Definition 2.7. The contractible loop space L of M is defined to be all the contractible loops in M. i.e. L := {x ∈ C∞(R/Z, M) | x is contractible}.
Denote D := {z ∈ C : |z| ≤ 1} to be the closed unit disk.
So for x ∈ L, there exists an extension u : D → M such that u(ei2πt) = x(t).
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 31
We will assume throughout that ω vanish over the second homotopy group π2(M) of M. i.e.
Z
S2
v∗ω = 0 (2.3)
for any smooth v : S2 → M, noting that this integral depend only on the homotopy class of v. This assumption is needed for the well-definedness of the action functional on M and is also crucial for the compactness of the so called moduli space. A symplecic manifold with this condition is called aspherical and we will denote this condition as ω(π2(M)) = 0.
Definition 2.8. The action functional A = AH : L → R is defined by
AH(x) := − Z
D
u∗ω + Z 1
0
Ht(x(t))dt, where u : D → M is an extension of x to the unit disk.
Remark 2.9. A is well defined by the following reason. Suppose u1, u2 both extends x, then we can “glue” the two maps along their boundary to get a map from S2 to M. More precisely, let S2 ⊂ R3 be the unit sphere and let π : (x, y, z) 7→ (x, y) by the projection onto the x− y plane. Define v : S2 → M by
v(p) =
u1(π(p)) if p is on the upper hemisphere, u2(π(p)) if p is on the lower hemisphere.
Then v is a well defined continuous map and R
S2u∗1ω − u∗2ω = R
S2v∗ω = 0. Therefore A is a well defined function.
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 32
It turns out that Arnold conjecture is easier to prove in the so called monotone case. Here we will make an even stronger assump-tion. Let J be an almost complex structure which is compatible with ω, i.e. J ∈ C∞(End(TM)), J2 = −I and
g(ξ, η) = hξ, ηi := ω(ξ, Jη), ξ, η ∈ TxM, (2.4) defines a Riemannian metric on M. Then by the symmetry of g, both ω and g are J-invariant, i.e. hJξ, Jηi = hξ, ηi and ω(Jξ, Jη) = ω(ξ, η). Such J exists in abundance and in fact the space J of all compatible almost complex structures of M is contractible (see for example [23]). Then (T M, J) is a complex vector bundle over M with first Chern class c1 = c1(T M, J) ∈ H2(M, Z). c1 is indepen-dent of the choice of J as we can join two such complex structures J1, J2 by a path and thus induce an isomorphism between (T M, J1) and (T M, J2) as complex vector bundle.
We will assume throughout, as in (2.3), that c1 vanishes on π2(M):
Z
S2
v∗c1 = 0 (2.5)
for any v : S2 → M. This assumption, denoted by c1(π2(M)) = 0, is needed to give a well-defined Maslov type index for the critical
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 33
point of A and thus a grading of the Floer homology groups. Let us state again our assumptions.
Assumption 2.10. We will assume throughout that both ω and c1
vanishes on π2(M):
Z
S2
v∗ω = 0 and Z
S2
v∗c1 = 0 for any v ∈ C∞(S2, M).
Remark 2.11. 1. Floer [11] actually proved the Arnold conjec-ture in the more general case where M is monotone. This means
Z
S2
v∗c1 = c Z
S2
v∗ω
for any v : S2 → M, where c is a positive constant. The weakly monotone case was proved by Hofer-Salamon [17] and Ono [26]. The general case was proved by Fukaya-Ono [14], Liu-Tian [21] and Ruan [29].
2. In the monotone case, by rescaling ω if necessary, R
S2v∗ω ∈ Z for any v : S1 → M. The argument in remark 2.9 shows that A is a well defined circle-valued function.
L is a very large space and is not a finite dimensional manifold (except when M is a point). For x∈ L, we define a “tangent vector”
ξ to be a vector field on x, i.e. ξ(t) ∈ Tx(t)M. In other words, ξ is a section of the induced bundle x∗T M. Fix ξ, let ys = y(s,·) be a one-parameter variation of contractible loop such that ∂y∂s(0, t) = ξ(t)
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 34
and y(0, t) = x(t). Explicitly, we can choose y(s, t) = expx(t)(s · ξ) (M is a compact Riemannian manifold once J is chosen). Let u : D→ M be an extension of x in the sense that u(ei2πt) = x(t). Note that we can extend y(s,·) by the “gluing” map y|S1×[0,s]#u for each s.
Then
dA(x)ξ = d ds
s=0
(− Z
D
u∗ω− Z s
0
Z 1 0
y∗ω + Z 1
0
Ht(y(s, t))dt)
= d
ds s=0
( Z s
0
Z 1 0
ω(∂y
∂t, ∂y
∂s)dt ds + Z 1
0
Ht(y(s, t))dt)
= Z 1
0
(ω(dx
dt, ξ) + dHt(ξ))dt.
Therefore dA(x) = 0 ⇔ ω( ˙x, ·) = −dHt. i.e. ˙x(t) = Xt(x(t)).
So critical points of the action functional correspond to the con-tractible periodic solutions to the Hamiltonian equation. Define
J := {ω-compatible almost complex structure on M}
and let J ∈ J . Let g be the induced Riemannian metric. Let ξ, η ∈ TxL, so ξ(t), η(t) ∈ Tx(t)M. We define a Riemannian metric
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 35
on L by
hξ, ηi :=
Z 1 0
g(ξ(t), η(t))dt.
Then
hgrad A, ξi = dA(ξ)
= Z 1
0
(ω( ˙x, ξ) + dHt(ξ))dt
= Z 1
0
(ω(J ˙x, Jξ) +h∇Ht, ξi)dt (∇ is the gradient w.r.t. g)
= Z 1
0 hJ ˙x + ∇Ht, ξidt.
So
grad A(x)(t) = J ˙x(t) +∇Ht(x(t)). (2.6) A negative gradient flow line of A is u : R → L, s 7→ us(·) such that
du
ds = −grad A(us).
By the above calculations, regarding u = u(s, t) := us(t) : R × (R/Z) → M, u is given by the partial differential equation
∂u
∂s = −J∂u
∂t − ∇Ht(u) i.e. ∂u
∂s + J(u)∂u
∂t +∇H(t, u) = 0. (2.7) We denote the left hand side of the above equation by
∂(u) = ∂H,J(u) := ∂u
∂s + J(u)∂u
∂t +∇H(t, u).
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 36
Remark 2.12. 1. The equation (2.7) can also be written as
∂u
∂s + J(∂u
∂t − Xt) = 0.
This is because J∇Ht = Xt as −dHt = h−∇Ht,·i = ω(−∇Ht, J·) = ω(J∇Ht,·) = ω(Xt,·).
2. If u(s, t) ≡ x(t) satisfying (2.7) is independent of s, then x(t) is a critical point of A as grad A(x) = −∂u∂s = 0, thus it is a periodic solution for (2.1).
If Ht ≡ constant, then (2.7) becomes
∂u
∂s + J∂u
∂t = 0,
that means u is a J-holomorphic curve. (A J-holomorphic curve is a map u from a Riemann surface (Σ, i) to an almost complex manifold (M, J) such that J ◦ du = du ◦ i, see [15]) 3. Finally if H(t, x) = H(x) is independent of t, then for those
solutions u = u(s) to (2.7) which is also independent of t sat-isfies
du
ds = −∇H(u). (2.8)
i.e. it satisfies the gradient flow equation for H. This observa-tion will be useful to relate the Morse homology with the Floer homology as we will see later in this section (see also section 4.5).
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 37
We are going to apply Morse-type theory to study the gradient flow line of A and more importantly, understand how the behavior of the critical points of A relates with the topology of M. However there are some problems preventing us from directly applying the classical Morse theory to the action functional.
In finite dimensional Morse theory, every gradient flow line of a Morse function f on a compact manifold M “begins” and “ends” at a critical point. More precisely, if γ(t) is a gradient flow line then
t→±∞lim γ(t) exists and the two limits are both critical points. However this is not true for the symplectic Floer theory. Actually this is true only when u is bounded.
Definition 2.13. Let u ∈ C∞(R × S1, M). The energy of u is defined by
E(u) := 1 2
Z 1 0
Z ∞
−∞
(
∂u
∂s
2
+
∂u
∂t − Xt(u)
2
)ds dt.
u is said to be bounded if E(u) < ∞.
Theorem 2.14. Suppose u = u(s, t) ∈ C∞(R × (R/Z), M) is a contractible solution of (2.7). Then E(u) < ∞ if and only if there exists x± ∈ P (H) such that
s→±∞lim u(s, t) = x±(t), (2.9) the limits being uniform in t.
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 38
Figure 2.1: Flow line of symplectic action.
Proof. We prove “⇐”. For u satisfying (2.9), E(u) = 1
2 Z 1
0
Z ∞
−∞
(
∂u
∂s
2
+
∂u
∂t − Xt(u)
2
)ds dt
= 1 2
Z ∞
−∞
Z 1 0
(
∂u
∂s
2
+
J∂u
∂t +∇Ht(u)
2
)dt ds
=
Z ∞
−∞
Z 1 0
∂u
∂s
2
dt ds
=
Z ∞
−∞
du ds
2
ds where || · || is the norm in L
=
Z ∞
−∞hdu
ds,−grad Aids
=
Z ∞
−∞
d
ds(−A(us))ds
= A(x−)− A(x+) < ∞. (2.10) We will prove the converse in theorem 4.2.
Suppose u solves (2.7) and E(u) < ∞, then u is called a bounded solution of (2.7) and denote the space of all bounded solutions of (2.7) by M. Given x± ∈ P (H), we also define
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 39
Definition 2.15. The moduli space of bounded solutions M(x−, x+) =M(x−, x+; H, J)
:= {u ∈ C∞(R× S1, M) | u solves (2.7) and lims
→±∞u(s, t) = x±(t)}
= {u ∈ C∞(R× S1, M) | ∂(u) = 0 and lim
s→±∞u(s, t) = x±(t)}.
In view of theorem 2.14, it is clear that
M = [
x,y∈P (H)
M(x, y).
Theorem 2.14 suggests that suggests that instead of considering the flow of A on the loop space L, one should look at the space of bounded energy solutions of its gradient flow equation.
Remark 2.16. If u ∈ M(x−, x+) with E(u) = 0, then ∂u∂s ≡ 0 and so u = u(t) satisfies dudt = Xt(u), therefore u(t) = x−(t) = x+(t) ∈ P (H). In particular if x− 6= x+ and M(x−, x+) 6= φ then the proof of theorem 2.14 shows that A(x−) > A(x+). This is analogous to the fact in classical Morse theory that the value of a Morse function f on M decreases strictly along a non-constant gradient flow line.
Remark 2.17. M(x−, x+) minimizes E among all curves with
bound-Chapter 2. Symplectic Fixed Points and Arnold Conjecture 40
ary conditions (2.9), this follows from E(u) = 1
2 Z ∞
−∞
Z 1 0
(
∂u
∂s
2
+
J∂u
∂t +∇H
2
)dt ds
= 1 2
Z ∞
−∞
Z 1 0
(
∂u
∂s + J∂u
∂t +∇H
2
− 2h∂u
∂s, J∂u
∂t +∇Hi)dt ds
= 1 2
Z ∞
−∞
Z 1 0
∂u
∂s + J∂u
∂t + ∇H
2
dt ds + Z ∞
−∞hdu
ds,−grad A ◦ ui)ds
= 1 2
Z ∞
−∞
Z 1 0
∂u
∂s + J∂u
∂t + ∇H
2
dt ds + A(x−)− A(x+).
There is a natural R-action on M(x−, x+) given by r · u(s, t) = u(r+s, t) for r ∈ R. Define the moduli space of unparametrized bounded solutions
M(xˆ −, x+) := M(x−, x+)/R.
We study M(x, y; H, J) locally by linearizing ∂ at u ∈ M(x, y) to get a differential operator F (u). More precisely, differentiating equation (2.7) in the direction of a vector field ξ ∈ C∞(u∗T M) on u leads to the first order linear differential operator F (u) = D∂(u):
F (u)ξ = ∇sξ + J(u)∇tξ + (∇ξJ(u))∂u
∂t +∇ξ∇Ht(u)
where ∇ denotes the covariant derivative with respect to the Rie-mannian metric given by (2.4). It turns out that if x, y are non-degenerate then F (u) is a Fredholm operator with a finite index between two appropriate Sobolev spaces. We want to smoothly ex-tend ∂ as a section of a Banach bundle over a Banach manifold, so thatM(x−, x+) is the zero section of ∂ and argue that 0 is a regular
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 41
value of ∂ if we choose a suitable J. After this has been done we can apply implicit function theorem to conclude that M(x−, x+) is a smooth manifold with the dimension near u given by the index of F (u).
Definition 2.18. (H, J) is called a regular pair if 1. All contractible x∈ P (H) are non-degenerate, and
2. If x± ∈ P (H) are contractible and u ∈ M(x−, x+), then F (u) is surjective.
Due to an infinite dimensional version of Sard’s theorem by Smale [35] (see also [13], [33] for the details), we have a transversality result:
Proposition 2.19. There is a dense subset of smooth almost com-plex structure Jreg ⊂ C∞(End(TM)) such that for all J ∈ Jreg and u ∈ M, F (u) is onto, i.e. (H, J) is regular.
So by implicit function theorem and by choosing a regular pair, we have
Theorem 2.20. For a regular pair (H, J), M(x−, x+; H, J) is a fi-nite dimensional smooth manifold for x± ∈ P (H) and the dimension of M(x−, x+) is given by the index:
dimM(x−, x+) = index F (u).
Chapter 2. Symplectic Fixed Points and Arnold Conjecture 42
We will see that index F (u) can be in turn calculated by the difference of a Maslov-type index µ:
index F (u) = µ(x−)− µ(x+).
The Maslov index µ : P (H) → Z associates an integer to every contractible periodic solution x ∈ P (H) of (2.1). The Maslov index will play the role of grading the Floer homology groups just like the Morse index does in Morse homology.