Quilted Floer trajectories with constant components

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by  Katrin Wehrheim

QUILTED FLOER TRAJECTORIES WITH CONSTANT COMPONENTS
KATRINWEHRHEIMANDCHRIS T. WOODWARD
1
1
0 Abstract. Wefilla gap in theproof of thetransversality result forquilted Floer trajec-
2 toriesin[6]byaddressingtrajectoriesforwhichsomebutnotallcomponentsareconstant.
n Namely we show that for generic sets of split Hamiltonian perturbations and split almost
a complexstructures,themodulispacesofparametrizedquiltedFloertrajectoriesofagiven
J indexaresmoothofexpecteddimension. AnadditionalbenefitofthegenericsplitHamil-

9 tonianperturbationsisthattheyperturbthegivencyclicLagrangiancorrespondencesuch
1 that any geometric composition of its factors is transverseand henceimmersed.

]
G
1. Introduction
S
.
h Quilted Floer homology is defined for a cyclic generalized Lagrangian correspondence
t L, that is, a sequence of symplectic manifolds M ,M ,...,M ,M with M = M for
a 0 1 r r+1 0 r+1
m r ≥ 0, and a sequence of Lagrangian correspondences
[ L ⊂ M−×M , L ⊂ M−×M , ..., L ⊂ M−×M .
01 0 1 12 1 2 r(r+1) r r+1
1
For the purpose of transversality arguments we do not need any monotonicity assumptions
v
0 asin[6], sothroughoutwemerely assumethatallLagrangian correspondencesarecompact.
7 Quilted Floer homology HF(L) can be defined as the standard Floer homology of a pair of
7 Lagrangians in the product manifold M−×M ×M−×...×M , given by products of the
3 0 1 2 r
1. Li(i+1). (Forevenroneaddsadiagonaltothesequencebeforemakingthisconstruction.) As
such,HF(L)dependsonthechoiceofaHamiltonianfunctionandalmostcomplexstructure
0
1 on this product manifold, which generically would not be of split form - i.e. induced by a
1 tuple of Hamiltonian functions and almost complex structures on each symplectic manifold
:
v M . The quilted definition of HF(L) in [6] on the one hand generalizes this construction
j
Xi by allowing a choice of widths δ = (δj > 0)j=0,...,r of the strips mapping to each Mj. On
the other hand, we claim in [6] that the quilted Floer complex can be constructed (in
r
a particular transversality can be achieved) for Hamiltonians and almost complex structures
of split type. That is, we restrict our choice of perturbation data to a tuple of Hamiltonian
functions and a tuple of almost complex structures in the complete metric spaces
H (δ) := ⊕r C∞([0,δ ]×M ,R), J (δ):= ⊕r C∞([0,δ ],J(M ,ω )),
t j=0 j j t j=0 j j j
where J(M ,ω ) is the space of smooth ω -compatible almost complex structures on M .
j j j j
While this split form is not necessary for the strip shrinking analysis in [8], it is par-
ticularly helpful for constructing relative invariants (such as the functor associated to a
correspondence in [9]) from more complicated quilted surfaces, which cannot be inter-
preted as single surface mapping to a product manifold. Unfortunately, the transversal-
ity proof in [6] for the quilted Floer trajectory spaces for generic split perturbation data
H ∈ H (δ) and J ∈ J (δ) has a significant gap: It fails to explicitly discuss trajectories
t t
u= (u : R×[0,δ ]→ M ) for which somebutnot all components areconstant. This
j j j j=0,...,r
intermediate situation is not an easy combination of the two extreme cases (all components
nonconstant, or all components constant) as we seem to claim in [6].
1
2 KATRINWEHRHEIMANDCHRIST.WOODWARD
Results. In Theorem 3.2 below we complete the proof of the transversality claimed in [6]
by working with a more specific set of generic split Hamiltonian perturbations which may
be of independent interest. In Theorem 2.3 and Corollary 2.4 we find a dense open subset
of H (δ) for any given cyclic Lagrangian correspondence such that, after perturbation by
t
one of those split Hamiltonian diffeomorphisms, any geometric composition of its factors is
transverseandhenceimmersed. StartingfromsuchaHamiltonian perturbation,weobserve
that quilted Floer trajectories with constant components induce quilted Floer trajectories
for a shorter cyclic generalized Lagrangian correspondence, given by a localized version of
geometric composition across the constant strips. Using this point of view, we are able to
find generic sets of split almost complex structures for which quilted Floer trajectories with
constant components are regular, as well. In fact, we show that quilted Floer trajectories
with constant components are very rare as summarized in Remark 3.3 and sketched below.
Idea of Proof. A key role in the proof is played by certain families of isotropic subspaces
which arise in the proof of transversality for the universal moduli space of almost complex
structures and Floer trajectories. The elements of the cokernel of the linearized opera-
tor of the universal moduli space are tuples of −J -holomorphic sections η of u∗TM for
i i i i
i = 0,...,r with Lagrangian seam conditions determined by the tangent spaces of L .
i(i+1)
Ignoring Hamiltonian perturbations for simplicity, the problem of constant components oc-
curs for example when some u is constant (with value say x ∈ M ) but the adjacent
j j j
components u ,u are non-constant. Then variations in the almost complex structures
j−1 j+1
prove vanishing of η ,η , and hence η : R×[0,δ ] → T M is −J -holomorphic with
j−1 j+1 j j xj j j
boundary conditions in
Λ (s):= Pr T L ∩({0}×T M ) ⊂ T M ,
j TxjMj (uj−1(δj−1,s),xj) (j−1)j xj j xj j
Λ′j(s):= PrTxjMj(cid:0)T(xj,uj+1(0,s))Lj(j+1)∩(TxjMj ×{0}) (cid:1) ⊂ TxjMj.
ThespacesΛj(s),Λ′j(s)areis(cid:0)otropicspacesvaryingwiths ∈R,des(cid:1)pitethefactthatuj ≡ xj
is constant. We can now proceed differently in three non-exclusive cases.
(a) The easiest case is to assume that Λ (s),Λ′(s) are s-independent. We may then
j j
enlarge these isotropic spaces to constant Lagrangian subspaces and deduce that η
j
lies in the kernel of an operator ∂ +A, where A is an s-independent self-adjoint
s
operator and invertible (since by choice of H the generators of the Floer complex
are cut out transversally). We then deduce vanishing of η from the general fact
j
that operators of this form ∂ +A are isomorphisms.
s
(b) An intermediate case occurs when Λ (s) or Λ′(s) fails to be Lagrangian (i.e. have
j j
maximal dimension) for some s ∈ R. For example, if L is the graph of a
(j−1)j
symplectomorphisms, then the intersection Λ is trivial. We show that this case of a
j
quilted Floer trajectory with constant component does not occur for generic (J ) .
i i6=j
(c) The most difficult case occurs when Λ (s) and Λ′(s) are non-constant families of
j j
Lagrangian subspaces. We show that for generic H the locus on which such varying
Lagrangiansubspacesarepossibleisofpositivecodimensioninthespaceofboundary
values(u (δ ,s),u (0,s)). Thenweagain excludethiscaseforgeneric(J ) .
j−1 j−1 j+1 i i6=j
Thus, for generic Hamiltonian perturbations H and almost complex structures J we in
fact show a splitting property for any quilted Floer trajectory with constant components,
namely along the seam (u (s,δ ),u (s,0)) ∈L we have TL = Λ ×Λ , where
j j j+1 j(j+1) j(j+1) j j+1
Λ ⊂ TM is a constant Lagrangian subspace given as above, and Λ is the s-dependent
j j j+1
projection of TL ∩({0}×TM ). For a precise statement see Remark 3.3.
j(j+1) j+1
QUILTED FLOER TRAJECTORIES WITH CONSTANT COMPONENTS 3
Thearguments in case (b) and (c) crucially rely on the following interpretation of quilted
Floer trajectories with constant components as quilted Floer trajectories for a generalized
Lagrangian correspondence obtained by a local version of geometric composition. If u =
(u ,...,u )is a solution with u ≡ x as above, then (u ,...,u ,u ,...,u ) is a quilted
0 r j j 0 j−1 j+1 r
Floer trajectory for the generalized correspondence (L ,...,L ◦L ,...,L ).
01 (j−1)j j(j+1) r(r+1)
We show in Theorem 2.3 that, after a generic Hamiltonian perturbation of L, any geomet-
ric composition L ◦L is an immersed Lagrangian correspondence. It becomes
(j−1)j j(j+1)
embedded if we restrict to values in M near x . Hence (u ,...,u ,u ,...,u ) can be
j j 0 j−1 j+1 r
viewed as quilted Floer trajectory for a smooth generalized Lagrangian correspondence.
We showed in [8] that transversality for this composed correspondence implies transver-
sality for the original correspondence for sufficiently small widths δ > 0. Here we extend
j
this transversality to solutions with constant u for arbitrary δ > 0 and generic perturba-
j j
tion data H,J.
Alternative approaches. It is perhaps worth remarking that all of the correspondences
intended as applications in [6, 10, 11] fit into the easiest case (a) described above since
these Lagrangian correspondences L ⊂ M− × M are quasisplit in the following sense:
01 0 1
The intersection (T M × {0}) ∩ T L is independent of x and the intersection
x0 0 (x0,x1) 01 1
({0}×T M )∩T L isindependentofx . ExamplesaresplitcorrespondencesL ×L ,
x1 1 (x0,x1) 01 0 0 1
graphs of symplectomorphisms, correspondences arising from fibered coisotropics, and the
embedded geometric composition of any two quasisplit correspondences. If all Lagrangian
correspondences are quasisplit then the simple argument in case (a) above completes the
transversality argument for the universal moduli space in [6].
Note however that one can easily construct Lagrangian correspondences that are not
quasisplit by applying a nonsplit Hamiltonian diffeomorphism of M−×M to a split corre-
0 1
spondence L ×L .
0 1
Anotherpossibilityforachievingtransversality atquiltedFloertrajectories withconstant
components is to introduce nonsplit perturbations as in [4] and [3]. However, in order to
implement this perturbation scheme for more general relative quilt invariants, one would
have to replace each seam with seam condition L ⊂ M × M by a novel triple seam
ij i j
condition pairing the two patches in M and M via a diagonal with one boundaryof a strip
i j
in M ×M , whose other boundary has boundary conditions in L . In that setup we may
i j ij
use non-split perturbations on the strip.
We thank Max Lipyanski for pointing out the question of constant components.
2. Hamiltonian perturbations of generalized Lagrangian correspondences
Given a cyclic generalized Lagrangian correspondence L = (L ) , widths δ =
j(j+1) j=0,...,r
(δ >0) , and a tuple of Hamiltonian functions H = (H ) ∈ H (δ), the genera-
j j=0,...,r j j=0,...,r t
tors of the quilted Floer complex are tuples of Hamiltonian chords,
x˙ (t) = X (x (t)),
I(L,H) := x = x : [0,δ ]→ M j Hj j .
( j j j j=0,...,r (cid:12)(xj(δj),xj+1(0)) ∈ Lj(j+1))
(cid:12)
(cid:0) (cid:1) (cid:12)
(cid:12)
(cid:12)
4 KATRINWEHRHEIMANDCHRIST.WOODWARD
They are canonically identified, via x 7→ (x (δ ),x (0),x (δ ),...,x (δ ),x (0)), with the
0 0 1 1 1 r r 0
fiber product
× L × L ...× L
φH0 01 φH1 12 φHr r(r+1)
δ0 δ1 δr
:= L(cid:0) ×L ×...×L ∩ gra(cid:1)ph(φH1)×graph(φH2)×...×graph(φH0) T,
01 12 r(r+1) δ1 δ2 δ0
where φ(cid:0)Hj is the time δ Hamilton(cid:1)ian(cid:0)flow of H and (...)T denotes the exchange of(cid:1)factors
δj j j
M ×...×M ×M → M ×M ×...×M . Inthissettingweprovedin[6]thatHamiltonians
1 0 0 0 1 0
of split type suffice to achieve transversality for the generators. We now strengthen this
result to achieve transversality for all partial fiber products.
Convention 2.1. Here and in the following we use indices j ∈ N modulo r +1. A pair
of indices j < j′ denotes a pair j,j′ ∈ N with j < j′ ≤ j +r+1. A pair of indices j ⊳j′
denotes a pair j,j′ ∈ N with j +1 < j′ ≤ j +r+1, that is, with at least one other index
between j and j′.
For any proper subset I ⊂ {0,...,r} let IC ⊂ {0,...,r} be its complement. Then a
consecutive pair of indices j < j′ ∈ IC (resp. j⊳j′ ∈IC) denotes a pair j < j′ (resp. j⊳j′)
as above such that j,j′ ∈ IC and {j +1,...,j′−1} ⊂ I.
Definition 2.2. For any pair of indices j ⊳j′ we define the partial fiber product
Lj(j+1)×Hj+1 L(j+1)(j+2)...×Hj′−1 L(j′−1)j′
:= Lj(j+1)×...×L(j′−1)j′ ∩ Mj ×graph(φHδjj++11)×...×graph(φHδjj′−′−11)×Mj′ .
We trivi(cid:0)ally extend this notation(cid:1) to(cid:0)the case j′ = j +1 by Ljj′ = Lj(j+1). For a ge(cid:1)neral
proper subset of indices I ⊂ {0,...,r} we then define the partial fiber product
×I,HL := Lj(j+1)×Hj+1 ...×Hj′−1 L(j′−1)j′
consec.Yj<j′∈IC
tobetheproductoftheabovefiberproductsforeachconsecutivepairofindicesj <j′ ∈ IC.
We viewtheintersection I(L,H)= × Lasthefullfiberproductcase I = {0,...,r}.
{0,...,r},H
Theorem 2.3. There is a dense open subset Ham∗(L) ⊂ H (δ) so that for every H ∈
t
Ham∗(L) the defining equations for × L for any I ⊂ {0,...,r} are transversal.
I,H
Proof. Each of the fiber products under consideration is of the following form: It is the set
of tuples (m′,m ,m′,...,m′,m )∈ L ×...×L satisfying
0 1 1 r r+1 01 r(r+1)
(1) φHi(m )= m′ ∀i∈ I.
δi i i
It suffices to find a dense open subset of regular Hamiltonians for each of these problems,
since the intersection of finitely many dense open subsets remains dense and open. So we
fix some choice of I ⊂ {0,...,r} and consider the universal moduli Muniv space of data
(H ,...,H ,m′,m ,...,m′,m ) satisfying (1), where now each H has class Cℓ for some
0 r 0 1 r r+1 j
ℓ > dimM . It is cut out by the diagonal values of the Cℓ-map
i∈IC i
r
P
L ×L ...×L × Cℓ([0,δ ]×M ) −→ M ×M ,
01 12 r(r+1) k k j j
k=0 j∈I
M M
(m′,...,m ,H ,...,H ) 7−→ (φHi(m ),m′) .
0 r+1 0 r δi i i i∈I
QUILTED FLOER TRAJECTORIES WITH CONSTANT COMPONENTS 5
The linearized equations for Muniv are
(2) v′ −DφHi(h ,v ) = 0∈ TM ∀i∈ I.
i δi i i i
for vi ∈TmiMi, vi′ ∈ Tm′iMi, and hi ∈ Cℓ([0,δi]×Mi). The map
Cℓ([0,δ ]×M ) → T M , h 7→ DφHi(h ,0)
i i φHδii(mi) i i δi i
is surjective, which shows that the product of the operators on the left-hand side of (2) is
also surjective. So by the implicit function theorem Muniv is a Cℓ Banach manifold, and
we consider its projection to ⊕r Cℓ([0,δ ]×M ). This is a Fredholm map of class Cℓ and
k=0 k k
index dimM (in particular 0 for the full intersection I = {0,...,r}). Hence, by the
i∈IC i
Sard-Smaletheorem,thesetofregularvalues(whichcoincideswiththesetoffunctionsH =
(H ,..P.,H )such thattheperturbedintersection is transversal) is densein ⊕r Cℓ([0,δ ]×
0 r k=0 k
M ). Moreover, the set of regular values is open for each ℓ > dimM . Indeed, by
k i∈IC i
the compactness of L ×L ...×L , a C1-small change in H leads to a small change
01 12 r(r+1)
P
in perturbed intersection points, with small change in the linearized operators.
Now, by approximation of C∞-functions with Cℓ-functions, the set of regular values in
⊕r C∞([0,δ ]×M ) is densein theCℓ-topology for all ℓ > dimM , and hencedense
k=0 k k i∈IC i
in the C∞-topology. Finally, the set of regular smooth H is open in the C∞-topology as a
special case of the C1-openness. P (cid:3)
We now reformulate this Theorem by using the Hamiltonian flows of H to perturb the
Lagrangian correspondences and then applying a geometric composition in some factors.
Corollary 2.4. For H ∈Ham∗(L) the perturbed generalized correspondence
L′ := L′ := Id ×φHj+1 L
Mj δj+1 j(j+1) j=0,...,r
(cid:16) (cid:17)
(cid:0) (cid:1)
has the following intersection and composition properties:
(a) The generalized intersection
I(L′,0) = (L′ ×...×L′ )∩(∆ ×...×∆ )T
01 r(r+1) M1 M0
is transverse and canonically identified with I(L,H).
(b) For any proper subset I ⊂ {0,...,r} the partial fiber product × L′ is cut out
I,0
transversally (and canonically identified with × L). It is a product of the trans-
I,H
verse intersections
L′ × L′ ...× L′
j(j+1) ∆j+1 (j+1)(j+2) ∆j′−1 (j′−1)j′
:= L′j(j+1)×L′(j+1)(j+2)...×L′(j′−1)j′ ∩ Mj ×∆Mj+1...∆Mj′−1 ×Mj′
= (cid:0)IdMj ×φHδjj++11 ×IdMj+1...×φHδjj′−′−11(cid:1)×φ(cid:0)Hδjj′′ Lj(j+1)×Hj+1 ...×Hj′−1 L(cid:1)(j′−1)j′
f(cid:0)or consecutive pairs of indices j < j′ ∈ IC.(cid:1)(cid:0) (cid:1)
(c) By a direct generalization of [6, Lemma 2.0.5], the projection
ΠMj×Mj′ : L′j(j+1)×∆j+1 ...×∆j′−1 L′(j′−1)j′ −→ Mj ×Mj′
is an immersion onto the geometric composition L′j(j+1)◦...◦L′(j′−1)j′ ⊂ Mj×Mj′.
6 KATRINWEHRHEIMANDCHRIST.WOODWARD
We will in particular be interested in this composition near a fixed point in M ×...×
j+1
Mj′−1 given by the components of an intersection point in I(L′,0). For any such point
there is a neighbourhoodU ⊂ Mj+1×...×Mj′−1 such that the projection ΠMj×Mj′ embeds
L′j(j+1)×∆j+1 L′(j+1)(j+2)...×∆j′−1 L′(j′−1)j′ ∩ Mj ×U ×Mj′ into Mj ×Mj′. This is a
localized version oftheembeddedgeometric composition (as studiedin[6])oftheperturbed
(cid:0) (cid:1) (cid:0) (cid:1)
Lagrangians. We will be using the following analogue perturbed geometric composition of
the unperturbed Lagrangian correspondences.
Definition 2.5. For a proper subset I ⊂ {0,...,r} and x ∈ I(L,H) we define the lo-
cally composed cyclic Lagrangian correspondence LI,H,x between the underlying manifolds
H,x
(Mj)j∈IC to be the cyclic sequence consisting of Ljj′ ⊂ Mj×Mj′ for each consecutive pair
of indices j < j′ ∈ IC, given by
LHjj,′x := ΠMj×Mj′ Lj(j+1)×Hj+1 L(j+1)(j+2)...×Hj′−1 L(j′−1)j′ ∩U˜x,j,j′
for U˜x,j,j′ := Mj×Ux,j,j′×(cid:0)M(cid:0)j′, whereUx,j,j′ is a neighbourhoodof xj+1((cid:1)0),xj+1(δ(cid:1)j+1),...,
xj′−1(0),xj′−1(δj′−1) such that ΠMj×Mj′ is injective on the inters(cid:0)ection.
Remark 2.6. Given a(cid:1) regular tuple of Hamiltonian functions H ∈ Ham∗(L) as in Theo-
rem 2.3 and a proper subset I ⊂ {0,...,r}, let δI := (δ ) and HI := (H ) . Then
j j∈IC j j∈IC
the transversality assertions of Theorem 2.3 moreover imply that for any x ∈ I(L,H) the
intersection I(LI,H,x,HI) is transverse. It contains (x ) , and no other points if the
j j∈IC
neighbourhoods Ux,j,j′ are chosen sufficiently small.
In preparation for the analysis of quilted Floer trajectories with constant components,
H,x
we next study the lift from Ljj′ to Mj+1×Mj′−1 and its connection with the intersections
TLj(j+1)∩({0}×TMj+1)andTL(j′−1)j′∩(TMj′−1×{0}). Apriori,thelatterareisomorphic
to collections of isotropic subspaces of TMj+1 resp. TMj′−1 parametrized by Lj(j+1) resp.
L(j′−1)j′. As mentioned in the introduction, a first step is to understand the locus where
H,x
these subspaces are Lagrangian, and how they may vary along L . For that purpose we
jj′
introduce the following notation.
Definition 2.7. Let j ⊳j′ be a pair of indices.
(a) We denote by Sjj′ ⊂ Lj(j+1)×L(j′−1)j′ the set of points q = (qj,qj+1,qj′−1,qj′) for
which
Λ(j+1)(j′−1)(q):= Tq(Lj(j+1)×L(j′−1)j′)∩Tq({qj}×Mj+1×Mj′−1×{qj′}
induces a Lagrangian subspace in Tqj+1Mj+1 × Tqj′−1Mj′−1 (with the app(cid:1)ropriate
signs on the symplectic forms).
(b) Given moreover H ∈ Ham∗(L), x ∈ I(L,H), we denote by
H,x
Pjj′ : Ljj′ −→ Mj+1×Mj′−1
thecompositionoftheliftfromLjHj,′x to Lj(j+1)×Hj+1...×Hj′−1L(j′−1)j′ ∩U˜x,j,j′ and
the projection to the second and penultimate component, i.e. to a neighbourhood of
(cid:0) (cid:1)
(xj+1(0),xj′−1(δj′−1)).
The following Proposition shows that the set Sjj′ can equivalently be defined as the
locus where the linearized Lagrangian correspondences split, and that this splitting locus is
closely related to the vanishing of DPjj′.
QUILTED FLOER TRAJECTORIES WITH CONSTANT COMPONENTS 7
Proposition 2.8. The following holds for any pair of indices j⊳j′.
(a) Λ(j+1)(j′−1)(q) is Lagrangian if and only if we have splittings
T(qj,qj+1)Lj(j+1) =Λj ×Λj+1, T(qj′−1,qj′)L(j′−1)j′ = Λj′−1×Λj′
into Lagrangian subspaces
Λ = Pr T L ∩(T M ×{0}) ,
j TMj (qj,qj+1) j(j+1) qj j
Λj+1 = PrTMj+1(cid:0)T(qj,qj+1)Lj(j+1)∩({0}×Tqj+1M(cid:1)j+1) ,
Λj′−1 = PrTMj′−1 T(cid:0)(qj′−1,qj′)L(j′−1)j′ ∩(Tqj′−1Mj′−1×{0(cid:1)}) ,
Λj′ = PrTMj′(cid:0)T(qj′−1,qj′)L(j′−1)j′ ∩({0}×Tqj′Mj′) . (cid:1)
(b) The subset Sjj′ ⊂Lj(j+1)(cid:0)×L(j′−1)j′ is compact. (cid:1)
(c) For any H ∈ Ham∗(L), x ∈ I(L,H) the linearization D(qj,qj′)Pjj′ : T(qj,qj′)LjHj,′x →
TPjj′(qj,qj′)(Mj+1×Mj′−1) is trivial iff (qj,Pjj′(qj,qj′),qj′)∈ Sjj′.
Proof. By definition, Λ(j+1)(j′−1)(q) induces the (automatically isotropic) subspace Λj+1×
Λj′−1 ⊂ TMj+1 × TMj′−1. It is Lagrangian iff both factors have maximal dimension,
i.e. are Lagrangian. Moreover, since TL is Lagrangian, maximal dimension of its
j(j+1)
subspace Λ directly implies maximal dimension of Λ , and vice versa, and analogously
j+1 j
with TL(j′−1)j′. This proves (a).
Thesplittingconditionsin(a)canbephrasedasintersections havingmaximaldimension,
hence are preserved in a limit, so occur on a closed subset of Lj(j+1) × L(j′−1)j′. This
shows that Sjj′ is closed, so (b) follows directly from the compactness of all Lagrangian
submanifolds involved.
Givenapoint(qj,qj+1,qj′′−1,qj′)= (qj,Pjj′(qj,qj′),qj′)thereexistqj′+1,qj+2,...,qj′′−2,qj′−1
such that (qj,qj+1,qj′+1,...,qj′−1,qj′′−1,qj′) ∈ Lj(j+1) ×Hj+1 ...×Hj′−1 L(j′−1)j′ ∩U˜x,j,j′.
H,x
Now D(qj,qj′)Pjj′ is the composition of the lift f(cid:0)rom T(qj,qj′)Ljj′ to (cid:1)
T(qj,qj+1)Lj(j+1)×...×T(qj′′−1,qj′)L(j′−1)j′
(3) ∩ TqjMj ×graph(dφHδjj++11(qj+1))×...×graph(dφδHjj′−′−11(qj′−1))×Tqj′Mj′
and the projection to Tqj+1Mj+1 × Tqj′′−1Mj′−1. Hence D(qj,qj′)Pjj′ ≡ 0 if and only if
(3) is a subset of TqjMj × {0}×Tqj′+1Mj+1 ×...×Tqj′−1Mj′−1 ×{0} ×Tqj′Mj′. On the
other hand, our choice of H guarantees that the projection of (3) to TqjMj ×Tqj′Mj′ is
H,x
injective withLagrangian image T(qj,qj′)Ljj′ . Soif D(qj,qj′)Pjj′ ≡ 0, then bothintersections
T(qj,qj+1)Lj(j+1)∩{0}×Tqj+1Mj+1 and T(qj′′−1,qj′)L(j′−1)j′∩Tqj′′−1Mj′−1×{0} have maximal
dimension. As above that implies (qj,Pjj′(qj,qj′),qj′) ∈ Sjj′. Conversely, if the latter is
true, i.e. both T(qj,qj+1)Lj(j+1) = Λj×Λj+1 and T(qj′′−1,qj′)L(j′−1)j′ = Λj′−1×Λj′ are of split
form, then the intersection with the graphs in (3) cannot allow for any nonzero vectors in
Λj+1 or Λj′−1, and hence D(qj,qj′)Pjj′ ≡ 0. (cid:3)
In the following section we will “generically“ exclude quilted Floer trajectories with con-
stantcomponentsofthefollowingtwotypes: Firstly, thosealongwhoseseamvalueswehave
DPjj′ 6≡ 0 somewhere; secondly, those along whose seam values DPjj′ ≡ 0 but Λ(j+1)(j′−1)
varies. This will only leave quilted Floer trajectories with constant components, for which
transversality follows fromtransversality forthemodulispaceofthelocally composedcyclic
8 KATRINWEHRHEIMANDCHRIST.WOODWARD
Lagrangian correspondence. These arguments require the following understanding of the
structure of the split form set Sjj′, the variation of the intersection Λ(j+1)(j′−1), and the
H,x
intersection of Sjj′ with lifts of the local compositions Ljj′ .
Theorem 2.9. The following intersection properties hold for any pair of indices j ⊳j′.
(a) For any q ∈ Sjj′ there exists an open neighbourhood Vq ⊂ Lj(j+1) × L(j′−1)j′ and
smoothfunctionsG :V → Rforn = 1,...,N := (dimMj+dimMj′)(dimMj+1+dimMj′−1)
n q 4
such that
N
Sjj′ ∩Vq = G−n1(0).
n=1
\
Moreover, if γ :(−ǫ,ǫ) → Sjj′∩ Mj×{qj+1}×{qj′−1}×Mj′ is a smooth path with
dGn(γ) ≡ 0 for all n = 1,...N, then Λ(j+1)(j′−1)(γ(t)) is constant in t ∈ (−ǫ,ǫ) as
(cid:0) (cid:1)
subspace of Tqj+1Mj+1×Tqj′−1Mj′−1.
(b) Fix a finite open cover Sjj′ ⊂ q∈Sjj′ Vq by subsets as in (b) with Sjj′ ⊂ Sjj′ finite.
Then there is a dense open subSset Hjj′(L) ⊂Ham∗(L) such that the following holds:
For every H ∈ Hjj′(L), x ∈ I(L,H), q ∈ Sjj′, and 1 ≤ n ≤ N the function
H,x H,x
Gq,n :Vq,n → R, (zj,zj′) 7→ Gn(zj,Pjj′(zj,zj′),zj′) defined on the open set
VqH,n,x := (zj,zj′)∈ LHjj,′x (zj,Pjj′(zj,zj′),zj′)∈ Vq, dGn(zj,Pjj′(zj,zj′),zj′) 6= 0
is tran(cid:8)sverse to 0. (cid:12) (cid:9)
(cid:12)
Proof. Given q = (qj,qj+1,qj′−1,qj′) ∈ Sjj′ we have splittings
T L = Λ0×Λ0 ⊂ T M ×T M ,
(qj,qj+1) j(j+1) j j+1 qj j qj+1 j+1
T(qj′′−1,qj′)L(j′−1)j′ = Λ0j′−1×Λ0j′ ⊂ Tqj′−1Mj′−1×Tqj′Mj′
into products of Lagrangian subspaces. With this we have Λ(j+1)(j′−1)(q) = {0}×Λ0j+1 ×
Λ0 ×{0} and permutation of factors provides an isomorphism
j′−1
Tq(Lj(j+1)×L(j′−1)j′) ∼= K0×Λ0 with K0 := Λ0j ×Λ0j′, Λ0 := Λ0j+1×Λ0j′−1.
Now there exists an open neighbourhood Wq ⊂ Mj × Mj + 1 × Mj′−1 × Mj′ of q that
is symplectomorphic to an open subset of K0 × Λ0 × K × Λ ∗ such that V := W ∩
0 0 q q
(Lj(j+1) ×L(j′−1)j′) corresponds to graphdF for some s(cid:0)mooth fu(cid:1)nction F : K0 ×Λ0 → R,
restricted to an open subset. We will identify K0 ∼= K0∗ ∼= Rk, k = 21(dimMj +dimMj′)
and Λ0 ∼= Λ∗0 ∼= Rℓ, ℓ = 21(dimMj+1 +dimMj′−1) and use coordinates (x,y) ∈ K0 ×Λ0.
Then the intersection Λ(j+1)(j′−1) at some point (x,y,∇K0F(x,y),∇Λ0F(x,y)) is spanned
bythosevectors 0,b,0, ℓ b ∂2F forwhich ℓ b ∂2F = 0forκ = 1,...,k.
i=1 i∂yi∂yl l=1,...,ℓ i=1 i∂yi∂xκ
It is Lagrangian(cid:0)iff its ra(cid:0)nPk is the max(cid:1)imal ℓ.(cid:1)Hence Sjj′P∩Vq is the zero set of
∂2F
(G ) := : V −→ Rℓk = RN.
n n=1,...,N q
∂y ∂x
(cid:18) i κ(cid:19)i=1,...,ℓ,κ=1,...,k
For the second part of (a) we consider a path γ(t) = (x(t),0,∇ F(x(t),0),0) given by a
K0
smooth path x : (−ǫ,ǫ) → K such that ∂F(x,0) ≡ 0, ∂2F (x,0) ≡ 0, and in particular
0 ∂yi ∂yi∂xκ
QUILTED FLOER TRAJECTORIES WITH CONSTANT COMPONENTS 9
∂ G (x,0) = ∂3F (x,0) ≡ 0foralll,i,κ. Thelatter guarantees that ∂2F (x(t),0)
yl n≃(i,κ) ∂yl∂yi∂xκ ∂yi∂yl
and hence Λ(j+1)(j′−1)(γ(t)) is independent of t.
Approaching (b), note that we may reformulate the claim as transversal intersection of
Lj(j+1)×Hj+1...×Hj′−1L(j′−1)j′ with the zero set of G˜n(z):= Gn(zj,zj+1,zj′′−1,zj′) on the
open set
(cid:26)z = (zj,zj+1,zj′+1,...,zj′′−1,zj′) ∈ Lj(j+1)×...×L(j′−1)j′(cid:12) (dzGj,nz(jz+j,1z,jz+j′′1−,1z,j′z′−j′1),∈zj′V)q6=, 0 (cid:27).
(cid:12)
The universal moduli space of regularity m ∈ N for this pro(cid:12)blem is the preimage of {0}×
(cid:12)
∆ ×...∆ of the map
Mj+1 Mj′−1
j′−1 j′−1
Lj(j+1)×L(j+1)(j+2)...×L(j′−1)j′ × Cm([0,δi]×Mi) −→ R× Mi ×Mi
i=j+1 i=j+1
M M
given by
(zj,...,zj′,Hj+1,...,Hj′−1) 7−→ Gn(zj,zj+1,zj′′−1,zj′),(φHδii(zi),zi′)i=j+1,...,j′−1 .
Hence the universal moduli space is a(cid:0)Cm manifold if at every solution the operator (cid:1)
(vj,...,vj′,hj+1,...,hj′−1)7−→ dGn(vj,vj+1,vj′′−1,vj′),(vi′ −DφHδii(hi,vi))i=j+1,...,j′−1
is onto. Here surjectivity in the fir(cid:0)stcomponent is guaranteed by the condition dG˜ (z)6=(cid:1)0,
n
and in the second component already h 7→ DφHi(h ,0) is surjective as in Theorem 2.3.
i δi i
Now as before the implicit function theorem and Sard-Smale theorem, using m > dimM +
j
dimMj′−1to satisfy theindex condition, providea densesubsetof ij=′−j1+1Cm([0,δi]×Mi)
for which G˜n : Lj(j+1) ×Hj+1 ... ×Hj′−1 L(j′−1)j′ ∩ {dG˜n 6= 0} →LR is transverse to 0.
Since this contains the lift of GH,x : VH,x → R, we find a dense open set of regular
(cid:0) q,n q,n (cid:1)
Hamiltonians of class Cm for any given q ∈ Sjj′, 1 ≤ n ≤ N, x ∈ I(L,H), and sufficiently
largem ∈N. Finally, C1-small changes inH lead tosmallchanges inintersection points and
the linearized operators, hence we obtain open dense sets of regular values, and may take
countable intersections to find a dense open subset Hjj′(L) ⊂ Ham∗(L) of regular smooth
Hamiltonians. (cid:3)
3. Quilted Floer trajectories with constant components
Given a cyclic generalized Lagrangian correspondence L, widths δ, a regular tuple of
Hamiltonian functions H ∈ Ham∗(L), we now consider the Floer trajectories for some
choice of almost complex structures J = (J ) ∈ J (δ). For any pair x−,x+ ∈ I(L,H)
j j=0,...,r t
of generators and index k ∈ Z, the moduli space of quilted Floer trajectories
Mk(x−,x+;L,J):= u = u : R×[0,δ ]→ M (4),(5),(6),IndD ∂ = k /R
j j j j=0,...,r u J
is the space modulo sim(cid:8)ultan(cid:0)eous R-shift of tuples(cid:1)of pertu(cid:12)rbed holomorphic strips (cid:9)
(cid:12)
(4) ∂ u =∂ u +J ∂ u −X (u ) =0 ∀j = 0,...r,
Jj,Hj j s j j t j Hj j
satisfying the seam conditions (cid:0) (cid:1)
(5) (u (s,δ ),u (s,0)) ∈ L ∀j = 0,...r, s ∈ R
j j j+1 j(j+1)
10 KATRINWEHRHEIMANDCHRIST.WOODWARD
as well as uniform limits
(6) lim u (s,·) = x± ∀j = 0,...,r.
j j
s→±∞
Moreover, we fixed the index of the linearized operator – as explained in the following. By
standard local action arguments, any such solution also has finite energy, and exponential
decay analysis as in [8] shows that any solution is of Sobolev regularity W1,p relative to the
limits for any p > 2in the following sense: We trivially extend x± to maps R×[0,δ ] → M ,
j j j
thenthereexistsR > 0suchthatu (±s,t)takesvaluesinanexponentialballaroundx±(s,t)
j j
for ±s > R, and such that for each j = 0,...,r
(7) (s,t) 7→ exp−1 (u (±s,t)) ∈ W1,p([R,∞)×[0,δ ],x±∗TM ).
x±(t) j j j j
j
With this, the(cid:0)modulispace can beidentifi(cid:1)ed withtheR-quotient of thezeroset of asection
∂ :B → E of a Banach bundle, where
J
B := u = u ∈ W1,p(R×[0,δ ],M ) (5),(7) ,
j loc j j j=0,...,r
E → B is the Banach b(cid:8)undle(cid:0)with fibers E = ⊕r Lp(R(cid:1) ×[0,δ(cid:12)],u∗TM(cid:9)), and ∂ : B → E
u j=0 j(cid:12) j j J
is the (R-invariant) Cauchy-Riemann operator ∂ (u) = ∂ u . In [6] we proved
J Jj,Hj j j=0,...,r
that ∂ is a Fredholm section, and in the definition of the moduli space Mk(x−,x+) we fix
J (cid:0) (cid:1)
the Fredholm index of its linearization D ∂ : T B → E . In order to achieve transversality
u J u u
of the section s, we now restrict ourselves to a further dense open subset of Hamiltonian
perturbations, as constructed in Section 2.
Definition 3.1. Given a cyclic generalized Lagrangian correspondence L and widths δ, let
Hreg(L)= Hjj′(L) ⊂ Ht(δ)
j,j′
\
be the intersection over all pairs of indices j ⊳ j′ of the dense open subsets of regular
Hamiltonians for some choices of covers of Sjj′ as in Theorem 2.9.
We now prove the main result.
Theorem 3.2. For any cyclic generalized Lagrangian correspondence L and any choice of
widthsδ andregularHamiltoniansH ⊂ H (L), thereexistsacomeagre1 subsetJ (L;H) ⊂
reg reg
J (δ) such that for all J ∈ J (L;H), x± ∈ I(L,H), and k ∈ Z the Cauchy-Riemann sec-
t reg
tion ∂ :B → E defined above is transverse to the zero section.
J
Remark 3.3. In fact, we prove that for generic perturbation data H ⊂ Ham∗(L) and J ∈
Jreg(L;H) any solution u ∈ Mk(x−,x+;L,J) with some constant components has split
t
linearized seam conditions between constant and nonconstant components in the following
sense: If ∂ u 6≡ 0 and u (s,t) = x (t), then T L = Λ (s)× Λ
s j j+1 j+1 (uj(s,δj),xj+1(0)) j(j+1) j j+1
splits into two families of Lagrangian subspaces
Λ (s) = Pr T L ∩(T M ×{0}) ,
j TMj (uj(s,δj),xj+1(0)) j(j+1) uj(s,δj) j
Λj+1 = PrTMj+(cid:0)1 T(uj(s,δj),xj+1(0))Lj(j+1)∩({0}×Txj+1(0)M(cid:1)j+1) ,
1Asubsetofatopologicalspace(cid:0)iscomeagreifitistheintersectionofcountablymanyop(cid:1)endensesubsets.
Manyauthorsinsymplectictopologywouldusetheterm“Bairesecondcategory”,whichhoweverinclassical
Bairetheory[5,Chapter7.8]denotesmoregenerallysubsetsthatarenotmeagre,i.e.notthecomplementof
acomeagre subset. Baire’s Theorem applies tocompletemetric spacessuch asthespaces ofsmooth almost
complex structuresconsidered here,and implies that every comeagre set is dense.