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A standard gluing argument shows that KΩ is a chain map.
It is easy to construct a homotopy P #
K between ΦΩ L1#L2 ◦ M# and KΩ . In fact, it is enough
to consider the space of pairs (α, u), where α is a positive number and u is a solution of the
Floer equation on [0, +∞[×[−1, 1] converging to x for s → +∞, and such that the curve t ↦→
π ◦ u(0, 2t − 1) belongs to the evolution at time α of
Γ � W u (γ1; −grad g1ËΩ L1 ) × W u (γ2; −grad g2ËΩ L2 ) � ,
by flow of −gradËΩ . Here Γ is the concatenation map defined in section 2.6. More precisely,
L1#L2
set
M #
K (γ1,
� �
�
γ2; x) := (α, u) � α > 0, u ∈ C ∞ ([0, +∞[×[−1, 1], T ∗ M) solves (32),
π ◦ u(s, −1) = π ◦ u(s, 1) = q0 ∀s ≥ 0, lim u(s, 2t − 1) = x(t), uniformly in t ∈ [0, 1],
s→+∞
π ◦ u(0, 2 · −1) ∈ φ Ω� u
α Γ(W (γ1, −gradËΩ
) × W L1 u (γ2, −gradËΩ
)) L2 ��
,
where φ Ω s denotes the flow of −gradËΩ L1#L2 . For a generic choice of g1, g2, H1, and H2, M #
K (γ1, γ2; x)
- if non-empty - is a smooth manifold of dimension
dimM #
K (γ1, γ2; x) = m Ω (γ1; L1) + m Ω (γ2; L2) − µ Ω (x; H1#H2) + 1,
and these manifolds carry coherent orientations. The energy estimate is again (52). By counting
the elements of the zero-dimensional manifolds, we obtain a homomorphism
P #
K : � M(ËΩ
, g1) ⊗ M(ËΩ
, g2) L1 L2 �
k → F Ω k+1 (H1#H2, J).
A standard gluing argument shows that P #
K is a chain homotopy between ΦΩ L1#L2 ◦ M# and K Ω .
The homotopy P K Υ between KΩ and ΥΩ ◦ (ΦΩ L1 ⊗ ΦΩ ) is defined by counting solutions of the
L2
Floer equation on a one-parameter family of Riemann surfaces ΣK Υ (α), obtained by removing a
point from the closed disk. More precisely, given α > 0 we define ΣK Υ (α) as the quotient of the
disjoint union [0, +∞[×[−1, 0] ⊔ [0, +∞[×[0, 1] under the identification
(s, 0 − ) ∼ (s, 0 + ) for s ≥ α.
This object is a Riemann surface with boundary: its complex structure at each interior point and
at each boundary point other than (α, 0) is induced by the inclusion, whereas the holomorphic
coordinate at (α, 0) is given by the map
{ζ ∈�|Re ζ ≥ 0, |ζ| < ǫ} → Σ K Υ (α), ζ ↦→ α + ζ2 ,
where the positive number ǫ is smaller than 1 and √ α. Given γ1 ∈ P(L1), γ2 ∈ P(L2), and
x ∈ P(H1#H2), we consider the space of pairs (α, u) where α is a positive number, and u(s, t)
is a solution of the Floer equation on ΣK Υ (α) which converges to x for s → +∞, lies above some
element in the unstable manifold of γ1 for s = 0 and −1 ≤ t ≤ 0, lies above some element in
the unstable manifold of γ2 for s = 0 and 0 ≤ t ≤ 1, and lies above q0 at all the other boundary
points. More precisely, M K Υ (γ1, γ2; x) is the set of pairs (α, u) where α is a positive number and
u ∈ C∞ (ΣK Υ (α), T ∗M) is a solution of
satisfying the boundary conditions
∂J,H(u) = 0,
π ◦ u(s, −1) = π ◦ u(s, 1) = q0, ∀s ≥ 0,
π ◦ u(s, 0 − ) = π ◦ u(s, 0 + ) = q0, ∀s ∈ [0, α],
π ◦ u(0, · − 1) ∈ W u (γ1, −gradËΩ L1 ), π ◦ u(0, ·) ∈ W u (γ2, −gradËΩ L2 ),
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