise we could choose a sequence ζn such that

‖ζn‖ = 1 and

‖(dyφ1H − id)(ζn)‖ < 1

n;

by compactness of the unit ball this would have a convergent subsequence,which would contradict the non-degeneracy of H.

Now consider the Taylor expansion of φ1H to first order at y:

φ1H(y+ h) = y+ dyφ1H(h) +O(‖h‖2)

Expanding (φ1H − id)(y+ h), we deduce that

‖(φ1H − id)(y+ h)‖ = ‖(dyφ1H − id)(h)‖+O(‖h‖2) ≥ ε‖h‖+O(‖h‖2).

For ‖h‖ sufficiently small, the higher-order terms will be too small to cancelout the term ε‖h‖, hence φ1H has no fixed points other than y in some neigh-bourhood of y. Since the fixed points of φ1H are precisely the initial values ofthe elements of P(H) by Lemma 3.1, this completes the proof.

This lemma implies that for any x ∈ P(H) there are only finitely manyy ∈ P(H) with A(y) ≤ A(x). But if A(y) > A(x) then M(x, y) is empty byLemma 3.3. Therefore ∂x is always a finite sum, and so the map ∂ is well-defined.

Next, we of course wish to prove that ∂2 = 0. This is a consequence of thefollowing theorem:

Theorem 3.36. If x, z ∈ P(H) and µ(x) −µ(z) = 2, then the closure of M(x, z) isa compact 1-dimensional manifold with boundary, and

∂M(x, z) =⋃

y∈P(H)µ(x)−µ(y)=1

M(x, y)× M(y, z).

Moreover, the induced orientation of the boundary gives ([u], [v]) ∈ M(x, y)×M(y, z) the sign ε([u])ε([v]).

This is a consequence of Floer’s Gluing Theorem; we refer to [9, pp. 32–35]for more details. A compact 1-manifold with boundary is just a finite disjointunion of circles and closed intervals, and so counting boundary points withsigns from an orientation always gives zero. Applying this to M(x, z) as above,we get ∑

y∈P(H)µ(x)−µ(y)=1

∑[u]∈cM(x,y)

∑[v]∈cM(y,z)

ε([u])ε([v]) = 0.

But the left-hand side is ∑y∈P(H)

µ(x)−µ(y)=1

n(x, y)n(y, z)

which is precisely the coefficient of z in ∂2x. Thus the coefficient in ∂2x of everyz ∈ P(H) is zero, i.e. ∂2x = 0.

37

This means that (CF∗(H), ∂(H, J)) is a chain complex; it is called the Floercomplex. We define the Floer homology groups HF∗(H, J) to be the homologygroups of this complex.

The following invariance results, which we state without proof, hold forHF∗(H, J):

Proposition 3.37. Changing the orientations of the manifolds M(x−, x+) in away that is consistent with gluing gives an isomorphic Floer chain complex.

Theorem 3.38. If J, J ′ are almost complex structures in Jreg(H), and ‖J− J‖ < j0,‖J ′ − J‖ < j0, then there exists an isomorphism

φJJ ′ : (CF∗(H), ∂(H, J))−→(CF∗(H), ∂(H, J ′)).

This isomorphism is unique up to chain homotopy, and if J ′′ is another suitablealmost complex structure, then φJJ ′′ and φJ ′J ′′ φJJ ′ are chain homotopic.

Theorem 3.39. IfH,H ′ are non-degenerate Hamiltonians on T×T∗M satisfying(H1) and (H2), and J ∈ Jreg(H)∩ Jreg(H

′) with ‖J− J‖ < j0, then there is a chainhomotopy equivalence

ψHH ′ : (CF∗(H), ∂(H, J)) −→ (CF∗(H′), ∂(H ′, J)),

unique up to chain homotopy. If H ′′ is another suitable Hamiltonian, thenψHH ′′ and ψH ′H ′′ ψHH ′ are chain homotopic.

Thus HF∗(H, J) is independent of H and J, and so is an invariant of T∗M.For proofs of these results, see [2, pp. 284-290].

38

4 Construction of the Lagrangian Morse Homology

In this section we briefly discuss the Morse homology of a 1-periodic Lagrangian,which we will relate to Floer homology in the next section; further details canbe found in [2, pp. 290–297].

A Lagrangian is just a smooth function

L : TM→ R,

possibly time-dependent. If L is a Lagrangian, its fibre derivative is the mapFL : TM → T∗M which sends v ∈ TM to dv(L|TpM); this is a smooth fibre-preserving map. Using FL we can pull back the standard symplectic form ω

on T∗M to the Lagrange 2-form ωL = FL∗ω on TM. We define the action of Lto be A : TM→ R,

A(v) = (FL(v))(v),

and the energy of L to be E = A − L. Then a Lagrangian vector field for L is avector field YL on TM such that

ιYLωL = ωL(YL, –) = −dE.

We say that a Lagrangian L is regular if FL has only regular values (or,equivalently, is a local diffeomorphism), and that L is hyperregular if FL is adiffeomorphism. It is easily seen that ωL is a symplectic form if and only if Lis regular, so if L is regular there exists a unique Lagrangian vector field for L.

Now let τ : TM→ R be the standard projection, and set

TV(q,v)TM = kerd(q,v)τ,

with orthogonal complement TH(q,v) (with respect to the same metric 〈–, –〉 as inthe last section). Denote by ∇HH, ∇HV = ∇VH and ∇VV the splitting of theHessian of the Levi-Civita connection into vertical and horizontal parts. Wewill consider 1-periodic Lagrangians L : T × TM → R satisfying the followingtwo conditions:

(L1) There is a constant l0 > 0 such that for all (t, v) ∈ T× TM,

∇HHL(t, v) ≥ l0I

as a quadratic form.

(L2) There is a constant l1 ≥ 0 such that for all (t, v) ∈ T× TM,

‖∇HHL(t, v)‖ ≤ l1,

‖∇HVL(t, v)‖ ≤ l1(1+ ‖v‖),

‖∇VVL(t, v)‖ ≤ l1(1+ ‖v‖2).

These conditions correspond to the conditions (H1) and (H2) for a Hamiltonianunder the Legendre transform, as we will note in the next section. It can beshown that (L1) and (L2) are independent of the choice of metric on M, andthat (L1) implies that L is hyperregular for all t as well as bounded below.

39

Since L is hyperregular, it has a Lagrangian vector field YL. The integralcurves y : (a, b) → TM of YL can be shown to be of the form (q, q) whereq : (a, b)→ TM solves the differential equation

∇t(∇HL(t, q(t), q(t))) = ∇VL(t, q(t), q(t)), (3)

with ∇t the covariant derivative along q. We let P(L) be the set of 1-periodicsolutions of (3), and we assume that L is non-degenerate, meaning that forevery x ∈ P(L) the differential at q(0) of the time-1 integral flow map of YLdoes not have eigenvalue 1.

We can then define the Lagrangian action functional E : Λ1(M)→ R to be

E(q) =

∫10

L(t, q(t), q(t))dt.

This is smooth on Λ1(M), and it can be shown that the critical points of E areprecisely the elements of P(L). Moreover, the non-degeneracy condition for Lholds if and only if all these critical points q are non-degenerate, meaning thatthe Hessian Hess(E, q) does not have 0 as an eigenvalue. If L is non-degenerate,we define the Morse index m(q) of q ∈ P(L) to be the number of negativeeigenvalues of Hess(E, q).

We can now appeal to the following result:

Fact. If a functional f of class Ch, h ≥ 2, on a Riemannian Hilbert manifold(M, 〈〈–, –〉〉) satisfies

(i) all critical points of f are non-degenerate and have finite Morse index,

(ii) f is bounded below,

(iii) (M, 〈〈–, –〉〉) is complete,

(iv) f satisfies the Palais-Smale condition on (M, 〈〈–, –〉〉), meaning that if qn isa sequence in M such that f(qn) is bounded and ‖∇f(qn)‖ is infinitesimal,then qn is compact,

then we can associate to f a chain complex of abelian groups, the Morse com-plex of f, whose homology is isomorphic to the singular homology of M.

The conditions (L1) and (L2), and the non-degeneracy of L, imply that E

satisfies (i)–(iv) on (Λ1(M), 〈〈–, –〉〉), where 〈〈–, –〉〉 is the metric defined by

〈〈ξ, η〉〉 =

∫10

(〈∇tξ,∇tη〉+ 〈ξ, η〉)dt.

We will now briefly sketch how the Morse complex for E is actually constructed:LetCMk(L) be the free abelian group generated by all x ∈ P(L) withm(x) =

k. Suppose that g is a metric on Λ1(M) which is complete, uniformly equiva-lent to 〈〈–, –〉〉, and such that E satisfies the Palais-Smale condition with respectto g. Then let ∇gE be the gradient of E with respect to g, and let ψtg be thelocal integral flow of −∇gE. Its rest points are the critical points of E, and E isstrictly decreasing along the non-constant orbits of ψtg.

40

We define the stable and unstable manifolds of x ∈ P(L) to be, respectively,

Ws(x) = p ∈ Λ1(M) : limt→∞ψtg(p) = x,

Wu(x) = p ∈ Λ1(M) : limt→−∞ψtg(p) = x.

It can be shown that these are smooth manifolds with dimWu(x) = codimWs(x) =m(x), and that Wu(x) is orientable and Ws(x) coorientable (meaning that itsnormal bundle is orientable).

It is possible to choose the metric g to be a Morse-Smale metric, whichmeans that the manifolds Wu(x) and Ws(y) intersect transversely for all x,y ∈ P(L). If g is a Morse-Smale metric, and x, y ∈ P(L), then

(i) Ifm(x) > m(y),Wu(x)∩Ws(y) is a manifold of dimensionm(x) −m(y).

(ii) Wu(x) ∩Ws(x) consists of just the stationary solution x.

(iii) Ifm(x) ≤ m(y) and x 6= y, thenWu(x) ∩Ws(y) = ∅ .

We now fix an orientation for each Wu(x). This gives an orientation ofWu(x) ∩Ws(y) for all x, ywithm(x) −m(y) = 1.

It can be shown that the manifolds Wu(x) are all precompact in Λ1(M),and we can therefore define a boundary map in the same way as we did inthe construction of Floer homology: If m(x) −m(y) = 1, then Wu(x) ∩Ws(y)consists of finitely many flow lines; we define

N(x, y) = [Wu(x) ∩Ws(y)]/R

to be the set of flow lines. Each flow line u is given a sign ε(u) by the orienta-tion. Let ν(x, y) be the sum of ε(u) over flow lines u ∈ N(x, y); then we definethe boundary map ∂ = ∂(L, g) : CMk(L)→ CMk−1(L) by, for x ∈ P(L),

∂x =∑

y∈P(L)m(x)−m(y)=1

ν(x, y)y.

It can be shown that the set P(L) ∩ E ≤ a is finite for all a, and so the sumabove is always finite (asWu(x) ∩Ws(y) = ∅ if E(y) > E(x)).

The square ∂2 can be proved to equal 0, so (CM∗(L), ∂(L, g)) is a chain com-plex; its homology HM∗(L, g) is the Morse homology of L (or, more correctly,of E). Moreover, it can be shown that HM∗(L, g) is independent of the choiceof Morse-Smale metric g, and, by means of a cellular filtration, that HM∗(L, g)is isomorphic to the singular homology ofΛ1(M). Since the spacesΛ1(M) andΛ(M) are homotopy equivalent, this implies that

HM∗(L, g) ∼= H∗(Λ(M)).

41

5 The Isomorphism between the Floer and MorseHomologies

In this section we will outline the construction of an isomorphism betweenthe Morse homology of a 1-periodic Lagrangian L and the Floer homology ofthe Hamiltonian H corresponding to L under the Legendre transform, which wewill define below. Since the Floer homology is independent of the choice of H,this will imply that the Floer homology of T∗M is isomorphic to the singularhomology of Λ(M). We will generally not give detailed proofs in this section;more details can be found in [2, pp. 298–315].

5.1 Set-Up

We begin by defining the Legendre transform:

Definition 5.1. If L is a hyperregular Lagrangian, then the Legendre transformof L is the Hamiltonian

H = E (FL)−1 : T∗M→ R,

where E : TM→ R is defined by

E(v) = (FL(v))(v) − L(v),

as in the previous section.

If H is the Legendre transform of L, it is easily checked that (FL)∗YL = XL.Moreover, if L is a 1-periodic Lagrangian satisfying the conditions (L1) and(L2), then:

(i) H satisfies (H1) and (H2).

(ii) H is non-degenerate if and only if L is non-degenerate.

(iii) FL induces a bijection from P(L) to P(H) by

q 7→ (FL)(q, q).

(iv) If q ∈ P(L) and x = (FL)(q, q) is the corresponding element of P(H), thenm(q) = µ(x).

We now define the moduli spaces M+(q, x) of orbits connecting q ∈ P(L)to x ∈ P(H). Let J be a t-dependent 1-periodic almost complex structure inJreg(H), with ‖J − J‖∞ < j0, and let g be a metric on Λ1(M) that is uniformlyequivalent to 〈〈–, –〉〉 and a Morse-Smale metric for E. Also fix some r ∈ (2, 4].For q ∈ P(L), x ∈ P(H), we then let M+(q, x) be the set of all maps u : [0,∞)×T→ T∗M such that:

(i) u is smooth on (0,∞)× T.

(ii) u is of classW1,r on (0, 1)× T.

42

(iii) u satisfies∂su− J(t, u)(∂tu− XH(t, u)) = 0

on (0,∞)× T.

(iv) The projection τ∗u(0, –) lies in the unstable manifoldWu(q).

(v) lims→∞ u(s, t) = x(t), with the limit uniform in t.

It can be shown that if u ∈M+(q, x), then

E(q) ≥ E(τ∗u(0, –)) ≥ A(u(0, –)) ≥ A(u(s, –)) ≥ A(x)

and E(q) = A(x) if and only if x and q correspond under the Legendre trans-form, in which case M+(q, x) consists of just the stationary solution x.

We will construct our isomorphism by showing that the spaces M+(q, x)are smooth precompact orientable manifolds of dimensionm(q) − µ(x), muchin the same way as we constructed the boundary map for Floer homology fromthe spaces M(x−, x+).

5.2 Finite-Dimensionality

In this section we outline the proof of the finite-dimensionality of the mani-folds M+(q, x); this is very similar to the proof of finite-dimensionality for themoduli spaces of Floer homology. We proceed as follows:

Let B+ be the space of all maps u : [0,∞)× T→ T∗M such that:

(i) τ∗u(0, –) ∈Wu(q).

(ii) There exists an s0 ≥ 0 such that

u(s, t) = expx(t)(ζ(s, t))

for all (s, t) ∈ (s0,∞)× T, where ζ is a W1,r-section of the vector bundlex∗(T T∗M)→ (s0,∞)× T.

B+ is a smooth Banach manifold, and TuB+ is isomorphic to the set of W1,r-sections w of u∗(T T∗M) with

du(0,–)τ∗(w(0, –)) ∈ Tτ∗u(0,–)W

u(q).

Let W+ be the Banach bundle over B+ whose fibre W+u at u is the space of

Lr-sections of u∗(T T∗M). Then we can show that M+(q, x) is the set of zerosof the smooth section ∂+

J,H : B+ →W+ defined by

∂+J,H(u) = ∂su− J(t, u)(∂tu− XH(t, u)).

For u ∈M+(q, x) we define the operatorD+u : TuB+ →W+

u to be the composite

TuB+du∂

+J,H−−−−−→ T(u,0)W

+ ∼= TuB+ ⊕W+u

projection−−−−−→W+u .

We observe that D+u is surjective if and only if ∂+

J,H is transverse to the zerosection at u.

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Theorem 5.2. Let Jreg(H, g) be the set of all almost complex structures J ∈Jreg(H) such that for all q ∈ P(L) and x ∈ P(H), the section ∂+

J,H is transverseto the zero section everywhere. Then Jreg(H, g) is dense in J(ω).

See [2, pp. 312–313] for a discussion of this result; it allows us to choose ourJ to lie in the set Jreg(H, g). The difficult part of the proof is then to show thatD+u is a Fredholm operator of index m(q) − µ(x) for all u ∈ M+(q, x). Given

this, the proof is completed by the following steps (the proofs are essentiallythe same as those of the corresponding results in section 3.2):

1. Define F+u : TuB+ →W+

u by

F+u (ξ) = Ψ+

u(ξ)−1∂+J,H(expu(ξ)),

where Ψ+u(ξ) : TuT

∗M → Texpu(ξ)T∗M is parallel transport for the Levi-

Civita connection along the curve t 7→ expu(tξ).

2. Prove that D+u = d0F

+u .

3. Every u ∈ B+ near u can be expressed uniquely as expu(ξ) for someξ ∈ TuB+ near 0. Under this identification elements u of M+(q, x) near ucorrespond to the zeros ξ of F+

u near 0. Show that dξF+u is surjective and

a Fredholm operator for such ξ.

4. This implies that 0 is a regular value of F+u |V , where V is a sufficiently

small neighbourhood of 0. Apply the Banach Space Implicit FunctionTheorem, Theorem 2.21, to deduce that X := (F+

u |V)−1(0) is a smoothmanifold. As dξF+

u is Fredholm for all ξ ∈ X, we also get that X is finite-dimensional with dimension

indd0F+u = indD+

u = m(q) − µ(x).

Since expu is a diffeomorphism near 0, expu(X) is also a smooth mani-fold of dimensionm(q) − µ(x). As this is an open neighbourhood of u inM+(q, x) and u was arbitrary, M+(q, x) is a smooth manifold of dimen-sionm(q) − µ(x), which is what we wanted to prove.

We will not discuss the proof that D+u is a Fredholm operator of index

m(q) − µ(x) here; it can be found in [2, pp. 300-308].

5.3 Compactness

In this section we will briefly discuss the proof of precompactness for the mani-folds M+(q, x) in theC∞

loc-topology. The proof is similar to the proof of precom-pactness for M(x−, x+) — we deduce the precompactness of M+(q, x) fromthat of the set U+(a, b, c), which we define to be the set of all u : [0,∞) × T →T∗M such that:

(i) u is smooth on (0,∞)× T.

(ii) u is of classW1,r on (0, 1)× T.

(iii) u solves the negative gradient equation (2) on (0,∞).

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(iv) a ≤ A(u(s, –)) ≤ b for all s ∈ [0,∞).

(v) ‖τ∗u(0, –)‖W1−1/r,r ≤ c. (This actually requires an extension of the notionof weak derivatives to non-integral orders, which we will not define here.)

We begin with a result analogous to Proposition 3.27:

Proposition 5.3. Assume ‖J‖∞ < ∞. For all real numbers a < b and c thereexists a constant C > 0 such that for every u = (q, p) ∈ U+(a, b, c),

‖p‖L2(I×T) ≤ C|I|1/2,

‖∇p‖L2(I×T) ≤ C(|I|1/2 + 1),

The proof is almost the same as that of Proposition 3.27, and is given in [2,pp. 275-279]. From this we can deduce the following result, analogous to The-orem 3.28.

Theorem 5.4. If r > 2, then there exists a j1 such that if ‖J− J‖∞ < j1, then forall real numbers a < b, c the set U+(a, b, c) is bounded in the L∞-norm.

A proof of this can be found in [2, p. 282]. Next, we deduce that the gradi-ents of elements of U+(a, b, c) are uniformly bounded if ‖J−J‖∞ < j1, by show-ing that otherwise we get a J-holomorphic sphere in T∗M; as mentioned above,this is impossible sinceω is exact. By an elliptic bootstrapping argument anal-ogous to the proof of Proposition 3.31 we get bounds for all the derivatives andit follows that U+(a, b, c) is precompact in the C∞

loc-topology.Now if u ∈ M+(q, x), then τ∗u(0, –) lies in the unstable manifold Wu(q)

of q. This is precompact in the W1,2-topology and hence τ∗u(0, –) for u ∈M+(q, x) is bounded in the W1,2-norm. As r ≤ 4 if can be shown that W1,2

embeds continously in W1−1/r,r, so we get a W1−1/r,r-bound K, say. ThusM+(q, x) is contained in U+(A(x),E(q), K) and therefore it is precompact.

Corollary 5.5. Ifm(q) = µ(x), then M+(q, x) is a finite set of points.

This follows by exactly the same argument as that of the proof of Corol-lary 3.34.

5.4 The Isomorphism

To construct our isomorphism, we wish to count the elements of M+(q, x) withsigns when m(q) = µ(x). To do so, we need to orient the manifolds M+(q, x).We will not discuss the proof that this is possible; the details can be found in [2,pp. 309–311].

We are now ready to define the isomorphism: For q ∈ P(L) and x ∈ P(H)with m(q) = µ(x), let ε(u) be the sign of u ∈M+(q, x) induced by the orienta-tion. Define n+(q, x) to be ∑

u∈M+(q,x)

ε(u).

Then the isomorphism Θ : (CM∗(L), ∂(L, g))→ (CF∗(H), ∂(H, J)) is defined by,for q ∈ P(L),

Θ(q) =∑

x∈P(H)µ(x)=m(q)

n+(q, x)x.

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This sum is always finite, since there are only finitely many x ∈ P(H) withA(x) ≤ E(q), and M+(q, x) = ∅ if A(x) > E(q). Note also that n+(q, x) = ±1 ifx and q correspond under the Legendre transform.

Next, we need to prove that Θ is a chain map, i.e. that

Θ∂(L, g) = ∂(H, J)Θ.

An analogue of Floer’s Gluing Theorem (see [2, pp. 311–312, 313–314]) impliesthe following:

Theorem 5.6. If q ∈ P(L), x ∈ P(H), and m(q) − µ(x) = 1, then the closure ofM+(q, x) is a compact 1-dimensional manifold with boundary, and

∂M+(q, x) =⋃

q∈P(L)m(q)−m(q)=1

N(q, q)×M+(q, x)

∪⋃

x∈P(H)µ(x)−µ(x)=1

M+(q, x)× M(x, x).

Moreover, the orientation induced on the boundary is such that by countingpoints in the boundary with signs given by this orientation (which gives 0 forany compact 1-manifold) we find that∑

q∈P(L)m(q)−m(q)=1

∑u∈N(q,q)

∑v∈M+(q,x)

ε(u)ε(v)

=∑

x∈P(H)µ(x)−µ(x)=1

∑w∈M+(q,x)

∑[z]∈cM(x,x)

ε(w)ε([z]).

The left-hand side here is precisely the coefficient of x in Θ∂q, and the right-hand side is the coefficient of x in ∂Θq. Thus, as x was arbitrary, Θ∂q = ∂Θq,and hence Θ is indeed a chain map.

Proposition 5.7. Θ is an isomorphism.

Proof. Order the generators ofCM∗(L) andCF∗(H) by increasing action, but sothat elements that correspond under the Legendre transform are in the sameposition. Recall that n+(q, x) = 0 for q, x with E(q) ≤ A(x) unless q and xcorrespond, in which case n+(q, x) = ±1. So with respect to these bases, Θ is a(possibly infinite) lower-triangular matrix with entries ±1 along the diagonal.It is easy to see that such a matrix is invertible over Z, and so Θ has an inverse,which must also be a chain map. Thus Θ is a chain complex isomorphism.

Since HM∗(L, g) is isomorphic to H∗(Λ(M)), it follows from this result thatthe Floer homology HF∗(H, J) is also isomorphic to H∗(Λ(M)).

In fact, the induced map Θ : HM∗(L, g) → HF∗(H, J) is an isomorphismof rings: There is a natural product on Floer homology, known as the pair-of-pants product, and on the singular homology of the free loop space Λ(M) wecan construct the so-called Chas-Sullivan loop product. This induces a producton the Morse homology of L, and Θ maps this to the pair-of-pants product.This result was proved by Abbondandolo and Schwarz, and a sketch of theproof, together with the definitions of the product structures, is given in [1].

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References

[1] Alberto Abbondandolo and Matthias Schwarz, Notes on Floer homologyand loop space homology, Morse Theoretic Methods in Non-Linear Anal-ysis and Symplectic Topology (P. Biran, O. Cornea, and F. Lalonde,eds.), NATO Science Series II: Mathematics, Physics and Chemistry,Springer-Verlag, 2006, available online from http://www.dm.unipi.it/˜abbondandolo/publications/publications.html, pp. 75–108.

[2] , On the Floer homology of cotangent bundles, Communications onPure and Applied Mathematics 59 (2006), 254–316.

[3] Ralph Abraham and Jerrold E. Marsden, Foundations of mechanics, seconded., Addison-Wesley, 1987, available online at http://caltechbook.library.caltech.edu/103/.

[4] Andreas Floer, Helmut Hofer, and Dietmar Salamon, Transversality in el-liptic Morse theory for the symplectic action, Duke Mathematical Journal 80(1995), no. 1, 251–292, available online from http://www.math.ethz.ch/˜salamon/publications.html.

[5] Dusa McDuff and Dietmar Salamon, J-holomorphic curves and quantum co-homology, University Lecture Series, vol. 6, American Mathematical Soci-ety, Providence, Rhode Island, 1994, available online from http://www.math.ethz.ch/˜salamon/publications.html (page references toonline version).

[6] , Introduction to symplectic topology, Oxford University Press, 1999.

[7] , J-holomorphic curves and symplectic topology, Colloquium Publica-tions, vol. 52, American Mathematical Society, Providence, Rhode Island,2004.

[8] Dietmar Salamon, Morse theory, the Conley index and Floer homology, Bul-letin of the London Mathematical Society 22 (1990), no. 2, 113–140.

[9] , Lectures on Floer homology, Symplectic Geometry and Topology(Y. Eliashberg and L. Traynor, eds.), IAS/Park City Mathematics Series,vol. 7, American Mathematical Society, 1999, available online from http://www.math.ethz.ch/˜salamon/publications.html (page refer-ences to online version).

[10] Joa Weber, Three approaches towards Floer homology of cotangent bundles,Journal of Symplectic Geometry 3 (2005), no. 4, 671–701, math.SG/0602230 (page references to online version).

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