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)
Chapter 4. Floer Homology 77
Theorem 4.3 (Gromov compactness). Let (un)n∈N be a sequence in M(x, y). Then there is a subsequence (unk)k∈N and sequences of time (sik)k∈N, i = 0,· · · , m such that
klim→∞unk(s + sik, t) = ui(s, t) (4.2) where ui ∈ M(xi, xi+1), xi ∈ P (H), x0 = x, xm+1 = y. The conver-gence is uniform in all derivatives on compact subset (i.e. in Cloc∞ topology). If (H, J) is a regular pair, then
µ(xi) > µ(xi+1) for i = 0,· · · , m.
Remark 4.4. The convergence as in theorem 4.3 (4.2), m ≥ 1, is called a geometric convergence (or weak convergence) to-wards a broken trajectory of order m.
Figure 4.1: Convergence towards a broken trajectory.
The geometric convergence for a sequence of unparametrized trajec-tory ˆun = [un] ∈ ˆM(x, y) is defined analogously, and is denoted as
ˆ
un ⇀ (ˆu0,· · · , ˆum).
Chapter 4. Floer Homology 78
If µ(x)− µ(y) = 2 and the order m = 1 then we say it converges to a simply broken trajectory.
We emphasize here that in theorem 4.3, our assumption (2.3) that ω vanishes on π2(M) is essential, otherwise another kind of limiting behavior, called bubbling, would occur. The “⊃” part of theorem 4.2 has been proved in theorem 2.14. For “⊂”, we need the following two lemmas.
Denote the open disk of radius r by Br := {z ∈ C : |z| < r}.
Lemma 4.5. There exists a constant ε = ε(M, ω, H, J) > 0 such that for all solution u ∈ C∞(Br, M) of (2.7) with
Z
Br
∂u
∂s
2
dt ds ≤ ε, then
∂u
∂s(0)
≤ 1 + 8 πr2
Z
Br
∂u
∂s
2
dt ds.
By lemma 4.5, for u ∈ M, since
∂u∂t(s, t) ≤
∂u∂s(s, t)
+|∇H(t, u)|
and M is compact,
||∇u||L∞(R×S1) := sup
s∈R,t∈S1{max(∂u
∂s(s, t),∂u
∂t(s, t))} < ∞. (4.3) For u : R×S1 ∼= C/iZ → M, we can also regard u as u = u(s+it) ∈ C∞(C, M). Of course it does not really matter if we are concerning about || · ||L∞.
Chapter 4. Floer Homology 79
Lemma 4.6. Let Ω ⊂ C be an open domain. Then every sequence of solutions un ∈ C∞(Ω, M) of (2.7) such that
sup
n∈N||∇un||L∞(Ω) < ∞ has a subsequence converging in Cloc∞ topology.
Note that such (sub-) sequence un converges to some u ∈ M.
For if p = u(s0, t0) ∈ M, by choosing a chart around p to some relatively compact domain Ω ⊂ R2n and by restricting the domain of un, we can assume that un ∈ C∞(R× S1, Ω) such that
∂un
∂s + J(un)∂un
∂t +∇Ht(un) = 0.
Then for ε > 0,
∂u
∂s + J(u)∂u
∂t +∇Ht(u)
= (∂u
∂s + J(u)∂u
∂t +∇Ht(u))− (∂un
∂s + J(un)∂un
∂t +∇Ht(un))
< ǫ
for sufficiently large n by the convergence of un and uniform con-tinuity of H and J. Since ε > 0 is arbitrary we conclude that u satisfies (2.7).
The proofs of lemma 4.5 and lemma 4.6 can be found in [30].
Proof of theorem 4.2. We have to prove thatM = [
x,y∈P (H)
M(x, y).
Suppose not, then there exists u ∈ M, ε > 0 and a sequence
Chapter 4. Floer Homology 80
(sn, tn) ∈ R × S1 such that |sn| → ∞ and for all x ∈ P (H), n ∈ N, d(u(sn, tn), x(tn)) ≥ ε
where d denotes the distance in M induced by its Riemannian met-ric. Define un(s, t) := u(s + sn, t). Then by lemma 4.6 there exists a subsequence, which we still denote by un for convenience, converging to some v ∈ M. Assume also that tn converges to t0. Then
d(v(0, t0), x(t0)) ≥ ε (4.4) for any x ∈ P (H). Since |sn| → ∞, as R
R
R
S1
∂u∂s
2dt ds < ∞, Z T
−T
Z
S1
∂v
∂s
2
dt ds = lim
n→∞
Z T
−T
Z
S1
∂un
∂s
2
dt ds
= lim
n→∞
Z T
−T
Z
S1
∂u
∂s(s + sn, t)
2
dt ds = 0 for all T > 0. So ∂v∂s = 0 and hence v is independent of s, but then v(s, t) = x(t) for some x∈ P (H), this contradicts (4.4).
Proof of theorem 4.3. This proof is from [30].
Step 1. We first claim that sup
u∈M||∇u||L∞ < ∞
We prove by contradiction. Suppose the contrary, then there exists a sequence un ∈ M such that
cn := ||∇un||L∞ → ∞.
Chapter 4. Floer Homology 81
We assume the domain of un is C for convenience. So there exists zn = sn + itn such that
max
∂un
∂s (zn) ,
∂un
∂t (zn)
≥ cn 2.
Define vn(z) := un(zn+c1nz) and denote Br(z0) := {z ∈ C : |z−z0| <
r}. Then
|∇vn(0)| ≥ 1
2, ||∇vn||L∞ ≤ 1, (4.5)
∂vn
∂s + J(vn)∂vn
∂t + 1
cn∇H(vn, tn+ t cn
) = 0, and (4.6) Z
Bcn(0)
∂vn
∂s
2
= Z
B1(zn)
∂un
∂s
2
≤ 2E(un) (by definition 2.13)
≤ 2 max
x,y∈P (H)|A(x) − A(y)| (by (2.10)). (4.7) From (4.5) and by lemma 4.6, vn converges to some v ∈ C∞(C, M) such that
∇v(0) 6= 0, (4.8)
∂v
∂s + J(v)∂v
∂t = 0, and (4.9)
0 <
Z
C
∂v
∂s
2
< ∞. (4.10)
Define γr : S1 → M by γr(θ) := v(rei2πθ), observe that
| ˙γr(θ)| = 2πr
∂v
∂s(rei2πθ) . So
Z ∞
0
1
2πr|| ˙γr||2L2(S1)dr = Z ∞
0
1 2πr
Z 1
0 | ˙γr(θ)|2dθ dr = Z
C
∂v
∂s
2
< ∞.
Chapter 4. Floer Homology 82
As the length of γr, l(γr) = R1
0 | ˙γr| · 1 dθ ≤ (R1
0 | ˙γr|2dθ)12 = || ˙γr||L2
by the Cauchy-Schwarz inequality, by choosing a sufficiently large R > 0, l(γR) can be arbitrary small. Choose a symplectic chart h : U → R2n of M such that γr(S1) ⊂ U, h(U) is a bounded convex domain and h(γr(1)) = 0.
Define w : S2 = C∪ {∞} → M by
w(rei2πθ) :=
v(rei2πθ) r ≤ R h−1(Rrh◦ v(Rei2πθ)) r ≥ R.
Let ˜w(ρ, θ) := ρRγR(θ), ε > 0 and consider Z
S2−BR
w∗ω = − Z
B1 R
˜ w∗ω0
=
Z 1/R 0
Z 1 0
ω0(RγR, ρR ˙γR)dρ dθ
= R2
Z 1/R 0
Z 1 0
ρg(γR,−J ˙γR)dρ dθ
= R2
Z 1/R 0
Z 1 0
ρ|γR|| ˙γR|dρ dθ
≤ c · l(γR) < ε
for sufficiently large R where the constant c > 0 depends on h(U ) only. Therefore
Z
S2
w∗ω ≥ Z
BR
v∗ω − ε(R) = Z
BR
∂v
∂s
2
− ε(R) > 0
for sufficiently large R, the last equation follows from ω(∂v∂s,∂v∂t) = ω(∂v∂s, J∂v∂s) = |∂v∂s|2. This contradicts our assumption that ω(π2(M)) =
Chapter 4. Floer Homology 83
0. Our claim is proved.
Step 2. Since P (H) are isolated, there exists ε > 0 such that sup
t∈S1
d(x(t), y(t)) > 2ε for all x, y ∈ P (H). Given a sequence un
in M(x, y), clearly we can assume x 6= y, for otherwise x = y im-plies E(un) = A(x) − A(y) = 0 and so un ≡ x for all n by remark 2.16 and we have nothing to prove. Define
s1n := inf{s ∈ R : d(un(s, t), x(t)) > ε for some t ∈ S1}.
The sets which we are taking infimum at are all non-empty as
slim→∞un(s, t) = y(t) and s1n 6= −∞ by a similar reason, as lim
s→−∞un(s, t) = x(t). By step 1 and lemma 4.6, there is a subsequence (which for convenience taken to be itself), such that u1n(s, t) := un(s + s1n, t) converges to some u1 ∈ M. By theorem 4.2 (1), u1 ∈ M(x0, x1) for some x0, x1 ∈ P (H). Since d(u1(s, t), x(t)) ≤ ε for all s ≤ 0, x0 = x. Also x1 6= x, for if otherwise, u1 ≡ x(t) again by remark 2.16. But by the definition of u1n there exists t0 ∈ S1 such that d(u1(0, t0), x(t0)) ≥ ε, a contradiction. If x1 = y, then we are done.
Otherwise we prove by induction, i.e. we claim that if we have ui ∈ M(xi−1, xi), lim
n→∞un(s + sin, t) = ui(s, t) for i = 0,· · · , k with xk 6= y, then there exists uk+1 ∈ M(xk, xk+1) and a sequence sk+1n such that lim
n→∞un(s + sk+1n , t) = uk+1(s, t) for some xk+1 ∈ P (H), xk+1 6= xk.
Since uk ∈ M(xk−1, xk), there exists s0 such that if s ≥ s0, d(uk(s, t), xk(t)) <
ε for all t ∈ S1. Then for sufficiently large n, for all t ∈ S1,
Chapter 4. Floer Homology 84
d(ukn(s0, t), xk(t)) = d(un(s0 + skn, t), xk(t)) < ε. So by passing into subsequence, define
sk+1n := sup{s : s ≥ skn+ s0, d(un(s, t), xk(t)) < ε for all t∈ S1}.
Then without loss of generality the sequence uk+1(s, t) := un(s + sk+1n ) converges to uk+1 ∈ M by step 1 and lemma 4.6. We claim that uk+1 ∈ M(xk, xk+1) with xk+1 6= xk.
The sequence sk+1n − skn → ∞ for otherwise [s0, sk+1n − skn] is con-tained in a compact interval [s0, s1], then for each t, uk+1n (0, t) = ukn(sk+1n − skn, t) will converge to a point in uk([s0, s1] × S1), so d(uk+1(0, t), xk(t)) < ε. But by our construction there exists t0 ∈ S1 such that d(uk+1(0, t0), xk(t0)) ≥ ε, a contradiction. So skn− sk+1n + s0 → −∞ and lim
n→∞uk+1n (skn− sk+1n + s0, t) = lim
n→∞ukn(s0, t) = uk(s0, t) with d(uk(s0, t), xk(t)) < ε. It follows that lim
s→−∞uk+1(s, t) = xk(t).
xk+1 6= xk by the same reason as before.
By remark 2.16, if xk 6= xk+1 and uk ∈ M(xk, xk+1), A(xk) >
A(xk+1), i.e. the action decreases. Since P (H) is finite, this process must terminate to arrive at xm+1 = y, m < ∞.
Finally if (H, J) is regular, as xi 6= xi+1 for all i, the moduli space M(xi, xi+1) ∋ ui is at least one-dimensional and it follows from the dimension formula (4.1) dimM(xi, xi+1) = µ(xi)− µ(xi+1) that µ(xi) > µ(xi+1).
If un does not converge to broken trajectories (of order m ≥ 1), then its convergence is stronger:
Chapter 4. Floer Homology 85
Proposition 4.7. Let (un)n∈N be a sequence in M(x, y) converging to u ∈ M(x, y) in Cloc∞ sense,
nlim→∞un(s, t) = u(s, t).
Then it also converges in C∞ sense, i.e. it converges uniformly in all derivatives.
The idea is that for fixed ends x and y, the non-degeneracy of x and y implies uniform exponential convergence of the ends of the trajectories. Away from the two ends, the uniform convergence is ensured by theorem 4.3.
Proposition 4.8. Let (H, J) be a regular pair and x, y ∈ P (H) with µ(x)− µ(y) = 1, then the 0-dimensional manifold ˆM(x, y) is compact. i.e. it consists of finite number of points. In other words the set of trajectories between x and y is finite (modulo shifting).
Proof. Let un ∈ M(x, y). Then by theorem 4.3, without loss of generality we can assume that
nlim→∞un(s, t) = u(s, t)
in Cloc∞ sense where u(s, t) ∈ M(x, y) (since µ(x) − µ(y) = 1, it cannot converge to a broken trajectory.) By proposition 4.7, un
converges uniformly to u and thus [un] → [u]. Therefore ˆM(x, y) is compact.
Chapter 4. Floer Homology 86
We have to analyze the one-dimensional moduli space of un-parametrized trajectories of relative index 2 in order to prove that the boundary operator really defines Floer homology. We need a gluing construction due to Floer.
The gluing construction can be considered as a converse of Gro-mov’s compactness (theorem 4.3), which states that any sequence un ∈ M(x, y) not converging in M(x, y) must converge (up to a subsequence) to a broken trajectory of some order m. The gluing construction tells us that we can reverse this process, i.e. we can
“glue” a broken trajectory of order m (u0,· · · , um) ∈ M(x0, x1) ×
· · · × M(xm, xm+1) together to get a trajectory in M(x0, xm+1), up to m gluing parameters (which in some sense measures how close the resulting trajectory with each ui is). For simplicity we will only give the statement for gluing a simply broken trajectory, which is sufficient in our treatment.
Proposition 4.9 (Floer’s Gluing). (Unparametrized version) Let (H, J) be regular and K ⊂ ˆM(x, y) × ˆM(y, z) be a compact subset. Then there exists a constant ρ0 = ρ0(K) and a gluing map
# : K × [ρ0,∞) → M(x, z)ˆ (ˆu, ˆv, ρ) 7→ ˆu#ρvˆ such that
1. # is an embedding;
Chapter 4. Floer Homology 87
2. ˆu#ρv converges to the broken trajectory (ˆˆ u, ˆv) geometrically as ρ → ∞ (see remark 4.4),
ˆ
u#ρˆv ⇀ (ˆu, ˆv)
3. for a sequence (ˆun)n∈N of unparametrized trajectories in ˆM(x, z) converging geometrically to the simply broken trajectory (ˆu, ˆv), then for sufficiently large n, ˆun lies within the range of #.
The details can be found in [11] and also [8]. The idea is that we can first “pre-glue” u and v at y to get an approximate solution u ˆ#ρv of (2.7) such that ||∂(u ˆ#ρv)||Lp ≤ e−cρ for large enough ρ where c = c(K) > 0. Explicitly this can be done by
u ˆ#ρv(s, t) :=
u(s + ρ, t) , s≤ −1
expy(t)(β(−s)ξ(s + ρ, t) + β(s)ζ(s − ρ, t)) , s ∈ [−1, 1]
u(s− ρ, t) , s ≥ 1.
where ξ, ζ are defined such that u(s, t) = expy(t)(ξ(s, t)) for s≥ ρ0−1 and v(s, t) = expy(t)(ζ(s, t)) for s ≤ −ρ0 + 1; β ∈ C∞(R, [0, 1]) is non-decreasing such that β(s) = 0 for s ≤ 0 and β(s) = 1 for s ≥ 1. One then uses the Picard’s method (see [8] lemma 4.2) to prove the existence of a vector field ξ = ξ(u, v, ρ) on w :=
u ˆ#ρv with ||ξ||W1,p ≤ e−cρ and such that expwξ is in M(x, z).
We then define ˆu#ρv := [expˆ wξ] ∈ ˆM(x, z). Of course this also gives the parametrized version of the gluing of u and v, and up to
Chapter 4. Floer Homology 88
reparametrizations of u, v and u#ρv, u#ρv is uniquely defined by u and v.
The techniques of gluing combined with Gromov’s compactness are useful to prove several important results as we will see later.