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transverse to the stable manifold of each γ2 ∈ P Ω (L) in Λ 1 (M). When m Ω (γ2) = m Λ (γ1) − n,
the set
W u (γ1; −gradËΛ L ) ∩ W s (γ2; −gradËΩ L )
consists of finitely many oriented points, which determine the integer ni! (γ1, γ2). These integers
are the coefficients of a chain map
Mi! : M∗(ËΛ L , g Λ ) → M∗−n(ËΩ L , g Ω ),
which in homology induces the homomorphism i!.
2.6 Morse theoretical interpretation of the Pontrjagin product
In the last section we have described the vertical arrows of diagram (8), as well as a preferred left
inverse of the top-right vertical arrow. The top horizontal arrow has already been described at
the end of section 2.4. There remains to describe the middle and the bottom horizontal arrows,
that is the loop product and the Pontrjagin product. This section is devoted to the description
of the latter product. The following Propositions 2.5, 2.7 and 2.8 are consequences of the general
statements in Sections 2.1–2.4
Given two Lagrangians L1, L2 ∈ C∞ ([0, 1] × TM) such that L(1, ·) = L2(0, ·) with all the time
derivatives, we define the Lagrangian L1#L2 ∈ C∞ ([0, 1] × TM) as
�
2L1(2t, q, v/2) if 0 ≤ t ≤ 1/2,
L1#L2(t, q, v) =
(23)
2L2(2t − 1, q, v/2) if 1/2 ≤ t ≤ 1.
The curve γ : [0, 1] → M is a solution of the Lagrangian equation (18) with L = L1#L2 if and
only if the rescaled curves t ↦→ γ(t/2) and t ↦→ γ((t+1)/2) solve the corresponding equation given
by the Lagrangians L1 and L2, on [0, 1].
In view of the results of section 2.3, we wish to consider the functionalËΩ L1 ⊕ËΩ L2 on Ω 1 (M, q0)×
Ω 1 (M, q0), ËΩ L1 ⊕ËΩ L2 (γ1, γ2) =ËΩ L1 (γ1) +ËΩ L2 (γ2),
and the functionalËΩ L1#L2 on Ω 1 (M, q0). The concatenation map
Γ : Ω 1 (M, q0) × Ω 1 (M, q0) → Ω 1 (M, q0)
is nowhere a submersion, so condition (9) for the triplet (Γ,ËΩ L1 ⊕ËΩ L2 ,ËΩ L1#L2 ) requires that the
image of Γ does not meet the critical set ofËΩ L1#L2 , that is
γ ∈ P Ω (L1#L2) =⇒ γ(1/2) �= q0. (24)
Notice that (24) allows L1 and L2 to be equal, and actually it allows them to be also autonomous
(however, it implies that q0 is not a stationary solution, so they cannot be the Lagrangian associated
to a geodesic flow).
Assuming (24), condition (10) is automatically fulfilled. Moreover, if g1, g2 are metrics on
Ω 1 (M, q0), we have that for every γ1 ∈ P Ω (L1), γ2 ∈ P Ω (L2),
Γ(W u ((γ1, γ2); −grad g1×g2ËΩ L1 ⊕ËΩ L2 )) ∩ crit(ËΩ L1#L2 ) = ∅.
By Remark 2.3, there is no need to perturb the metric g1 ×g2 on Ω 1 (M, q0) ×Ω 1 (M, q0) to achieve
transversality, and we arrive at the following description of the Pontrjagin product.
Let L1, L2 be Lagrangians such that L(1, ·) = L2(0, ·) with all the time derivatives, satisfying
(L0) Ω , (L1), (L2), and (24), such that also L1#L2 satisfies (L0) Ω . Let g1, g2, g be complete
metrics on Ω 1 (M, q0) such that −grad g1ËΩ L1 , −grad g2ËΩ L2 , −grad gËΩ L1#L2 satisfy the Palais-Smale
and the Morse-Smale condition. Fix an arbitrary orientation for the unstable manifolds of each
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