Wrapped Floer cohomology and Lagrangian correspondences
We study Lagrangian correspondences between Liouville manifolds and construct functors between wrapped Fukaya categories. The study naturally brings up the question on comparing two versions of wrapped Fukaya categories of the product manifold, which we prove quasi-isomorphism on the level of wrapped Floer cohomology. To prove representability of these functors constructed from Lagrangian correspondences, we introduce the geometric compositions of Lagrangian correspondences under wrapping, as new classes of objects in the wrapped Fukaya category, which we prove to represent the functors by establishing a canonical isomorphism of quilted version of wrapped Floer cohomology under geometric composition.
- 1 Introduction
- 2 Geometry of Liouville manifolds and Lagrangian submanifolds
- 3 Wrapped Floer cohomology
- 4 Wrapped Floer cohomology for Lagrangian correspondences
- 5 Functoriality in Lagrangian correspondences
- 6 Geometric compositions and representability
- 7 From split Hamiltonians to admissible Hamiltonians
Lagrangian Floer theory [FOOO1] has laid the foundation from the symplectic geometry side for Kontsevich’s celebrated Homological Mirror Symmetry Conjecture, which predicts certain equivalence between A-model category (the Fukaya category) and B-model category (derived category of coherent sheaves) for a mirror pair of Calabi-Yau manifolds. The story has been extended to more general situations, including Fano manifolds, varieties of general type as symplectic manifolds (for example the genus two curve in [Seidel]), and certain non-compact symplectic manifolds (the punctured spheres in [Abouzaid-Auroux-Efimov-Katzarkov-Orlov]). The corresponding B-model categories are typically triangulated category of singularities of a Landau-Ginzburg potential.
Funtoriality properties are well established for B-model categories, usually defined in algebro-geometric terms, for example derived pushforward/pullback (of sheaves) associated to morphisms between the base varieties. Further thought on homological mirror symmetry should provide some clue to understanding functorial properties of Fukaya categories, and possibly proving functoriality of the homological mirror symmetry itself. However, there are not many morphisms between symplectic manifolds realized by maps in the usual sense, except for the very restrictive classes: symplectomorphisms, coverings and embeddings. To understand functoriality of A-model categories, it is necessary to change the viewpoint from morphisms to correspondences; the natural sources are Lagrangian correspondences between symplectic manifolds.
In [Wehrheim-Woodward2], Wehrheim and Woodward study functors of Fukaya categories associated to Lagrangian correspondences, in the case of compact monotone symplectic manifolds and compact monotone Lagrangian correspondences. One of their main results is that any compact monotone Lagrangian correspondence from to gives rise to an -functor from the compact monotone Fukaya category of to the dg-category of -modules over (technically, on the level of cohomology categories). It is expected that Lagrangian correspondences should give rise to -functors between compact unobstructed Fukaya categories, improving the above-mentioned result. Moreover, with a lot of recent developments, the story is expected to be extended to full generality for all unobstructed compact Lagrangian correspondences. Part of the main ideas are discussed in [Fukaya2], by using Lagrangian Floer theory for immersed Lagrangians, which is developed in [Akaho-Joyce].
In the case of certain non-compact symplectic manifolds which satisfy appropriate convexity conditions, there is a variant of Fukaya category, called the wrapped Fukaya category, whose objects also include certain class of non-compact Lagrangian submanifolds, and whose morphisms spaces. This is introduced in [Abouzaid-Seidel], [Abouzaid1] for Liouville manifolds. The objects of the wrapped Fukaya category of a Liouville manifold are closed exact Lagrangian submanifolds in the compact part of , as well as non-compact exact Lagrangian submanifolds that are invariant under the Liouville flow over the cylindrical ends, called conical Lagrangian submanifolds. Compared to the Fukaya category of closed Lagrangian submanifolds in , the wrapped Fukaya category captures additional information about Reeb dynamics on the contact manifold at infinity. From the standpoint of homological mirror symmetry, the wrapped Fukaya category seems more suitable for the role of A-model category for a non-compact symplectic manifold (see [Abouzaid-Auroux-Efimov-Katzarkov-Orlov]) which is mirror to coherent sheaves on the mirror variety/family.
Also, there are many interesting affine varieties that are related by quotients or birational transformations. Some of these affine varieties appear as the divisor complements of log Calabi-Yau pairs [Auroux], [Ganatra-Pomerleano], [Gross-Hacking-Keel], [Pascaleff], for which the structure of the symplectic cohomology and the wrapped Fukaya category can be understood well to some extend. This motivates us to find a symplectic analogue of the relations among these affine varieties in terms of Lagrangian correspondences, and try to understand how their wrapped Fukaya categories are related.
In this series of reseach, we begin by concerning the foundational matters regarding -functors between wrapped Fukaya categories from the viewpoint of Lagrangian correspondences. The very first step is to adapt the quilted Floer cohomology [Wehrheim-Woodward1], [Wehrheim-Woodward2] developed by Wehrheim-Woodward to our setting. For each admissible Lagrangian correspondence in the sense of wrapped Floer theory (Definition 3.9), the principle is that we should be able to construct a canonical -functor between wrapped Fukaya categories. To carry out the construction, we introduce a quilted version of wrapped Floer cohomology, whose definition is a straightforward modification of quilted Floer cohomology [Wehrheim-Woodward1] with large Hamiltonian perturbation. However, there is one essential difference: wrapped Floer theory in the product manifold involves certain class of non-compact Lagrangian submanifolds, and there are certain issues with what objects can be and should be included. Such problems require careful inspection and will be main portion of our discussion.
In this paper, we focus on foundational matters, while staying on the cohomology-level. Chain-level refinements and the full -structures will be dealt with in the upcoming work [Gao]. Further applications will also be discussed there.
1.1. Floer theory in product manifolds
The story of Lagrangian correspondences begins with studying Floer theory in product manifolds. There are several technical issues with wrapped Floer theory for the product symplectic manifold . First, the standard definition of wrapped Floer cohomology depends crucially on the convexity property of the symplectic manifold. However, the product of two convex symplectic manifolds might no longer be convex. It is the reason for which we consider only Liouville manifolds, which behave nicely under products - the product of two Liouville manifolds is again a Liouville manifold, and is therefore convex at infinity. Second, for the usual wrapped Floer theory to work, we need to make a choice of a cylindrical end for . There is a natural choice, as observed in [Oancea], which will be described in section 2.2. Third, the sum of the chosen two admissible Hamiltonian functions , which we call the split Hamiltonian, is a priori not admissible. There is a similar issue with the product almost complex structure . This is the main obstacle to studying Floer theory in the product Liouville manifold, and has been an annoying issue for quite a while. One of the main results of this paper is to prove well-definedness of wrapped Floer cohomology in the product manifold and show the following invariance property:
Suppose are admissible Lagrangian submanifolds of , i.e. they are either product Lagrangian submanifolds, or conical Lagrangian submanifolds with respect to the cylindrical end . The wrapped Floer cohomology with respect to the split Hamiltonian and the product almost complex structure
and the one defined using an admissible admissible and an admissible almost complex structure with respect to the cylindrical end
are both well-defined.
Moreover, given a split Hamiltonian and a product almost complex structure , there is an admissible Hamiltonian and an admissible almost complex structure with respect to the cylindrical end , as well as a cochain map
which respects the action filtration and induces an isomorphism on cohomology.
A problem of the same nature was considered in [Oancea] in which he studied the Künneth formula for symplectic cohomology. We adapt his strategy and extend the argument to the case involving Lagrangian submanifolds. However, there is a small technical difference. In [Oancea] the definition of symplectic cohomology uses linear Hamiltonians, so the split Hamiltonian is indeed admissible; however, symplectic cohomology defined using linear Hamiltonians involves a limit, and it is not clear that the double iterated limit is equivalent to a single limit. For the purpose of defining a quilted version of wrapped Floer cohomology and carrying out the chain level construction of -structure in the wrapped Fukaya category, it is more convenient to work with quadratic Hamiltonians, which do not involve taking limits. But the difficulty is transferred to the non-admissibility of split Hamiltonians, which we resolve in this paper. In addition to that, we also study multiplication structures on wrapped Floer cohomology and prove the isomorphism in Theorem 1.1 intertwines multiplication structures.
The action-restriction map preserves the multiplication structure on wrapped Floer cohomology. More concretely, there exists a cochain map
of degree such that the following diagram
is homotopy commutative, with the cochain homotopy between the two possible compositions precisely given by .
An immediate corollary is the Künneth formula for wrapped Floer cohomology:
Let and be conical Lagrangian submanifolds. Then the wrapped Floer cohomology
is well-defined with respect to any admissible Hamiltonian and any admissible almost complex structure with respect to the cylindrical end of . And there is a quasi-isomorphism
where the differential on the left-hand side is the tensor product differential.
The statement of Künneth formula can be improved to an -version, identifying the -tensor product of wrapped Fukaya categories with that of the product manifold , under additional assumptions. In particular, all the -structure maps are suitably packaged, and this equivalence respects such structures. Such improvement will be discussed in [Gao].
One important output of the study of wrapped Floer theory in the product is a quilted version of wrapped Floer cohomology for Lagrangian correspondences, discussed in section 4. The quilted wrapped Floer cohomology is defined for the usual class of Lagrangian submanifolds and an admissible Lagrangian correspondence . It allows us to build an -functor from to , the dg-category of left -modules over , associated to an admissible Lagrangian correspondence . While leaving the -structures in the upcoming paper [Gao], we carry out the construction on the level of cohomology categories in section 5.
Associated to each admissible Lagrangian correspondence , quilted wrapped Floer cohomology gives rise to a functor
to the category of left-modules over . Moreover, the construction is functorial in the product manifold , meaning that this can be improved to a functor
to the category of functors. In addition, both functors are cohomologically unital.
Attempting to obtain a functor to the honest wrapped Fukaya category , we ask for representability of these functors, in the sense of [Fukaya1]. The natural candidate to represent the module-valued functor is the geometric composition of Lagrangian correspondences. However, the geometric composition is not in general a conical Lagrangian submanifold/correspondence even if we assume it is embedded. Thus well-definedness of the wrapped Floer cohomology of the geometric composition is an essential difficulty. In section 6.2 we will explain well-definedness of wrapped Floer cohomology of the geometric compositions, and prove isomorphism of wrapped Floer cohomology under geometric composition.
Suppose we are given Lagrangian submanifolds , as well as an admissible Lagrangian correspondence . Suppose that the geometric composition is a properly embedded Lagrangian submanifold of . Then:
There is a well-defined wrapped Floer cohomology group , whose underlying cochain complex and differential are defined as usual;
There is a quasi-isomorphism
Based on these results, we are going to prove the representability of the module-valued -functors associated to Lagrangian correspondences in the upcoming work [Gao], in which certain non-compact immersed Lagrangian submanifolds are taken care of, and chain-level -structures are constructed. This, combined with an -analogue of Yoneda lemma, will imply that we may regard these -functors as landing in , up to quasi-isomorphism, as long as we allow certain immersed Lagrangian submanifolds as objects.
Naturally, pushing the story one step further, we may want to study the compositions of these functors as we have introduced geometric compositions of Lagrangian correspondences. However, unlike in the case of compact Lagrangian submanifolds, wrapped Floer theory in multiple products of Liouville manifolds encounters more serious difficulty regarding classes of objects to be included, which have not yet been overcome and can be topics of future research.
This work is part of a project during the author’s PhD studies at Stony Brook University. The author is grateful for his advisor Kenji Fukaya for guidance and support, sharing his ideas in Lagrangian correspondences and motivating the project. The author would also like to thank Mark McLean for sharing his knowledge on symplectic cohomology, and helpful conversations regarding certain technical issues in this work.
2. Geometry of Liouville manifolds and Lagrangian submanifolds
2.1. Liouville manifolds
Let be a Liouville manifold, that is, is symplectic, and the vector field defined by
generates a complete expanding flow.
For practical purposes, we assume that is the completion of a Liouville domain . That is, inside there is an open submanifold whose closure is a compact manifold with boundary, such that points outward near , and the flow of identifies with the positive symplectization , where is equipped with the contact form .
By abuse of notation, we write , and also
Denote by the radial coordinate on , so that over the cylindrical end, . In such a case, we shall sometimes call the interior part of (only in this paper), although this is not a standard terminology.
Let us make the following assumption on the Liouville form :
All periodic Reeb orbits of on are non-degenerate.
This is a generic condition, and for such a choice of -form , there will be only finitely many Reeb chords on shorter than any given constant.
2.2. Product Liouville manifolds
As noted before, the product of two Liouville manifolds is again a Liouville manifold, carrying the product symplectic form , the product Liouville form , and the product Liouville vector field . Suppose that we have chosen cylindrical ends for and individually. Then there is a natural choice of cylindrical end for , as observed in [Oancea].
We consider the following three subsets of :
We still denote the radial coordinate on by , and that on by . Then can be thought of as functions on these regions (via pull back). Let be a hypersurface that is transverse to the Liouville vector field , such that:
for some . The reason that we choose to be bigger than is because we do not want the hypersurface to have corners (note has a corner because it is the boundary of the product of two manifolds with boundary).
Let be the time- Liouville flow on , and be the time- Liouville flow on . Let denote the compact part that separates from the non-compact part of . Define a cylindrical end of by the following parametrization:
2.3. Lagrangian submanifolds
The Lagrangian submanifolds we are going to consider are either exact closed Lagrangian submanifolds in the interior of or non-compact exact Lagrangian submanifolds that intersect transversely with boundary being Legendrian submanifolds in , such that over the cylindrical end, is of the form . We call such a non-compact Lagrangian submanifold a conical Lagrangian submanifold. These two kinds of Lagrangian submanifolds are said to be admissible. We will mostly focus on conical Lagrangian submanifolds since only they involve non-trivial wrapping.
More generally, we will allow some additional Lagrangian submanifolds in the wrapped Fukaya category, and also call them admissible, in the sense that wrapped Floer cohomology for those Lagrangian submanifolds are well-defined. See section 6.2.
For a conical Lagrangian submanifold , we choose a primitive for the restriction of to , i.e. . Since vanishes on , we can choose such that on the cylindrical end of the function is locally constant, in particular independent of the radial coordinate .
2.4. Spin structures and gradings
The coefficients for the wrapped Floer cohomology groups of a pair of Lagrangian submanifolds will only be if we do not impose any conditions on the Lagrangian submanifolds. In order to have coefficients being , we need to study orientations on various moduli spaces of pseudoholomorphic disks used in the definition of Floer cochain complexes. It is by now standard that a choice of spin structures on relevant Lagrangian submanifolds determines coherent orientations on all moduli spaces of (inhomogeneous) pseudoholomorphic disks. For a proof, see Chapter 8 of [FOOO2] for the compact case, and section 11 of [Seidel] for the exact case.
The relevant Floer cohomology groups a priori carry only a -grading. If we desire a -grading, we need a trivialization of the square of the anti-canonical bundle of , as well as gradings on Lagrangian submanifolds. A choice of gradings on relevant Lagrangian submanifolds determines -valued gradings on generators of various Floer cochain complexes, i.e. integral lift of Maslov indices of Hamiltonians chords between a pair of Lagrangian submanifolds. For this matter, also see section 11 of [Seidel].
Let us make the following assumption on the Lagrangian submanifolds in consideration:
Under Assumption 2.3, admits both spin structures and gradings. We will fix a choice of spin structure and grading for every Lagrangian submanifold that we are going to look at, and will not repeat this later. The reader is referred to [Seidel] or [Abouzaid1] for a discussion on the orientations on the moduli spaces of inhomogeneous pseudoholomorphic disks and the signs they determine.
2.5. Hamiltonian functions
We will work with a restricted class of Hamiltonian functions which have rigid behaviour near infinity.
A (time-independent) Hamiltonian is called admissible, if over the cylindrical end , depends only on the radial coordinate, and is quadratic in the radial coordinate outside a compact set containing the interior , i.e. takes the form for for some number . Denote by the space of admissible Hamiltonians.
The reason that we use time-independent Hamiltonians to setup wrapped Floer theory is that it simplifies many estimates required for proving compactness results for the relevant moduli spaces, in particular when we define the quilted version of wrapped Floer cohomology (sections 3.4, 4.3), and introduce geometric compositions into wrapped Floer theory (section 6.2).
In practice, we may use a Hamiltonian that is quadratic for , and is -small in the interior of , takes values in there with the -norm of in the interior of also being in . Additionally, on a collar neighborhood of in , in particular at the level , we require the Hamiltonian takes values between . Moreover, we require the derivative of with respect to be less than , namely does not grow too fast in . The additional requirements on admissible Hamiltonians will not affect the resulting Floer theory, up to quasi-isomorphism, by a standard continuation argument. This works because it suffices to interpolate the two almost complex structures on a compact set, so the usual -estimates hold.
These additional conditions imply the following lemma, the proof of which is a straightforward calculation.
Let be an admissible Hamiltonian satisfying the additional requirements described above, and its Hamiltonian vector field. The restriction of on each level hypersurface is times the Reeb vector field. And for each time- -chord , we have
If lies in the interior of , then
If is a Reeb chord on the level hypersurface for some , then
If is a Reeb chord on the level hypersurface for some , then
Now let be two conical Lagrangian submanifolds of . We may also make the following assumption on the admissible Hamiltonian that we are going to work with, which can be achieved in a generic situation.
All time-one chords of from to are non-degenerate.
2.6. Almost complex structures
One of the technical advantages of Liouville manifolds in studying pseudoholomorphic curves and studying Floer theory is that they are convex for a natural class of almost complex structures. These almost complex structures are compatible with the symplectic structures, and over the cylindrical end transform the Liouville vector field to Reeb vector fields on level hypersurfaces , making as a convex boundary of . Furthermore, the restriction of such an almost complex structure to the hyperplane distribution of the contact structure on each level hypersurface induces an almost complex structure compatible with the symplectic structure on that hyperplane distribution induced by the differential of the contact form. We call these almost complex structures of contact type. This can be made concise by saying that is of contact type if it is compatible with the symplectic structure and over the cylindrical end it satisfies
In fact, this condition can be loosen to the one which only requires the almost complex structure to be of contact type away from a compact set. The Floer theory defined using almost complex structures of the latter type will be quasi-isomorphic to the one defined using the previous ones, by a standard continuation argument, for the reason similar to that with Hamiltonians. We call this larger class of almost complex structures admissible, and denote the space of all admissible almost complex structures on by .
2.7. Floer’s equation over the strip
Suppose we are given an admissible Hamiltonian as well as a one-parameter family of almost complex structures of contact type parametrized by . For conical Lagrangian submanifolds of , and two time-one chords of from to , we can consider Floer’s equation:
In the third equation above, we require the convergence is exponentially fast in . This is equivalent to the condition that the solutions have finite energy. We denote by the set of solutions to Floer’s equation as above, and call it the (parametrized) moduli space of inhomogeneous pseudoholomorphic strips (which we also call Floer trajectories) between and . Since the equation is invariant under translation in the -variable, there is an -action on , which is free whenever the solutions are not constant maps. We denote by the quotient , and call it the (unparametrized) moduli space. The following lemma regarding transversality is standard. However, since we are using a time-independent Hamiltonian , there does not seem to be a clear proof in the literature, so we include one here.
Suppose that . For a generic one-parameter family of admissible almost complex structures , the moduli space is a smooth manifold whose dimension is equal to . If , for a generic one-parameter family of admissible almost complex structures and a generic one-parameter family of Hamiltonians, the moduli space is a smooth manifold also.
There are three cases to consider.
The moduli space consists of trivial solutions, i.e. those satisfying identically. By a traditional trick of Gromov, this corresponds to a constant pseudoholomorphic section of some locally trivial Hamiltonian fibration with respect to a distinguished almost complex structure determined by and . For these solutions, the linearized Cauchy-Riemann operator is the standard Cauchy-Riemann operator (perturbed by a constant vector) with linear Lagrangian boundary conditions, which is then surjective because the domain has genus zero.
One of is a Hamiltonian chord in the interior of , and is non-trivial. In this case, due to the convexity, has to have image entirely contained in the interior of , in particular the other chord also has to be inside the interior. Over there we are freely allowed to perturb , so is not difficult to show that the linearization of the universal Cauchy-Riemann operator (including the almost complex structure as a variable) is surjective. To set it up, let and consider the universal Cauchy-Riemann operator as a section of the Banach bundle , where
and the fiber of over is
The inhomogeneous Cauchy-Riemann operator is then regarded as a section, denoted by , which sends to , where is the fixed complex structure on the domain . The linearization of this is then the following Fredholm operator:
where is the tangent space of the space of compatible almost complex structures, and is the inhomogeneous Cauchy-Riemann operator for the single and . We want to prove that the universal linearized operator is surjective. Suppose the contrary. Then there exists a nonzero , which is automatically smooth by elliptic regularity, and vanishes at most at a discrete set of points, such that
But since is not a trivial solution, vanishes at most at a discrete set of points on the domain. In particular, we can choose such that at some point which is not in the union of these two discrete sets,
This is a contradiction, which implies that is surjective. Considering the projection , we get the conclusion by applying Sard-Smale theorem.
Both are Reeb chords contained in level hypersurfaces of the cylindrical end , and they are contained in different levels, say and . In this case, we need a SFT-type transversality argument. Over the cylindrical end the almost complex structure can be assumed to be of contact type, which means that it is the direct sum of the almost complex structure on the contact hyperplane distribution of the contact manifold with the standard conjugate complex structure on , where the first factor is the Reeb direction. Therefore, writing we can split this equation to a SFT-type system of equations:
And there are boundary conditions for : the two boundary components lie in the Lagrangian subbundles of which are the tangent bundles of the Legendrian submanifolds and respectively.
Suppose now . Then the first equation is non-trivial, and is a Cauchy-Riemann equation on the contact hyperplane distribution with Lagrangian boundary conditions, and can be any almost complex structure compatible with the symplectic form on , hence the linearized operator can be made surjective by a generic choice of . Note this is true even if the projection is constant (namely when is contained in the Reeb direction), because that reduces to the usual linear Cauchy-Riemann equation with linear Lagrangian boundary conditions.
In order to prove that the linearization of the full equation can be made surjective by generic choice of , we need to show that the component of the linearization in the -direction is indeed regular, because writing down the linearization explicitly shows that there is nothing to perturb. Let us try to solve the equation
Let us write as the decompositon according to the decomposition of the tangent bundle of . As indicated above, the component in can be made surjective, so we obtain some solution of Sobolev class , which is in fact in for any by elliptic bootstrapping. The remaining component of the equation has the form:
where are functions of depending only on the contact structure on , as well as on the point where we linearize the equation, and are in fact polynomial in , but do not depend on the derivatives of it. For , decays exponentially in , because of the Sobolev condition. By elementary method in PDE (for example using Fourier series), the only solution has exponential growth with the same rate at both and , and hence is not of class for any .
As a consequence, the unparametrized moduli space is a smooth manifold of dimension whenever the -action is free, otherwise the parametrized moduli space consists of constant maps and is regarded as empty.
We introduce a (partial) compactification of by adding strata consisting of broken strips:
Because the symplectic manifold is exact, and the Lagrangian submanifolds are exact, there cannot be any sphere bubbling or disk bubbling. The maximum principle implies that this is indeed a compactification. However, the method of proof we are going to use is slightly different, though essentially equivalent.
For a generic one-parameter family of admissible almost complex structures , is a compact manifold with boundary and corners of dimension . The codimension one boundary stratum is covered by images of the natural inclusions
This is a direct consequence of the action-energy equality, which expresses the energy of a pseudoholomorphic strip between two chords in terms of the action of these two chords:
It follows that there cannot be pseudoholomorphic strips escaping to infinity, because the energy of the pseudoholomorphic strips between given two chords is fixed by the action of them. On the other hand, once and are fixed, there can only be finitely many non-trivial broken strips, because each non-trivial pseudoholomorphic strip picks up some energy which is positive and uniformly bounded from below by a constant which depends only on the background geometry - the symplectic manifold, the Lagrangian submanifolds and the Hamiltonian function. Moreoever, there cannot be sphere bubbles or disk bubbles as mentioned before. ∎
However, the set of -chords between and is in general infinite. But still we have the following finiteness result, which also follows from the action-energy equality.
For each time-one -chord from to , the compactified moduli space is empty for all but finitely many chords .
The above two lemmata are the key geometric ingredients that ensure finiteness in the definition of Floer’s differential, the statements and most of the ideas of which, except the SFT-type argument, are learned from [Abouzaid1], [Abouzaid-Seidel].
3. Wrapped Floer cohomology
3.1. Basic definition
Let us briefly recall the definition of wrapped Floer cohomology, basically taken from [Abouzaid1]. We include it here for the convenience of the reader, and use it to fix notations. Given a pair of admissible Lagrangian submanifolds , we define a graded -module:
where is the set of all time-one -chords from to , and is the canonical orientation line associated to the chord , (see [Seidel]).
This as a graded module is in fact independent of . We then define a differential on by counting isolated elements in the moduli space of inhomogeneous pseudoholomorphic strips. More precisely, we consider the case where . Using the orientation on the moduli space determined by the chosen spin structures on the Lagrangian submanifolds, we obtain an isomorphism . We denote by the induced map on orientation lines, and define
By Lemma 2.9, the sum is finite. The fact that squares to zero comes from the structure of the boundary of one-dimensional moduli space, described in Lemma 2.8, plus a standard gluing argument. We call the resulting cohomology group the wrapped Floer cohomology of with respect to and denote it by . The wrapped Floer cochain complex does not depend on admissible up to canonical chain homotopy up to higher chain homotopies, the proof of which uses continuation maps. Thus the wrapped Floer cohomology is independent of admissible pairs .
From now on, when we define various kinds of moduli spaces, we always start with one-parameter families of almost complex structures that are of contact type. Instead of emphasizing this point every time, we will only denote them as , etc. But these symbols should be understood as one-parameter families of almost complex structures parametrized by , unless otherwise specified.
3.2. Multiplicative structure
We review the multiplicative structure on wrapped Floer cohomology. The chain level operation (which is in general not necessarily associative)
is defined to be the composition of
where the first map is defined by counting inhomogeneous pseudoholomorphic triangles, and the second map is induced by conjugating by the Liouville flow, which is an isomorphism. Both maps are cochain maps. See [Abouzaid1] for the original construction of all -structure maps, but let us review some of the definitions to fix terminology and notations we shall use.
Let be the time- Liouville flow. We first consider the map (3.5), whose definition uses pseudoholomorphic triangles. Let be a disk with three boundary punctures where is a negative puncture while the other two are positive punctures. We then need to choose a smooth function which is near and is near . We call such a function a time-shifting function. Let denote the space of admissible Hamiltonians on , and the space of admissible almost complex structures on . The definition of the inhomogeneous pseudoholomorphic triangles also involves additional data as described below.
A Floer datum for consists of
A basic one-form on , which is closed () and vanishes on the boundary of , and whose pullback by is if , and if ;
A -dependent family of admissible Hamiltonians on , i.e. a smooth map , whose pullback under agrees with if , and with if ;
A -dependent family of admissible almost complex structures on , i.e. a smooth map , whose pullback under agrees with if , and with if .
Let be the moduli space of inhomogeneous pseudoholomorphic maps which satisfies the following conditions:
In the above equation, the -part of the differential is defined with respect to the complex structure on and the -parametrized family of almost complex structures on . We have the following transversality result :
For a generic family of admissible almost complex structures as part of the Floer datum defined in Definition 3.2, the moduli space is a smooth manifold of dimension .
If one of the chords lie in the interior of , then the proof argues with the linearization of the universal Cauchy-Riemann operator, in which is regarded as a variable. It is not difficult to show that this universal Fredholm operator is surjective, essentially because we are allowed to perturb in the interior of in an arbitrary way (unlike the case that has to be of contact-type over the cylindrical end), then a Sard-Smale argument finishes the proof. We omit the detail here, which is in fact the same as the proof given for Lemma 2.7.
If all the chords lie in the cylindrical end where the almost complex structures are only allowed to be perturbed in the space of admissible (or more restrictively contact-type) almost complex structures, it is not clear a priori the universal Fredholm operator is still surjective. For this we need a SFT-type as in the proof of Lemma 2.7. Since all the three chords lie in the cylindrical end, the inhomogeneous pseudoholomorphic triangle must be also contained in the cylindrical end, because the collar neighborhood of in the interior of is concave. So in the inhomogeneous Cauchy-Riemann equation that satisfies, the almost complex structure is of contact type, which means that it is the direct sum of the almost complex structure on the contact hyperplane distribution of the contact manifold with the standard conjugate complex structure on , where the first factor is the Reeb direction. Therefore, writing we can split this equation to a SFT-type system of equations:
Here is the projection to the contact hyperplane distribution, and is the projection to the Reeb direction. Following the same kind of SFT-type argument as in the proof of Lemma 2.7 by going to strip-like ends, we obtain the desired conclusion. In this case, that is the domain is a disk with three or more boundary punctures, the proof even also works directly for dimension two. This is because we have used a generic domain-dependent family of almost complex structures that can vary freely away from the strip-like ends, and consequently multiply-covered solutions will never appear. ∎
Since there cannot be sphere bubbles or disk bubbles, the moduli space has a natural compactification whose codimension one boundary stratum is covered by
is a compact manifold, because we have a similar result to Lemma 2.9, which in this case says that for a fixed pair , the moduli space is empty for all but finitely many .
3.3. The cohomological unit
It is well-known that cohomology-level multiplication on wrapped Floer cohomology has a unit. For the convenience of the reader, and also for the purpose of using the representative explicitly later in the proof of Theorem 1.5, we include a description here. To save notations we consider the case of a single Lagrangian submanifold , and the multiplication:
The unit for the multiplication is the image of under the canonical map
called the Piunikhin-Salamon-Schwarz (PSS) homomorphism, which is defined as follows. Consider the disk with one negative boundary puncture . Choose a strip-like end near the puncture. Choose a family of Hamiltonians depending on , which agrees with over the strip-like end, and vanishes in a neighborhood of , and moreover away from this neighborhood is admissible for all . Also, choose a family of admissible almost complex structures depending on which agrees with over the strip-like end. We will see that admissibility condition is in fact irrelevant. Given an -chord , and a locally finite chain (we fix a smooth triangulation of at the beginning) which can be taken to be represented by an oriented piecewise linear submanifold, we consider the parametrized inhomogeneous Cauchy-Riemann equation:
for smooth maps . Note that although does not extend across the point , the Hamiltonian vanishes in a neighborhood of in , so the above equation still makes sense. Alternatively, one may use a different one-form on , which is only sub-closed, and vanishes on the boundary. This is a minor issue which will not affect the definition of the cochain map, up to homotopy. Let be the moduli space of solutions to the equation (3.12). The following is a transversality result for this moduli space, whose proof is similar to that of Lemma 2.7.
For a generic family of almost complex structures , the moduli space is a piecewise linear manifold of dimension .
The only difference from the conclusion by standard transversality argument is that our moduli space is only a piecewise linear manifold. This is because without the condition , the resulting moduli space is a smooth manifold, and we perturb the almost complex structures further such that the evaluation map from that moduli space at the boundary marked point is transverse to the piecewise linear submanifold . Of course, if were represented by a smooth submanifold, the moduli space would be a smooth manifold. But there are obstructions to representing integral homology classes by smooth submanifolds. ∎
If we are only concerned with the cohomological unit, we only have to consider the case where is the (locally finite) fundamental chain of , which is represented by the smooth manifold itself. In that case, the resulting moduli space is a smooth manifold, after generic perturbation.
Using a Morse complex for certain Morse function on , or pseudocycles to compute the cohomology would resolve the problem of the moduli space not a priori being a smooth manifold. But piecewise linear manifolds are fine for our purpose, because we will only use zero-dimensional and one-dimensional moduli spaces.
There is a natural compactification of the moduli space whose codimension one stratum is covered by:
It can be proved that is a compact piecewise linear manifold with boundary and corners, but the proof is somewhat cumbersome and we do not quite need it. We only need the result for zero-dimensional and one-dimensional moduli spaces, which is straightforward. In particular, the zero-dimensional moduli space is compact, where .
We also have the following finiteness result.
For any locally finite chain of , the compactified moduli space is empty for all but finitely many -chords .
Note that for any , is empty unless the relative homotopy class of the map is trivial, i.e. the same as a constant map , in the space of paths in with the two ends in . Reeb chords on the level hypersurfaces do not have trivial relative homotopy class. Therefore the possible outputs can only be Hamiltonians chords contained in the interior of , and there are finitely many of them. ∎
We consider the zero-dimensional moduli space , namely the case . Each