PDF (1016 kB)

The L 2 -negative gradient equation for the Hamiltonian action functional�H is the Floer equation
∂J,H(u) := ∂su + J(u)[∂tu − XH(t, u)] = 0, (32)
for u = u(s, t), s ∈Ê, t inÌor in [0, 1]. A generic choice of the Hamiltonian H ∈ C∞ (Ì×T ∗M) makes the space of solutions of the Floer equation on the cylinder,
M Λ �
∂ (x, y) = u ∈ C ∞ (Ê×Ì, T ∗ �
M) | ∂J,H(u) = 0 and lim u(s, t) = x(t), lim u(s, t) = y(t)
s→−∞ s→+∞
a manifold of dimension µ Λ (x) − µ Λ (y), for every x, y ∈ PΛ (H). Similarly, a generic choice of
H ∈ C∞ ([0, 1] × T ∗M) makes the space of solutions of the Floer equation on the strip,
M Ω �
∂ (x, y) = u ∈ C ∞ (Ê×[0, 1], T ∗ M) | ∂J,H(u) = 0, u(s, 0), u(s, 1) ∈ T ∗ q0M ∀s ∈Ê, and
�
lim u(s, t) = x(t), lim u(s, t) = y(t)
s→−∞ s→+∞
a manifold of dimension µ Ω (x) − µ Ω (y), for every x, y ∈ P Ω (H). Here generic means for a countable
intersection of open and dense subsets of the space of smooth time-dependent Hamiltonians
satisfying (H0), (H1), and (H2), with respect to suitable topologies (see [FHS96] for transversality
issues). In particular, the perturbation of a given Hamiltonian H satisfying (H0), (H1), (H2) can
be chosen in such a way that the discrete set P(H) is unaffected.
3.3. Remark. As it is well-known, transversality can also be achieved for a fixed Hamiltonian
by perturbing the almost complex structure J in a time-dependent way. In order to have good
compactness properties for the spaces M∂ one needs the perturbed almost complex structure J1
to be C 0 -close enough to the metric one J defined by (31) (see [AS06b, Theorem 1.14]). Other
compactness issues in this paper would impose further restrictions on the distance between J1 and
J. For this reason here we prefer to work with the fixed almost complex structure J, and to achieve
transversality by perturbing the Hamiltonian.
The manifolds M∂(x, y) can be oriented in a coherent way. Assumptions (H1) and (H2) imply
that these manifolds have nice compactifications. In particular, when µ(x) − µ(y) = 1, M∂(x, y)
consists of finitely many one-parameter families of solutions σ ↦→ u(· + σ, ·), each of which comes
with a sign ±1, depending whether its orientation agrees or not with the orientation determined
by letting σ increase. The algebraic sum of these numbers is an integer n Λ ∂ (x, y), or nΩ ∂
(x, y). If we
let Fk(H) denote the free Abelian group generated by the elements x ∈ P(H) of index µ(x) = k,
the above coefficients define the homomorphism
∂ : Fk(H) → Fk−1(H), x ↦→ �
n∂(x, y)y,
y∈P(H)
µ(x)=k−1
which turns out to be a boundary operator. The resulting chain complexes F Λ (H, J) and F Ω (H, J)
are the Floer complexes associated to the periodic orbits problem, and to the fixed-ends problem.
If we change the metric on M - hence the almost complex structure J - and the orientation data,
the Floer complex F(H, J) changes by an isomorphism. If we change the Hamiltonian H, the new
Floer complex is homotopically equivalent to the old one. In particular, the homology of the Floer
complex does not depend on the metric, on H, and on the orientation data. This fact allows us to
denote this graded Abelian group as HF Λ ∗ (T ∗ M), in the periodic case, and HF Ω ∗ (T ∗ M), in the
fixed-ends case. Actually, the homology of F Λ (H, J) is isomorphic to the singular homology of the
free loop space Λ(M), while the homology of F Ω (H, J) is isomorphic to the singular homology of
the based loop space Ω(M, q0),
HF Λ k (T ∗ M) ∼ = Hk(Λ(M)), HF Ω k (T ∗ M) ∼ = Hk(Ω(M, q0)).
26