Fredholm Index

Chapter 3. Fredholm Theory 57

Definition 3.10. For a contractible periodic solution x ∈ P (H) of (2.1), define the Maslov index of x by

µ(x) := µ(Ψ).

Remark 3.11. If we choose two trivializations Φ, Φ^′, by proposition 3.8, they are homotopic and so induces homotopic paths Ψ, Ψ^′ ∈ LP^∗ , by proposition 3.7, Ψ and Ψ^′ have the same Maslov index.

Therefore µ(x) is independent of choice of Φ and hence Ψ.

Chapter 3. Fredholm Theory 58

Consider for p ≥ 2, the operator F : W^1,p(R×S¹, R²ⁿ) → L^p(R× S¹, R²ⁿ) defined by

F ξ := ∂ξ

∂s + J0

∂ξ

∂t + Sξ, (3.6)

where S^T = S = S(s, t) ∈ M(2n, R) is a continuous matrix valued function on R× S¹ such that

S^± := lim

s→±∞S(s, t)

exists with uniform convergence in t. When S = 0, F is called the Cauchy-Riemann operator. This operator is considered because we will show (see the proof of theorem 3.16) that by applying a pertur-bation (which does not affect the Fredholm property) if necessary, the operator F (u) in (3.1) is of this form after a trivialization.

There is a one-one correspondence between S and Ψ(s, t)∈ Sp(2n, R) defined by the differential equation







∂Ψ

∂t = J0SΨ, Ψ(s, 0) = I2n×2n.

(3.7)

Ψ(s, t)∈ Sp(2n, R) as for fixed s, d

dt(Ψ^TJ0Ψ) = Ψ^TS^TJ₀^TJ0Ψ + Ψ^TJ0J0SΨ = 0

and Ψ(s, 0) ∈ Sp(2n, R) for all s. It turns out that Ψ converges uniformly in t as s → ±∞. Denote

Ψ^±(t) := lim

s→±∞Ψ(s, t).

Chapter 3. Fredholm Theory 59

Proposition 3.13 ([33] theorem 4.1). If Ψ^± ∈ LP^∗, then 1. F is a Fredholm operator, and

2. index F = µ(Ψ⁻)− µ(Ψ⁺).

The proof uses the following lemma (see for example [24], [34]).

Lemma 3.14. Let X, Y, Z be Banach spaces, F : X → Y is a bounded linear operator and K : X → Z is compact. Suppose there exists c > 0 such that for all ξ ∈ X,

||ξ||^X ≤ c(||F ξ||^Y +||Kξ||^Z).

Then F has a closed range and its kernel is finite dimensional. F with this property is said to be semi-Fredholm.

The Fredholm property of F is easiest to prove for p = 2, which will be done here. For p > 2, see [31]. We will assume p = 2 from now on.

Proof of Proposition 3.13 (1). Let X := R× (R/Z) = R × S¹. The proof of (1) consists of four steps.

Step 1.

Suppose S(s, t) = S(t), Ψ(s, t) = Ψ(t) are both independent of s.

We claim that there exists c > 0 such that

||ξ||W^1,2(X) ≤ c||F ξ||L²(X).

Chapter 3. Fredholm Theory 60

Define the symmetric operator A : W^1,2(S¹, R²ⁿ) → L²(S¹, R²ⁿ) by Aξ(t) = J0dξ

dt + S(t)ξ(t).

Then

ker A 6= 0 ⇔ ∃ξ(t) 6= 0, Aξ = 0

⇔ ξ(t) = Ψ(t)ξ(0) by uniqueness of solution to (3.7)

⇔ Ψ(1)ξ(0) = ξ(0) 6= 0 i.e. 1 ∈ σ(Ψ(1))

⇔ Ψ /∈ LP^∗. (3.8)

Therefore A is invertible onto its range and so exists c0 > 0 such that

||ξ||W^1,2(S¹) ≤ c0||Aξ||L²(S¹).

Identify Cⁿ with R²ⁿ and consider the Fourier transform F : L²(R× S¹, R²ⁿ) → L²(R× S¹, R²ⁿ) defined by

(Fξ)(ω, t) := 1

√2π Z _∞

−∞

exp(−iωs)ξ(s, t)ds.

Then F is an isometry. Also denote Fξ by ˆξ. Then F(^∂ξ_∂s) = iω ˆξ and so

F(∂ξ

∂s + Aξ) = iω ˆξ + A ˆξ. (3.9) So now

||ξ||²W^1,2(X) =

Z _∞

−∞

(        

∂ξ

∂t        

2 L²(S¹)

+        

∂ξ

∂s        

2 L²(S¹)

+||ξ||²L²(S¹))ds

=

Z _∞

−∞

(          

∂ ˆξ

∂t          

2

L²(S¹)

+||iω ˆξ||²L²(S¹)+||ˆξ||²L²(S¹))dω

=

Z _∞

−∞

(||ˆξ||W^1,2(S¹) + ω²||ξ||²L²(S¹))dω. (3.10)

Chapter 3. Fredholm Theory 61

Consider

||ˆξ||W^1,2(S¹) ≤ c0||Aˆξ||L²(S¹) ≤ c0||Aˆξ + iω ˆξ||L²(S¹) (3.11) for ω ∈ R as A is symmetric and ||Aˆξ + iω ˆξ||²_L²_(S¹₎ = ||Aˆξ||²_L²_(S¹₎ + ω²||ˆξ||²_L²_(S¹₎. Also,

|ω| · ||ˆξ||²L²(S¹) ≤ ||hˆξ, iω ˆξ + A ˆξi||L²(S¹)

≤ ||ˆξ||L²(S¹)||iω ˆξ + A ˆξ||L²(S¹)

⇒ |ω| · ||ˆξ||L²(S¹) ≤ ||iω ˆξ + A ˆξ||L²(S¹). (3.12) So by (3.11) and (3.12), (3.10) becomes

||ξ||²W^1,2(X) ≤ (c²0+1) Z _∞

−∞||Aˆξ+iω ˆξ||L²(S¹)dω = c||F ξ||²L²(X) by (3.9) where c = c(S).

Step 2.

For general S and hence F , we claim that there exists sufficiently large T > 0, c = c(T ) > 0 such that for all ξ ∈ W^1,2(X, R²ⁿ,) with ξ|[−T,T ] = 0, then

||ξ||W^1,2(X) ≤ c||F ξ||L²(X). By step 1, there exists c^± such that

||ξ||W^1,2(X) ≤ c^±||F^±ξ||L²(X)

for the limit operators F^±ξ = ^∂ξ_∂s + J₀^∂ξ_∂t + S^±ξ.

Let c = max(c⁺, c⁻) and let ε > 0. Then there exists T > 0 such

Chapter 3. Fredholm Theory 62

that for all t,







|S(s, t) − S⁻(t)| < ε for s ≤ −T,

|S(s, t) − S⁺(t)| < ε for s ≥ T.

For ξ|[−T,T ] = 0, ξ = ξ⁻+ ξ⁺ where







ξ⁻|[−T,∞] = 0, ξ⁺|[−∞,T ] = 0.

||F^±ξ^±||L²(X) ≤ ||(F^±− F )ξ^±||L²(X) +||F ξ^±||L²(X)

≤ ε||ξ^±||L²(X) +||F ξ^±||L²(X) as F^±− F = S^±

∴ ||ξ||W^1,2(X) = ||ξ⁻||W^1,2(X) +||ξ⁺||W^1,2(X)

≤ c(||F⁻ξ⁻||L²(X)+ ||F⁺ξ⁺||L²(X))

≤ c(||F ξ⁻||L²(X) +||F ξ⁺||L²(X) + ε(||ξ⁻||L²(X)+ ||ξ⁺||L²(X)))

= c(||F ξ||L²(X) + ε||ξ||L²(X))

≤ c(||F ξ||L²(X) + ε||ξ||W^1,2(X))

∴ (1− cε)||ξ||W^1,2(X) ≤ c||F ξ||L²(X).

Take ε < 1/c and for the corresponding T = T (ε), then if ξ|[−T,T ] = 0,

||ξ||W^1,2(X) ≤ c

1− cε||F ξ||L²(X) = c^′||F ξ||L²(X). Step 3.

Claim: there exists c > 0, a Banach space Z, a compact operator K : X → Z such that

k|ξ||W^1,2(X) ≤ c(||F ξ||L²(X)+||Kξ||Z). (3.13)

Chapter 3. Fredholm Theory 63

This estimate shows that F is semi-Fredholm by lemma 3.14. Now let T = T (S) be given by step 2. Let ξ ∈ W^1,2(X, R²ⁿ) with support in [−T, T ] × S¹. Denote XT := [−T, T ] × S¹. Then in this case

       

∂ξ

∂s        

2 L²(X)

+        

∂ξ

∂t        

2 L²(X)

=        

∂ξ

∂s + J0

∂ξ

∂t        

2 L²(X)

. (3.14)

To see this it suffices to assume n = 1, so ξ = (f, g) and L.H.S. = R.H.S. + 2

Z

R

Z

S¹

(∂f

∂s

∂g

∂t − ∂g

∂s

∂f

∂t)dt ds.

Integration by parts gives Z

R

Z

S¹

∂g

∂s

∂f

∂tdt ds = − Z

R

Z

S¹

f ∂²g

∂s∂tdt ds.

On the other hand, by integrating in s first and applying by parts, since f = 0 for s > T ,

Z

S¹

Z

R

∂f

∂s

∂g

∂tds dt = − Z

R

Z

S¹

f ∂²g

∂s∂tdt ds.

From

   

∂ξ

∂s + J0

∂ξ

∂t + 2Sξ    

2

≥ 0, 

  

∂ξ

∂s + J₀∂ξ

∂t    

2

+h∂ξ

∂s + J₀∂ξ

∂t, Sξi + |Sξ|² ≥ 1 2    

∂ξ

∂s + J₀∂ξ

∂t    

2

− |Sξ|². So

Z

X |F ξ|²ds dt = Z

XT

(    

∂ξ

∂s + J0

∂ξ

∂t + Sξ    

2

)ds dt

≥ Z

XT

(1 2    

∂ξ

∂s + J0

∂ξ

∂t    

2

− |Sξ|²)ds dt

≥ 1 2

Z

XT

(    

∂ξ

∂s + J0

∂ξ

∂t    

2

− ˜c|ξ|²)ds dt

Chapter 3. Fredholm Theory 64

where ˜c = sup

XT

|S(s, t)|. Hence

||ξ||²W^1,2(X) =        

∂ξ

∂s + J0

∂ξ

∂t        

2 L²(X)

+ ||ξ||²L²(X) by (3.14)

≤ (˜c − 1) ||ξ||² + 2||F ξ||².

⇒ ||ξ||_W^1,2_(X) ≤ c(||ξ||_L²_(X) +||F ξ||_L²_(X)). (3.15) Now let β ∈ C^∞(R, [0, 1]) be a cutoff function such that

β(s) =







1 for |s| ≤ T − 1, 0 for |s| ≥ T.

Then

||F (βξ)||L²(X) = ||β(∂ξ

∂s + J0∂ξ

∂t + Sξ) + ˙βξ||L²(X)

≤ ||F ξ||L²(X)+ c||ξ||L²(XT) for some c > 0. (3.16) Similarly,

||F ((1 − β)ξ)||L²(X) ≤ ||F ξ||L²(X) + c||ξ||L²(XT) for some c > 0.

(3.17) By Rellich compact embedding, the following composition is a com-pact operator,

K : W^1,2(X, R²ⁿ) restriction

−→ W^1,2(XT, R²ⁿ) ^compact֒→ L²(XT, R²ⁿ) Kξ := ξ|XT.

Chapter 3. Fredholm Theory 65

Therefore

||ξ||W^1,2(X)

≤||βξ||W^1,2(X) +||(1 − β)ξ||W^1,2(X)

≤c¹(||βξ||L²(X) +||F (βξ)||L²(X)) + c2||F ((1 − β)ξ)||L²(X) by step 2 and (3.15)

≤c(||F ξ||L²(X) +||Kξ||^Z) by (3.16), (3.17) Step 4.

We have shown F has finite dimensional kernel and has a closed range. So the cokernel of F satisfies the isomorphism

cokerF ∼= R(F )^⊥ ∼= ker F^∗. Observe that the adjoint operator F^∗ is given by

F^∗ξ =−∂ξ

∂s + J0

∂ξ

∂t + Sξ.

The previous steps can be carried out for F^∗ and so ker F^∗ and hence cokerF is also finite dimensional. This shows that F is Fredholm.

For the proof of the second part of proposition 3.13, we have to study the so called spectral flow of a family of operators. But first of all we have the following lemma which allows us to consider particular nice kind of Fredholm operator as in (3.6) given by some S which is easier to analyze. Suppose now we have another symmetric family of matrices ˜S(s, t) as before with

s→±∞lim

S(s, t) = S˜ ^±(t).

Chapter 3. Fredholm Theory 66

Then by proposition 3.13, ˜F := _∂s^∂ + J0 ∂

∂t + ˜S is clearly also Fred-holm. Denote the space of Fredholm operator from W^1,2(X, R²ⁿ) to L²(X, R²ⁿ) by F. Proposition 3.13 tells us that the operator of the form FS := _∂s^∂ + J0 ∂

∂t + S ∈ F. Define Σ := {F = FS ∈ F} all the Fredholm operators of this form. We define FS˜ to be equivalent to FS if

s→±∞lim

S = S˜ ^± = lim

s→±∞S

and denote by ΘS the equivalence class of FS in Σ. Then

Lemma 3.15. ΘS is contractible within Σ as a subspace in F.

Proof. Take any FS0 ∈ Σ and let Θ = ΘS0. Define the homotopy to be

H : [0, 1]× Θ → Θ by (τ, FS) 7→ F(1−τ)S⁰+τ S. For all τ ∈ [0, 1], F(1−τ)S⁰+τ S ∈ Θ as lim_s

→±∞(1−τ)S⁰+τ S = lim

s→±∞S0 = S₀^±. It is also easy to check that it is continuous.

The significance of the above lemma is that the index map µ : Σ → Z is constant when restricted to the equivalence class Θ^S. In other words, µ(FS) depends only on the endpoints S^± of S. Now consider a continuous family of operator

A(s) : W^1,2(R/Z, R²ⁿ) → L²(R/Z, R²ⁿ) for s ∈ R defined by

A(s)ξ(t) := J0

dξ

dt + S(s, t)ξ(t).

Chapter 3. Fredholm Theory 67

This is a family of symmetric operator defined on (a dense subset of) L²(R/Z, R²ⁿ).

These symmetric operators have a discrete spectrum consisting of real eigenvalues, each having finite multiplicity (see [33]). Also, the eigenvalues of A(s) occur in continuous families λ_j(s) for j ∈ Z counted with multiplicity. The limit operator A^± = lim

s→±∞A(s) is invertible by (3.8). The Fredholm index of (3.6) is then given by the spectral flow of A (see [3], [28]), which roughly speaking measures the algebraic increase of eigenvalues of A(s) flowing from negative to positive, as s goes form −∞ to ∞. More precisely,

index F = #{j : λj(−∞) < 0 < λj(∞)}−#{j : λj(−∞) > 0 > λj(∞)}.

Now we want to prove that the spectral flow agrees with µ(Ψ⁻) − µ(Ψ⁺). This would prove the remaining part of proposition 3.13.

Proof of Proposition 3.13 (2). In each homotopy class ofLP^∗, there exists a path of the form Ψ(t) ∈ Sp(2n, R) = exp(J⁰St) ∈ Sp(2n, R) with Ψ(1) = W^±, where S is a constant real symmetric matrix.

More precisely, recall that by proposition 3.7, each homotopy class in LP^∗ is characterized by the Maslov index µ(Ψ) = k.

For odd n− k, by decomposing R²ⁿ as (R²)ⁿ, choose

S =





0 log 2 log 2 0



⊕

n−1

M

j=1





mjπ 0 0 mjπ



 =

n

M

j=1

Sj (3.18)

where m1 = n− k − 2 and mj = −1 otherwise.

Chapter 3. Fredholm Theory 68

For even n− k, choose

S =





−π 0 0 −π



⊕

n−1

M

j=1





mjπ 0 0 mjπ



 =

n

M

j=1

S_j (3.19)

where m1 = n− k − 1 and mj = −1 otherwise.

We will prove that for these S, the induced Ψ(t) = exp(J0St) has µ(Ψ) = k and Ψ(1) = W^±, in particular Ψ ∈ LP^∗. Thus each path in LP^∗ is homotopic to exactly one of these Ψ. Indeed if Ψ(1) = W^±, then by definition 3.6 of µ and the product and de-terminant property in proposition 3.2, µ(Ψ) =

n

X

j=1

µ(Ψ_j), where Ψj(t) := exp(J0Sjt). By lemma 3.15, index F depends only on the endpoints S^±, which in particular can be chosen in the form of (3.18) or (3.19). So now let

S(s) = β(s)S⁺+ (1− β(s))S⁻

where S^± is of the above form and β ∈ C^∞(R, [0, 1]) is an non-decreasing smooth function such that

β(s) =







1 if s ≥ 1 0 if s ≤ −1

We now study the spectral flow for A(s) given by this S(s). It also suffices to decompose the matrix Ψ(s, t) = exp(J0S(s)t) ∈ Sp(2n, R) into 2 × 2 blocks. So we can assume n = 1. There are three cases.

Chapter 3. Fredholm Theory 69

Case (i)

S⁻ =





−k⁻π 0 0 −k⁻π



, S⁺ =





−k⁺π 0 0 −k⁺π



 where k^± are odd.

Then

Ψ^±(t) = exp(J0S^±t) =





cos(k^±πt) − sin(k^±πt) sin(k^±πt) cos(k^±πt)



∈ U(1) By the determinant property in proposition 3.2, ρ(Ψ^±) = e^ik±πt, so Ψ(1) =−I = W⁺ and µ(Ψ^±) = k^±. Now consider

λ ∈ σ(A(s)) ⇔ J0

dξ

dt + S(s)ξ = λξ for some ξ 6= 0

⇔ dξ

dt = J0(S(s)− λI)ξ

⇔ ξ(t) = exp(J⁰(S(s)− λI)t)ξ(0) for some ξ(0)6= 0

⇔ 1 ∈ σ(exp(J0(S(s)− λI))) as ξ(1) = ξ(0) 6= 0.

S(s) =





−ω(s) 0

0 −ω(s)



 where ω(s) = (β(s)k⁺+ (1− β(s))k⁻)π.

So

exp(J₀(S(s)− λI)t) =





cos(ω(s) + λ)t − sin(ω(s) + λ)t sin(ω(s) + λ)t cos(ω(s) + λ)t



. This implies 1∈ σ(exp(J⁰(S(s)−λI))) ⇔ ω(s)+λ ∈ 2πZ. Therefore the family of eigenvalues of A(s) are exactly

λj(s) = −ω(s) + 2πj where j ∈ Z

Chapter 3. Fredholm Theory 70

and each λj(s) occurs in multiplicity 2. As ω(s) varies monotonically from k⁻π to k⁺π and k^± are odd, there are exactly ^|k−−k₂ ^+| values (not counting multiplicities) of j such that 0 ∈ σ(A(s)) for some s, i.e. these eigenvalues of A(s) will cross the zero. (For example if k⁻ = −3, k⁺ = 1, then all these j are j = 0, 1). If k⁻ > k⁺, then λj(s) increase when s goes from −∞ to ∞, otherwise if k⁻ < k⁺ they decrease. Thus the spectral flow of A(s) is k⁻− k⁺.

Similar argument shows that for case (ii) where

S⁻ =





0 log 2 log 2 0



 and S⁺ =





−π 0 0 −π



, we have µ(Ψ⁻) = 0 and µ(Ψ⁺) = 1.

There is only one eigenvalue λ(s) = (1− β(s)) log 2 − β(s)π of A(s) crossing zero (with constant eigenfuction ξ(t) ≡ (1, 1)). Thus the spectral flow is -1 which agrees with µ(Ψ⁻)− µ(Ψ⁺).

The remaining case (iii) is the same as case (ii) except the roles of S⁺ and S⁻ are switched. Thus by reversing time s we get the same conclusion.

Consider the more general operator F : W^1,2(R× (R/Z), R²ⁿ) → L²(R× (R/Z), R²ⁿ) by

F ξ = ∂ξ

∂s + J0∂ξ

∂t + (A + S)ξ (3.20)

where S is symmetric as before and A = A(s, t) ∈ M(2n, R) is

Chapter 3. Fredholm Theory 71

continuous matrix valued and is skew symmetric for all s, t. Suppose

s→±∞lim A(s, t) = 0

uniformly in t. Then this operator F is a compact perturbation of (3.6) and so is also Fredholm of the same index.

Now recall that for a pair of non-degenerate solutions x^± ∈ P (H) of (2.1) and u ∈ C^∞(R× S¹, M) such that

s→±∞lim u(s, t) = x^±(t), (3.21)

s→±∞lim

∂u

∂t = ˙x^±(t) and lim

s→±∞

∂u

∂s = 0 (3.22)

where all limits are uniform in t, the linear operator F (u) : W^1,2(u^∗T M) → L²(u^∗T M) is defined by

F (u)ξ := ∇sξ + J(u)∇tξ + (∇ξJ(u))∂u

∂t +∇ξ∇Ht(u). (3.23) Theorem 3.16. Suppose u : R × S¹ → M satisfies (3.21), (3.22) for a pair of non-degenerate solutions x^± ∈ P (H) of (2.1). Then F (u) is Fredholm and its index is given by

index F (u) = µ(x⁻)− µ(x⁺).

Proof. The key is to use a compact perturbation if necessary, and using local coordinates, alter the operator to the form in (3.6). Then we can apply proposition 3.13 to get the index as the difference of the respective Maslov indices.

Chapter 3. Fredholm Theory 72

By altering u if necessary, we can assume that

u(s, t) =







x⁻(t) if x ≤ −1 x⁺(t) if x ≥ 1.

This would not change the Fredholm index. By proposition 3.8, there is a symplectic orthogonal trivialization

Φ = Φ(s, t) : R²ⁿ → Tu(s,t)M

In this local coordinates, the operator F (u) is represented by F := Φ⁻¹F (u)Φ : W^1,2(R× S¹, R²ⁿ) → L²(R× S¹, R²ⁿ)

F = ∂

∂s + J0 ∂

∂t + (S + A),

where S and A denotes the symmetric and anti-symmetric part of the matrix given byhZi, F (u)Zji and Zi = Zi(s, t) ∈ Tu(s,t)M are the orthonormal frames given by the trivialization Φ. (A side remark:

the asymptotic operators J0 ∂

∂t + S(±∞) is the Hessian of A^H at x^± in the trivialization Φ(±∞), so F can be regarded as one-parameter family of operators which is asymptotically symmetric.) By direct calculations,

Aij = hZi,∇sZji = −Aji and

Sij = hZi, (∇ZjJ)∂u

∂t +∇Zj∇Ht + J∇tZji = Sji. (3.24) As Zi(s, t) ≡ Zi(±1, t) for ±s ≥ 1, A(s, t) = 0 for |s| ≥ 1. So by a compact perturbation we can assume that A(s, t) = 0 for all s, t.

Chapter 3. Fredholm Theory 73

So the operator becomes

F = ∂

∂s + J₀ ∂

∂t + S.

Let S^± = S(±, t), Φ^±(t) = Φ(±1, t) and Ψ^±(t) ∈ Sp(2n, R) be given by (3.5). It remains to show Ψ^± satisfies (3.7). Clearly we only have to show it for Ψ⁺, so to simplify the notations, denote S = S⁺, Φt = Φ(t) = Φ⁺(t), x0 = x⁺(0), Ψt = Ψ⁺(t), x(t) = x⁺(t) and dψt = dψt(x⁺(0)). Let v ∈ R²ⁿ. By definition,

Φ_tΨ_tv = dψ_tΦ₀v. (3.25) Applying covariant derivative with respect to t on R.H.S.,

∇t(dψtΦ0v) = ∇t∇s=0(ψt ◦ expx0(sΦ0v))

= ∇s=0∇t(ψt ◦ expx0(sΦ0v))

= ∇dψtΦ0vX(x(t))

= ∇ΦtΨtvX.

So differentiating (3.25) gives

ΦtΨ˙tv + (∇tΦt)Ψtv = ∇ΦtΨtvX.

Chapter 3. Fredholm Theory 74

Consider

Φ_tJ₀Ψ˙ _tv = JΦ_tΨ˙_tv (as Φ_tJ₀ = JΦ_t)

= J(∇^ΦtΨtvX − (∇^tΦt)Ψtv)

= J(∇^ΦtΨtv(J∇H) − (∇^tΦt)Ψtv)

= J(∇ΦtΨtvJ)∇H − ∇ΦtΨtv(∇H) − J(∇tΦt)Ψtv

= −∇ΦtΨtvJ(J∇H) − ∇ΦtΨtv(∇H) − J(∇tΦt)Ψtv

= −(∇ΦtΨtvJ)(X)− ∇ΦtΨtv(∇H) − J(∇tΦt)Ψtv

= −ΦtS(t)Ψ_tv by (3.24).

Therefore

Ψ˙^±(t) = J0S^±Ψ^±(t).

By proposition 3.13,

index F (u) = index F = µ(Ψ⁻)− µ(Ψ⁺) = µ(x⁻)− µ(x⁺).

Chapter 4

Floer Homology

In this chapter, we will look more closely at the Floer homology groups of Hamiltonian function for symplectic manifolds. We will provide some details of the proofs of the invariance of Floer ho-mology and the isomorphism between Floer hoho-mology and singular homology of M.

4.1 Transversality

Recall that M is the set of bounded solution to (2.7) and (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.

Proposition 4.1. There is a dense subset of smooth almost complex structure Jreg ⊂ C^∞(End(TM)) such that for all J ∈ Jreg and u ∈

75

Chapter 4. Floer Homology 76

M, F (u) is onto, i.e. (H, J) is regular.

The proof can be found in [9] and [11]. It uses a result of Smale [35], which generalizes the Sard’s theorem into the infinite dimen-sional case.

Using this result, by choosing (H, J) to be regular, and using the implicit function theorem, it follows that there is a neighborhood of u in M(x, y) which is diffeomorphic to a neighborhood of zero in ker F (u). Furthermore, by theorem 3.16, we have

dimM(x, y) = index F (u)

= µ(x)− µ(y). (4.1)

在文檔中 Floer Homology on Symplectic Manifolds (頁 64-83)