A wrapped Fukaya category of knot complement
This is the first of a series of two articles where we construct a version of wrapped Fukaya category of the cotangent bundle of the knot complement of a compact 3-manifold , and do some calculation for the case of hyperbolic knots . For the construction, we use the wrapping induced by the kinetic energy Hamiltonian associated to the cylindrical adjustment on of a smooth metric defined on . We then consider the torus as an object in this category and its wrapped Floer complex where is a tubular neighborhood of . We prove that the quasi-equivalence class of the category and the quasi-isomorphism class of the algebra are independent of the choice of cylindrical adjustments of such metrics depending only on the isotopy class of the knot in .
In a sequel [BKO], we give constructions of a wrapped Fukaya category for hyperbolic knot and of algebra directly using the hyperbolic metric on , and prove a formality result for the asymptotic boundary of .
Key words and phrases:Knot complement, wrapped Fukaya category, Knot Floer algebra, horizontal -estimates
- 1 Introduction
- 2 Geometric preliminaries
- 3 A choice of one-form on
- I Construction of a wrapped Fukaya category for
- II -estimates for the moduli spaces
Idea of using the conormal lift of a knot (or a link) in or as a Legendrian submanifold in the unit cotangent bundle has been exploited by Ekholm-Etynre-Ng-Sullivan [EENS] in their construction of knot contact homology and proved that this analytic invariants recovers Ng’s combinatorial invariants of the knot which is an isomorphism class of certain differential graded algebras [Ng]. On the other hand, Floer homology of conormal bundles of submanifolds of a compact smooth manifold in the full cotangent bundle were studied as a quantization of singular homology of the submanifold (see [Oh2], [KO]). Such a construction has been extended in the level by Nadler-Zaslow [NZ], [N] and also studied by Abbondandolo-Portaluri-Schwarz [APS] in its relation to the singular homology of the space of cords of the submanifold.
The present article is the first of a series of two articles where we construct a version of wrapped Fukaya category of of the knot-complement , which is noncompact, as an invariant of knot (or more generally of links) and do some computation of the invariant for the case of hyperbolic knot by relating the (perturbed) pseudoholomorphic triangles in to the hyperbolic geodesic triangles of the base .
For our purpose of investigating the effect on the topology of of the special metric behavior such as the existence of a hyperbolic metric in the complement , it is important to directly deal with the cotangent bundle of the full complement equipped with the wrapping induced by the kinetic energy Hamiltonian of a metric on . On the other hand, we would like to emphasize that the space may not be convex in the sense of [EG] in general towards the direction of horizontal infinity, i.e., towards the direction of the knot in . In particular, it may not carry a Liouville structure when equipped with the tautological Liouville one form. Because of these reasons, to carry out necessary analysis of the relevant perturbed Cauchy-Riemann equation, we need to impose certain tame behavior of the associated Hamiltonian and almost complex structure at infinity of . Because of noncompactness of , the resulting category a priori depends on the Liptschitz-equivalence class of such metrics modulo conformal equivalence and requires a choice of such equivalence class in the construction.
In the present paper, we will consider the restricted class of Hamiltonians that asymptotically coincide with the kinetic energy Hamiltonian denoted by associated Riemannian metric on the complement and the Sasakian almost complex structure associated to the metric . We will need a suitable ‘tameness’ of the metric near the end of complement so that a uniform horizontal bound holds for the relevant Cauchy-Riemann equation with any given tuple of Lagrangian boundary conditions from the given collection of admissible Lagrangians. As long as such a bound is available, one can directly, construct a wrapped Fukaya category using such a metric. It turns out that such a horizontal bound can be proved in general only for the metric with suitable tame behavior such as those with cylindrical ends or with certain type of homogeneous behavior at the end of like a complete hyperbolic metric. We will consider the case of hyperbolic knot in a sequel [BKO] to the present paper. It turns out that is convex at infinity in the sense that there is a -pluri-subharmonic exhaustion function in a neighborhood at infinity of when is a hyperbolic knot. We refer to [BKO] for the explanation of this latter property of the hyperbolic knot.
1.1. Construction of wrapped Fukaya category
The main purpose of the present paper is to define a Fukaya-type category canonically associated to the knot complement . To make our definition of wrapped Fukaya category of knot complement flexible enough, we consider a compact oriented Riemannian manifolds without boundary. (We remark that the case of hyperbolic knot does not belong to this case because the hyperbolic metric on cannot be smoothly extended to the whole .)
For this purpose, we take a tubular neighborhood of and decompose
and equip a cylindrical metric on with . We call such a metric a cylindrical adjustment of the given metric on , and denote by the cylindrical adjustment of on . An essential analytical reason why we take such an cylindrical adjustment of the metric and its associated kinetic energy Hamiltonian is because it enables us to obtain the horizontal -estimates for the relevant perturbed Cauchy-Riemann equation with various boundary conditions. To highlight importance and nontriviality of such -estimates, we collect all the proofs of relevant -estimates in Part 2.
Denote the associated kinetic energy Hamiltonian of on by
The first main theorem is a construction the following functor between two different choices of various data involved in the construction of .
Let be a tubular neighborhood of be given. For any two smooth metrics on , denote by the associated cylindrical adjustments thereof as above. Then there exist a natural quasi-equivalence
for any pair such that . Furthermore its quasi-isomorphism class depends only on the isotopy type of the knot independent of the choice of tubular neighborhoods and other data.
We denote by
any such category . Each category will be constructed on using the kinetic energy Hamiltonian and the Sasakian almost complex structure on of on : In general a Sasakian almost complex structure associated to the metric on a Riemannian manifold is given by
under the splitting via the Levi-Civita connection of .
Then the proof of Theorem 1.1 is relied on construction of various operators, functors and homotopies in the current context of Floer theory on . The general strategy of such construction is by now standard in Floer theory. (See, for example, [Se1].) In fact, our construction applies to any arbitrary tame orientable 3-manifold with boundary and similar computational result applies when the 3-manifold admits a complete hyperbolic metric of finite volume. Our construction is given in this generality. (See Theorem 9.3 for the precise statement.)
In the proofs of Theorems 1.1 and 10.2, we need to construct various functors and homotopies between them which enter in the invariance proofs. We adopt the definitions of them given in [Lef] for the constructions of the functor and the homotopy directly using the continuations of either Hamiltonians or of Lagrangians or others. Construction of functors appear in the literature in various circumstances, but we could not locate a literature containing geometric construction of an homotopy in the sense of [Lef, Se1] for the case of geometric continuations such as Hamiltonian isotopies. (In [FOOO1, FOOO2, F], the notion of homotopy is defined via a suspension model, the notion of pseudo-isotopy, of the chain complex.) Because of these reasons and for the convenience of readers, we provide full details, in the categorical context, of the construction homotopy associated to a continuation of Hamiltonians in Section 8. Our construction of functor is the counterpart of the standard Floer continuation equation also applied to the higher maps with . It turns out that actual constructions of the associated homotopy as well as of functors are rather subtle and require some thought on the correct moduli spaces that enter in definitions of functors and homotopies associated to geometric continuations. (See Subsection 7.2 and Section 8 for the definitions of relevant moduli spaces.) We also call readers’ attention to Savelyev’s relevant construction in the context of -category in [Sa].
1.2. Construction of Knot Floer algebra
For a concrete computation we do in [BKO], we focus on a particular object in this category canonically associated to the knot. For given tubular neighborhood of , we consider the conormal
and then the wrapped Fukaya algebra of the Lagrangian in . We remark that the wrapped Fukaya algebra of in for a closed manifold can be described by purely topological data arising from the base space, more specifically that of the space of paths attached to in . (See [APS] for some relevant result.) So it is important to consider as an object for to get more interesting knot invariant.
On the other hand, since we restrict the class of our Hamiltonians to that of kinetic energy Hamiltonian associated to a Riemannian metric on , the pair is not a nondegenerate pair but a clean pair in that the set of Hamiltonian chords contains a continuum of constant chords valued at points of .
We denote by the set of Hamiltonian chords of attached to a Lagrangian submanifold in general. We have
where the subindex of in the right hand side denotes the length of the geodesic associated to the Hamiltonian chords of . We note that and the component is clean in the sense of Bott.
where is a cochain complex of , e.g., the de Rham complex and associate an algebra following the construction from [FOOO1].
Let be a tubular neighborhood of and let . The algebra
can be defined, and its isomorphism class does not depend on the various choices involved such as tubular neighborhood and the metric on .
Due to the presence of Morse-Bott component of constant chords, there are two routes toward construction of wrapped Floer complex, which we denote by One is to take the model
where is a chain complex of such as the singular chain complex as in [FOOO1] or the de Rham complex, and the other is to take
where the fiberwise translation of by , suitably interpolated with away from the zero section, for a sufficiently -small compactly supported Morse function such that . We refer readers to [BKO, Section 2] for the detailed explanation on the latter model.
The isomorphism class of independent of then provides a knot-invariant of in for an arbitrary knot . To emphasize the fact that we regard as an object in the cotangent bundle of a knot complement , not as one in the full cotangent bundle , we denote the cohomology group of as follows.
Definition 1.5 (Knot Floer algebra).
We denote the cohomology of by
which carries a natural product arising from map. We call this Knot Floer algebra of .
By letting the torus converge to the ideal boundary of , we may regard as the wrapped Floer cohomology of the ‘ideal boundary’ of the hyperbolic manifolds , which is the origin of the notation we are adopting.
In a sequel [BKO], we introduce a reduced version of the algebra, denoted by , by considering the complex generated by non-constant Hamiltonian chords, and prove that for the case of hyperbolic knot this algebra can be also directly calculated by considering a horo-torus and the wrapped Floer complex of the hyperbolic metric although cannot be smoothly extended to the whole manifold . We also prove a formality result of its structure for any hyperbolic knot . The following is the main result we prove in [BKO].
Theorem 1.6 (Theorem 1.6 [Bko]).
Suppose is a hyperbolic knot on . Then we have an (algebra) isomorphism
for all integer . Furthermore the reduced cohomology for all .
One crucial new ingredient in the proofs of the above theorems is to establish the horizontal estimates of solutions of the perturbed Cauchy-Riemann equation
mainly for the energy Hamiltonian . For this purpose, we have to require the one-form to satisfy ‘co-closedness’
for the inclusion map . This brings the question whether such one-forms exist in the way that the choice is compatible with the gluing process of the relevant moduli spaces. We explicitly construct such a by pulling back a one form from a slit domain. This choice of one form is consistent under the gluing of moduli spaces.
Roughly speaking, this -estimates guarantees that under the above preparation, whenever the test objects are all contained in , the images of the solutions of the Cauchy-Riemann equation is also contained in , i.e., they do not approach the knot . We refer to Part 2 for the proofs of various estimates needed for the construction. Furthermore for the proof of independence of the category under the choice smooth metric , one need to construct an functor between them for two different choices of . Because the maximum principle applies only in the increasing direction of the associated Hamiltonian from to , we have to impose the monotonicity condition
1.4. Further perspectives
Putting the main results of the present paper in perspective, we may regard that our category is a version of partially wrapped Fukaya category on with the ‘stop’ given by
for . This is a 3-dimensional coisotropic submanifold of the asymptotic contact boundary
of which is of 5 dimension. In this regard, our construction can be put into the framework of partially wrapped Fukaya categories. The way how we avoid this stop is by attaching the cylindrical end on along its boundary and considering the kinetic energy Hamiltonian of a cylindrical adjustment of a smooth metric defined on . See Remark 2.3 for the difference between the behaviors of the two Hamiltonian vector fields put in the neighborhood of our coisotropic stop and that associated to Liouville sectors that was introduced by Sylvan [Sy] and Ganatra-Pardon-Shende [GPS]. We note that the resulting wrapped Fukaya categories depend on the type of wrapping imposed near the knot as usual for the general partially wrapped Fukaya categories depending on the wrapping around the stop.
We would also like to compare our approach with that in the arXiv version of [ENS, §6.5] Ekholm-Ng-Shende. They considered a wrapped Fukaya category on the Weinstein manifold denoted by to that is obtained by attaching a punctured handle to along the unit conormal bundle of the knot and altering the Liouville vector field of along . Our approach directly working with , which is well-adapted to the metric structures on the base manifold, will be important for our later purpose of studying hyperbolic knots in a sequel [BKO].
A similar construction of algebra for the conormal of as above can be carried out for the ‘conormal’ of or the micro-support of the characteristic function , which is given by
where is the zero section restricted to and is equipped with boundary orientation of . Combined with the construction of the wrapped version of the natural restriction morphisms constructed in [Oh3], this construction would give rise to morphisms and a natural functor
where and for two tubular neighborhoods of for . Here denotes the Yoneda image of the Lagrangian in general. (Since we will not use this construction in this series, we will leave further discussion elsewhere not to further lengthen the paper.)
On the other hand it is interesting to see that when we are given a exhaustion sequence , the union is a limit of the above mentioned conormal as on in the Gromov-Hausdorff sense. Furthermore the canonical smoothing of given in [KO, Theorem 2.3] can be made to converge to the -family Lagrangian surgery of denoted by in [AENV], [ENS] which consider the case . While construction of Lagrangian requires some geometric restriction such as being fibered (see [AENV, Lemma 6.12]), construction of exact Lagrangian smoothing of can be done for arbitrary knot . It would be interesting to see what the ramification of this observation will be.
In the literature on symplectic geometry, Hamiltonian dynamics, contact geometry and the physics literature, there are various conventions used which are different from one another one way or the other. Because many things considered in the present paper such as the energy estimates, the estimates applying the maximum principle and construction of the Floer continuation map depend on the choice of various conventions, we highlight the essential components of our convention that affect their validity.
The major differences between different conventions in the literature lie in the choice of the following three definitions:
Definition of Hamiltonian vector field: On a symplectic manifold , the Hamiltonian vector field associated to a function is given by the formula
Compatible almost complex structure: In both conventions, is compatible to if the bilinear form is positive definite.
Canonical symplectic form: On the cotangent bundle , the canonical symplectic form is given by
In addition, we would like to take
as our basic perturbed Cauchy-Riemann equation on the strip. Since we work with the cohomological version of Floer complex, we would like to regard this equation as the positive gradient flow of an action functional as in [FOOO2]. This, under our convention laid out above, leads us to our choice of the action functional associated to Hamiltonian on given by
which is the negative of the classical action functional. With this definition, Floer’s continuation map is defined for the homotopy of Hamiltonian for which the inequality is satisfied, i.e., in the direction for which the Hamiltonian is increasing. (See Section 7 for the relevant discussion.)
For the kinetic energy Hamiltonian , we have
where is the Hamiltonian chord associated to the geodesic and is the energy of with respect to the metric .
Acknowledgement: Y. Bae thanks Research Institute for Mathematical Sciences, Kyoto University for its warm hospitality. Y.-G. Oh thanks H. Tanaka for his interest in the present work and explanation of some relevance of Savelyev’s work [Sa] to our construction of homotopy associated to the homotopy of Floer continuation maps.
2. Geometric preliminaries
In this section, we consider the cotangent bundle with the canonical symplectic form
which is nothing but , where is the Liouville one-form . (Our convention of the canonical symplectic form on the cotangent bundle is different from that of [Se2].) Then the radial vector field
In particular its flow satisfies
Therefore is convex at infinity in the sense of [EG].
Let us consider a Riemannian metric of . We denote by
the ‘raising’ and the ‘lowering’ operations associated to the metric . Then we also equip with the metric with respect to the splitting
induced from the Levi-Civita connection of on .
2.1. Tame manifolds and cylindrical adjustments
Let us consider a noncompact tame 3-manifold , which means that there exists an exhaustion , a sequence of compact manifolds with such that
and each is homeomorphic to . It is convenient to consider a compact manifold with boundary such that is homeomorphic to . Let us choose a neighborhood of inside , called an end of . Then there is a homeomorphism , where is called the asymptotic boundary of .
A typical example of tame manifold is a knot complement with an ambient closed 3-manifold . In this case, we can take an exhaustion by simply choosing a sequence of nested tubular neighborhoods of a knot .
Now we focus on the knot complement, i.e. is a 2-dimensional torus , specially denoted by . Although most of statements in this article also work for arbitrary tame 3-manifolds in general, the results in the sequel [BKO] requires the torus boundary condition of in order to exploit hyperbolic geometry using complete hyperbolic metric of finite volume.
We can also consider a more general Riemannian metric of possibly incomplete. For example, if we consider a knot complement , a natural choice of such a metric is a restriction of a smooth metric of a closed ambient manifold .
Definition 2.1 (Cylindrical adjustment).
We define the cylindrical adjustment of the metric on with respect to the exhaustion (4.1) by
for some , which is suitably interpolated on which is fixed.
Here is the coordinate for for the following decomposition
We call the metric an asymptotically cylindrical adjustment of on , which are all complete Riemannian metrics of . We denote in the above decomposition equipped with the cylindrical metric .
We remark that the following property of asymptotically cylindrical adjustments of smooth metrics on restricted to will be important in Section 9 later.
Suppose there is given an exhaustion (2.4). For any choice of two smooth Riemannian metrics and defined on , for every pair of and are Lipschitz equivalent, i.e., there is a constant such that
Let be a tubular neighborhood of . By choosing sufficiently large , we may assume and both are cylindrical outside .
Using the normal exponential map of , we can parameterize the tubular neighborhood of by where parameterizes the knot and is the polar coordinates of at for the metric . Similarly we denote by the coordinates associated to .
More explicitly, we take a normal frame along along . We denote by the normal exponential map along of the metric and by that of . Using this frame, we have an embedding of defined by
and in a similar way. Then the composition map
is well-defined and smooth if we choose a smaller whose size depends only on . In particular, we have
on for some constant . This in particular proves
We recall that the cylindrical adjustment of is defined
in coordinates with the coordinate change and and . Exactly the same formula holds for the cylindrical adjustment of in the coordinates replaced by , those with primes. In particular all the metric coefficients appearing in the formulae are exactly the same for both adjustments. This proves
for all ’s if we choose the tubular neighborhood of sufficient small. This finishes the proof. ∎
It may be worthwhile to examine the behavior of Hamiltonian vector field on as approach to . For this purpose, let be the coordinate system on . The metric can be written as
Define a cylindrical adjustment with respect to the coordinate with for . Ignoring , the associate Hamiltonian of the cylindrical adjustment (for ) is given by
We highlight the last two summands which makes the associated Hamiltonian flow rotates around the torus with higher and higher speed as around the knot . This asymptotic behavior is different from that of the Hamiltonian vector field used in defining the partially wrapped Fukaya category whose stop is given by the Liouville sector whose horizontal component is parallel to the radial vector field .
The above discussion and construction of wrapped Fukaya category can be extended to a general tame manifold whose end may have more than one connected component, as long as we fix a Lipschitz-equivalence class of metrics on . An example of such is the complement where is a link.
2.2. Kinetic energy Hamiltonian and almost complex structures
We now fix a complete Riemannian manifold with cylindrical end. We denote the resulting Rimanninan manifold by .
We denote by the space of smooth functions that satisfies
for a sufficiently large , where is the dual metric of . We call such map on admissible Hamiltonian with respect to the metric . The Hamiltonian vector field on is defined to satisfy
for a given .
We next describe the set of adapted almost complex structures. From now on, we use instead of , when there is no danger of confusion. For each given , the level set
is a hypersurface of contact type in , and the domain
becomes a Liouville domain in the sense of [Se2].
For each , the level set admits a contact from
by the restriction. We denote the associated contact distribution by
Note that the corresponding Reeb flow is nothing but a reparametrization of the -geodesic flow on .
If we denote by the cylindrical coordinates given by
Including the zero section, we have decomposition
To highlight the metric dependence, we use instead of , respectively. On with metric , we have the natural splitting
where is a vector field which generates the -geodesic flow on . We recall that we have a canonical almost complex structure on associated to a metric on defined by
in terms of the splitting with respect to the Levi-Civita connection of . We call this the Sasakian almost complex structure on .
Definition 2.5 (Admissible almost complex structure).
We say an almost complex structure on is -admissible if on and on for the Sasakian almost complex structure associated to for a metric decomposition with with . Denote by the set of -admissible almost complex structures.
All admissible almost complex structures satisfy the following important property which enters in the study of Floer theoretic construction on Liouville manifolds in general.
Let be the energy Hamiltonian on associated metric on the base given above. An almost complex structure on is called contact type if it satisfies .
Appearance of the negative sign here is because our convention of the canonical symplectic form on the cotangent bundle is . Compared with the convention of [AS], our choice of the one-form therein is .
3. A choice of one-form on
In Abouzaid-Seidel’s construction of wrapped Floer cohomology given in [AS, Section 3.7], [A1], they start from a compact Liouville domain with contact type boundary and consider the perturbed Cauchy-Riemann equation of the type
Because of the compactness assumption, their setting does not directly apply to our current cotangent bundle where the tame base manifold is noncompact: To perform analytical study of the relevant moduli spaces, the first step is to establish suitable -estimate.
3.1. Co-closed and sub-closed one-forms
We first recall Abouzaid-Seidel’s construction of what they call a sub-closed one-form in the context of Liouville domain with compact contact-type boundary. For each given , let us consider a Riemann surface of genus zero with -ends. This is isomorphic to the closed unit disk minus boundary points in the counterclockwise way. Each end admits a holomorphic embedding
preserving the boundaries and satisfying , for . We call the distinguished point at infinity a root.
For a given weight satisfying
a total sub-closed one-form is considered in [A2] satisfying
This requirement is put to establish both (geometric) energy bound and (vertical) -bound.
It turns out that in our case of where the base is noncompact, sub-closedness of is not enough to control the behavior of the Floer moduli space of solutions of (3.1): we need the following restriction
3.2. Construction of one-forms
For each given equipped with weight datum satisfying the balancing condition (3.2), we will choose a one-form on that satisfies
For this purpose, we first consider the slit domain representation of the conformal structure whose explanation is in order.
and its gluing along the inclusions of the following rays
for some , . In other words, for a collection , the glued domain becomes
Here, for , means that there exists such that . We may regards
Then there is a conformal mapping
with respect to satisfying that
Let be a connected boundary component of between and , for . Then