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When µ Θ (y) = µ Λ (x1) + µ Λ (x2) − n, ME(x1, x2; y) is a finite set of oriented points, and
we denote by nE(x1, x2; y) the algebraic sum of the corresponding orientation signs. Similarly,
when µ Λ (z) = µ Θ (y), MG(y, z) is a finite set of oriented points, and we denote by nG(y, z) the
algebraic sum of the corresponding orientation signs. These integers are the coefficients of the
homomorphisms
E : F Λ h (H1) ⊗ F Λ k (H2) → F Θ h+k−n (H1 ⊕ H2), x1 ⊗ x2 ↦→ �
nE(x1, x2; y)y,
y∈P Θ (H1⊕H2)
µ Θ (y)=h+k−n
G : F Θ k (H1 ⊕ H2) → F Λ k (H1#H2), y ↦→ �
z∈P Λ (H1#H2)
µ Λ (z)=k
nG(y, z)z.
A standard gluing argument shows that these homomorphisms are chain maps. The main result
of this section states that the pair-of-pants product on T ∗ M factors through the Floer homology
of figure-8 loops:
3.11. Theorem. The chain maps
Υ Λ , G ◦ E : � F Λ (H1, J) ⊗ F Λ (H2, J) �
are chain homotopic.
k
= �
j+h=k
F Λ j (H1, J) ⊗ F Λ h (H2, J) → F Λ k−n(H1#H2, J)
In order to prove the above theorem, we must construct a homomorphism
P Υ GE : � F Λ (H1, J) ⊗ F Λ (H2, J) �
k → F Λ k−n+1 (H1#H2, J),
such that
(Υ Λ − G ◦ E)(α ⊗ β) = ∂ Λ J,H1#H2 ◦ P Υ GE (α ⊗ β) + P Υ � Λ
GE ∂ α ⊗ β + (−1) J,H1 h α ⊗ ∂ Λ J,H2β� , (47)
for every α ∈ F Λ h (H1) and β ∈ F Λ j (H2). The chain homotopy P Υ GE is defined by counting solutions
of the Floer equation on a one-parameter family of Riemann surfaces with boundary Σ Υ GE (α),
α ∈]0, +∞[, obtained by removing two open disks from the pair-of-pants.
More precisely, given α ∈]0, +∞[, we define ΣΥ GE (α) as the quotient of the disjoint union
(Ê×[−1, 0]) ⊔ (Ê×[0, 1] under the identifications
�
− (s, −1) ∼ (s, 0 )
(s, 0 + ) ∼ (s, 1)
if s ≤ 0,
�
(s, −1) ∼ (s, 1)
(s, 0− ) ∼ (s, 0 + )
if s ≥ α.
This object is a Riemann surface with boundary, with the holomorphic coordinates (42) and (43)
at (0, −1) ∼ (0, 0 − ) and at (0, 0 + ) ∼ (0, 1), with the holomorphic coordinates (36) and (44)
(translated by α) at (α, 0) and at (α, −1) ∼ (α, 1). The resulting object is a Riemann surface with
three cylindrical ends, and two boundary circles.
Given x1 ∈ P Λ (H1), x2 ∈ P Λ (H2), and z ∈ P Λ (H1#H2), we define M Υ GE (x1, x2; z) to be
the space of pairs (α, u), with α > 0 and u ∈ C ∞ (Σ Υ GE (α), T ∗ M) solution of
∂J,H(u) = 0,
with boundary conditions
� πu(s, −1) = πu(s, 0 − ) = πu(s, 0 + ) = πu(s, 1),
u(s, 0 − ) − u(s, −1) + u(s, 1) − u(s, 0 + ) = 0,
and asymptotic conditions
∀s ∈ [0, α],
lim
s→−∞ u(s, t − 1) = x1(t), lim
s→−∞ u(s, t) = x2(t), lim u(s, 2t − 1) = z(t),
s→+∞
uniformly in t ∈ [0, 1]. The following result is proved in section 5.10.
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