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2.2 Functoriality
Let (M1, g1) and (M2, g2) be complete Riemannian Hilbert manifolds, and let f1 ∈ F(M1, g1),
f2 ∈ F(M2, g2) be such that −gradf1 and −gradf2 satisfy the Morse-Smale condition. Denote
by φ 1 and φ 2 the corresponding negative gradient flows.
Let ϕ : M1 → M2 be a smooth map. We assume that
each y ∈ crit(f2) is a regular value of ϕ; (9)
x ∈ crit(f1), ϕ(x) ∈ crit(f2) =⇒ m(x; f1) ≥ m(ϕ(x); f2). (10)
The set of critical points of f2 is discrete, and in many cases (for instance, if ϕ is a Fredholm
map) the set of regular values of ϕ is generic (i.e. a countable intersection of open dense sets, by
Sard-Smale theorem [Sma65]). In such a situation, condition (9) can be achieved by arbitrary
small (in several senses) perturbations of ϕ or of f2. Also condition (10) can be achieved by an
arbitrary small perturbation of ϕ or of f2, simply by requiring that the image of the set of critical
points of f1 by ϕ does not meet the set of critical point of f2.
By (9) and (10), up to perturbing the metrics g1 and g2, we may assume that
∀x ∈ crit(f1), ∀y ∈ crit(f2), ϕ| W u (x;−grad f1) is transverse to W s (y; −gradf2). (11)
Indeed, by (9) and (10) one can perturb g1 in such a way that if p ∈ W u (x; −gradf1) and ϕ(p) is
a critical point of f2 then rankDϕ(p)| TpW u (x) ≥ m(ϕ(p); f2). The possibility of perturbing g2 so
that (11) holds is now a consequence of the following fact: if W is a finite dimensional manifold
and ψ : W → M2 is a smooth map such that for every p ∈ W with ψ(p) ∈ crit(f2) there holds
rankDψ(p) ≥ m(ϕ(p); f2), then the set of metrics g2 on M2 such that the map ψ is transverse
to the stable manifold of every critical point of f2 is generic in the set of all metrics, with many
reasonable topologies 5 .
The transversality condition (11) ensures that if x ∈ crit(f1) and y ∈ crit(f2), then
W(x, y) := W u (x; −gradf1) ∩ ϕ −1 (W s (y; −gradf2))
is a submanifold of dimension m(x; f1) − m(y; f2). If W u (x; −gradf1) is oriented and the normal
bundle of W s (y; −gradf2) in M2 is oriented, the manifold W(x, y) carries a canonical orientation.
In particular, if m(x; f1) = m(y; f2), W(x, y) is a discrete set, each of whose point carries an
orientation sign ±1. The transversality condition (11) and the fact that W u (x; −gradf1) has
compact closure in M1 imply that the discrete set W(x, y) is also compact, so it is a finite set and
we denote by nϕ(x, y) ∈�the algebraic sum of the corresponding orientation signs. We can then
define the homomorphism
Mkϕ : Mk(f1, g1) → Mk(f2, g2), Mkϕx = �
nϕ(x, y)y,
y∈critk(f2)
for every x ∈ critk(f1).
We claim that M∗ϕ is a chain map from the Morse complex of (f1, g1) to the Morse complex of
(f2, g2), and that the corresponding homomorphism in homology coincides - via the isomorphism
described in section 2.1 - with the homomorphism ϕ∗ : H∗(M1) → H∗(M2).
Indeed, let us fix small open neighborhoods Ui(x), i = 1, 2, for each critical point x ∈ crit(fi),
such that the sequence of open sets
�
= φ i ([0, +∞[×Ui(x)), k ∈Æ, i = 1, 2,
U k i
x∈crit(fi)
m(x;fi)≤k
5Here one is interested in finding perturbations of g1 and g2 which are so small that −gradf1 and −grad f2
still satisfy the Morse-Smale condition and the corresponding Morse complexes are unaffected (for instance, because
the perturbed flows are topologically conjugated to the original ones). This can be done by considering
Banach spaces of Ck perturbations of g1 and g2 endowed with a Whitney norm, that is something like
�h� = P
1≤j≤k supp∈M ǫ(p)|D
i jh(p)|, for a suitable positive function ǫ : Mi →]0,+∞[. We shall not specify
these topologies any further, and we shall always assume that the perturbations are so small that the good properties
of the original metrics are preserved.
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