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2.7. Proposition. The homomorphism
Mk(ËΛ L1 ⊕ËΛ L2 , g1 × g2) → Mk−n(ËΘ L1⊕L2 , g Θ ), γ − ↦→ �
is a chain map, and it induces the Umkehr map e! in homology.
γ + ∈P Θ (L1⊕L2)
m Θ (γ + )=k−n
ne! (γ− , γ + )γ + ,
By composing this homomorphism with the Morse theoretical version of the exterior homology
product described in section 2.3, that is the isomorphism
we obtain the homomorphism
Mj(ËΛ L1 , g1) ⊗ Mh(ËΛ L2 , g2) → Mj+h(ËΛ L1 ⊕ËΛ L2 , g1 × g2),
M! : Mj(ËΛ L1 , g1) ⊗ Mh(ËΛ L2 , g2) → Mj+h−n(ËΘ L1⊕L2 , g Θ ).
Let us describe the homomorphism
Γ∗ : Hk(Θ 1 (M)) → Hk(Λ 1 (M)),
induced by the concatenation map Γ. Let L1, L2 be Lagrangians such that L(1, ·) = L2(0, ·) with
all the time derivatives, satisfying (L1), (L2). We assume that (L1, L2) satisfies (L0) Θ and L1#L2
satisfies (L0) Λ . We would like to apply the results of section 2.2 to the functionalsËΘ L1⊕L2 on
Θ 1 (M) andËΛ L1#L2 on Λ 1 (M). The map Γ : Θ 1 (M) → Λ 1 (M) is nowhere a submersion, so
condition (9) for the triplet (Γ,ËΘ L1⊕L2 ,ËΛ L1#L2 ) requires that Γ(Θ 1 (M)) does not contain critical
points ofËΛ L1#L2 , that is
γ ∈ P Λ (L1#L2) =⇒ γ(1/2) �= γ(0). (26)
Assuming (26), conditions (9) and (10) are automatically fulfilled. Therefore, the discussion of
section 2.2 implies that we can find complete metrics g Θ and g Λ on on Θ 1 (M) and Λ 1 (M) such
that −grad g ΘËΘ L1⊕L2 and −grad g ΛËΛ L1#L2 satisfy the Palais-Smale and the Morse-Smale condition,
and that the restriction of Γ to the unstable manifold
W u (γ − ; −grad g ΘËΘ 1
L1⊕L2 )
of every critical point γ − = (γ − 1 , γ− 2 ) ∈ PΘ (L1 ⊕ L2) is transverse to the stable manifold
W s (γ + ; −grad g ΛËΛ L1#L2 )
of every critical point γ + ∈ P Λ (L1#L2). Fix arbitrary orientations for the unstable manifolds of
every critical point ofËΘ L1⊕L2 andËΛ L1#L2 . When m Λ (γ + ) = m Θ (γ − ), the intersection
� (α1, α2) ∈ W u (γ − ; −gradËΘ L1⊕L2 ) | Γ(α1, α2) ∈ W s (γ + ; −gradËΛ L1#L2 ) � ,
is a finite set of oriented points. If we denote by nΓ(γ − , γ + ) the algebraic sum of these orientation
signs, we have the following:
2.8. Proposition. The homomorphism
MΓ : Mk(ËΘ L1⊕L2 , g Θ ) → Mk(ËΛ L1#L2 , g Λ ), γ − ↦→ �
γ + ∈P Λ (L1#L2)
m Λ (γ)=k
nΓ(γ − , γ + )γ + ,
is a chain map, and it induces the homomorphism Γ∗ : Hk(Θ 1 (M)) → Hk(Λ 1 (M)) in homology.
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