Quilted Floer cohomology
Abstract.
We generalize Lagrangian Floer cohomology to sequences of Lagrangian correspondences. For sequences related by the geometric composition of Lagrangian correspondences we establish an isomorphism of the Floer cohomologies. This provides the foundation for the construction of a symplectic category as well as for the definition of topological invariants via decomposition and representation in the symplectic category. Here we give some first direct symplectic applications: Calculations of Floer cohomology, displaceability of Lagrangian correspondences, and transfer of displaceability under geometric composition.
Contents
1. Introduction
Lagrangian Floer cohomology associates to a pair of compact Lagrangian manifolds a chain complex whose differential counts pseudoholomorphic strips with boundary values in the given Lagrangians. In this paper we generalize Floer cohomology to include compact Lagrangian correspondences. Recall that if and are symplectic manifolds, then a Lagrangian correspondence from to is a Lagrangian submanifold , where . These were introduced by Weinstein [44] in an attempt to create a symplectic category with morphisms between not necessarily symplectomorphic manifolds. So we also denote a Lagrangian correspondence by . With this notation we can view a pair of Lagrangian submanifolds as sequence of Lagrangian correspondences from the point via back to the point. This is a special case of a cyclic sequence of Lagrangian correspondences
for which we will define a quilted Floer cohomology
(1) 
The quilted differential counts tuples of pseudoholomorphic strips whose boundaries match up via the Lagrangian correspondences, for . These tuples are examples of pseudoholomorphic quilts with the strips thought of as patches and the boundary matching conditions thought of as seams. The theory of quilts is developed in higher generality in [39]. In this paper, we next investigate the effect of geometric composition on Floer cohomology. The geometric composition of two Lagrangian correspondences , is
(2) 
In general, this will be a singular subset of . However, if we assume transversality of the intersection , then the restriction of the projection to is automatically a Lagrangian immersion. We will study the class of embedded geometric compositions, for which in addition is injective, and hence is a smooth Lagrangian correspondence. If the composition is embedded, then we obtain under suitable monotonicity assumptions a canonical isomorphism
(3) 
For the precise monotonicity and admissibility conditions see Section 5.4. The proof proceeds in two steps. First, we allow for varying widths of the pseudoholomorphic strips defining the differential. Section 5.3 of this paper shows that Floer cohomology is independent of the choice of widths. (These domains are not conformally equivalent due to the identification between boundary components that is implicit in the seam conditions.) The second (hard analytic) part is to prove that with the width sufficiently close to zero, the tuples of holomorphic strips with seam conditions in are in onetoone correspondence with the tuples of holomorphic strips with seam conditions in . This analysis is completely analogous to [38], where we establish the bijection for the Floer trajectories of the special cyclic sequence when . The monotonicity assumptions are crucial for this part since the exclusion of a novel “figure eight bubble” in [38] hinges on a strict energyindex proportionality.
In section 6 we provide a number of new tools for the calculation of Floer cohomology (and hence detection of nondisplaceability), arising as direct consequences of (3) or from a conjectural generalization of Perutz’ long exact Gysin sequence [26]. These are meant to exemplify the wide applicabilty and reach of the basic isomorphism (3). We believe that it should have much more dramatic consequences once systematically employed. As first specific example of direct consequences of (3) we confirm the calculation of Cho [4] for the Clifford torus in , and we calculate some further Floer cohomologies in using reduction at pairs of transverse level sets. Next, we prove Hamiltonian nondisplaceability of the Lagrangian sphere arising from reduction at the level set of an action on containing . The latter will be deduced from the nontriviality of together with our isomorphism
and the fact that (since contains an element that acts as the identity on ). Finally, we generalize this nondisplaceability result to the Lagrangian embedding of , which arises from the monotone level set of an action on for .
Another consequence of our results is a general prescription for defining topological invariants by decomposing into elementary pieces. For example, let be a compact manifold and a Morse function, which induces a decomposition into elementary cobordisms by cutting along noncritical level sets . First one associates to each a monotone symplectic manifold , and to each with a smooth monotone Lagrangian correspondence (taking and to be points.) Second, one checks that the basic moves described by Cerf theory (cancellation or change of order of critical points) change the sequence of Lagrangian correspondences by replacing adjacent correspondences with an embedded composition, or viceversa. In other words, the equivalence class of sequences of Lagrangian correspondences by embedded compositions does not depend on the choice of the Morse function . Then the results of this paper provide a groupvalued invariant of , by taking the Floer homology of the sequence of Lagrangian correspondences. For example, in [42, 43] we investigate the theory which uses as symplectic manifolds the moduli spaces of flat bundles with compact structure group on threedimensional cobordisms containing tangles. Conjecturally this provides a symplectic construction of Donaldson type gauge theoretic invariants: instanton Floer homology, it’s higher rank version (though not strictly defined in the gauge theoretic setting), and the tangle invariants defined by KronheimerMrowka [14] from singular instantons. The same setup is used to give alternative constructions of Heegard Floer homology [15] and SeidelSmith invariants [27].
Even more generally, in [40] the quilted Floer cohomology groups provide the morphism spaces in our construction of a symplectic category which contains all Lagrangian correspondences as morphisms, and where the composition is equivalent to geometric composition, if the latter is embedded.
This setup in turn is used by AbouzaidSmith [1] to prove mirror symmetry for the torus and deduce various further symplectic consequences.
Notation and Organization: We will frequently refer to the assumptions (M12), (L13), and (G12) that can be found on pages 4.1  4.1.
Section 2 is a detailed introduction to Lagrangian correspondences, geometric composition, and sequences of correspondences, which also provides the basic framework for the sequels [40, 39] to this paper. In Section 3 we generalize gradings to Lagrangian correspondences and establish their behaviour under geometric composition, so that the isomorphism (3) becomes an isomorphism of graded groups. Section 4 provides a review of monotonicity and Floer cohomology and gives a first definition of the Floer cohomology (1) by building a pair of Lagrangians in the product . The latter is however unsatisfactory since it does not provide an approach to the isomorphism (3). Section 5 gives the general definition of quilted Floer cohomology (1) and finalizes the proof of the isomorphism (3). Finally, Section 6 gives a number of direct symplectic applications of the isomorphism (3).
We thank Paul Seidel and Ivan Smith for encouragement and helpful discussions, and we are very grateful to the meticulous referee for help with cleaning up the details.
2. Lagrangian correspondences
Let be a smooth manifold. A symplectic form on is a closed, nondegenerate twoform . A symplectic manifold is a smooth manifold equipped with a symplectic form. If and are symplectic manifolds, then a diffeomorphism is a symplectomorphism if . Let denote the category whose objects are symplectic manifolds and whose morphisms are symplectomorphisms. We have the following natural operations on .

(Duals) If is a symplectic manifold, then is a symplectic manifold, called the dual of .

(Sums) If are symplectic manifolds, then the disjoint union equipped with the symplectic structure on and on , is a symplectic manifold. The empty set is a unit for the disjoint union.

(Products) Let be symplectic manifolds, then the Cartesian product is a symplectic manifold. (Here denotes the projections.) The symplectic manifold , consisting of a single point, is a unit for the Cartesian product.
Clearly the notion of symplectomorphism is very restrictive; in particular, the symplectic manifolds must be of the same dimension. A more flexible notion of morphism is that of Lagrangian correspondence, defined as follows [45, 44, 12]. Let be a symplectic manifold. A submanifold is isotropic, resp. coisotropic, resp. Lagrangian if the orthogonal complement satisfies resp. resp. .
Definition 2.0.1.
Let be symplectic manifolds. A Lagrangian correspondence from to is a Lagrangian submanifold .
Example 2.0.2.
The following are examples of Lagrangian correspondences:

(Trivial correspondence) The one and only Lagrangian correspondence between and any other is .

(Lagrangians) Any Lagrangian submanifold can be viewed both as correspondence from the point to and as correspondence from to the point.

(Graphs) If is a symplectomorphism then its graph
is a Lagrangian correspondence.

(Fibered coisotropics) Suppose that is a coisotropic submanifold. Then the null distribution is integrable, see e.g. [18, Lemma 5.30]. Suppose that is in fact fibrating, that is, there exists a symplectic manifold and a fibration such that is the pullback . Then
maps to a Lagrangian correspondence.

(Level sets of moment maps) Let be a Lie group with Lie algebra . Suppose that acts on by Hamiltonian symplectomorphisms generated by a moment map . (That is is equivariant and the generating vector fields satisfy .) If acts freely on , then is a smooth coisotropic fibered over the symplectic quotient , which is a symplectic manifold. Hence we have a Lagrangian correspondence
The symplectic twoform on is the unique form on satisfying .
Definition 2.0.3.
Let be symplectic manifolds and , Lagrangian correspondences.

The dual Lagrangian correspondence of is

The geometric composition of and is
Geometric composition and duals of Lagrangian correspondences satisfy the following:

(Graphs) If and are symplectomorphisms, then

(Zero) Composition with always yields , that is for any Lagrangian correspondence we have

(Identity) If is a Lagrangian correspondence and are the diagonals, then

(Associativity) If are Lagrangian correspondences, then
The geometric composition can equivalently be defined as , the image under the projection of
Here denotes the diagonal. is an immersed Lagrangian submanifold if intersects transversally. In general, the geometric composition of smooth Lagrangian submanifolds may not even be immersed. We will be working with the following class of compositions, for which the resulting Lagrangian correspondence is in fact a smooth submanifold, as will be seen in Lemma 2.0.5 below.
Definition 2.0.4.
We say that the composition is embedded if is cut out transversally (i.e. ) and the projection is an embedding. (For compact Lagrangians it suffices to assume that is injective, by Lemma 2.0.5 below.)
By some authors (e.g. [12]) geometric composition of Lagrangian correspondences is more generally defined under clean intersection hypotheses. This extension is not needed in the present paper, because the quilted Floer cohomology is invariant under Hamiltonian isotopy, and after such an isotopy transversality may always be achieved. However, transverse intersection only yields an immersed^{1}^{1}1One can not necessarily remove all selfintersections of the immersed composition by Hamiltonian isotopy on one correspondence. A basic example is the composition of transverse Lagrangian submanifolds . Identifying the projection maps the (finite) intersection to a point. Lagrangian correspondence, as the following Lemma shows.
Lemma 2.0.5.
Let , be Lagrangian correspondences such that the intersection is transverse. Then the projection is an immersion.
In particular, if the Lagrangians are compact, the intersection is transverse, and the projection is injective, then the composition is embedded.
Proof.
The proof essentially is a special case of the fact that the geometric composition of linear Lagrangian correspondences is always well defined (i.e. yields another linear Lagrangian correspondence), see e.g. [12, Section 4.1].
Fix a point then we need to check that for the projection restricted to . In fact, we will show that
(4) 
which is zero by transversality. To simplify notation we abbreviate , , and . Now (4) follows as in [12, Section 4.1]. For completeness we recall the precise argument: We identify
(5) 
where and similarly . On the other hand, we use the symplectic complements with respect to on to identify
(6) 
where , similarly , and we used the equivalence
Now the two vector spaces in (5) and (6) are identified by the dualities and , which follow from the Lagrangian property of resp. ,
This proves (4) and hence finishes the proof that is an immersion. Finally, note that an injective immersion of a compact set is automatically an embedding. ∎
Remark 2.0.6.
Suppose that the composition is embedded.

By the injectivity, for every there is a unique solution to . Due to the transversality assumption, this solution is given by a smooth map .

If both fundamental groups and have torsion image in the respective ambient space, then has torsion image in . (Any loop lifts to a loop with . By assumption, some multiple cover is the boundary of a map . Hence the same cover is contractible.)
2.1. Generalized Lagrangian correspondences
A simple resolution of the composition problem (that geometric composition is not always defined) is given by passing to sequences of Lagrangian correspondences and composing them by concatenation. In [40] we employ these to define a symplectic category containing all smooth Lagrangian correspondences as composable morphisms, yet retaining geometric composition in cases where it is well defined.
Definition 2.1.1.
Let be symplectic manifolds. A generalized Lagrangian correspondence from to consists of

a sequence of any length of symplectic manifolds with and ,

a sequence of compact Lagrangian correspondences with for .
Definition 2.1.2.
Let from to and from to be two generalized Lagrangian correspondences. Then we define composition
as a generalized Lagrangian correspondence from to . Moreover, we define the dual
as a generalized Lagrangian correspondence from to .
We conclude this subsection by mentioning special cases of generalized Lagrangian correspondences. The first is the case , which we will want to view separately as a cyclic correspondence, without fixing the “base point” .
Definition 2.1.3.
A cyclic generalized Lagrangian correspondence consists of

a cyclic sequence of symplectic manifolds of any length ,

a sequence of compact Lagrangian correspondences with for .
The second special case is , which generalizes the concept of Lagrangian submanifolds. Namely, note that any Lagrangian submanifold can be viewed as correspondence .
Definition 2.1.4.
Let be a symplectic manifold. A generalized Lagrangian submanifold of is a generalized Lagrangian correspondence from a point to . That is, consists of

a sequence of any length of symplectic manifolds with a point and ,

a sequence of compact Lagrangian correspondences .
3. Gradings
The purpose of this section is to review the theory of graded Lagrangians and extend it to generalized Lagrangian correspondences. It can be skipped at first reading.
Following Kontsevich and Seidel [31] one can define graded Lagrangian subspaces as follows. Let be a symplectic vector space and let be the Lagrangian Grassmannian of . An fold Maslov covering for is a covering associated to the Maslov class in . (More precisely, the mod reduction of the Maslov class defines a representation . Let be the universal cover of , then the fold Maslov covering associated to the given representation is the associated bundle .) A grading of a Lagrangian subspace is a lift to .
Remark 3.0.1.

For any basepoint we obtain an fold Maslov cover given as the homotopy classes of paths with base point , modulo loops whose Maslov index is a multiple of . The covering is . The base point has a canonical grading given by the constant path . Any path between basepoints induces an identification .

For the diagonal we fix a canonical grading and orientation as follows. (This choice is made in order to obtain the degree identity in Lemma 3.0.8 (d).) We identify the Maslov coverings and by concatenation of the paths^{2}^{2}2 The first path arises from the canonical path between and in . The second path can be understood as the graphs of the symplectomorphisms on . For this graph converges to the split Lagrangian ; for the graph is the diagonal.
(7) where is an compatible complex structure on (i.e. and is symmetric and positive definite). In particular, this induces the canonical grading on the diagonal with respect to any Maslov covering , by continuation. Any identification induced by a path in maps the graded diagonal to the graded diagonal, since the product of any loop has Maslov index . Similarly, we define a canonical orientation on by choosing any orientation on , giving the product the product orientation (which is well defined), and extending the orientation over the path (7). This is related to the orientation induced by projection of the diagonal on the second factor by a sign , where .
Let be a symplectic manifold and let be the fiber bundle whose fiber over is the space of Lagrangian subspaces of . An fold Maslov covering of is an fold cover whose restriction to each fiber is an fold Maslov covering . Any choice of Maslov cover for induces a onetoone correspondence between fold Maslov covers of and structures on . Here and is the fold covering group of associated to the mod reduction of the Maslov class in . (Explicitly, this is realized by using the identity as base point.) An structure on is an bundle together with an isomorphism . It induces the fold Maslov covering to the symplectic frame bundle of
The notions of duals, disjoint union, and Cartesian product extend naturally to the graded setting as follows. The dual of a Maslov covering is the same space with the inverted action. We denote this identification by
(8) 
For structures and the embedding
induces an structure on the product and an equivariant map
(9) 
covering the inclusion . The corresponding product of fold Maslov covers on is the fold Maslov covering
Combining this product with the dual yields a Maslov covering for which we can identify with
Finally, the inclusion lifts to a map
(10) 
with fiber . It is defined by combining the product (9) with the basic product of the linear Maslov cover .
Definition 3.0.2.

Let , be two symplectic manifolds equipped with fold Maslov covers and let be a symplectomorphisms. A grading of is a lift of the canonical isomorphism to an isomorphism , or equivalently, a lift of the canonical isomorphism of symplectic frame bundles to an isomorphism .

Let be a Lagrangian submanifold and be equipped with an fold Maslov cover. A grading of is a lift of the canonical section .
Remark 3.0.3.

The set of graded symplectomorphisms forms a group under composition. In particular, the identity on has a canonical grading, given by the identity on .

Given a oneparameter family of symplectomorphisms with , we obtain a grading of by continuity.

Any choice of grading on the diagonal induces a bijection between gradings of a symplectomorphism and gradings of its graph with respect to the induced Maslov cover . Indeed, the graph of the grading, with structure group , . The graded diagonal descends under the associated fiber bundle construction with to a section of lifting . Moreover, this construction is equivariant for the transitive action of on both the set of gradings of and the set of gradings of . is a principal bundle over
We will refer to this as the canonical bijection when using the canonical grading in Remark 3.0.1. In particular, the diagonal in has a canonical grading induced by the canonical bijection from the canonical grading of the identity on .

Any grading of a Lagrangian submanifold induces a grading of via the diffeomorphism .

Given graded Lagrangian submanifolds , the product inherits a grading from (10).

Given a graded symplectomorphism and a graded Lagrangian submanifold , the image inherits a grading by composition .
Example 3.0.4.

Let be the bundle whose fiber over is the space of oriented Lagrangian subspaces of . Then is a fold Maslov covering. A grading of a Lagrangian is equivalent to an orientation on .

By [31, Section 2], any symplectic manifold with and minimal Chern number admits an fold Maslov covering iff divides . Any Lagrangian with minimal Maslov number admits a