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Let A0, A1, A2 be non-degenerate paths with respect to W0, Wk, and Wk+k ′, respectively. Then
orientations oW (A0, A1) and oW ′(A1, A2) of det(DW (A0, A1)) and det(DW ′(A1, A2)), respectively,
determine in a canonical way a glued orientation
oW (A0, A1)#oW ′(A1, A2)
of det(DW #W ′(A0, A2)). The construction is analogous to the one described in [FH93, section
3]. This way of gluing orientations is associative. A coherent orientation is a set of orientations
oW (A − , A + ) for each choice of compatible data such that
oW #W ′(A0, A1) = oW (A0, A1)#oW ′(A1, A2),
whenever the latter glued orientation is well-defined. The proof of the existence of a coherent
orientation is analogous to the proof of Theorem 12 in [FH93].
The choice of such a coherent orientation in this linear setting determines orientations for all
the nonlinear objects we are interested in, and such orientations are compatible with gluing. As
mentioned above, the fact that we are dealing with the cotangent bundle of an oriented manifold
makes the step from the linear setting to the nonlinear one easier. The reason is that we can
fix once for all special symplectic trivializations of the bundle x ∗ (TT ∗ M), for every solution x of
our Hamiltonian problem. In fact, one starts by fixing an orthogonal and orientation preserving
trivialization of (π ◦x) ∗ (TM), and then considers the induced unitary trivialization of x ∗ (TT ∗ M).
Let u be an element in some space M(x, y), consisting of the solutions of a Floer equation on
the strip Σ which are asymptotic to two Hamiltonian orbits x and y and satisfy suitable jumping
co-normal boundary conditions. Then we can find a unitary trivialization of u ∗ (TT ∗ M) which
converges to the given unitary trivializations of x ∗ (TT ∗ M) and y ∗ (TT ∗ M). We may use such a
trivialization to linearize the problem, producing a Fredholm operator in DS ,W (A − , A + ). Here
A − , A + are determined by the fixed unitray trivializations of x ∗ (TT ∗ M) and y ∗ (TT ∗ M). The
orientation of the determinant bundle over DS ,W (A − , A + ) then induced an orientation of the
tangent space of M(x, y) at u, that is an orientation of M(x, y). See [AS06b, section 1.4] for more
details.
When the manifold M is not orientable, one cannot fix once for all trivializations along the
Hamiltonian orbits, and the construction of coherent orientations requires understanding the effect
of changing the trivialization, as in [FH93, Lemma 15]. The Floer complex and the pair-of-pants
product are still well-defined over integer coefficients, whereas the Chas-Sullivan loop product
requires�2 coefficients.
5.10 Linearization
In this section we recall the nonlinear setting which allows to see the various spaces of solutions of
the Floer equation considered in this paper as zeroes of sections of Banach bundles. By showing
that the fiberwise derivatives of such sections are conjugated to linear perturbed Cauchy-Riemann
operators on a strip with jumping conormal boundary conditions, we prove that these spaces of
solutions are generically manifolds, and we compute their dimension. We treat with some details
the case of M Ω Υ , the space of solutions of the Floer equation on the holomorphic triangle, defining
the triangle product on Floer homology. The functional setting for the other spaces of solutions
is similar, so in the other cases we mainly focus on the dimension computation.
The space M Ω Υ . Let us consider the model case of M Ω Υ (x1, x2; y), where x1 ∈ PΩ (H1), x2 ∈
PΩ (H2), and y ∈ PΩ (H1#H2) (see section 3.3). This is a space of solutions of the Floer equation
on the Riemann surface with boundary ΣΩ Υ , described as a strip with a slit in section 3.2.
Let us fix some p ∈]2, +∞[, and let us consider the space W Ω Υ = W Ω Υ (x1, x2; y) of maps
u : ΣΩ Υ → T ∗M mapping the boundary of ΣΩ Υ into T ∗ q0M, which are of Sobolev class W 1,p on
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