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1.1. Remark. The loop product was defined by Chas and Sullivan in [ChS99], by using intersection
theory for transversal chains. The definition we use here is due to Cohen and Jones [CJ02].
See also [ChS04, Sul03, BCR06, Coh06, CHV06, Ram06, Sul07] for more information and for
other interpretations of this product.
2 Morse theoretical descriptions
2.1 The Morse complex
Let us recall the construction of the Morse complex for functions defined on an infinite-dimensional
Hilbert manifold. See [AM06] for detailed proofs. Let M be a (possibly infinite-dimensional)
Hilbert manifold, and let g be a complete Riemannian metric on M. Let F(M, g) be the set of
C 2 functions f : M →Êsuch that:
(f1) f is bounded below;
(f2) each critical point of f is non-degenerate and has finite Morse index;
(f3) f satisfies the Palais-Smale condition with respect to g (that is, every sequence (pn) such
that (f(pn)) is bounded and �Df(pn)� → 0 has a converging subsequence).
Here � · � is the norm on T ∗M induced by the metric g.
Let f ∈ F(M, g). We denote by crit(f) the set of critical points of f, and by critk(f) the
set of critical points of Morse index m(x) = k. Let φ be the local flow determined by the vector
field −gradf (our assumptions imply that the domain of such a flow contains [0, +∞[×M, and
that each orbit converges to a critical point of f for t → +∞). The stable (respectively unstable)
manifold of a critical point x,
W s (x) :=
�
� �
p ∈ M | lim φ(t, p) = x resp. W
t→+∞ u (x) :=
�
�� p ∈ M | lim φ(t, p) = x
t→−∞
is a submanifold of codimension (resp. dimension) m(x). Actually, it is the image of an embedding
of V + (Hess f(x)) (resp. V − (Hess f(x))), the positive (resp. negative) eigenspace of the Hessian of
f at x. Our assumptions imply that the closure of W u (x) in M is compact.
The vector field −gradf is said to satisfy the Morse-Smale condition if for every x, y ∈ crit(f)
the intersection W u (x) ∩W s (y) is transverse. The Morse-Smale condition can be achieved by perturbing
the metric g. Such perturbations can be chosen to be arbitrarily small in many reasonable
senses, in particular the perturbed metric can be chosen to be equivalent to the original one, so
that f satisfies the Palais-Smale condition also with respect to the new metric.
Let us assume that −gradf satisfies the Morse-Smale condition. Then we can choose an open
neighborhood U(x) of each critical point x so small that the increasing sequence of positively
invariant open sets
U k := �
φ([0, +∞[×U(x)),
x∈crit(f)
m(x)≤k
is a cellular filtration 3 of U ∞ := �
k∈ÆU k , meaning that Hj(U k , U k−1 ) = 0 if j �= k. Actually,
Hk(U k , U k−1 ) is isomorphic to the free Abelian group generated by the critical points of Morse
index k, which we denote by Mk(f). Indeed, Hk(U k , U k−1 ) is generated by the relative homology
3 Actually here one needs a stronger transversality condition, namely that every critical point x is not a cluster
point for the union of all the unstable manifolds of critical points of Morse index not exceeding m(x), other than
x. This assumption follows from the standard Morse-Smale condition provided that there are finitely many critical
point with any given Morse index. Since the latter condition automatically holds on sublevels of f, the Morse
complex can be defined under the standard Morse-Smale assumption by a direct limit argument on sublevels. See
[AM06] for full details.
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