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5.9 Coherent orientations
As noticed in [AS06b, section 1.4], the problem of giving coherent orientations for the spaces of
maps arising in Floer homology on cotangent bundles is somehow simpler than in the case of a
general symplectic manifolds, treated in [FH93]. This fact remains true if we deal with Cauchy-
Riemann type operators on strips and half-strips with jumping conormal boundary conditions.
We briefly discuss this issue in the general case of nonlocal boundary conditions on the strip, the
case of the half-strip being similar (see [AS06b, section 3.2]).
We recall that the space Fred(E, F) of Fredholm linear operators from the real Banach space E
to the real Banach space F is the base space of a smooth real non-trivial line-bundle det(Fred(E, F)),
with fibers
det(A) := Λ max (ker A) ⊗ (Λ max (cokerA)) ∗ , ∀A ∈ Fred(E, F),
where Λmax (V ) denotes the component of top degree in the exterior algebra of the finite-dimensional
vector space V (see [Qui85]).
Let us recall the setting from section 5.8. We fix the data k ≥ 0, S = {s1, . . . , sk, s1+i, . . ., sk+
i}, with s1 < · · · < sk, and W = (W0, . . . , Wk), where W0, . . . , Wk are linear subspaces ofÊn ×Ên ,
such that Wj−1 is partially orthogonal to Wj, for j = 1, . . .,k. Let A ± : [0, 1] → Sym(Ên ) be
continuous paths of symmetric matrices such that the linear problems
� w ′ (t) = iA − (t)w(t),
(w(0), Cw(1)) ∈ N ∗ W0,
� w ′ (t) = iA + (t)w(t),
(w(0), Cw(1)) ∈ N ∗ Wk,
have only the trivial solution w = 0. Such paths are refereed to as non-degenerate paths (with
respect to W0 and Wk, respectively). Fix some p > 1, and let DS ,W (A − , A + ) be the space of
operators of the form
∂A : X 1,p
S ,W (Σ,�n ) → X p
S (Σ,�n ), u ↦→ ∂u + Au,
where A ∈ C0 ± (Ê×[0, 1], L(Ê2n
,Ê2n )) is such that A(±∞, t) = A (t) for every t ∈ [0, 1]. By
Theorem 5.23, DS ,W (A− , A + ) is a subset of Fred(X 1,p
S ,W (Σ,�n p
), XS (Σ,�n )). It is actually a
convex subset, so the restriction of the determinant bundle to DS ,W (A− , A + ) - that we denote
by det(DS ,W (A− , A + )) - is trivial.
Let S be the family of all subsets of Σ consisting of exactly k pairs of opposite boundary
points. It is a k-dimensional manifold, diffeomorphic to an open subsets ofÊk . An orientation
of det(DS ,W (A− , A + )) for a given S in S uniquely determines an orientation for all choices of
S ′ ∈ S. Indeed, the disjoint unions
�
X 1,p
S ,W (Σ,�n
�
), X p
S (Σ,�n
),
S ∈S
define locally trivial Banach bundles over S, and the operators ∂A define a Fredholm bundlemorphism
between them. Since S is connected and simply connected, an orientation of the
determinant space of this operator between the fibers of a given point S induces an orientation
of the determinant spaces of the operators over each S ′ ∈ S.
The space of all Fredholm bundle-morphisms between the above Banach bundles induced by
operators of the form ∂A with fixed asymptotic paths A− and A + is denoted by DW (A− , A + ). An
orientation of the determinant bundle over this space of Fredholm bundle-morphisms is denoted
by oW (A− , A + ).
Let W = (W0, . . . , Wk), W ′ = (Wk, . . . , Wk+k ′) be vectors consisting of consecutively partially
orthogonal linear subspaces ofÊn ×Ên , and set
S ∈S
W #W ′ := (W0, . . .,Wk+k ′).
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