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3.12. Proposition. For a generic choice of H1 and H2, M Υ GE (x1, x2; z) - if non-empty - is a
manifold of dimension
dimM Υ GE(x1, x2; z) = µ Λ (x1) + µ Λ (x2) − µ Λ (z) − n + 1.
The projection (α, u) ↦→ α is smooth on M Υ GE (x1, x2; z). These manifolds carry coherent orientations.
Energy estimates together with (H1) and (H2) again imply compactness. When µ Λ (z) =
µ Λ (x1)+µ Λ (x2)−n+1, M Υ GE (x1, x2; z) is a finite set of oriented points. Denoting by nΥ GE (x1, x2; z)
the algebraic sum of the corresponding orientation signs, we define the homomorphism
P Υ GE : F Λ h (H1) ⊗ F Λ k (H2) → F Λ h+k−n+1 (H1#H2), x1 ⊗ x2 ↦→ �
n Υ GE (x1, x2; z)z.
Then Theorem 3.11 follows from the following:
z∈P Λ (H1#H2)
µ Λ (z)=h+k−n+1
3.13. Proposition. The homomorphism P Υ GE is a chain homotopy between ΥΛ and G ◦ E.
The proof of the above result is contained in section 6.3.
3.6 The homomorphisms C, Ev, and I!
Let us define the Floer homological counterparts of the the homomorphisms
c∗ : Hk(M) → Hk(Λ(M)), ev∗ : Hk(Λ(M)) → Hk(M).
Let f be a smooth Morse function on M, and assume that the vector field −gradf satisfies the
Morse-Smale condition. Let H ∈ C ∞ (Ì×T ∗ M) be a Hamiltonian satisfying (H0) Λ , (H1), (H2).
We shall define two chain maps
C : Mk(f, 〈·, ·〉) → Fk(H, J), Ev : Fk(H, J) → Mk(f, 〈·, ·〉).
Given x ∈ crit(f) and y ∈ PΛ (H), consider the following spaces of maps
�
MC(x, y) = u ∈ C ∞ ([0, +∞[×Ì, T ∗ �
�
M) �∂J,H(u) = 0, π ◦ u(0, t) ≡ q ∈ W u (x) ∀t ∈Ì,
lim u(s, t) = y(t) uniformly in t ∈Ì�
,
s→+∞
and
�
MEv(y, x) = u ∈ C ∞ (] − ∞, 0] ×Ì, T ∗ �
�
M) �∂J,H(u) = 0, u(0, t) ∈ÇM ∀t ∈Ì,
u(0, 0) ∈ W s (x), lim u(s, t) = y(t) uniformly in t ∈Ì�
,
s→−∞
whereÇM denotes the zero section in T ∗ M.
The following result is proved in section 5.10.
3.14. Proposition. For a generic choice of H, MC(x, y) and MEv(y, x) are manifolds with
dimMC(x, y) = m(x) − µ Λ (y), MEv(y, x) = µ Λ (y) − m(x).
These manifolds carry coherent orientations.
If u belongs to MC(x, y) or MEv(y, x), the fact that u(0, ·) takes value either on the fiber of
some point q ∈ M or on the zero section of T ∗M implies that
�H(u(0, ·)) = −
� 1
0
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H(t, u(0, t))dt.