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The orbits of −grad(f1 ⊕f2) are just pairs of orbits of −gradf1 and −gradf2, so the Morse-Smale
condition holds. If we fix orientations for the unstable manifold of each critical point of f1, f2, and
we endow the unstable manifold of each (x1, x2) ∈ crit(f1 ⊕ f2),
W u ((x1, x2)) = W u (x1) × W u (x2),
with the product orientation, we see that the boundary operator in the Morse complex of (f1 ⊕
f2, g1 × g2) is given by
∂(x1, x2) = (∂x1, x2) + (−1) m(x1) (x1, ∂x2), ∀xi ∈ crit(fi), i = 1, 2.
We conclude that the Morse complex of (f1 ⊕ f2, g1 × g2) is the tensor product of the Morse
complexes of (f1, g1) and (f2, g2). So, using the natural homomorphism from the tensor product of
the homology of two chain complexes to the homology of the tensor product of the two complexes,
we obtain the homomorphism
HjM(f1, g1) ⊗ HkM(f2, g2) → Hj+kM(f1 ⊕ f2, g1 × g2). (12)
We claim that this homomorphism corresponds to the exterior product homomorphism
Hj(M1) ⊗ Hk(M2) ×
−→ Hj+k(M1 × M2), (13)
via the isomorphism between Morse homology and singular homology described in section 2.1.
Indeed, the cellular filtration in M1 × M2 can be chosen to be generated by small product
neighborhoods of the critical points,
W ℓ =
�
(x1,x2)∈crit(f1⊕f2)
m(x1)+m(x2)=ℓ
φ 1 ([0, +∞[×U1(x1)) × φ 2 ([0, +∞[×U2(x2)) = �
j+k=ℓ
U j
1 × Uk 2 .
By excision and by the Kunneth theorem, together with the fact that we are dealing with free
Abelian groups, one easily obtains that
WℓW = Hℓ(W ℓ , W ℓ−1 ) ∼ = �
j+k=ℓ
Hj(U j
1 , Uj−1 1 ) ⊗ Hk(U k 2 , Uk−1 2 ),
and that the boundary homomorphism of the cellular filtration W is the tensor product of the
boundary homomorphisms of the cellular filtrations U1 and U2. Passing to homology, we find that
(12) corresponds to the exterior homology product
Hj(U ∞ 1 ) ⊗ Hk(U ∞ 2 ) ×
−→ Hj+k(U ∞ 1 × U ∞ 2 ) = Hj+k(W ∞ ),
by the usual identification of the cellular complex to the Morse complex induced by a choice of
orientations for the unstable manifolds. But since the inclusion U ∞ 1 ֒→ M1 and U ∞ 2 ֒→ M2 are
homotopy equivalences, we conclude that (12) corresponds to (13).
2.4 Intersection products
Let M0 be a Hilbert manifold, and let π : E → M0 be a smooth n-dimensional oriented real
vector bundle over M0. It is easy to describe the Thom isomorphism
τ : Hk(E, E \ M0) ∼ =
−→ Hk−n(M0), α ↦→ τE ∩ α,
in a Morse theoretical way (τE ∈ H n (E, E \ M0) denotes the Thom class of the vector bundle E).
Indeed, let g0 be a complete metric on M0 and let f0 ∈ F(M0, g0) be such that −gradf0
satisfies the Morse-Smale condition. The choice of a Riemannian structure on the vector bundle
E determines a class of product-type metrics g1 on the manifold E, and a smooth function
f1(ξ) := f0(π(ξ)) − |ξ| 2 , ∀ξ ∈ E.
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