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4.5 The right-hand square is homotopy commutative
In this section we prove that the chain maps ΦΛ L1#L2 ◦ MΓ and G ◦ ΦΘ are both homotopic to
L1⊕L2
a third chain map, named KΘ . This fact implies that the right-hand square in the diagram (53)
commutes up to chain homotopy.
The chain map KΘ is defined by using the following spaces of solutions of the Floer equation
on the half-cylinder for the Hamiltonian H1#H2: given γ ∈ PΘ (L1 ⊕ L2) and x ∈ PΛ (H1#H2),
set
M Θ �
K (γ; x) := u ∈ C ∞ ([0, +∞[×Ì, T ∗ �
�
M) �∂J,H1#H2(u) = 0,
π ◦ u(0, ·) ∈ Γ � W u (γ; −grad gΘËΘ L1⊕L2 ) � �
, lim u(s, ·) = x uniformly in t .
s→+∞
By Theorem 3.2 in [AS06b] (or by the arguments of section 5.10), the space M Θ K (γ; x) is a smooth
manifold of dimension
dimM Θ K(γ; x) = m Θ (γ) − µ Λ (x),
for a generic choice of g Θ , H1, and H2. These manifolds carry coherent orientations.
Compactness follows from the energy estimate
�
]0,+∞[×Ì|∂su(s, t)| 2 dsdt ≤ËL1⊕L2(γ) −�H1#H2(x),
implied by (49). Counting the elements of the zero-dimensional spaces, we define a homomorphism
K Θ : Mj(ËΘ L1⊕L2 , g Θ ) → F Λ j (H1#H2, J),
which is shown to be a chain map.
It is easy to construct a chain homotopy P Γ K between ΦΛ L1#L2 ◦MΓ and K Θ by considering the
space
M Γ �
K(γ; x) := (α, u) ∈]0, +∞[×C ∞ ([0, +∞[×Ì, T ∗ �
�
M) �∂J,H1#H2(u) = 0,
φ Λ −α (π ◦ u(0, ·)) ∈ Γ�W u (γ; −grad gΘËΘ ) L1⊕L2 � �
, lim u(s, ·) = x uniformly in t .
s→+∞
where φ Λ s denotes the flow of −gradËΛ L1#L2 on Λ 1 (M). As before, we find that generically
M Γ K (γ1, γ2; x) is a manifold of dimension
dimM Γ K (γ; x) = mΛ (γ) − µ Θ (x) + 1.
Compactness holds, so an algebraic count of the zero-dimensional spaces produces the homomorphism
P Γ K : Mj(ËΘ L1⊕L2 , g Θ ) → F Λ j+1(H1#H2, J).
A standard gluing argument shows that P Γ K is the required homotopy.
Finally, the construction of the chain homotopy P K G between KΘ and G ◦ ΦΘ is based
L1⊕L2
on the one-parameter family of Riemann surfaces ΣK G (α), α > 0, defined as the quotient of the
disjoint union [0, +∞[×[−1, 0] ⊔ [0, +∞[×[0, 1] under the identifications
(s, 0 − ) ∼ (s, 0 + ) and (s, −1) ∼ (s, 1) for s ≥ α.
This object is a Riemann surface with boundary, the holomorphic structure at (α, 0) being given
by the map
{ζ ∈�|Re ζ ≥ 0, |ζ| < ǫ} → Σ K G (α), ζ ↦→ α + ζ2 ,
50