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this intersection is a finite set of points, each of which comes with an orientation sign ±1. Denoting
by ne! (x, y) the algebraic sum of these signs, we conclude that the homomorphism
Mk(f, g) → Mk−n(f0, g0), x ↦→ �
y∈crit(f0)
m(y;f0)=k−n
ne! (x, y)y, ∀x ∈ critk(f),
is a chain map of degree −n, and that it induces the Umkehr map e! in homology (by the identification
of the homology of the Morse complex with singular homology described in section 2.1).
Let us conclude this section by describing the Morse theoretical interpretation of the intersection
product in homology. Let M be a finite-dimensional oriented manifold, and consider the
diagonal embedding e : ∆M ֒→ M ×M, which is n-codimensional and co-oriented. The intersection
product is defined by the composition
and it is denoted by
Hj(M) ⊗ Hk(M) ×
−→ Hj+k(M × M) e!
−→ Hj+k−n(∆M) ∼ = Hj+k−n(M),
• : Hj(M) ⊗ Hk(M) −→ Hj+k−n(M).
The above description of e! and the description of the exterior homology product × given in section
2.3 immediately yield the following description of •. Let g1, g2, g3 be complete metrics on M, and
let fi ∈ F(M, gi), i = 1, 2, 3, be such that −gradfi satisfies the Morse-Smale condition. The
non-degeneracy conditions (14) and (15), necessary to represent e!, are now
x ∈ crit(f1) ∩ crit(f2) =⇒ m(x; f1) + m(x; f2) ≥ n, (16)
x ∈ crit(f1) ∩ crit(f2) ∩ crit(f3) =⇒ m(x; f1) + m(x; f2) ≥ m(x; f3) + n. (17)
These conditions are implied for instance by the generic assumption that f1 and f2 do not have
common critical points. We can now perturb the metrics g1, g2, and g3 on M in such a way that
for every triplet xi ∈ crit(fi), i = 1, 2, 3, the intersection
W u ((x1, x2); −grad g1×g2 f1 ⊕ f2) ∩ a(W s (x3; −grad g3 f3)),
a : M → M × M being the map a(p) = (p, p), is transverse in M × M, hence it is an oriented
submanifold of ∆M of dimension m(x1; f1) + m(x2; f2) − m(x3; f3) − n. By compactness and
transversality, when m(x3; f3) = m(x1; f1) + m(x2; f2) − n, this intersection, which can also be
written as
{(p, p) ∈ W u (x1; −gradf1) × W u (x2; −gradf2) | p ∈ W s (x3; −gradf3)} ,
is a finite set of points, each of which comes with an orientation sign ±1. Denoting by n•(x1, x2; x3)
the algebraic sum of these signs, we conclude that the homomorphism
Mj(f1, g1) ⊗ Mk(f2, g2) → Mj+k−n(f3, g3), x1 ⊗ x2 ↦→ �
n•(x1, x2; x3)x3,
x3∈crit(f3)
m(x3;f3)=j+k−n
where x1 ∈ critj(f1), x2 ∈ critk(f2), is a chain map of degree −n from the complex M(f1, g1) ⊗
M(f2, g2) to M(f3, g3), and that it induces the intersection product • in homology (by the identification
of the homology of the Morse complex with singular homology described in section 2.1).
2.5 Lagrangian action functionals, and Morse theoretical interpretation
of the homomorphisms c∗, ev∗, and i!
We are now in a good position to describe the homomorphisms appearing in diagram (8) in a
Morse theoretical way. The first thing to do is to replace the Banach manifolds Ω(M, q0), Λ(M),
and Θ(M) by the Hilbert manifolds
Ω 1 (M, q0) := � γ ∈ Λ 1 � 1 1,2
(M) | γ(0) = q0 , Λ (M) := W (Ì, M),
Θ 1 (M) := � (γ1, γ2) ∈ Λ 1 (M) × Λ 1 (M) | γ1(0) = γ2(0) � ,
16