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and converging to Hamiltonian orbits x− and x + for s → −∞ and s → +∞. The results of section
5 imply that for a generic choice of the Hamiltonian H the space M is a manifold of dimension
dimM = µ Q0 (x − ) − µ Qk
k�
+
(x ) − (dim Qj−1 − dimQj−1 ∩ Qj).
Here µ Q0 (x − ) and µ Qk (x + ) are the Maslov indices of the Hamiltonian orbits x − and x + , with
boundary conditions (x − (0), −x − (1)) ∈ N ∗ Q0, (x + (0), −x + (1)) ∈ N ∗ Qk, suitably shifted so that
in the case of a fiberwise convex Hamiltonian they coincide with the Morse indices of the corresponding
critical points γ − and γ + of the Lagrangian action functional on the spaces of paths
satisfying (γ − (0), γ − (1)) ∈ Q0 and (γ + (0), γ + (1)) ∈ Qk, respectively. Similar formulas hold for
problems on the half-strip.
Cobordism arguments. The main results of this paper always reduce to the fact that certain
diagrams involving homomorphism defined either in a Floer or in a Morse theoretical way should
commute. The proof of such a commutativity is based on cobordism arguments, saying that a given
solution of a certain Problem 1 can be “continued” by a unique one-parameter family of solutions of
a certain Problem 2, and that this family of solutions converges to a solution of a certain Problem
3. In many situations such a statement can be proved by the classical gluing argument in Floer
theory: One finds the one-parameter family of solutions of Problem 2 by using the given solution
of Problem 1 to construct an approximate solution, to be used as the starting point of a Newton
iteration scheme which converges to a true solution. When this is the case, we just refer to the
literature. However, we encounter three situations in which the standard arguments do not apply,
one reason being that we face a Problem 2 involving a Riemann surface whose conformal structure
is varying with the parameter: this occurs when proving that the pair-of-pants product factorizes
through the figure-8 Floer homology (subsection 3.5), that the Pontrjagin product corresponds
to the triangle product (subsection 4.2), and that the homomorphism e! ◦ × corresponds to its
Floer homological counterpart (subsection 4.4). We manage to reduce the former two statements
to the standard implicit function theorem (see subsections 6.3 and 6.4). The proof of the latter
statement is more involved, because in this case the solution of Problem 2 we are looking for
cannot be expected to be even C 0 -close to the solution of Problem 1 we start with. We overcome
this difficulty by the following algebraic observation: In order to prove that two chain maps
ϕ, ψ : C → C ′ are chain homotopic, it suffices to find a chain homotopy between the chain maps
ϕ ⊗ ψ and ψ ⊗ ϕ, and to find an element ǫ ∈ C0 and a chain map δ from the complex C ′ to
the trivial complex (�, 0) such that δ(ϕ(ǫ)) = δ(ψ(ǫ)) (see Lemma 4.6 below). In our situation,
the chain homotopy between ϕ ⊗ ψ and ψ ⊗ ϕ is easier to find, by using a localization argument
and the implicit function theorem (see subsection 6.5). This argument is somehow reminiscent of
an alternative way suggested by Hofer to prove standard gluing results in Floer homology. The
construction of the element ǫ and of the chain map δ is presented in subsection 4.4, together
with the proof of the required algebraic identity. This is done by considering special Hamiltonian
systems, having a hyperbolic equilibrium point.
The main results of this paper were announced in [AS06a]. Related results concerning the
equivariant loop product and its interpretation in the symplectic field theory of unitary cotangent
bundles have been announced in [CL07].
Outlook. An immediate question raised by the main result of this paper whether other product
structures in classical homoloy theories for path and loop spaces can also be constructed naturally
on chain level in Floer theory. The answer appears to be affirmative for all so far considered
structures. In a following paper [AS08] we give the explicit construction of the cup-product in
all path and loop space cases, a direct proof of the Hopf algebra structure on HF Ω (T ∗ M) based
on Floer chain complex morphisms, and we also construct the counterpart of the very recently
introduced product structure on relative cohomology
j=1
H k (Λ(M), M) × H l (Λ(M), M) → H k+l+n−1 (Λ(M), M)
6