A wrapped Fukaya category of knot complement
AA WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT
YOUNGJIN BAE, SEONHWA KIM, YONG-GEUN OH
Abstract.
This is the first of a series of two articles where we construct aversion of wrapped Fukaya category WF ( M \ K ; H g ) of the cotangent bundle T ∗ ( M \ K ) of the knot complement M \ K of a compact 3-manifold M , and dosome calculation for the case of hyperbolic knots K ⊂ M . For the construction,we use the wrapping induced by the kinetic energy Hamiltonian H g associ-ated to the cylindrical adjustment g on M \ K of a smooth metric g defined on M . We then consider the torus T = ∂N ( K ) as an object in this category andits wrapped Floer complex CW ∗ ( ν ∗ T ; H g ) where N ( K ) is a tubular neigh-borhood of K ⊂ M . We prove that the quasi-equivalence class of the categoryand the quasi-isomorphism class of the A ∞ algebra CW ∗ ( ν ∗ T ; H g ) are in-dependent of the choice of cylindrical adjustments of such metrics dependingonly on the isotopy class of the knot K in M .In a sequel [BKO], we give constructions of a wrapped Fukaya category WF ( M \ K ; H h ) for hyperbolic knot K and of A ∞ algebra CW ∗ ( ν ∗ T ; H h )directly using the hyperbolic metric h on M \ K , and prove a formality resultfor the asymptotic boundary of ( M \ K, h ). Contents
1. Introduction 22. Geometric preliminaries 93. A choice of one-form β on Σ 13 Part 1. Construction of a wrapped Fukaya category for M \ K A ∞ structure 196. Construction of A ∞ structure map 217. Construction of A ∞ functor 24 Date : January 2, 2019; Revision on March 10, 2019.
Key words and phrases.
Knot complement, wrapped Fukaya category, Knot Floer algebra,horizontal C -estimates.SK and YO are supported by the IBS project IBS-R003-D1. YO is also partially supportedby the National Science Foundation under Grant No. DMS-1440140 during his residence at theMathematical Sciences Research Institute in Berkeley, California in the fall of 2018. YB waspartially supported by IBS-R003-D1 and JSPS International Research Fellowship Program. a r X i v : . [ m a t h . S G ] M a r YOUNGJIN BAE, SEONHWA KIM, YONG-GEUN OH
8. Homotopy of A ∞ functors 339. Independence of choice of metrics 4210. Construction of Knot Floer algebra HW ( ∂ ∞ ( M \ K )) 44 Part 2. C -estimates for the moduli spaces C estimates for m -components 4612. Vertical C estimates for m -components 4813. C estimates for moving Lagrangian boundary 50Appendix A. Energy identity for Floer’s continuation equation 54Appendix B. Gradings and signs for the moduli spaces 55References 601. Introduction
Idea of using the conormal lift of a knot (or a link) in R or S as a Legendriansubmanifold in the unit cotangent bundle has been exploited by Ekholm-Etynre-Ng-Sullivan [EENS] in their construction of knot contact homology and proved that thisanalytic invariants recovers Ng’s combinatorial invariants of the knot which is anisomorphism class of certain differential graded algebras [Ng]. On the other hand,Floer homology of conormal bundles of submanifolds of a compact smooth manifoldin the full cotangent bundle were studied as a quantization of singular homology ofthe submanifold (see [Oh2], [KO]). Such a construction has been extended in the A ∞ 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 thesubmanifold.The present article is the first of a series of two articles where we construct aversion of wrapped Fukaya category of T ∗ ( M \ K ) of the knot-complement M \ K , which is noncompact , as an invariant of knot K (or more generally of links)and do some computation of the invariant for the case of hyperbolic knot K ⊂ M by relating the (perturbed) pseudoholomorphic triangles in T ∗ ( M \ K ) to thehyperbolic geodesic triangles of the base M \ K . (See [BKO] for the latter.)For our purpose of investigating the effect on the topology of M \ K of thespecial metric behavior such as the existence of a hyperbolic metric in the com-plement M \ K , it is important to directly deal with the cotangent bundle of thefull complement M \ K equipped with the wrapping induced by the kinetic energyHamiltonian of a metric on M \ K . To carry out necessary analysis of the relevantperturbed Cauchy-Riemann equation, we need to impose certain tame behavior ofthe associated Hamiltonian and almost complex structure at infinity of M \ K .Because of noncompactness of M \ K , the resulting A ∞ category a priori dependson the Liptschitz-equivalence class of such metrics modulo conformal equivalenceand requires a choice of such equivalence class in the construction.In the present paper, we will consider the restricted class of Hamiltonians thatasymptotically coincide with the kinetic energy Hamiltonian denoted by H g as-sociated Riemannian metric g on the complement and the Sasakian almost com-plex structure J g associated to the metric g . We will need a suitable ‘tameness’of the metric near the end of complement M \ K so that a uniform horizontal C bound holds for the relevant Cauchy-Riemann equation with any given tuple WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 3 ( L , . . . , L k ) of Lagrangian boundary conditions from the given collection of ad-missible Lagrangians. As long as such a C bound is available, one can directly,construct a wrapped Fukaya category using such a metric. It turns out that such ahorizontal bound can be proved in general only for the metric with suitable tamebehavior such as those with cylindrical ends or with certain type of homogeneousbehavior at the end of M \ K like a complete hyperbolic metric. We will considerthe case of hyperbolic knot K ⊂ M in a sequel [BKO] to the present paper. Itturns out that T ∗ ( M \ K ) is convex at infinity in the sense that there is a J h -pluri-subharmonic exhaustion function in a neighborhood at infinity of T ∗ ( M \ K )when K is a hyperbolic knot. We refer to [BKO] for the explanation of this latterproperty of the hyperbolic knot.1.1. Construction of wrapped Fukaya category WF ( M \ K ) . The main pur-pose of the present paper is to define a Fukaya-type category canonically associatedto the knot complement M \ K . To make our definition of wrapped Fukaya categoryof knot complement flexible enough, we consider a compact oriented Riemannianmanifolds ( M, g ) without boundary. (We remark that the case of hyperbolic knot K ⊂ M does not belong to this case because the hyperbolic metric on M \ K cannotbe smoothly extended to the whole M .)For this purpose, we take a tubular neighborhood N ( K ) ⊂ M of K and decom-pose M \ K = N cpt ∪ ( N ( K ) \ K ) (1.1)and equip a cylindrical metric on N ( K ) \ K ∼ = [0 , ∞ ) × T K with T K = ∂N ( K ). Wecall such a metric a cylindrical adjustment of the given metric g on M , and denoteby g the cylindrical adjustment of g on N ( K ) \ K . An essential analytical reasonwhy we take such an cylindrical adjustment of the metric and its associated kineticenergy Hamiltonian is because it enables us to obtain the horizontal C -estimates for the relevant perturbed Cauchy-Riemann equation with various boundary con-ditions. To highlight importance and nontriviality of such C -estimates, we collectall the proofs of relevant C -estimates in Part 2. As a function on T ∗ ( M \ K ), thekinetic energy of such a metric on M \ K , blows up outside the zero section as oneapproaches to the knot K .Denote the associated kinetic energy Hamiltonian of g on T ∗ ( M \ K ) by H g = 12 | p | g . (1.2)The first main theorem is a construction the following A ∞ functor between twodifferent choices of various data involved in the construction of WF ( M \ K ; H g ). Theorem 1.1.
Let N ( K ) be a tubular neighborhood of K be given. For any twosmooth metrics g, g (cid:48) on M , denote by g , g (cid:48) the associated cylindrical adjustmentsthereof as above. Then there exist a natural A ∞ quasi-equivalence Φ gg (cid:48) : WF ( M \ K ; H g ) → WF ( M \ K ; H g (cid:48) ) for any pair g, g (cid:48) such that Φ gg = id . Furthermore its quasi-equivalence classdepends only on the isotopy type of the knot K independent of the choice of tubularneighborhoods and other data. We denote by WF ( M \ K ) YOUNGJIN BAE, SEONHWA KIM, YONG-GEUN OH any such A ∞ category WF ( M \ K ; H g ). Each A ∞ category WF ( M \ K ) willbe constructed on T ∗ ( M \ K ) using the kinetic energy Hamiltonian H g and theSasakian almost complex structure J g on T ∗ ( M \ K ) of g on M \ K : In general aSasakian almost complex structure J h associated to the metric h on a Riemannianmanifold ( N, h ) is given by J h ( X ) = X (cid:91) , J h ( α ) = − α (cid:93) (1.3)under the splitting T ( T ∗ N ) (cid:39) T N ⊕ T ∗ N via the Levi-Civita connection of h .Then the proof of Theorem 1.1 is relied on construction of various A ∞ opera-tors, A ∞ functors and A ∞ homotopies in the current context of Floer theory on T ∗ ( M \ K ). The general strategy of such construction is by now standard in Floertheory. (See, for example, [Se1].) In fact, our construction applies to any arbi-trary tame orientable 3-manifold with boundary and similar computational resultapplies 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 state-ment.) Remark 1.2.
In the proofs of Theorems 1.1 and 1.3, we need to construct vari-ous A ∞ functors and A ∞ homotopies between them which enter in the invarianceproofs. We adopt the definitions of them given in [Lef] for the constructions of the A ∞ functor and the A ∞ homotopy directly using the continuations of either Hamil-tonians or of Lagrangians or others. Construction of A ∞ functors appear in theliterature in various circumstances, but we could not locate a literature containinggeometric construction of an A ∞ homotopy in the sense of [Lef, Se1] for the case ofgeometric continuations such as Hamiltonian isotopies. (In [FOOO1, FOOO2, F],the notion of A ∞ homotopy is defined via a suspension model, the notion of pseudo-isotopy, of the chain complex.) Because of these reasons and for the convenience ofreaders, we provide full details, in the categorical context, of the construction A ∞ homotopy associated to a continuation of Hamiltonians in Section 8. Our construc-tion of A ∞ functor is the counterpart of the standard Floer continuation equationalso applied to the higher m k maps with k ≥
2. It turns out that actual construc-tions of the associated A ∞ homotopy as well as of A ∞ functors are rather subtleand require some thought on the correct moduli spaces that enter in definitions of A ∞ functors and homotopies associated to geometric continuations. (See Subsec-tion 7.2 and Section 8 for the definitions of relevant moduli spaces.) We also callreaders’ attention to Savelyev’s relevant construction in the context of ∞ -categoryin [Sa].1.2. Construction of Knot Floer algebra.
For a concrete computation we doin [BKO], we focus on a particular object in this category WF ( M \ K ) canonicallyassociated to the knot. For given tubular neighborhood N ( K ) of K , we considerthe conormal L = ν ∗ T, T = ∂N ( K )and then the wrapped Fukaya algebra of the Lagrangian L in T ∗ ( M \ K ). Weremark that the wrapped Fukaya algebra of ν ∗ T in T ∗ M for a closed manifold M can be described by purely topological data arising from the base space, morespecifically that of the space of paths attached to T in M . (See [APS] for somerelevant result.) So it is important to consider L as an object for T ∗ ( M \ K ) to getmore interesting knot invariant. WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 5
On the other hand, since we restrict the class of our Hamiltonians to that ofkinetic energy Hamiltonian H g associated to a Riemannian metric g on M \ K ,the pair ( ν ∗ T, H g ) is not a nondegenerate pair but a clean pair in that the set ofHamiltonian chords contains a continuum of constant chords valued at points of T ∼ = T .We denote by X ( L ; H g ) = X ( L, L ; H g ) the set of Hamiltonian chords of H g attached to a Lagrangian submanifold L in general. We have X ( L ; H g ) = X ( L ; H g ) (cid:97) X < ( L ; H g )where the subindex of X in the right hand side denotes the action of the Hamiltonianchord of H g which is the negative of the length of the corresponding geodesic. (See(1.8) below for the explanation on the relevant convention used in the present pa-per.) It is shown in [BKO, Proposition 2.1] that the constant component X ( L ; H g )of the critical set of the action functional A g is clean in the sense of Bott and sodiffeomorphic to T as a smooth manifold. We take CW ( ν ∗ T, ν ∗ T ; T ∗ ( M \ K ); H g ) = C ∗ ( T ) ⊕ Z { X < ( L ; H g ) } where C ∗ ( T ) is any model of cochain complex of T , e.g., C ∗ ( T ) = Ω ∗ ( T ) thede Rham complex and associate an A ∞ algebra following the construction from[FOOO1]. Theorem 1.3.
Let N ( K ) be a tubular neighborhood of K and let T = ∂ ( N ( K )) .The A ∞ algebra ( CW ( ν ∗ T, T ∗ ( M \ K ); H g ) , m ) , m = { m k } ≤ k< ∞ can be defined, and its isomorphism class does not depend on the various choicesinvolved such as tubular neighborhood N ( K ) and the metric g on M . Remark 1.4.
Due to the presence of Morse-Bott component of constant chords,there are two routes toward construction of wrapped Floer complex, which wedenote by CW g ( ν ∗ T, T ∗ ( M \ K )) : One is to take the model CW ( ν ∗ T, ν ∗ T ; T ∗ ( M \ K ); H g ) = C ∗ ( T ) ⊕ Z (cid:104) X < ( L ; H h ) (cid:105) where C ∗ ( T ) is a chain complex of T such as the singular chain complex as in[FOOO1] or the de Rham complex, and the other is to take CW ( ν ∗ k T, ν ∗ k T ; T ∗ ( M \ K ); H g ) ∼ = Z (cid:104) Crit k (cid:105) ⊕ Z (cid:104) X < ( L ; H h ) (cid:105) where ν ∗ k T the fiberwise translation of ν ∗ T by dk , suitably interpolated with ν ∗ T away from the zero section, for a sufficiently C -small compactly supported Morsefunction k : M \ K → R such that ν ∗ T (cid:116) Image dk . We refer readers to [BKO,Section 2] for the detailed explanation on the latter model.The isomorphism class of CW g ( T, M \ K ) independent of g then provides aknot-invariant of K in M for an arbitrary knot K . To emphasize the fact that weregard L = ν ∗ T as an object in the cotangent bundle of a knot complement M \ K ,not as one in the full cotangent bundle T ∗ M , we denote the cohomology group of CW g ( T, M \ K ) as follows. Definition 1.5 (Knot Floer algebra) . We denote the cohomology of CW g ( T, M \ K )by HW ( ∂ ∞ ( M \ K )) YOUNGJIN BAE, SEONHWA KIM, YONG-GEUN OH which carries a natural product arising from m map. We call this Knot Floeralgebra of K ⊂ M .By letting the torus converge to the ideal boundary of M \ K , we may regard HW ( ∂ ∞ ( M \ K )) as the wrapped Floer cohomology of the ‘ideal boundary’ of thehyperbolic manifolds M \ K , which is the origin of the notation ∂ ∞ ( M \ K ) we areadopting.In a sequel [BKO], we introduce a reduced version of the A ∞ algebra, denoted by (cid:103) CW d ( ∂ ∞ ( M \ K )), by considering the complex generated by non-constant Hamil-tonian chords, and prove that for the case of hyperbolic knot K ⊂ M this algebracan be also directly calculated by considering a horo-torus T and the wrapped Floercomplex CW ( ν ∗ T, T ∗ ( M \ K ); H h ) of the hyperbolic metric h although h cannotbe smoothly extended to the whole manifold M . We also prove a formality resultof its A ∞ structure (cid:103) CW ( ν ∗ T, T ∗ ( M \ K ); H h ) for any hyperbolic knot K . Thefollowing is the main result we prove in [BKO]. Theorem 1.6 (Theorem 1.6 [BKO]) . Suppose K is a hyperbolic knot on M . Thenwe have an (algebra) isomorphism HW d ( ∂ ∞ ( M \ K )) ∼ = HW d ( ν ∗ T ; H h ) for all integer d ≥ . Furthermore the reduced cohomology (cid:103) HW d ( ∂ ∞ ( M \ K )) = 0 for all d ≥ . C estimates. One crucial new ingredient in the proofs of the above theoremsis to establish the horizontal C estimates of solutions of the perturbed Cauchy-Riemann equation ( du − X H ⊗ β ) (0 , J = 0 (1.4)mainly for the energy Hamiltonian H g ( q, p ) = | p | g . For this purpose, we haveto require the one-form β to satisfy ‘co-closedness’ d ( β ◦ j ) = 0in addition to the usual requirement imposed in [AS]. Together with the well-knownrequirement of subclosedness for the vertical C estimates [AS], we need to require β to satisfy dβ ≤ , d ( β ◦ j ) = 0 , i ∗ β = 0 (1.5)for the inclusion map i : ∂ Σ → Σ. This brings the question whether such one-formsexist in the way that the choice is compatible with the gluing process of the relevantmoduli spaces. We explicitly construct such a β by pulling back a one form froma slit domain . This choice of one form is consistent under the gluing of modulispaces.Roughly speaking, this C -estimates guarantees that under the above prepara-tion, whenever the test objects L , . . . , L k are all contained in W i = T ∗ N i , theimages of the solutions of the Cauchy-Riemann equation is also contained in W i ,i.e., they do not approach the knot K . We refer to Part 2 for the proofs of various C estimates needed for the construction. Furthermore for the proof of independenceof the A ∞ category WF g ( M \ K ) under the choice smooth metric g , one need toconstruct an A ∞ functor between them for two different choices of g, g (cid:48) . Because WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 7 the maximum principle applies only in the increasing direction of the associatedHamiltonian from H g to H g (cid:48) , we have to impose the monotonicity condition g ≥ g (cid:48) or equivalently H g (cid:48) ≤ H g . Further perspective.
Putting the main results of the present paper in per-spective, we compare our category WF ( M \ K ) is a version of partially wrappedFukaya category on T ∗ M with the ‘stop’ given byΛ ∞ K := ∂ ∞ ( T ∗ M ) | K ⊂ ∂ ∞ ( T ∗ M )for Λ K = T ∗ M | K . This is a 3-dimensional coisotropic submanifold of the asymp-totic contact boundary ∂ ∞ ( T ∗ M ) = (cid:91) q ∈ M ∂ ∞ ( T ∗ q M )of T ∗ M which is of 5 dimension. In this regard, our construction can be comparedwith the partially wrapped Fukaya categories as follows.Our category avoids the entire colostropic submanifold Λ K ⊂ T ∗ M while par-tially wrapped Fukaya category with relevant coisotropic stop avoids Λ ∞ K ⊂ ∂ ∞ ( T ∗ M )only asymptotically. In this regard, M. Abouzaid pointed out that Sylvan’s par-tially wrapped Fukaya category with stop Λ ∞ K can be regarded as a bulk defor-mation of our category via coisotropic submanifold of codimension 2. (See [Se3]for the usage of such bulk deformation via a complex divisor . We also refer to[FOOO3] for a general theory of bulk deformations by toric divisors .) The wayhow we avoid this stop is by attaching the cylindrical end on M \ N ( K ) along itsboundary ∂ ( M \ N ( K )) = ∂N ( K ) and considering the kinetic energy Hamiltonianof a cylindrical adjustment g of a smooth metric g defined on M . As mentionedbefore, this Hamiltonian blows up as we approach to K off the zero section, whichwill prevent the image of any finite energy solution of perturbed Cauchy-Riemannequation (1.4) from touching the fiber of T ∗ M restricted to K . Our approach di-rectly working with T ∗ ( M \ K ), which is well-adapted to the metric structures onthe base manifold, will be important for our later purpose of studying hyperbolicknots in a sequel [BKO].We would also like to mention that in the arXiv version of [ENS, § W K to K that is obtained by attaching a punctured handle to DT ∗ R along the unit conormal bundle of the knot and altering the Liouville vector fieldof T ∗ M along ν ∗ K . It would be interesting to compare our category with theirs.A similar construction of A ∞ algebra for the conormal ν ∗ T of T as above canbe carried out for the ‘conormal’ ν ∗ N cpt of N cpt := N \ N ( K ) or the micro-support of the characteristic function χ N cpt , which is given by ν ∗ N cpt = o N cpt (cid:97) ν ∗− ( ∂N cpt )where o N cpt is the zero section restricted to N cpt , T = ∂N cpt is equipped withboundary orientation of N cpt and ν ∗− ( ∂N cpt ) is the negative conormal of ∂N cpt .Combined with the construction of the wrapped version of the natural restrictionmorphisms constructed in [Oh3], this construction would give rise to A ∞ morphisms YOUNGJIN BAE, SEONHWA KIM, YONG-GEUN OH (cid:91) ν ∗ N i → (cid:100) ν ∗ T and a natural A ∞ functor (cid:91) ν ∗ N (cid:47) (cid:47) (cid:15) (cid:15) (cid:91) ν ∗ N (cid:15) (cid:15) (cid:91) ν ∗ T (cid:47) (cid:47) (cid:91) ν ∗ T (1.6)where N i = M \ N i ( K ) and T i = ∂N i for two tubular neighborhoods N ( K ) ⊂ N ( K ) of K for i = 1 ,
2. Here (cid:98) L denotes the Yoneda image of the Lagrangian L ingeneral. (Since we will not use this construction in this series, we will leave furtherdiscussion elsewhere not to further lengthen the paper.)On the other hand it is interesting to see that when we are given a exhaustionsequence N ⊂ N ⊂ · · · N i ⊂ · · · , the union M ∪ ν ∗ K is a limit of the abovementioned conormal ν ∗ N i as i → ∞ on T ∗ ( M \ K ) in the Gromov-Hausdorff sense.Furthermore the canonical smoothing of ν ∗ N i given in [KO, Theorem 2.3] can bemade to converge to the S -family Lagrangian surgery of o M ∪ ν ∗ K denoted by M K in [AENV], [ENS] which consider the case M = S . While construction ofLagrangian M K requires some geometric restriction K such as being fibered (see[AENV, Lemma 6.12]), construction of exact Lagrangian smoothing of ν ∗ K can bedone for arbitrary knot K . It would be interesting to see what the ramification ofthis observation will be. Remark 1.7.
It is reasonable to expect that our category is equivalent to the limitof the wrapped Fukaya categories W ( T ∗ N i ) defined in [GPS] for the symplecticmanifolds with boundary, or Liouville sectors X i = T ∗ N i , ∂X i = T ∗ N i | ∂N i :While we put wrapping through by ambient Hamiltonian flows, [GPS] put wrappingon the Lagrangian itself but considered the unperturbed Cauchy-Riemann equationwith positively moving boundary condition and localization trick of category.1.5. Conventions.
In the literature on symplectic geometry, Hamiltonian dynam-ics, contact geometry and the physics literature, there are various conventions usedwhich are different from one another one way or the other. Because many thingsconsidered in the present paper such as the energy estimates, the C estimatesapplying the maximum principle and construction of the Floer continuation mapdepend on the choice of various conventions, we highlight the essential componentsof our convention that affect their validity.The major differences between different conventions in the literature lie in thechoice of the following three definitions: • Definition of Hamiltonian vector field:
On a symplectic manifold(
P, ω ), the Hamiltonian vector field associated to a function H is givenby the formula ω ( X H , · ) = dH ( resp. ω ( X H , · ) = − dH ) , • Compatible almost complex structure:
In both conventions, J is com-patible to ω if the bilinear form ω ( · , J · ) is positive definite. • Canonical symplectic form:
On the cotangent bundle T ∗ N , the canon-ical symplectic form is given by ω = dq ∧ dp, (resp. dp ∧ dq ) . WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 9
In addition, we would like to take ∂u∂τ + J (cid:18) ∂u∂t − X H ( u ) (cid:19) = 0 (1.7)as our basic perturbed Cauchy-Riemann equation on the strip. Since we work withthe cohomological version of Floer complex, we would like to regard this equationas the positive gradient flow of an action functional A H as in [FOOO2]. This,under our convention laid out above, leads us to our choice of the action functionalassociated to Hamiltonian H on T ∗ N given by A H ( γ ) = − (cid:90) γ ∗ θ + (cid:90) H ( t, γ ( t )) dt, which is the negative of the classical action functional. With this definition, Floer’scontinuation map is defined for the homotopy of Hamiltonian s (cid:55)→ H s for whichthe inequality ∂H s ∂s ≥ increasing . (See Section 7 for the relevant discussion.)For the kinetic energy Hamiltonian H = H g ( x ), we have A H ( γ c ) = − E g ( c ) (1.8)where γ c is the Hamiltonian chord associated to the geodesic c and E g ( c ) is theenergy of c with respect to the metric g . Acknowledgement:
Y. Bae thanks Research Institute for Mathematical Sciences,Kyoto University for its warm hospitality. Y.-G. Oh thanks H. Tanaka for his inter-est in the present work and explanation of some relevance of Savelyev’s work [Sa] toour construction of A ∞ homotopy associated to the homotopy of Floer continuationmaps. The authors also thank M. Abouzaid for making an interesting commenton the relationship between our wrapped Fukaya category and Sylvan’s partiallywrapped Fukaya category with stop given by Λ ∞ K in Fukaya’s 60-th birthday con-ference in Kyoto in February 2019.2. Geometric preliminaries
In this section, we consider the cotangent bundle W = T ∗ N with the canonicalsymplectic form ω = (cid:88) i =1 dq i ∧ dp i which is nothing but ω = − dθ , where θ is the Liouville one-form θ = (cid:80) p i dq i . (Ourconvention of the canonical symplectic form on the cotangent bundle is differentfrom that of [Se2].) Then the radial vector field Z = (cid:88) i =1 p i ∂∂p i (2.1)satisfies Z (cid:99) ω = − θ, L Z ω = ω (2.2)In particular its flow φ t satisfies ( φ t ) ∗ ω = e t ω . (2.3)Therefore T ∗ N is convex at infinity in the sense of [EG]. Let us consider a Riemannian metric g of N . We denote by (cid:93) : T ∗ N → T N, (cid:91) : T ∗ N → T N the ‘raising’ and the ‘lowering’ operations associated to the metric g . Then wealso equip T ∗ N with the metric (cid:98) g = g ⊕ g (cid:91) with respect to the splitting T ( T ∗ N ) = T N ⊕ T ∗ N induced from the Levi-Civita connection of g on N .2.1. Tame manifolds and cylindrical adjustments.
Let us consider a non-compact tame N , which means that there exists an exhaustion { N i } , asequence of compact manifolds N i with (cid:83) i N i = N such that N ⊂ int( N ) ⊂ N ⊂ · · · ⊂ N i ⊂ int( N i +1 ) ⊂ · · · (2.4)and each N i +1 \ int( N i ) is homeomorphic to ∂N i × [0 , N (cid:48) such that N is homeomorphic to Int N (cid:48) . Letus choose a neighborhood U = U ( ∂N (cid:48) ) of ∂N (cid:48) inside N (cid:48) , called an end of N . Thenthere is a homeomorphism φ : U → ∂N × (0 , + ∞ ], where φ ( ∂N (cid:48) ) = ∂N × { + ∞} iscalled the asymptotic boundary of N .A typical example of tame manifold is a knot complement N = M \ K with anambient closed 3-manifold M . In this case, we can take an exhaustion by simplychoosing a sequence of nested tubular neighborhoods of a knot K .Now we focus on the knot complement, i.e. ∂N cpt is a 2-dimensional torus T , specially denoted by T i . Although most of statements in this article also workfor arbitrary tame 3-manifolds in general, the results in the sequel [BKO] requiresthe torus boundary condition of N cpt in order to exploit hyperbolic geometry usingcomplete hyperbolic metric of finite volume.We can also consider a more general Riemannian metric g of N possibly incom-plete. For example, if we consider a knot complement N = M \ K , a natural choiceof such a metric is a restriction of a smooth metric g = g M of a closed ambientmanifold M . Definition 2.1 (Cylindrical adjustment) . We define the cylindrical adjustment g of the metric g on M with respect to the exhaustion (4.1) by g = (cid:40) g on N (cid:48) i da ⊕ g | ∂N i on N i \ K (2.5)for some i , which is suitably interpolated on N i \ N (cid:48) i which is fixed.Here a is the coordinate for [0 , + ∞ ) for the following decomposition N = N cpt ∪ T (cid:0) T × [0 , + ∞ ) (cid:1) . We call the metric g an asymptotically cylindrical adjustment of g on N , whichare all complete Riemannian metrics of N . We denote N end = T × [0 , + ∞ ) in theabove decomposition equipped with the cylindrical metric g | T i ⊕ da .We remark that the following property of asymptotically cylindrical adjustments g of smooth metrics g on M restricted to M \ K will be important in Section 9later. WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 11
Proposition 2.2.
Suppose there is given an exhaustion (2.4) . For any choice oftwo smooth Riemannian metrics g and g (cid:48) defined on M , denote by g and g (cid:48) thecylindrical adjustment made on N i for a same i . for every pair of g and g (cid:48) areLipschitz equivalent, i.e., there is a constant C = C ( g, g (cid:48) ) ≥ such that C g ≤ g (cid:48) ≤ Cg on M \ K .Proof. Let N ( K ) be a tubular neighborhood of K . By choosing sufficiently large i , we may assume M \ N i ⊂ N ( K ) and both g , g (cid:48) are cylindrical outside N i .Using the normal exponential map of K , we can parameterize the tubular neigh-borhood T ( K ) of K by ( r, θ, ϕ ) where ϕ ∈ S parameterizes the knot K and ( r, θ )is the polar coordinates of N x K at x ∈ K for the metric g . Similarly we denote by( r (cid:48) , θ (cid:48) , ϕ (cid:48) ) the coordinates associated to g (cid:48) .More explicitly, we take a normal frame { X, Y } along K along K . We denoteby exp g, ⊥ the normal exponential map along K of the metric g and by exp g (cid:48) , ⊥ thatof g (cid:48) . Using this frame, we have an embedding of ι g : D ( (cid:15) ) × S M defined by ι g ( r, θ, ϕ ) = exp g, ⊥ x ( φ ) ( r (cos θX ( x ( ϕ )) + sin θY ( x ( ϕ ))))and ι g (cid:48) in a similar way. Then the composition map ι − g (cid:48) ◦ ι g : D ( (cid:15) ) × S → D ( (cid:15) (cid:48) ) × S is well-defined and smooth if we choose a smaller (cid:15) = (cid:15) ( g, g (cid:48) ) whose size dependsonly on g, g (cid:48) . In particular, we have (cid:107) D ( ι − g (cid:48) ◦ ι g ) − id (cid:107) C < C = C ( g, g (cid:48) )on D ( (cid:15) ) × S for some constant C >
0. This in particular provessup (cid:26) g (cid:48) ( v, v ) g ( v, v ) (cid:12)(cid:12)(cid:12) v ∈ T ( M ) | N ( K ) , (cid:107) v (cid:107) g = 1 (cid:27) < C (cid:48) = C (cid:48) ( g, g (cid:48) ) . We recall that the cylindrical adjustment of g is defined g ∼ (cid:40) e − a da + e − a θ + dϕ for 0 ≤ r < (cid:15) (cid:15) ( da + dθ ) + dϕ for a ≥ i in coordinates with the coordinate change r = e − a and (cid:15) = e − i +1 / and (cid:15) = e − i . Exactly the same formula holds for the cylindrical adjustment of g (cid:48) in thecoordinates replaced by ( a (cid:48) , θ (cid:48) , ϕ (cid:48) ), those with primes. In particular all the metriccoefficients appearing in the formulae are exactly the same for both adjustments.This provessup (cid:26) g (cid:48) ( v, v ) g ( v, v ) (cid:12)(cid:12)(cid:12) v ∈ T ( M \ K ) | ι − g ( N ( K ) \ K ) , (cid:107) v (cid:107) g = 1 (cid:27) < C (cid:48) = C (cid:48) ( g, g (cid:48) ) . for all i ’s if we choose the tubular neighborhood N ( K ) of K sufficient small. Thisfinishes the proof. (cid:3) Remark 2.3. (1) It may be worthwhile to examine the behavior of Hamilton-ian vector field X H g ( q, p ) on M \ K as q approach to K . For this purpose,let ( r, θ, ϕ ) be the coordinate system on N ( K ). The metric g can be writtenas g = dr + r dϕ + dθ + o ( r ) . Define a cylindrical adjustment with respect to the coordinate ( a, θ, ϕ ) with r = e − a for a ∈ [0 , ∞ ). Ignoring o ( r ), the associate Hamiltonian of thecylindrical adjustment (for r = 1) is given by H g = 12 (cid:0) p a + p ϕ + p θ (cid:1) = 12 (cid:0) r p r + p ϕ + p θ (cid:1) . Therefore X H g = p a ∂∂a + p ϕ ∂∂ϕ + p θ ∂∂θ = − rp r ∂∂p r + r p r ∂∂r + p ϕ ∂∂ϕ + p θ ∂∂θ . We highlight the last two summands which makes the associated Hamilton-ian flow rotates around the torus with higher and higher speed as | p ϕ | + | p θ | → ∞ around the knot K . This asymptotic behavior is different fromthat of the Hamiltonian vector field used in defining the partially wrappedFukaya category whose stop is given by the Liouville sector T ∗ ( ∂N ( K )) ⊂ T ∗ ( M \ N ( K )) whose horizontal component is parallel to the radial vectorfield ∂∂r .(2) The above discussion and construction of wrapped Fukaya category can beextended to a general tame manifold N whose end may have more thanone connected component, as long as we fix a Lipschitz-equivalence class ofmetrics on N . An example of such N is the complement M \ L where L isa link.2.2. Kinetic energy Hamiltonian and almost complex structures.
We nowfix a complete Riemannian manifold with cylindrical end. We denote the resultingRiemannian manifold by (
N, g ). Definition 2.4.
We denote by H = H ( T ∗ N ) the space of smooth functions H : T ∗ N → R that satisfies H ( q, p ) = 12 | p | g (cid:91) on { ( q, p ) : q ∈ N, | p | g (cid:91) ≥ R } ∪ T ∗ N end for a sufficiently large R >
0, where g (cid:91) is the dual metric of g . We call such map on T ∗ N admissible Hamiltonian with respect to the metric g . The Hamiltonian vectorfield X H on T ∗ N is defined to satisfy X H (cid:99) ω = dH for a given H ∈ H i .We next describe the set of adapted almost complex structures. From now on,we use g instead of g (cid:91) , when there is no danger of confusion. For each given R > Y R := {| p | g = R } = H − ( R / T ∗ N , and the domain W R := {| p | g ≤ R } becomes a Liouville domain in the sense of [Se2].For each R >
0, the level set i R : Y R (cid:44) → T ∗ N admits a contact from λ R := − i ∗ R θ WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 13 by the restriction. We denote the associated contact distribution by ξ R = ker λ R ⊂ T Y R . Note that the corresponding Reeb flow is nothing but a reparametrization of the g -geodesic flow on Y R .If we denote by ( r, y ) the cylindrical coordinates given by r = | p | g T ∗ N \ { } ∼ = ST ∗ N × R + . Including the zero section, we have decomposition T ∗ N ∼ = W ∪ Y (cid:0) [0 , ∞ ) × Y (cid:1) . To highlight the metric dependence, we use Y g , λ g , ξ g instead of Y , λ , ξ R , respec-tively. On T ∗ N \ N with metric g , we have the natural splitting T ( T ∗ N \ N ) ∼ = R · ∂∂r ⊕ T Y g ∼ = Span (cid:26) (cid:101) X g , ∂∂r (cid:27) ⊕ ξ g ∼ = R ⊕ ξ g , where (cid:101) X g is a vector field which generates the g -geodesic flow on Y g . We recall thatwe have a canonical almost complex structure J g on T ∗ N associated to a metric on N defined by J g ( X ) = X (cid:91) , J g ( α ) = − α (cid:93) (2.6)in terms of the splitting T ( T ∗ N ) = T N ⊕ T ∗ N with respect to the Levi-Civitaconnection of g . We call this J g the Sasakian almost complex structure on T ∗ N . Definition 2.5 (Admissible almost complex structure) . We say an almost complexstructure J on T ∗ N is g -admissible if J = J g on T ∗ N end and on { ( q, p ) | | p | g ≥ R } for the Sasakian almost complex structure J g associated to g for a metricdecomposition N = N cpt ∪ N end with N end ∼ = [0 , ∞ ) × T with T = ∂N cpt . Denoteby J g the set of g -admissible almost complex structures.All admissible almost complex structures satisfy the following important prop-erty which enters in the study of Floer theoretic construction on Liouville manifoldsin general. Definition 2.6.
Let H g be the energy Hamiltonian on T ∗ N associated metric g onthe base N given above. An almost complex structure J on T ∗ N is called contacttype if it satisfies ( − θ ) ◦ J = dH g . Remark 2.7.
Appearance of the negative sign here is because our convention ofthe canonical symplectic form on the cotangent bundle is ω = − dθ . Comparedwith the convention of [AS], our choice of the one-form λ therein is λ = − θ .3. A choice of one-form β on ΣIn Abouzaid-Seidel’s construction of wrapped Floer cohomology given in [AS,Section 3.7], [A1], they start from a compact
Liouville domain with contact typeboundary and consider the perturbed Cauchy-Riemann equation of the type( du − X H ⊗ β ) (0 , J = 0 . (3.1)Because of the compactness assumption, their setting does not directly apply to ourcurrent cotangent bundle T ∗ N where the tame base manifold N is noncompact: Toperform analytical study of the relevant moduli spaces, the first step is to establishsuitable C -estimate. Co-closed and sub-closed one-forms.
We first recall Abouzaid-Seidel’sconstruction of what they call a sub-closed one-form in the context of Liouvilledomain with compact contact-type boundary. For each given k ≥
1, let us con-sider a Riemann surface (Σ , j ) of genus zero with ( k + 1)-ends. This is isomorphicto the closed unit disk D minus k + 1 boundary points z = { z , . . . , z k } in thecounterclockwise way. Each end admits a holomorphic embedding (cid:40) (cid:15) : Z + := { τ ≥ } × [0 , → Σ; (cid:15) i : Z − := { τ ≤ } × [0 , → Σ , for i = 1 , . . . , k preserving the boundaries and satisfying lim τ →±∞ (cid:15) (cid:96) = z (cid:96) , for (cid:96) = 0 , . . . , k . We callthe distinguished point at infinity z a root .For a given weight w = { w , . . . , w k } satisfying w = w + · · · + w k , (3.2)a total sub-closed one-form β ∈ Ω (Σ) is considered in [A2] satisfying (cid:40) dβ ≤ (cid:15) (cid:96) ) ∗ β = w (cid:96) dt. This requirement is put to establish both (geometric) energy bound and (vertical) C -bound.It turns out that in our case of T ∗ ( M \ K ) where the base is noncompact , sub-closedness of β is not enough to control the behavior of the Floer moduli space ofsolutions of (3.1): we need the following restriction d ( β ◦ j ) = 0 (3.3)in addition to the sub-closedness. We refer readers to Lemma 11.1 in Subsection11 for the reason why such a condition is needed.3.2. Construction of one-forms β . For each given k ≥ w = { w , . . . , w k } satisfying the balancing condition (3.2), we will choose aone-form β on Σ that satisfies dβ ≤ i ∗ β = 0 for the inclusion map i : ∂ Σ → Σ dβ = 0 near ∂ Σ d ( β ◦ j ) = 0( (cid:15) j ) ∗ β = w j dt on a subset of Z ± where ± τ (cid:29)
0. (3.4)For this purpose, we first consider the slit domain representation of the conformalstructure (Σ , j ) whose explanation is in order.Consider domains Z = { τ + √− t ∈ C | τ ∈ R , t ∈ [ w − w , w ] } ; Z = { τ + √− t ∈ C | τ ∈ R , t ∈ [ w − w − w , w − w ] } ;... Z k = { τ + √− t ∈ C | τ ∈ R , t ∈ [0 , w k ] } , WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 15 and its gluing along the inclusions of the following rays R (cid:96) = { τ + √− t (cid:96) ∈ C | τ ≥ s (cid:96) , t (cid:96) = w − ( w + · · · + w (cid:96) ) } ; j (cid:96) − : R (cid:96) (cid:44) → Z (cid:96) , j (cid:96) + : R (cid:96) (cid:44) → Z (cid:96) +1 , for some s (cid:96) ∈ R , (cid:96) = 1 , . . . , k −
1. In other words, for a collection s = { s , . . . , s k − } ,the glued domain becomes Z w ( s ) = Z (cid:97) Z (cid:97) · · · (cid:97) Z k / ∼ . Here, for ( ζ, ζ (cid:48) ) ∈ Z (cid:96) × Z (cid:96) +1 , ζ ∼ ζ (cid:48) means that there exists r ∈ R (cid:96) such that ζ = j (cid:96) − ( r ) , ζ (cid:48) = j (cid:96) + ( r ). We may regards Z ⊂ { τ + √− t ∈ C | τ ∈ R , t ∈ [0 , w ] } , with ( k − S (cid:96) = { τ + √− t (cid:96) ∈ C | τ ≤ s (cid:96) , t (cid:96) = w − ( w + · · · + w (cid:96) ) } where (cid:96) = 1 , . . . , k − ϕ : Σ = D \ { z , . . . , z k } → Z w ( s )with respect to s satisfying that(1) Let ∂ Σ (cid:96) be a connected boundary component of D \ { z , . . . , z k } between z (cid:96) and z (cid:96) +1 , for (cid:96) ∈ Z k +1 . Then ϕ ( ∂ Σ (cid:96) ) = w √− R for (cid:96) = 0; ϕ ( ∂ Σ (cid:96) ) = S (cid:96) for (cid:96) = 1 , . . . , k − ϕ ( ∂ Σ (cid:96) ) = R for (cid:96) = k. (2) A restriction ϕ | ˚Σ : ˚Σ → ˚ Z is a conformal diffeomorphism.(3) The following asymptotic conditions (cid:40) lim τ →−∞ ˚ ϕ − ( { τ } × ( w − (cid:80) (cid:96)j =1 w j , w − (cid:80) (cid:96) − j =1 w j )) = z (cid:96) for (cid:96) = 1 , . . . , k ;lim τ → + ∞ ˚ ϕ − ( { τ } × (0 , w )) = z . Let us denote such a slit domain by Z w ( s ) or simply Z if there is no confusion. w w w w w Figure 1.
An example of slit domainNow we consider dt ∈ Ω ( Z ) and we define the one-form β Lemma 3.1.
Define β = ϕ ∗ dt ∈ Ω (Σ) . (3.5) Then β satisfies all the requirements in (3.4) . Proof.
Observe that the holomorphic embedding provides a natural strip-like rep-resentation at each puncture. More precisely, we have (cid:40) ϕ ◦ (cid:15) : Z + → Z : ( τ, t ) (cid:55)→ ( τ + K, w t ); ϕ ◦ (cid:15) (cid:96) : Z − → Z : ( τ, t ) (cid:55)→ ( τ − K, w (cid:96) t + w − (cid:80) (cid:96)j =1 w j ) for (cid:96) = 1 , . . . , k for some K ∈ R + large enough. So it is direct to check that ( (cid:15) j ) ∗ β = w j dt and otherconditions in (3.5). In fact, this form satisfies the stronger condition begin closedrather than being sub-closed. Furthermore since ϕ is holomorphic, we compute β ◦ j = ( ϕ ∗ dt ) ◦ j = dtdϕ ◦ j = dtj ◦ dϕ = − dτ ◦ dϕ = − d ( τ ◦ ϕ )which is obviously closed. Therefore we have proved d ( β ◦ j ) = 0. This finishes theproof. (cid:3) Let us wrap up this section by recalling the gluing result of the slit domains.
Proposition 3.2.
Let k , k be positive integers, and let u = ( u , . . . , u k ) and v = ( v , . . . , v k ) be weight datum satisfying the balancing condition and u = v i for some i ≥ . Then there is a one-parameter gluing of Z ( u ) and Z ( v ) whichbecome a slit domain of w = u i v := ( v , . . . , v i − , u , . . . , u k , v i +1 , . . . , v k ) . Part Construction of a wrapped Fukaya category for M \ K In this part, we carry our construction of our wrapped Fukaya category of theknot complement M \ K .The standard Liouville structure given by the vector field Z λ = p ∂∂p = n (cid:88) i =1 p i ∂∂p i , n = dim N has noncompact contact type boundary ∂ ( DT ∗≤ N ) for N = M \ K . Because of this,construction of wrapped Floer cohomology for the sequence of conormal bundles ofthe type ν ∗ T with closed submanifold T ⊂ N meets a new obstruction arising fromthe noncompactness of N : One must examine the horizontal C bound away fromthe ‘infinity’ in N . Remark 3.3.
For the construction of a wrapped Fukaya category WF ( T ∗ N ), wedo not need to restrict ourselves to the case of knot complement but can considerthe general context of tame 3-manifolds M with several asymptotic boundaries.Since we will not use this general construction, we do not pursue it in this paper.4. Admissible Lagrangians and associated wrapped Floer complexes
We first describe the set of admissible Lagrangian submanifolds in such a man-ifold N for the set of objects of WF ( M \ K ). Since the base manifold N is non-compact, we need to restrict the class of Lagrangian manifolds as follows: Definition 4.1.
A Lagrangian submanifold L ⊂ ( W, ω ) is called admissible if(1) L is exact and embedded;(2) Its image under the projection π : W → N is compact in M \ K .(3) The relative first Chern class 2 c ( W, L ) vanishes on H ( W, L ) and the sec-ond Stiefel-Whitney class w ( L ) vanishes. WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 17 (4) Under the decomposition N ∼ = N cpt ∪ ([0 , ∞ ) × T ) and the associatedcylindrical adjustment g of g , there exists R > L ∩ {| p | g ≥ R } = Λ R · [ R, + ∞ )where Λ R = L ∩ {| p | g = R } given as in (2.5).For the purpose of controlling the horizontal C estimates of solutions of (3.1),we fix a compact exhaustion sequence N ⊂ N ⊂ · · · ⊂ N i ⊂ · · · be of N = M \ K .We also take another exhaustion N (cid:48) ⊂ N (cid:48) ⊂ · · · ⊂ N (cid:48) i ⊂ · · · such that N (cid:48) ⊂ N ⊂ N (cid:48) ⊂ N ⊂ · · · ⊂ N (cid:48) i ⊂ N i ⊂ · · · . (4.1)We mostly denote any such N i by N cpt when we do not need to specify the subindex.We let W ⊂ W ⊂ · · · ⊂ W i ⊂ · · · , W i = T ∗ N i be the exhaustion T ∗ ( M \ K ) = (cid:83) ∞ i =1 W i induced by (4.1). We have T ∗ N = T ∗ ( N i ∪ [0 , ∞ ) × ∂N i ) = W i ∪ T ∗ N end i . (4.2)(See Section B.1 for the implication of the condition (4).) For any two given ad-missible Lagrangian L and L , they are contained inside W i for some i .We consider a Hamiltonian H from H and define the set X ( H ; L , L ) = { x : [0 , → W | ˙ x ( t ) = X H ( x ( t )) , x (0) ∈ L , x (1) ∈ L } (4.3)of time-one Hamiltonian chords of H . It can be decomposed into X ( H ; L , L ) = (cid:97) i X i ( H ; L , L )where X k ( H ; L , L ) is the set of Reeb chords with degree k . By the bumpy metrictheorem [Ab] (or rather its version with free boundary condition), we may assumethat all Hamiltonian chords of the Hamiltonian H ∈ H g are nondegenerate by con-sidering a generic metric g , as long as we consider a countable family of Lagrangiansubmanifolds. Proposition 4.2.
Consider a decomposition N = N cpt ∪ N end as above assumethat N end is equipped with the cylindrical metric da ⊕ g | ∂N end . Denote W = T ∗ N cpt and let T ∗ N = W ∪ T ∗ N end be the associated decomposition. Suppose L , L ⊂ W .Then for all x ∈ X ( H ; L , L ) , we have Im x ⊂ W .Proof. We recall that the geodesic x satisfies the (elliptic) second order ODE ∇ t ˙ x =0. This implies that the a -component of x in N end satisfies d adt = 0. Applying theeasy maximum principle to a , a cannot have maximum at a point ( a, p ) ∈ N end .Since x (0) , x (1) ∈ W , this finishes the proof. (cid:3) We take the action functional A ( γ ) = − (cid:90) γ ∗ θ + (cid:90) H ( γ ( t )) dt + f ( γ (1)) − f ( γ (0)) (4.4)on the path spaceΩ(( L , L ) = { γ : [0 , → T ∗ N | γ (0) ∈ L , γ (1) ∈ L } . We alert the readers that this is the negative of the classical action functional.
We assign a Maslov index µ ( x ) to each chord x . See Section B.2. Consider thegraded module CW ( L , L ; H ) = (cid:77) i CW i ( L , L ; H ); CW k ( L , L ; H ) = (cid:77) µ ( x )= k Z (cid:104) x (cid:105) for H ∈ H g and x ∈ X k ( H ; L , L ). We would like to define a boundary map m on this module. Remark 4.3.
Here we adopt the notation CW ( L (cid:48) , L ; H ) for the complex associatedto the geometric path space and to the set of Hamiltonian chords from L to L (cid:48) which we denote by Ω( L, L (cid:48) ) and X ( H ; L, L (cid:48) ) respectively, following the notationalconvention of [FOOO1, FOOO2] in that CW ( L (cid:48) , L ; H ) is the cohomological complex ,not the homological one.We take a t -dependent family of almost complex structures { J t } t ∈ [0 , containedin admissible almost complex structures defined in Definition 2.5. For each givenpair x , x of Hamiltonian chords in CW ( L , L ; H ) for some τ ∈ R + , we considerthe moduli space (cid:102) M ( x ; x ) which consist of maps u : R × [0 , → W satisfying thefollowing perturbed J -holomorphic equation with the boundary and the asymptoticconditions given by ∂ τ u + J t ( ∂ t u − X H ) = 0 u ( R × { } ) ⊂ L , u ( R × { } ) ⊂ L u ( −∞ , t ) = x ( t ) , u (+ ∞ , t ) = x ( t ) . (4.5) Remark 4.4. (1) In the point of view of [Oh1], we may fix the Hamiltonian H and J and perturb the boundary Lagrangians to achieve this kind oftransversality result. This is the strategy that we adopt in the present paperand its sequel [BKO] . (2) Note that (4.5) is action increasing as τ → ∞ for the action functional(4.4). On the other hand we put the output at τ = + ∞ and the input at τ = −∞ . Combination of these two implies that our Floer complex is the cohomological version.For a generic choice of almost complex structures, the moduli space (cid:102) M ( x ; x )is a manifold of dimension µ ( x ) − µ ( x ). It admits a free R -action as long as x (cid:54) = x . We write M ( x ; x ) for the quotient space. Note that M ( x ; x ) is a setof oriented points, when µ ( x ) − µ ( x ) − o ( x ) → o ( x ) , where o ( x i ) is an orientation space associated to each Hamiltonian chord which isdescribed in Section B.2. Then the differential m : CW i ( L , L ) → CW i +1 ( L , L )is defined by counting rigid solutions of M ( x ; x ), m ( x ) = (cid:88) µ ( x )= µ ( x )+1 ( − µ ( x ) M ( x ; x ) x , where WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 19
For readers’ convenience, let us recall from [A2] a canonical isomorphism betweenthe two wrapped Floer complexes before and after conformal fiberwise rescaling.We denote by ψ λ is time- log ( λ ) Liouville flow on M , which can be expressed by ψ s ( r, y ) = ( sr, y ) with respect to the cylindrical coordinate discussed in Section2.2.For the simplicity of notation, when a pair ( L , L ) and the Hamiltonian H aregiven, we just write CW ∗ ( ψ ( L ) , ψ ( L )) for any of CW ∗ ( ψ ( L ) , ψ ( L ); ω, w H ◦ ψ, ψ ∗ J t ) for ψ = ψ w . Lemma 4.5.
Let g be the metric on N as above. Denote by H the kinetic energyHamiltonian on W of the metric g on N . Let ψ : W → W be a conformallysymplectic diffeomorphism satisfying ψ ∗ ω = w ω for some non-zero constant w .Then there is a canonical isomorphism CW ( ψ ) : CW ∗ ( L , L ) ∼ = CW ∗ ( ψ ( L ) , ψ ( L )) . Floer data for A ∞ structure Now consider a ( k + 1)-tuple of admissible Lagrangian L = ( L , . . . , L k )inside W i ⊂ W . Pick a ( k + 1)-tuple of Hamiltonian chords ( x ; x ) with x =( x , . . . , x k ) where x ∈ X ( H ; L , L k ); x i ∈ X ( H ; L i − , L i ) for i = 1 , . . . , k. (5.1)Recall the Riemann surface (Σ , j ) ∈ M k +1 is a genus zero disk ( k + 1) boundarypoints z = { z , . . . , z k } removed in a counter-clockwise way. Each end near z (cid:96) isequipped with a holomorphic embedding (cid:40) (cid:15) : Z + → Σ (cid:15) i : Z − → Σ for i = 1 , . . . , k (5.2)satisfying the boundary and the asymptotic conditions. We take these choices sothat they become consistent over the universal family S k +1 := { ( s, Σ , j ) | s ∈ Σ , (Σ , j ) ∈ M k +1 } → M k +1 and its compactification S k +1 → M k +1 as in [AS]. Usage of universal family will enter in a more significant way when weconsider construction of A ∞ homotopy later. Definition 5.1 (Floer data for A ∞ map) . A Floer datum D m = D m (Σ , j ) on astable disk (Σ , j ) ∈ M k +1 is the following:(1) Weights : A ( k + 1)-tuple of positive real numbers w = ( w , . . . , w k ) whichis assigned to the end points z satisfying w = w + · · · + w k . (2) One-form : β ∈ Ω (Σ) constructed in Section 3.2 satisfying (cid:15) j ∗ β agreeswith w j dt .(3) Hamiltonian : A map H : Σ → H i whose pull-back under (cid:15) j uniformlyconverges to H ( w j ) ◦ ψ w j near each z j for some H ∈ H i .(4) Almost complex structure : A map J : Σ → J ( T ∗ N ) whose pull-backunder (cid:15) j uniformly converges to ψ ∗ w j J t near each z j for some J t ∈ J i . (5) Vertical moving boundary : A map η : ∂ Σ → [1 , + ∞ ) which convergesto w j near each z j . Remark 5.2.
The condition (3) is automatically holds for any (fiberwise)globallyquadratic Hamiltonian such as the kinetic energy Hamiltonian that we are using inthe present paper. This is because of the equality Hw ◦ ψ w = H (5.3)for such a Hamiltonian. For the consistency with that of [A1], we leave the conditionas it is. On the other hand, to exploit this special situation, we extend the function η to whole Σ so that η ◦ (cid:15) i ( ∞ i , t ) ≡ w i . We then choose J so that ( ψ η ) ∗ J = J g . (5.4)Such a choice will be useful later in our study of C estimates.Two Floer data ( w , β , H , J , η ) and ( w , β , H , J , η ) are conformally equiv-alent if there exist a constant C > w = C w , β = Cβ , H = H ◦ ψ C C , J = ψ C ∗ J , η = Cη . Definition 5.3. A universal choice of Floer data D m for the A ∞ structure consistsof D m (Σ , j ) for every (Σ , j ) in M k for all k ≥ M k .(2) The data on ∂ M k is conformally equivalent the one on the lower strata M k × M k , k + k = k + 2.(3) The data are compatible with the gluing process in an infinite order.Let us consider a moduli space M w (Σ ,j ) ( x ; x ) = M ( x ; x ; D m (Σ , j )) which con-sists of a map u : Σ → T ∗ N satisfying a perturbed ( j, J )-holomorphic equationwith a vertical moving boundary condition and a shifted asymptotic condition: ( du − X H ⊗ β ) (0 , J = 0 ,u ( z ) ∈ ψ η ( z ) ( L i ), for z ∈ ∂ Σ between z i and z i +1 where i ∈ Z k +1 . u ◦ (cid:15) j ( −∞ , t ) = ψ w j ◦ x j ( t ) , for j = 1 , . . . , k.u ◦ (cid:15) (+ ∞ , t ) = ψ w ◦ x ( t ) . (5.5)Here precise meaning of the first equation is( du ( z ) − X H ( z ) ⊗ β ( z )) + J ( z ) ◦ ( du ( z ) − X H ( z ) ⊗ β ( z )) ◦ j = 0 . (5.6)An immediate but important observation is that the pull-back of (5.6) under (cid:15) j becomes the standard Floer equation ∂ τ u + J t (cid:16) ∂ t u − X Hwj ◦ ψ wj ( u ) (cid:17) = 0 , on the strip-like ends which can be obtained by applying ψ w j to (4.5). Now weconsider a parameterized moduli space M w ( x ; x ) = (cid:91) (Σ ,j ) ∈M k +1 M w (Σ ,j ) ( x ; x ) . Then by the standard transversality argument we have
WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 21
Lemma 5.4.
For a generic choice of universal Floer data D m , the moduli space M ( x ; x ) is a manifold of dimension µ ( x ) − (cid:80) ki =1 µ ( x i ) − k. Due to our construction of the one-forms β using the slit domain, consistency ofthe Floer data over all strata is automatic which is needed for the construction of A ∞ structures later.6. Construction of A ∞ structure map In this section, we would like to construct an A ∞ structure map m k : CW ∗ ( L , L ; H ) ⊗ · · · ⊗ CW ∗ ( L k , L k − ; H ) → CW ∗ ( L k , L ; H )[2 − k ] . We recall that the module CW ∗ ( L, L (cid:48) ; H ) is generated by time-one Hamiltonianchords of X H from L to L (cid:48) .The starting point of compactification of the moduli space relevant to the A ∞ Floer structure map is to establish a uniform energy bound and C estimates forthe solutions of perturbed pseudo holomorphic equations associated to the corre-sponding moduli spaces.6.1. Energy bound and C estimates. The uniform energy bound for the solu-tions of perturbed pseudo holomorphic equations is necessary for the compactifica-tion of the corresponding moduli spaces.Let L i be an admissible Lagrangian and f i : L → R be its potential function, i.e.,a function ι ∗ ( − θ ) = df i . We first recall the definition of action of x ∈ X ( H ; L i , L j ) A ( x ) = − (cid:90) x ∗ θ + (cid:90) H ( x ( t )) dt + f j ( x (1)) − f i ( x (0)) (6.1)from (4.4).The energy E ( u ) is defined by (cid:90) Σ | du − X H ( u ) ⊗ β | J for general smooth map u . For any solution u : Σ → T ∗ N of (5.5), we have thefollowing estimate E ( u ) = (cid:90) Σ u ∗ ω − u ∗ d H ∧ β ≤ (cid:90) Σ u ∗ ω − u ∗ d H ∧ β − (cid:90) Σ u ∗ H · dβ (6.2)= (cid:90) Σ u ∗ ω − d ( u ∗ H · β ) , (6.3)where the inequality in (6.2) comes from H ≥ β . By Stokes’theorem with the fixed Lagrangian boundary condition in (5.5) and β | ∂ Σ = 0 impliesthat (6.3) becomes A ( x ) − k (cid:88) j =1 A ( x j ) . The moduli spaces M w ( x ; x ) admit compactification with respect to the Floerdatum D m = D m (Σ , j ) on a stable disk (Σ , j ) ∈ M k +1 stated in Definition 5.1. Since our underlying manifold N is noncompact, the C -estimate for the J -holomorphic maps has to be preceded before starting the process of compactifi-cation. This is where the ‘co-closedness’ of β enters in an essential way which isneeded for an application of maximum principle.The following is the main proposition in this regard whose proof is postponeduntil Part 2. Proposition 6.1 (Horizontal C bound) . Let (Σ , j ) be equipped with strip-like endsat each puncture z i as before. Assume the one-form β is as in Definition 5.1. Let L = ( L , · · · , L k ) be ( k + 1) -pair of admissible Lagrangians in W (cid:96) ⊂ T ∗ N for some (cid:96) ∈ N , see Definition 4.1.Let x j ∈ X ( w j H ; L j − , L j ) for j = 1 , . . . , k and x ∈ X ( w H ; L , L k ) . Then forany solution u of (5.5) , we have Im u ⊂ W (cid:96) . (6.4)We also need to establish the vertical bound. Such a vertical bound is estab-lished in [AS, section 7c] using an integrated form of the strong maximum principle.Here, partially because similar arguments are needed for the (horizontally) movingboundary condition, we will provide a more standard argument of using the point-wise strong maximum principle in Part 2. Both conditions dβ ≤ i ∗ β = 0 willbe used in a crucial way similarly as in Abouzaid-Seidel’s proof Proposition 6.2 (Vertical C bound) . Let x = ( x , . . . , x k ) be given as in (5.1).Then max z ∈ Σ | p ( u ( z )) | ≤ ht ( x ; H, { L i } ) for any solution u of (5.5) . Remark 6.3.
In this remark, we summarize the standard arguments on how thecompactness arguments and dimension counting enter in the construction of mod-uli operator such as A ∞ maps. For the practical purpose, we restrict ourselvesto the cases of the moduli spaces M w ( x ; x ) of expected dimension one and zero.Consider a sequence of pseudo-holomorphic maps { u ν } ν ∈ N with a sequence of Floerdata {D ν } ν ∈ N defined on { (Σ ν , j ν ) } ν ∈ N diverging to one end. Let D ∞ be the cor-responding limit of the Floer datum which is defined over a broken stable disk(Σ ∞ , j ∞ ) = (Σ I , j I ) ∗ n +1 (Σ II , j II ). Here ∗ n +1 means 0-th puncture of Σ I and ( i +1)-th puncture of Σ II correspond to the breaking point. We already mentioned aboutthe C -estimate of the moduli space M w ( x ; x ). If the gradient of M w ( x ; x ) is notuniformly bounded, then we have sphere or disk bubbling phenomenon. But thesecannot happen since the symplectic manifold ( T ∗ N, ω ) and Lagrangian submanifold( L i , ι ∗ θ ) are exact. Then by Arzel`a-Ascoli theorem, there is a subsequence of { u ν } which C ∞ loc -converges to u ∞ with Floer datum D ∞ . The uniform energy estimateguarantees that the broken strip (or points) should be mapped to a Hamiltonianchord with a matching weight condition. So the boundary points correspond tobroken pseudo-holomorphic maps and each component of the broken map is a partof zero-dimensional moduli space. Note also that the moduli space M w ( x ; x ) ofdimension zero has empty boundary, and hence itself is a finite set.6.2. Definition of A ∞ structure map. Construction of the A ∞ structure map m = { m k } proceeds in two steps as in [A1]: WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 23
First we consider the moduli space M w ( x ; x ) which consist of solutions u : Σ → T ∗ N of (5.5) satisfying the asymptotic conditions( ψ w ( x ); ψ w ( x ) , . . . , ψ w k ( x k ))at each ends z = ( z ; z , . . . , z k ) of Σ, where (cid:40) ψ w ( x ) ∈ CW ∗ ( ψ w ( L k ) , ψ w ( L )); ψ w j ( x j ) ∈ CW ∗ ( ψ w j ( L j ) , ψ w j ( L j − )) for j = 1 , . . . , k for some Floer datum D = D m (Σ , j ) given in Definition 5.1. Here w = ( w ; w , . . . w k )is the given tuple of weights of asymptotic conformal shifting of boundary La-grangians.The count of such elements directly defines a map m k D : CW ∗ ( ψ w ( L ) , ψ w ( L )) ⊗ · · · ⊗ CW ∗ ( ψ w k ( L k ) , ψ w k ( L k − )) −→ CW ∗ ( ψ w ( L k ) , ψ w ( L ))[2 − k ] . Then we compose the tensor of vertical scaling maps CW ( ψ ) : CW ∗ ( L, L (cid:48) ) ∼ = CW ∗ ( ψ ( L ) , ψ ( L (cid:48) )) . in Lemma 4.5 and pre-compose its inverse to m k D , i.e., the map m k is defined by m k = ( CW ( ψ )) − ◦ m k D ◦ ( CW ( ψ )) ⊗ k (6.5)In conclusion, we have m k ( x ⊗ · · · ⊗ x k ) = (cid:88) x ( − † M w ( x ; x ) x , (6.6)where † = (cid:80) ki =1 i · µ ( x i ) and M w ( x ; x ). Remark 6.4.
Note that the A ∞ structure map { m k } k ∈ N is defined on the chaincomplex generated by time-1 Hamiltonian chords of the originally given H , while theintermediate maps m k D is defined on the complex generated by time-one Hamiltonianchords of the weighted Hamiltonian w i H . Now suppose that the moduli space M w ( x ; x ) is one dimensional. Then bythe standard argument in Gromov-Floer compactification, the boundary strata ∂ M ( x ; x ) consist of (cid:97) ¯ x M w ( x ; x ) × M w (¯ x ; x ) . Here, 0 ≤ n ≤ k − m and¯ x ∈ X ( | w | H ; L n + m , L n ) , | w | = w n +1 + · · · + w n + m ; x = ( x , . . . , x n , ¯ x, x n + m +1 , . . . , x k ) , w = w | x ; x = ( x n +1 , . . . , x n + m ) , w = w | x . We conclude the following relation. Here we follow the sign convention from [Se1],see also Section B.2.
Proposition 6.5.
The maps { m k } k ∈ N define an A ∞ structure i.e., satisfy (cid:88) m,n ( − ‡ m k − m +1 ( x , . . . , x n , m m ( x n +1 , . . . , x n + m ) , x n + m +1 , . . . , x k ) = 0 , where ‡ = ‡ n = (cid:80) ni =1 µ ( x i ) − n . We denote the resulting category by WF ( T ∗ ( M \ K ); H g ) . Construction of A ∞ functor In this section, we consider a pair of metrics g, g (cid:48) on M with g ≥ g (cid:48) .We shall construct a (homotopy) directed system of A ∞ functors F λ : WF ( M \ K ; H g ) → WF ( M \ K ; H g (cid:48) )for any monotone path λ : [0 , → C ( N ) with λ (0) = g, λ (1) = g (cid:48) . By Definition4.1 of the objects in the wrapped Fukaya category, there is a natural inclusion Ob ( WF ( M \ K ; H g )) (cid:44) → Ob ( WF ( M \ K ; H g (cid:48) )) . We take it as data of the maps Ob ( F λ ) : Ob ( WF ( M \ K ; H g )) → Ob ( WF ( M \ K ; H g (cid:48) )) . Note that even though L , L are objects of WF ( M \ K ; H g ) and hence of WF ( M \ K ; H g (cid:48) ), the corresponding morphism spaces could be different. In fact,the Hamiltonian used for WF ( M \ K ) is H g and that for WF ( M \ K ) is H g (cid:48) and sothe corresponding morphism spaces are CW ∗ ( L , L ; H g ) and CW ∗ ( L , L ; H g (cid:48) )respectively. Then we will show that the quasi-equivalence class of the A ∞ category WF ( T ∗ ( M \ K ); H g ) does not depend on g . We denote the A ∞ category by WF g ( M \ K ) := WF ( T ∗ ( M \ K ); H g )suppressing g from its notation.More precisely, we will construct a A ∞ functor f g (cid:48) g = { f kλ } with f kλ : CW ∗ ( L , L ; H g ) ⊗ . . . ⊗ CW ∗ ( L k , L k − ; H g ) → CW ∗ ( L k , L ; H g (cid:48) )associated to a homotopy of metrics from g to g (cid:48) . Such an A ∞ functor will beconstructed by an A ∞ version of Floer’s continuation map under the homotopy ofassociated Hamiltonians H g to H g (cid:48) . Remark 7.1.
It turns out that this construction of A ∞ morphism or A ∞ functorunder the change of Hamiltonians, at least in the form given in the present paper,has not been given in the existing literature as far as we are aware of. (See [Sa],however, for some construction which should be relevant to such a study.) It tookus some effort and time to arrive at the right definition we are presenting here. Atthe end of the day, our definition is motivated by the construction given in [FOOO1,Section 4.6] which defines an A ∞ morphism under its Hamiltonian isotopy in thecontext of singular chain complexes associated to a given Lagrangian submanifold L in the Morse-Bott context. WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 25
Moduli space of time-allocation stable curves.
In this subsection, werecall the moduli space, denoted by N k +1 in [FOOO2], of a decorated stable curvesof genus 0. This moduli space was used in the construction of A ∞ homomorphismstherein which we generalize for the construction of A ∞ functors. (See also [AS]for the similar moduli space named as the ‘popsicle moduli space’ for the moreelaborate version thereof.)Let M k +1 be the moduli space of (Σ; z ) where Σ is a genus zero bordered Rie-mann surface and z = ( z , . . . , z k ) are the boundary punctured points, orderedanti-clockwise, such that (Σ; z ) is stable. Let Σ = (cid:83) Σ i be the decomposition intothe irreducible components. The stability implies that there is no sphere compo-nent, since we do not put any interior marked points.We define a partial order on the set of irreducible components. Let { Σ α | α ∈ A } be the set of components of Σ and, by the definition of M k +1 , it admits a rootedtree structure, see Figure 7.1.21 in [FOOO2]. We assign a partial order ≺ on A with respect to the rooted tree structure as follows: Definition 7.2.
If every path joining Σ α to the rooted component Σ α , corre-sponding to z , intersect with Σ α , then we write α ≺ α .We recall the following definition of time-allocation. Definition 7.3 (Definition 7.1.53 [FOOO2]) . Let (Σ , ≺ ) be such a pair. We definethe time allocation ρ : A → [0 ,
1] to (Σ , ≺ ) so that if α i ≺ α j then ρ ( α i ) ≤ ρ ( α j ).See Figure 7.1.19 [FOOO2]. Definition 7.4 (Definition 7.1.54 [FOOO2]) . For k ≥
2, we define N k +1 to be theset of pairs ((Σ; z ) , ρ ) of (Σ; z ) ∈ M k +1 and its time allocation ρ .We equip N k +1 with the topology induced from that of M k +1 in an obviousway. We now describe the stratification of N k +1 . Following [FOOO2], we denote N = (Σ , z , ρ ) ∈ N k +1 . We consider the union of all irreducible components Σ i with ρ i = ρ ( i ) ∈ (0 , A of N ∈ N k +1 as follows: A = ρ − (0) (cid:116) ρ − ((0 , (cid:116) ρ − (1); (7.1) ρ − (0) = { m , . . . , m λ } ; ρ − ((0 , { f , . . . , f j } ; ρ − (1) = { m (cid:48) , . . . , m (cid:48) i } . Also denote Note that (cid:96) could be 0, and if ρ − (1) is non-empty then (cid:83) α ∈ ρ − (1) Σ α is connected and contains z . Definition 7.5.
The combinatorial type Γ = Γ( N ) of N is a ribbon graph withdecorations as follows: • Interior vertices V int of Γ correspond to the components set α ∈ A . • Exterior vertices V ext of Γ correspond to the punctured points z . • The matching condition of components determines interior edges of Γ. • An exterior edge is assigned when a component contains z . • The ribbon structure is determined by the cyclic order of the marked orsingular points on the boundary of each component. • There is a decomposition V int (Γ) = V f (Γ) (cid:116) V m (Γ) (cid:116) V m (cid:48) (Γ) (7.2)with respect to (7.1).We denote by G k +1 the set of such combinatorial types of N k +1 .Let N Γ = { N ∈ N k +1 : Γ( N ) = Γ } , then we have the decomposition N k +1 = (cid:71) Γ ∈ G k +1 N Γ . Let Γ be the graph that has only one interior vertex. For each k , there are threedifferent types of them as in (7.2). Lemma 7.6 (Lemma 7.1.55 [FOOO2]) . For any Γ ∈ G k +1 with | V (Γ) | > then N Γ is diffeomorphic to D | Γ | where | Γ | := k − − | V (Γ) \ V f (Γ) | . (7.3) Proposition 7.7 (Proposition 7.1.61 [FOOO2]) . N k +1 has a structure of smoothmanifold (with boundary or corners) that is compatible with the decomposition (ac-cording to the combinatorial types). It was shown in the proof of this proposition in [FOOO2], there exists a map I : M k +1 × [0 , → N k +1 that gives a homeomorphism onto the set of all N whose associated graph Γ hasonly one interior vertex and its time allocation ρ is constant.We then quote the following basic structure theorem of N k +1 from [FOOO2] Theorem 7.8 (Theorem 7.1.51 [FOOO2]) . For each k ∈ N , N k +1 carries thestructure of a cell complex which is diffeomorphic to k − dimensional disc D k − such that its boundary is decomposed to the union of cells described as follows: (1) M ( i +1 ,...,i + (cid:96) ) × N (1 ,...,i, ∗ ,i + (cid:96) +1 ,...,k ) . (2) m (cid:89) i =1 N ( (cid:96) i − +1 ,...,(cid:96) i ) × M m +1 , where (cid:96) , (cid:96) m = k , (cid:96) i < (cid:96) i +1 − .In the above, M ( i ,...,i k ) is M k with a new label ( i , . . . , i k ) on k punctures. Thesame for N . We remark that Case (1) contains the case i = 1, 1 + (cid:96) = k . This is the caseof M k +1 × N ∼ = M k +1 . It corresponds to the case when time allocation ρ is 1everywhere. Case (2) contains the case m = k . This is the case of N × ( M k +1 ) (cid:48) .It corresponds to the case when the time allocation ρ is 0 everywhere.Similarly as in the construction of A ∞ homomorphism given in Chapter 7 [FOOO2],we will use N k +1 for the construction of A ∞ functors.7.2. Definition of A ∞ functor and energy estimates. Firstly, consider cylin-drical adjustments g , g (cid:48) on N = M \ K of Riemannian metrics defined on M as introduced in Section 2.1. We assume g ≥ g (cid:48) and consider a homotopy λ : [0 , → C ( N ) of Riemannian metrics from g (cid:48) to g given by λ : r (cid:55)→ g ( r ) = g + r ( g (cid:48) − g ) WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 27 (or by any other homotopy r (cid:55)→ g ( r ) connecting λ (0) = g and g (cid:48) ). We denote by H λ the time-dependent Hamiltonian associated to λ defined by H λ ( s, x ) = H λ ( s ) ( x ).We will also impose additional monotonicity restriction g ≥ g (cid:48) or equivalently H g ≤ H g (cid:48) (7.4)and λ is monotone, which is needed, for example, for the energy estimates for theperturbed Cauchy-Riemann equation relevant to the definition of Floer’s continua-tion map for the wrapped Fukaya category of the cotangent bundle in general. Werefer to Remark 7.11 to see how this monotonicity condition enters in a crucial way. Remark 7.9.
For our purpose, we will obtain such an inequality by multiplyinga sufficiently large constant λ > g . Because the basemanifold M \ K is not compact, it is possible to achieve g (cid:48) ≤ λg everywhere on M \ K , only when g and g (cid:48) are Lipschitz-equivalent, i.e., when there is a constant C = C ( g, g ) (cid:48) > C g ≤ g (cid:48) ≤ Cg . This is precisely the reason why welook at those metrics g, g (cid:48) on M \ K that are smoothly extendable to whole M inthe present paper. We refer readers to Proposition 2.2 for the latter statement.For given λ = { g ( r ) } r ∈ [0 , , we consider the fiber bundles H λ → [0 , , H g ( r ) = H ( T ∗ N, (cid:98) g ( r ))and J λ → [0 , , J g ( r ) = J ( T ∗ N, (cid:98) g ( r )) . We will also use the elongation function χ : R × [0 ,
1] satisfying χ ( τ ) = (cid:40) τ ≤ τ ≥ , ≤ χ (cid:48) ≤ . The morphism of A ∞ functor F λ consists of the following data of A ∞ homomor-phism f kλ : CW ∗ ( L , L ; H g ) ⊗ · · · ⊗ CW ∗ ( L k , L k − ; H g ) → CW ∗ ( L k , L ; H g (cid:48) )[1 − k ] . In order to construct f k we need to use another moduli space of perturbed J -holomorphic curves of genus zero. Here we consider the case where components areperturbed J -holomorphic with respect to the compatible almost complex structuresand Hamiltonian functions which could vary on the components. We basically followthe construction described in [FOOO1, Section 4.6] in the current wrapped context.Note that each component Σ α is an element of M (cid:96) +1 for some (cid:96) ≥
1, and twocomponents have a matching asymptotic condition at most one point. Since ourunderlying manifold is exact and the Lagrangian submanifolds are exact, there isno possibility for sphere bubbles and disk bubbles.Now we are ready to give the definition of A ∞ functor. We start with listingthe data necessary for the construction. We would like to highlight that in theitem (3( ρ )) below, we put the time-dependent Hamiltonian H λ while we put thetime-independent Hamiltonian in Definition 5.1. Definition 7.10 (Floer data for A ∞ functor) . A Floer datum D f = D f (Σ , j ; ρ ) for(Σ , j ; ρ ) ∈ N k +1 is defined by replacing the properties (3), (4) in Definition 5.1 asfollows: (1) Weights : A ( k + 1)-tuple of non-negative integers w = ( w , . . . , w k ) whichis assigned to the marked points z satisfying w = w + · · · + w k . (2) One forms : A collection of one-forms β α ∈ Ω (Σ α ) constructed in Section3.2 satisfying (cid:15) j ∗ β agrees with w j dt .(3 ρ ) Hamiltonians : a collection H ρ of maps H α : Σ α → H ρ ( α ) assigned toeach irreducible component α ∈ A whose pull-back under (cid:15) j α uniformlyconverges to H jαλ ( w jα ) ◦ ψ w jα near each z j α for the path H j α λ : [0 , → H ρ ( α ) dictated as described above.(4 ρ ) Almost complex structures : J ρ which consist of maps J α : Σ α → J ρ ( α ) for each α ∈ A , whose pull-back under (cid:15) j α uniformly converges to φ ∗ w jα J j α near each z j α for the path J j α λ : [0 , → J ρ ( α ) associated to λ .(5) Vertical moving boundary : A map η : ∂ Σ → [1 , + ∞ ) which convergesto w j near each z j .A universal choice of Floer data D f for the A ∞ functor consists of the collectionof D f for every (Σ , j ; ρ ) ∈ (cid:96) k ≥ N k +1 satisfying the corresponding conditions as inDefinition 5.3 for N k +1 and its boundary strata.Now we consider the system of H ρ perturbed J ρ holomorphic maps { u α : Σ α → T ∗ N } α ∈ A for the data D f (Σ α , j α ; ρ ) and a ( k + 1)-pair of Hamiltonian chords x = ( x , . . . , x k ) ∈ X ( H g ; L , L ) × · · · × X ( H g ; L k − , L k ); y ∈ X ( H g (cid:48) ; L , L k )satisfying stability, shifted asymptotic conditions, and the perturbed J -holomorphicequation with respect to H s and J s for each parameter s as follows.With this preparation, we describe the structure of the collection of maps { u α } α ∈ A .As a warm-up, we first consider the case k = 1, i.e., with one input and oneoutput puncture whose domain is unstable. When the domain is smooth, it isisomorphic to R × [0 , H − , H + and homotopy { H s } s ∈ [0 , with H = H − , H = H + , we consider the non-autonomous Cauchy-Riemann equation (cid:40) ∂u∂τ + J χ ( τ ) (cid:0) ∂u∂t − X H χ ( τ ) ( u ) (cid:1) = 0 u ( τ, ∈ L, u ( τ, ∈ L (cid:48) . (7.5) Remark 7.11.
This appearance of non-autonomous Cauchy-Riemann equation inour construction of A ∞ functor is the reason why we imposed the monotonicityhypothesis (7.4) for the energy estimates. We refer to [Oh5] for the energy formula WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 29 for the non-autonomous equation: (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂τ (cid:12)(cid:12)(cid:12)(cid:12) J χ ( τ ) dt dτ = A H + ( z + ) − A H − ( z − ) − (cid:90) ∞−∞ χ (cid:48) ( τ ) (cid:18)(cid:90) ∂H s ∂s (cid:12)(cid:12)(cid:12) s = χ ( τ ) ( u ( τ, t )) dt (cid:19) dτ. (7.6)For readers’ convenience, we give its derivation in Appendix. It follows that wehave uniform energy bound for general Hamiltonians on non-compact manifolds provided the homotopy s (cid:55)→ H s is a monotonically increasing homotopy .Denote by N ( x (cid:48) ; x ) the moduli space of finite energy solution with u ( −∞ ) = x, u ( ∞ ) = x (cid:48) . We denote by N ( x (cid:48) ; x ) its compactification. An element of N ( x (cid:48) ; x )is a linear chain of maps elements u − , . . . , u − k − , u , u +1 , . . . , u + k + such that u − i ∈ M ( x − i − , x − i ) with x − i ∈ X ( H − ; L, L (cid:48) ) for 0 ≤ i ≤ k − , u + j ∈M ( x + j − , x + j ) x + j ∈ X ( H + ; L, L (cid:48) ) for 0 ≤ j ≤ k + and u ∈ N ( x − k − , x +0 ). We denotethe concatenation of such linear chain by u = ( u − , . . . , u − k − , u , u +1 , . . . , u + k + ) . (7.7)Then for given ordered sequence 0 ≤ ρ ≤ ρ ≤ . . . ≤ ρ λ ≤
1, we perform the aboveconstruction for each consecutive pair ( H ρ i , H ρ i +1 ) iteratively to form a ‘stair-case’chain which is a concatenation of the linear chains u i over i = 0 , . . . , (cid:96) . We denoteby N ( x (cid:48) ; x ) the set of such stair-case chains.Now we turn to the case k ≥
2. Let Σ α be an irreducible component of Σ. Denoteby { z α , . . . , z αk α } k α ≥ α which is the union of nodal pointsand marked points of Σ on Σ α . We denote by L αi the Lagrangian submanifold thatwe originally put around the vertex α in the chambers given by the dual graph ofΣ in the beginning. Denote by Σ α + the component attached to z α . Then we have α ≺ α + and so ρ ( α ) ≤ ρ ( α + ).We now describe the equation for u α on the strip-like region at z α . For given α, β , we consider the function homotopy s (cid:55)→ H ( α,β ) s defined by H ( α,β ) s = (1 − s ) H ρ ( α ) + sH ρ ( β ) or any homotopy { H s } s ∈ [0 , connecting H ρ ( α ) and H ρ ( β ) . In the strip-like regionnear z α on Σ α , we consider the non-autonomous Cauchy-Riemann equation on( −∞ , × [0 , ∂u∂τ + J χ ( τ α − τ ) (cid:18) ∂u∂t − X H ( α,α +) χ ( τα − τ ) ( u ) (cid:19) = 0for some τ α ≤ − z αi , we denote by Σ α − the componentattached to z iα at the nodal point z α − . Then we put the non-autonomous equationon [0 , ∞ ) ⊂ [0 , ∂u∂τ + J χ ( τ − τ αi ) (cid:18) ∂u∂t − X H ( α − ,α ) χ ( τ − ταi ) ( u ) (cid:19) = 0for some τ αi ≥ At each nodal point of Σ associated with α ≺ β , we insert a linear chain associ-ated to H − = H ρ ( α ) , H + = H ρ ( β ) of the type u − , . . . , u − k − , u on the strip-like regions of z αi with u − i as above but u is a map on [0 , ∞ ) × [0 , u , u +1 , . . . , u + k + on the strip-like regions of z β with u + j as above but u is a map on ( −∞ , × [0 , We put the chain at most one of the two regions.
In summary, we require the map u α on Σ α to satisfy(1) ( du α − X H α ⊗ β α ) (0 , J α = 0 (7.8)for each α ∈ A . Here we emphasize the fact that the Hamiltonian H α isautonomous away from the strip-like regions of each component. We donot exclude the case where the component is of the m -type, i.e., one thatsatisfies the autonomous equation of H ρ ( α ) . (2) u α ◦ (cid:15) j α ( −∞ , t ) = ψ w jα ◦ x j α ( t ), for the input punctures j α of Σ α .(3) u ◦ (cid:15) α (+ ∞ , t ) = ψ w α ◦ x α ( t ), for the output puncture 0 α of Σ α .(4) (cid:0) (Σ α , z ) , ( u α ) (cid:1) α ∈ A is stable. ρ ρ ρ ρ ≤ ≤ ≤ ≤ ≤ ρ w w w w w w w w w ρ Figure 2.
An example of slit domains for the A ∞ functorTo be able to construct the relevant compactified moduli space of solutions (7.8),we need the uniform energy bound and C estimates both of which require themonotonicity condition (7.4). The proof of the following energy bound is given bycombining the energy bound obtained in Section 6.1 and that of (7.6) in Remark7.11. Lemma 7.12.
For any finite energy solution of (7.8) , we have the energy identity E ( u ) ≤ A H + ( x ) − k (cid:88) j =1 A H − ( x j ) − (cid:88) α ∈ A (cid:90) ∞−∞ χ (cid:48) ( τ ) (cid:18)(cid:90) ∂H αλ χ ∂s (cid:12)(cid:12)(cid:12) s = χ ( τ ) ( u ( τ, t )) dt (cid:19) dτ (7.9) where the map λ χ : R ± → [0 , is the elongated path defined by λ χ ( τ ) = λ ( χ ( τ )) . In particular, if λ is a monotonically decreasing path, i.e, if ∂H λ ∂s ≥ , then we have E ( u ) ≤ A H + ( x ) − k (cid:88) j =1 A H − ( x j ) . WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 31
Then we denote the space of such a system of maps satisfying the above condi-tions by N k +1(Σ ,j ; ρ ) ( y ; x ) = N k +1 ( y ; x ; D f (Σ , j ; ρ )) . (7.10)Consider a parameterized moduli space N k +1 ( y ; x ) = (cid:71) (Σ ,j ; ρ ) ∈N k +1 N k +1(Σ ,j ; ρ ) ( y ; x ) . We have a forgetful map forget : N k +1 ( y ; x ) → N k +1 for k ≥ N k +1 ( y ; x ) = (cid:71) Γ ∈ G k +1 N Γ ( y ; x )where N Γ ( y ; x ) = forget − (Γ). Lemma 7.13.
For a generic choice of universal Floer data D f , the moduli space N Γ ( y ; x ) is a manifold of dimension µ ( y ) − k (cid:88) i =1 µ ( x i ) − | Γ | where | Γ | is given as in (7.3) . In particular, if V (Γ) = V f (Γ), i.e., there is no m or m (cid:48) component, then thedimension becomes µ ( y ) − k (cid:88) i =1 µ ( x i ) − k. The matrix coefficient of the A ∞ homomorphism f kλ : CW ∗ ( L , L ; H g ) ⊗ · · · ⊗ CW ∗ ( L k , L k − ; H g ) → CW ∗ ( L k , L ; H g (cid:48) )[1 − k ] . is defined by f kλ ( x ⊗ · · · ⊗ x k ) = (cid:88) y ( − ♠ N k +1 ( y ; x ) y, (7.11)where ♠ = (cid:80) kj =1 j · µ ( x j ) + k and N k +1 ( y ; x )) becomes zero unless µ ( y ) = k (cid:88) i =1 µ ( x i ) + 1 − k. Since the equation (7.8) for each given ρ ( α ) does not involve moving boundarycondition but only fixed boundary condition, the C -bound for (5.5) still applies toprove the following C -bounds.Let (Σ , j ) be equipped with strip-like ends (cid:15) k : Z ± → Σ at each puncture z k asbefore. Proposition 7.14 (Vertical C estimates for f -components) . Let x = ( x , . . . , x k ) be a ( k + 1) -tuple of Hamiltonian chords x j ∈ X ( H g ; L j − , L j ) . Then max z ∈ Σ | p ( u ( z )) | ≤ max r ∈ [0 , max { ht ( x ; H g ( r ) , { L i } ) | i = 0 , . . . , k } for any solution u of (7.8) . Proposition 7.15 (Horizontal C estimates for f -components) . Assume the one-form β satisfies (3.5) . Suppose L j ⊂ W λ for all j = 0 , . . . , k , and let x j ∈ X ( H g ; L j − , L j ) for j = 1 , . . . , k and x ∈ X ( H g (cid:48) ; L , L k ) be Hamiltonian chords.Then Im u ⊂ W λ (7.12) for any solution u of (7.8) . We now describe the structure of boundary (or codimension one) strata of N k +1 ( y ; x ). We first form the union N k +1 ( y ; x ) = (cid:71) (Σ ,j ; ρ ) ∈N k +1 N k +1(Σ ,j ; ρ ) ( y ; x ) . Then consider the asymptotic evaluation maps ev (cid:96) − : (cid:71) ( y ; x ) N k +1 ( y ; x ) → X ( H ρ ( z (cid:96) ) ; L (cid:96) − , L (cid:96) ) × [0 ,
1] for (cid:96) = 1 , . . . , kev : (cid:71) ( y ; x ) N k +1 ( y ; x ) → X ( H ρ ( z ) ; L , L k ) × [0 , ev (cid:96) − ( { u α } α ∈ A ) = ( u ◦ (cid:15) (cid:96) ( −∞ , · ) , ρ ( z (cid:96) )) for (cid:96) = 1 , . . . , kev ( { u α } α ∈ A ) = ( u ◦ (cid:15) (+ ∞ , · ) , ρ ( z ))where ρ ( z i ) ∈ [0 ,
1] is a ρ value of the component having the i -th end. We alsoconsider similar evaluation maps from ev (cid:96) − : (cid:71) ( y ; x ) M k +1 ( y ; x ) → X ( H g ; L (cid:96) − , L (cid:96) ) for (cid:96) = 1 , . . . , kev : (cid:71) ( y ; x ) M k +1 ( y ; x ) → X ( H g ; L , L k ) . Theorem 7.16.
Let g, g (cid:48) be a pair of smooth metrics on M satisfying g ≥ g (cid:48) ,and λ be a monotone homotopy between them with λ (0) = g, λ (1) = g (cid:48) . Then thedefinition (7.11) is well-defined and the maps { f kλ } satisfy the A ∞ functor relation.Proof. Let (cid:0) (Σ ( i ) , z ( i ) ) , ( u ( i ) α ) , ( ρ ( i ) α ) (cid:1) i ∈ N α ∈ A be a sequence of elements of N k +1 ( y ; x ),then(1) One of the component Σ ( i ) α splits into two components as i → + ∞ .(2) Two component Σ α and Σ α sharing one asymptotic matching conditionsatisfy lim ρ ( i ) α = lim ρ ( i ) α when i → + ∞ .(3) lim i → + ∞ ρ ( i ) α = 0.(4) lim i → + ∞ ρ ( i ) α = 1. WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 33 (See Figure 7.1.20 [FOOO2].)Both Type (1) and Type (2) above consists of the following fiber product N k +1 ( y ; x ) ev i + × ev − N k +1 ( ∗ ; x )with k + k = k and with opposite sign and so they cancel each other when µ ( y ) = (cid:80) ki =1 µ ( x i ) + 1 − k . Here for given x = ( x , x , . . . , x k ), x = ( x , . . . , x i − , ∗ , x i + k , . . . , x k + k ) , x = ( x i , . . . , x i + k − ) . Note that the case (3), (4) occur at leaf components and the root component of { Σ α } α ∈ A , which correspond to the two cases of domain degenerations described inTheorem 7.8 respectively.The case (3) contributes to the following terms (cid:88) k = k + k − n,k ( − ‡ +1 f k λ ( x , . . . , x n , m k ( x n +1 , . . . , x n + k ) , x n + k +1 , . . . , x k ) , (7.13)where ‡ = ‡ n = (cid:80) ni =1 µ ( x i ) − n . While the case (4) corresponds to (cid:88) k = k + ··· + k (cid:96) (cid:96),k ,...,k (cid:96) m (cid:96) ( f k λ ( x , . . . , x k ) , . . . , f k (cid:96) λ ( x k − k (cid:96) +1 , . . . , x k )) . (7.14)This verifies that { f k } k ∈ N satisfies the A ∞ homomorphism relation, i.e. (7.13)equals to (7.14). We refer to Chapter 7 Section 1 of [FOOO2] for the detail of thisproof for the case of A ∞ homomorphism but the same argument applies to the caseof A ∞ functors. (cid:3) Homotopy of A ∞ functors Now let λ , λ ∈ C ∞ ([0 , , C ( N )) be two paths of admissible metrics connecting g = λ (0) = λ (0) , g (cid:48) = λ (1) = λ (1)with g ≥ g (cid:48) . Denote by F = { f kλ } k ∈ N and F (cid:48) = { f (cid:48) kλ } k ∈ N be the corresponding A ∞ functors from WF ( M \ K ) to WF ( M \ K ) constructed in the previous subsection.Now suppose we are given a path Γ ∈ C ∞ ([0 , × [0 , , C ( N )) of paths connecting λ and λ . The main objective of this subsection is to construct an A ∞ homotopy,denoted by h = h Γ , between F and F (cid:48) .8.1. Definitions of A ∞ homotopy and composition. To motivate our con-struction of the relevant decorated moduli space below, we first recall the definitionof an A ∞ homotopy from [Lef], [Se1]. Definition 8.1 ( A ∞ homotopy) . Let ( A , m A ) , ( B , m B ) be two A ∞ categories, and F = { f k } k ∈ N , G = { g k } k ∈ N be two A ∞ functors from A to B . A homotopy H from F to G is a family of morphisms h k : A ⊗ k → B , k ≥ , of degree − k satisfying the equation that( f k − g k )( a , . . . , a k )= (cid:88) m,n ( − † h k − m +1 ( a , . . . , a n , m m A ( a n +1 , . . . , a n + m ) , a n + m +1 , . . . , a d )+ (cid:88) r,i (cid:88) s ,...,s r ( − ♣ m r B (cid:0) f s ( a , . . . , a s ) , . . . , f s i − ( . . . , a s + ··· + s i − ) , h s i ( a s + ··· + s i − +1 , . . . , a s + ··· + s i ) , g s i +1 ( a s + ··· + s i +1 , . . . ) , . . . , g s r ( a d − s r +1 , . . . , a d ) (cid:1) . (8.1)where s + · · · + s r = d and † = n (cid:88) i =1 µ ( a i ) − n, ♣ = s + ··· + s i − (cid:88) (cid:96) =1 µ ( a (cid:96) ) − i − (cid:88) (cid:96) =1 s λ . We will apply this definition of homotopy in the proof of consistency conditionfor the system (cid:0) WF ( M \ K ; H g ) , { m k } (cid:1) Φ λ −→ (cid:0) WF ( M \ K ; H g (cid:48) ) , { m k } (cid:1) to define a homotopy: For a triple g ≥ g (cid:48) ≥ g (cid:48)(cid:48) of metrics on M we have functors F λ , F λ (cid:48) , and F λ (cid:48)(cid:48) . We consider the composition of A ∞ functors and construct ahomotopy between F λ (cid:48) ◦ F λ and F λ (cid:48)(cid:48) .The composed A ∞ functor F λ (cid:48) ◦F λ which we recall consists of the following dataof A ∞ homomorphism:( F λ (cid:48) ◦ F λ ) d ( x , . . . , x d ) = (cid:88) r (cid:88) s ,...,s r f rλ (cid:48) ( f s λ ( x , . . . , x s ) , . . . , f s r λ ( x d − s r +1 , . . . , x d ))We consider two paths of cylindrical Riemannian metric between the concatenatedpath λ ∗ λ (cid:48) : [0 , → C ( N ) and the direct path λ (cid:48)(cid:48) connecting g and g (cid:48)(cid:48) . It easilyfollows from contractibility of the space of cylindrical Riemannian metrics we canconstruct a path (of paths) Γ : [0 , × [0 , → C ( N )between λ ∗ λ (cid:48) and λ (cid:48)(cid:48) . Here we use the coordinate ( r, s ) for [0 , × [0 , s isthe newly adopted one. Note that Γ( − , s ) =: λ s gives a homotopy between g and g (cid:48)(cid:48) for each s ∈ [0 , A ∞ homotopy associated to a geometrichomotopy. For each fixed s ∈ [0 , J rs = J ( T ∗ N, Γ( r, s )) , H rs = H ( T ∗ N, Γ( r, s )) . (8.2)Let us recall from Definition 7.2 that (Σ; z ; ρ ) ∈ N k +1 admits a partial order ≺ onthe set A of irreducible components α of Σ. Remark 8.2. (1) With the above preparation, it looks natural to consider theparameterized f -moduli space, which was called a timewise moduli space in[FOOO1, FOOO2] N k +1para ( y ; x ) = (cid:91) s ∈ [0 , { s } × N k +1 ρ ( s ) ( y ; x ) , (8.3) WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 35 where ρ ( s ) is a homotopy between ρ (0) , ρ (1) : A → [0 ,
1] which induces the A ∞ functors F λ (cid:48) ◦ F λ and F λ (cid:48)(cid:48) , respectively. It is fibered over [0 , ev [0 , : N k +1para ( y ; x ) → [0 , . By definition, each fiber of this map is compact. It is tempting to definethe homotopy by setting its matrix coefficient to be (cid:0) N k +1para ( y ; x ) (cid:1) when the moduli space associated to ( y ; x ) has its virtual dimension 0.However this definition will not lead to the relation required for the A ∞ homotopy described in Definition 8.1 because the one-dimensional com-ponents of the above defined moduli space has its boundary that is notconsistent with the homotopy relation.So we need to modify this moduli space so that the deformed homotopymap whose matrix coefficients are defined via counting the zero dimensionalcomponents of the modified moduli space. One requirement we impose inthis modification is that the strata the domain of each element of which isirreducible are unchanged from above. We will modify those strata whoseelements have nodal domains, i.e., not irreducible.(2) We would like to mention that this kind of modification process alreadyappeared in the definition of A ∞ map f where the time-order decoration ρ is added to the moduli space of bordered stable maps to incorporate thedatum of geometric homotopy λ : r (cid:55)→ g λ ( r ). We also recall the start-ing point of Fukaya-Oh-Ohta-Ono’s deformation theory of Floer homology[FOOO1] in which modifying the definition of Floer boundary map to curethe anomaly ∂ (cid:54) = 0.(3) A general categorial construction of A ∞ homotopy associated to Lagrangiancorrespondence is given as a natural transformation between two A ∞ func-tors using the quilted setting in [MWW]. It appears to us that to apply thisgeneral construction, we should lift a family of Hamiltonian to a Lagrangiancobordism arising as the suspension of the associated Lagrangian isotopy.We avoid using this general set-up and Lagrangian suspension but insteadquickly provide a direct and down-to-earth construction as a variation ofthe construction given in [FOOO2, Section 4.6].8.2. Timewise decorated time-allocation stable curves.
We recall the eval-uation map ev [0 , : N k +1para ( y ; x ) → [0 ,
1] which naturally induces a map A → [0 , α (cid:55)→ s ( α )where s ( α ) is the time in [0 ,
1] at which the irreducible component ((Σ α , z α ) , u α ) of((Σ , z ) , u ) is attached. For our modification of the above mentioned moduli space N k +1 w , para ( y ; x ), we assign a total order (cid:67) on the component set A of Σ which isdetermined by the rooted ribbon structure on (Σ , z ) ∈ M k +1 . This order will playan important role in establishing the homotopy relation (8.1) between F λ (cid:48)(cid:48) and F λ (cid:48) ◦ F λ . WF ( M \ K ; H g ) WF ( M \ K ; H g (cid:48)(cid:48) ) WF ( M \ K ; H g (cid:48) ) F λ F λ (cid:48)(cid:48) H F λ (cid:48) Motivated by this diagram and to achieve the A ∞ homotopy relation given inDefinition 8.1, we would like to attach the m -component only at ∂ [0 ,
1] = { , } inthe way that the one with time-allocation ρ = 0 at s = 0 and ρ = 1 at s = 1. Wenote that the components with ρ ( α ) = 0 , m -components.Recall from Definition 7.5 that Γ = Γ( N ) is the rooted ribbon tree associatedto N ∈ N k +1 or M k +1 . From now on, we canonically identify V ext (Γ) with theboundary punctures z = ( z , . . . , z k ) and V int (Γ) with the component A . Eachedge of Γ carries the natural orientation which flows into the root z .Each z j , j ≥ (cid:96) j = (cid:96) z z j from z to z j respecting the orientation of Γ (in the opposite direction). For an interior vertex v of Γ, we say v ∈ (cid:96) j if v ∈ V int ( (cid:96) j ) where V int ( (cid:96) j ) = { v ∈ V int (Γ) | v ∈ (cid:96) j } . We have an obvious order on each V int ( (cid:96) j ) given by the edge distance from theroot. Denote v α ∈ V int (Γ) the interior vertex corresponding to α ∈ A . Now define j tm : A → { , . . . , k } ; α (cid:55)→ min { j | v α ∈ (cid:96) j } . Definition 8.3.
Let Σ be a bordered stable curve of genus zero with k + 1 markedpoints with its combinatorial type T . Let α, β ∈ A . We say α (cid:67) β if one of thefollowing holds:(1) j tm ( α ) < j tm ( β ).(2) j tm ( α ) = j tm ( β ) and ρ ( α ) ≤ ρ ( β ).It is easy to check that (cid:67) defines a total order on A and depends only on theribbon structure of Γ. (We refer to [Sa, p.7] for more pictorial description of thisorder.) We denote the set of descendants of δ byDesc( δ ) := { α ∈ A | α (cid:67) δ, α (cid:54) = δ } . We now recall the universal family S k +1 → M k +1 . Each element v = (Σ , j ; σ ) ∈ S k +1 picks out a distinguished component δ ( v ) ∈ A that contains the point σ ∈ Σ. Definition 8.4 (Universal parametrization) . Let v ∈ S k +1 and δ ( v ) be the as-sociated component. We define an s -parameterized map s v : [0 , × A → [0 , s v ( s )( α ) = α ∈ Desc( δ ( v )); s if α = δ ( v );0 otherwise . (8.4)Note that the map s v is determined by the ribbon graph Γ( v ) and the distin-guished component δ ( v ). Indeed, s v is defined over S k +1 / ∼ where v ∼ v (cid:48) ⇐⇒ Γ( v ) = Γ( v (cid:48) ) , δ ( v ) = δ ( v (cid:48) ) . We are ready to describe a parameter space L k +1 defined over S k +1 / ∼ whichwill be used in the construction of A ∞ homotopy or 2-morphism. WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 37
Definition 8.5.
For each v = [Σ , j ; σ ] ∈ S k +1 / ∼ with k ≥
2, we define L k +1 to bethe set of quadruples (Σ , j ; ρ v ; s v ) , where • ρ v is a time allocation in Definition 7.3 satisfying ρ ( δ ( v )) (cid:54) = 0 or 1, • s v is a s -parameterized map defined in Definition 8.4.By extending Definition 7.5, let us denote a ribbon graph induced from L ∈ L k +1 by Γ( L ). The ribbon graph Γ( L ) has an additional decoration of δ ( v ). Note that δ ( v ) ∈ V f ( Γ ) by the condition of ρ v in Definition 8.5. Let L Γ = { L ∈ L k +1 | Γ( L ) = Γ } , then we have the decomposition L k +1 = (cid:91) Γ ∈ G k +1 L Γ , (8.5)where G k +1 be the set of combinatorial types of L k +1 .As analogues of Lemma 7.6 and Proposition 7.7 we have Lemma 8.6.
For any Γ ∈ G k +1 , L Γ is diffeomorphic to D | Γ | +1 .Proof. The space L Γ additionally has a datum of s -parameterized map s v comparedto N Γ , so the lemma follows. (cid:3) Proposition 8.7.
The parameter space L k +1 has a structure of smooth manifoldwith boundaries and corners, which are compatible with the decomposition (8.5).Moreover, there is a projection map L k +1 → [0 , , j ; ρ v ; s v ) (cid:55)→ s v ( s )( δ ( v )) = s which is a smooth submersion.Proof. The map factors through the map L k +1 → M k +1 × [0 ,
1] defined by(Σ , j ; ρ v ; s v ) (cid:55)→ (Σ , j ; s v ( s )( δ ( v ))) = (Σ , j ; s ) , so the submersion property follows. (cid:3) Lemma 8.8.
The boundary ∂ L k +1 is decomposed into the union of the followingthree types of fiber products: (1) M (cid:96) +1 (cid:0) N k +1 × · · · × L k i +1 × · · · × N k λ +1 (cid:1) , where (cid:80) (cid:96)i =1 k i = k . (2) L (cid:96) +1 M k − (cid:96) +1 where (cid:96) = 1 , . . . , k − . (3) { , } × N k +1 . Construction of A ∞ homotopy. We first describe the situation we are infor the purpose of constructing an A ∞ functor. Definition 8.9 (Floer data for A ∞ homotopy) . A Floer datum D h ( L ) for L =(Σ , j ; ρ ; s ) ∈ L k +1 is a quintuple( w , { β α } α ∈ A , H L = { H α } α ∈ A , J L = { J α } α ∈ A , η )defined by modifying a Floer datum for A ∞ functor in Definition 7.10 as follows: (1) A ( k + 1)-tuple of non-negative integers w = ( w , . . . , w k ) which is assignedto the marked points z satisfying w = w + · · · + w k . (2) A consistent choice of one-forms β α ∈ Ω (Σ α ), for each α ∈ A , constructedin Section 3.2 satisfying (cid:15) j ∗ β α agrees with w j dt .(3 ρ, s ) Hamiltonian perturbations H L which consist of maps H α : Σ α → H ρ ( α ) s v ( s,α ) for each α ∈ A , see (8.2). Its pull-back under the map (cid:15) j α uniformlyconverges to H ( w jα ) ◦ ψ w jα near each z j α for some H ∈ H ρ ( α ) s v ( s,α ) .(4 ρ, s ) Almost complex structures J L which consist of maps J α : Σ α → J ρ ( α ) s v ( s,α ) for each α ∈ A , whose pull-back under (cid:15) j α uniformly converges to φ ∗ w jα J near each z j α for some J ∈ J ρ ( α ) s v ( s,α ) , see also (8.2).(5) A map η : ∂ Σ → [1 , + ∞ ) which converges to w j near each z j .A universal choice of Floer data D h for the A ∞ homotopy consists of the collec-tion of D h for every (Σ , j ; ρ ; s ) ∈ (cid:96) k ≥ L k +1 satisfying the corresponding conditionsas in Definition 5.3 for L k +1 and its boundary strata. ρ ρ ρ ρ ≤ ≤ ≤ ≤ ≤ ρ w w w w w w w w w ρ s ≤ = = = ≤ Figure 3.
An example of slit domains for the A ∞ homotopyHere we would like to recall the readers that the current geometric circumstanceis that of 2-morphisms. The collection H = { H rs } ( r,s ) ∈ [0 , satisfies H s ≡ H g , H s ≡ H g λ , (8.6)see (8.2).For ( k + 1)-pair of Hamiltonian chords x = ( x , x , . . . , x k ) ∈ X ( H g ; L , L ) × · · · × X ( H g ; L k − , L k ); y ∈ X ( H λ ; L , L k ) , we consider a system of maps { u α : Σ α → T ∗ N } α ∈ A with respect to D h over L = (Σ , j ; ρ ; s ) ∈ L k +1 satisfying the following: WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 39 (1-a) For α ∈ A Σ except the distinguished component, u α satisfies( du α − X H α ⊗ β α ) (0 , J α = 0for H α ∈ H ρ ( α ) ∗ and J α ∈ J ρ ( α ) ∗ , where ∗ = 0 , s ( α ).(1-b) For the distinguished component δ ∈ A in Definition 8.4 and for each s ∈ [0 , u δ ( s ) satisfying( du δ ( s ) − X H δ ( s ) ⊗ β δ ) (0 , J δ ( s ) = 0 , where H δ ( s ) ∈ H ρ ( α ) s ( s,δ ) and J δ ( s ) ∈ J ρ ( α ) s ( s,δ ) .and the conditions (2)–(6) for N k +1(Σ ,j ; ρ ) ( y ; x ) in (7.10). For each s ∈ [0 , { u α } α ∈ A \{ δ } ∪ { u δ ( s ) } satisfying the above by L k +1 L ( s )( y ; x ) = L k +1 ( y ; x ; D h (Σ , j ; ρ ; s ; s )) . Now we consider parameterized moduli spaces given by L k +1 L ( y ; x ) = (cid:91) s ∈ [0 , L k +1 L ( s )( y ; x ); L k +1 ( s )( y ; x ) = (cid:91) L ∈L k +1 L k +1 L ( s )( y ; x ); L k +1 ( y ; x ) = (cid:91) L ∈L k +1 L k +1 L ( y ; x ) = (cid:91) s ∈ [0 , L k +1 ( s )( y ; x ) . Similarly as for N k +1 ( y ; x ), we have a forgetful map forget : L k +1 ( y ; x ) → L k +1 for k ≥ ev [0 , : L k +1 ( y ; x ) → [0 ,
1] defined by { u α } α ∈ A \{ δ } ∪ { u δ ( s ) } (cid:55)→ s . (8.7)Standard parameterized transversality then proves the following Lemma 8.10.
Suppose that L k +1 ( s )( y ; x ) with s = 0 , are transversal. Then fora generic choice Floer data D h for the A ∞ homotopy with fixed ends for s = 0 , ,the moduli space L k +1 ( y ; x ) is a manifold of dimension µ ( y ) − k (cid:88) i =1 µ ( x i ) + k. Again, when µ ( y ) = (cid:80) ki =1 µ ( x i ) − k , a map is defined by counting rigid solutionsof L k +1 ( y ; x ) with respect to the Floer data D h in Definition 8.9 h k D ( ψ w ( x ) ⊗ ψ w ( x ) ⊗ · · · ⊗ ψ w k ( x k )) = (cid:88) y ( − † (cid:0) L k +1 ( y ; x ) (cid:1) ψ w ( y ) , where † = (cid:80) ki =1 i · µ ( x i ). By the same identification as used in the definition m k and f k , we will obtain a A ∞ homotopy h k : CW ∗ ( L , L ; H g ) ⊗ · · · ⊗ CW ∗ ( L k , L k − ; H g ) → CW ∗ ( L k , L ; H g (cid:48) )[ − k ] . For the later purpose, we describe the structure of the zero dimensional com-ponent more carefully. Recall that the fiberwise dimension of L k +1 ( y ; x ) for thecurrent circumstance is − s = 0 , s = 1 are generic.Therefore there is no contribution therefrom. Therefore there are a finite numberof 0 < s < s < · · · < s j < (cid:0) L k +1 ( y ; x ) (cid:1) = j (cid:88) i =1 (cid:0) L k +1 ( s )( y ; x ) | s = s i (cid:1) . Furthermore, generically the associated moduli space L k +1 ( s )( y ; x ) | s = s i is mini-mally degenerate , i.e., the cokernel of the associated linearized operator has dimen-sion 1. (See [Lee] for the relevant parameterized gluing result.) Denote bySing I ⊂ [0 , A ∞ homotopy for themap { h k } k ∈ N we need to look up the codimension one strata of the moduli space L k +1 ( y ; x ). Especially we need to examine the structure of the boundary of one-dimensional components i.e., of those satisfying µ ( y ) − k (cid:88) i =1 µ ( x j ) + k = 1 . (8.8) Theorem 8.11.
The maps { h k } k ∈ N satisfies the homotopy relation (8.1) .Proof. We first recall the current geometric circumstance. We are given two Hamil-tonian H ∈ H ( (cid:98) g i ) and H ∈ H ( (cid:98) g λ ), and two paths r (cid:55)→ H ∗ ( r ) ∈ H ( (cid:98) g ( r )) with ∗ = 0 , s (cid:55)→ H s interpo-lating the two paths with s ∈ [0 , WF ( M \ K ; H g ) WF ( M \ K ; H g (cid:48)(cid:48) ) F λ (cid:48)(cid:48) ( H ( r )) F λ (cid:48)(cid:48) ( H ( r )) H ( H s ( r )) Let us consider a sequence (cid:16) s ( i ) δ , Γ( (cid:126)u ( i ) ) , (Σ ( i ) α , ρ ( i ) α ) α ∈ A (cid:17) i ∈ N (8.9)constructed from (cid:126)u ∈ L k +1 ( y ; x ) satisfying (8.8), i.e., when the fiberwise dimensionbecomes zero. Here s δ is the s -image of the distinguished component δ , see (8.7),and Γ( (cid:126)u ) is a ribbon graph with decorations defined as in Definition 7.5.Generically, the change of Γ( (cid:126)u ) occurs at finite number points s = s in [0 , II ⊂ [0 , s at which the fiber ev − , ( s ) contains an element that is minimally degenerate, i.e., the cokernel of the WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 41 linearized operator has dimension 1. We denote bySing
III ⊂ [0 , . Generically, the three subsets Sing I , Sing II , Sing
III are pairwise disjoint. We denotethe union by Sing([0 , s ).The sequence (8.9) with fixed type of domain configuration Γ(Σ), see Defini-tion 7.5. By choosing a subsequence, we may assume that the sequence s ( i ) δ isincreasing.Then we have the following possible scenarios:(1) s ( i ) δ converges to 0 < s <
1, one of the points in Sing([0 , s ).(2) lim i → + ∞ s ( i ) δ = 0.(3) lim i → + ∞ s ( i ) δ = 1.We examine the three cases more closely. The first case s ( i ) δ → s is furtherdivided into the following four different scenarios by Gromov-Floer compactness:(1-i) One of the component Σ ( i ) α with 0 < ρ ( i ) α < s = s as i → + ∞ .(1-ii) Two component Σ α and Σ α at s = s sharing one asymptotic matchingcondition satisfy 0 < lim ρ ( i ) α = lim ρ ( i ) α < s = s when i → + ∞ .(1-iii) lim i → + ∞ ρ ( i ) α = 0 .(1-iv) lim i → + ∞ ρ ( i ) α = 1.Each of these cases is the analog to the corresponding scenario of the study of f -moduli space N k +1 ( y ; x ) in the previous section. One difference is that the pa-rameter s = s is not transversal fiberwise but transversal as a parameterizedproblem. So among the two components α , α one is an m -component and theother is an h -component, i.e., a fiberwise f -component with minimal degeneracy.Then by the same argument as in N k +1 ( y ; x ) applied to the parametric case, thecase (1-i) and (1-ii) cancel out.For the convenience sake, we denote the associated structure maps by m ∗ , f ∗ ,and h ∗ respectively. A codimension one phenomenon of (1-iii) occurs when α is anindex for a m -component. We note that this component α is generically regularand the underlying zero-dimensional h -component containing α is also regular as aparameterized moduli space. Since at ρ = 0 ,
1, the Floer datum is constant over s ∈ [0 , s near s . This contradicts tothe fact that s = s is contained in the discrete set Sing([0 , s ) unless m -bubble isattached at s = 0 ,
1. Then it contributes the following terms (cid:88) m,n ( − † h k − m +1 ( x , . . . , x n , m mg ( x n +1 , . . . , x n + m ) , x n + m +1 , . . . , x d ) , (8.10)where † = (cid:80) ni =1 µ ( x i ) − n. For the case (1-iv), it follows from (8.4) that a codimension one strata can beobtained when α is an index for the root component and is different from the distinguished component δ . Then it corresponds to (cid:88) r,i (cid:88) s ,...,s r ( − ♣ m rg (cid:48)(cid:48) (cid:0) f s ( x , . . . , x s ) , . . . , f s i − ( . . . , x s + ··· + s i − ) , h s i ( x s + ··· + s i − +1 , . . . , x s + ··· + s i ) , g s i +1 ( x s + ··· + s i +1 , . . . ) , . . . , g s r ( x d − s r +1 , . . . , x d ) (cid:1) , (8.11)where ♣ = (cid:80) s + ··· + s i − (cid:96) =1 µ ( x (cid:96) ) − (cid:80) i − (cid:96) =1 s (cid:96) . Among many terms from the cases (2)and (3), all of them are cancelled out except the following two terms(2 (cid:48) ) lim i → + ∞ s ( i ) δ = 0, where δ is the minimum index with respect to (cid:67) .(3 (cid:48) ) lim i → + ∞ s ( i ) δ = 1, where δ is the maximum index with respect to (cid:67) .because of the total order (cid:67) and the definition of time-wise product s . These twocases give the following terms − f d ( x , . . . , x d ) , g d ( x , . . . , x d ) , (8.12)respectively.The algebraic relation coming from the combination of (8.10), (8.11), and (8.12)is nothing but the A ∞ homotopy relation (8.1). (cid:3) This concludes the proof of the following theorem.
Theorem 8.12.
Let M be a closed manifold and g, g (cid:48) be two metrics on M suchthat their cylindrical adjustments satisfy g ≥ g (cid:48) . Suppose λ and λ be homotopiesconnecting g and g (cid:48) and let Γ : [0 , → C ( N ) be a homotopy between λ and λ .Then there exists an A ∞ homotopy between F λ and F λ . We denote the resulting A ∞ category by WF g ( M \ K ) := WF ( T ∗ N, H g ) . (8.13)In the next section, we will prove the quasi-equivalence class of the A ∞ category WF ( T ∗ N ; H g ) is independent of the choice of metrics g .9. Independence of choice of metrics
In this section, we consider the wrapped Fukaya categories WF ( T ∗ N, g ) and WF ( T ∗ N, h ) constructed on the knot complement M \ K for two metrics g, h of M .Moreover, it follows from Proposition 2.2 that the two Riemannian metric g and h are Lipschitz equivalent , i.e., there exists a constant
C >
C h ≤ g ≤ Ch , (9.1)on M \ K .For the convenience of notation, we denote the kinetic energy Hamiltonian H g associated to g also by H ( g ) in this section.Next we prove the following equivalence theorem. Theorem 9.1.
Let ( g, h ) be a pair of Lipschitz equivalent metrics on an orientabletame manifold N . Then two induced wrapped Fukaya categories WF g ( T ∗ N ) and WF h ( T ∗ N ) are quasi-equivalent. WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 43
Proof.
By the Lipschitz equivalence (9.1), we have (cid:40) H ( g ) ≤ H ( h /C ); H ( h ) ≤ H ( g /C ) (9.2)recalling that the Hamiltonian is given by the dual metric.We have two A ∞ functors between induced wrapped Fukaya categoriesΦ : WF ( T ∗ N ; H ( h )) → WF ( T ∗ N ; H ( g /C ));Ψ : WF ( T ∗ N ; H ( g )) → WF ( T ∗ N, H ( h /C )) (9.3)which are defined by the standard C -estimates for the monotone homotopies. Notethat the morphism is naturally defined from the smaller metric to the bigger metric .Now consider the composition of the functorsΨ ◦ Φ : WF ( T ∗ N ; H ( h )) → WF ( T ∗ N ; H ( g /C ));Φ ◦ Ψ : WF ( T ∗ N ; H ( g )) → WF ( T ∗ N ; H ( g /C )) . These are homotopic to natural isomorphisms induced by the rescaling of metrics ρ C : WF ( T ∗ N ; H ( h )) → WF ( T ∗ N, H ( h /C )); η C : WF ( T ∗ N ; H ( g )) → WF ( T ∗ N ; H ( g /C )) , respectively. This proves that Φ and Ψ are quasi-equivalences. (cid:3) Remark 9.2.
Another class of metrics we will study in [BKO] is a complete hy-perbolic metric h on N = M \ K for hyperbolic knots K . In such a case, we can ex-ploit hyperbolic geometry to directly construct another A ∞ category WF ( ν ∗ T ; H h )without taking a cylindrical adjustment. Since h is not Lipschitz-equivalent to acylindrical adjustment g on M \ K of any smooth metric g on M , that categorymay not be quasi-equivalent to WF ( M \ K ) constructed in the present paper.9.1. Wrap-up of the construction of wrapped Fukaya category WF ( M \ K ) . In this section, we restrict ourselves to the case when N is a knot complement M \ K in a closed oriented 3-manifold M , and wrap up the proofs of the main theoremsstated in the introduction.We first note that one class of Lipschitz equivalent metrics on N considered inTheorem 9.1 is obtained by restricting any Riemannian metric g M to M . Note thatthese metrics on N is incomplete . Theorem 9.3.
Let M be a closed oriented -manifold equipped with a metric and let M \ K be a knot. Then there exists an A ∞ category WF ( M \ K ) whose isomorphismclass depends only on the isotopy type of K in M .Proof. Let g be a Riemannian metric on M and restrict the metric to M \ K . Bythe remark above, the quasi-isomorphism type of A ∞ category WF ( T ∗ N, H g ) doesnot depend on the choice of the metric.It remains to show the isotopy invariance of WF ( T ∗ N, H g ). More precisely, let K , K be isotopic to each other. Suppose φ t : M → M be an isotopy such that K = φ ( K ). Denote K t = φ t ( K ). We fix a metric g on M and consider the familyof metric g t := φ t ∗ g . We also fix a pair of precompact domains W ⊂ W (cid:48) ⊂ M \ K with smooth boundary and fix a cylindrical adjustment g (cid:48) of g outside W (cid:48) so that g (cid:48) = g | ∂W ⊕ dr on M \ W (cid:48) which is interpolated in W (cid:48) \ W to g . Then we considerthe isotopy of pairs ( M \ K t , g t ). We denote W t = φ t ( W ) and W (cid:48) t = φ t ( W (cid:48) ). Then choose a smooth family of cylindrical adjustments g (cid:48) t for g t outside W (cid:48) t ⊂ M \ K t with g t = φ t ∗ ( g ) on W t which are interpolated in between.Then we construct an A ∞ functorΦ : WF g ( T ∗ ( M \ K )) → WF φ ∗ g ( T ∗ ( M \ K ))which are given by L → φ ( L ) objectwise and whose morphism is defined by thesame construction performed in the previous construction of A ∞ functor. By con-sidering the inverse isotopy, we also haveΨ : WF ( T ∗ ( M \ K ); H g ) → WF ( T ∗ ( M \ K ); H g (cid:48) ) . Then we consider the isotopy which is a concatenation of φ t and its inverse isotopywhich is homotopic to the constant isotopy. Then the same construction of thehomotopy as the one in Section 7 can be applied to produce a A ∞ homotopybetween Ψ ◦ Φ and the identity functor.This proves WF ( T ∗ ( M \ K ) , H g ) is quasi-isomorphic to WF ( T ∗ ( M \ K ) , H φ ∗ g ),which finishes the proof. (cid:3) Construction of Knot Floer algebra HW ( ∂ ∞ ( M \ K ))In this section, we give construction of Knot Floer algebra mentioned in Defini-tion 1.5.Denote by G g ( T ) the energy of the shortest geodesic cord of T relative to themetric g . Then the following lemma is a standard fact in Riemannian geometrysince T is a compact smooth submanifold and g is of bounded geometry. Lemma 10.1.
Denote by G g ( T ) the infimum of the energy of non-constant geodesics.Then G g ( T ) > and all non-constant Hamiltonian chords γ of ( ν ∗ T ; H g ) have A H g ( γ ) = − E ( c γ ) ≤ − G g ( T ) . Now we perform the algebra version in the Morse-Bott setting of the constructiongiven in the previous sections associates an A ∞ algebra CW g ( L, T ∗ ( M \ K )) := C ∗ ( T ) ⊕ Z { X < ( H g ; ν ∗ T, ν ∗ T ) } where C ∗ ( T ) is chosen to be a cochain complex of T , e.g., the de Rham complexsimilarly as in [FOOO1, FOOO2]. We note that for 0 < (cid:15) < G g ( T ) is sufficientlysmall, we have X ≥− (cid:15) ( H g ; ν ∗ T, ν ∗ T ) ∼ = T and X < − (cid:15) ( H g ; ν ∗ T, ν ∗ T )is in one-one correspondence with the set of non-constant geodesic cords of ( T, g )and so CW < − (cid:15) ( L, T ∗ ( M \ K ); H g ) = Z (cid:104) X < ( H g ; ν ∗ T, ν ∗ T ) (cid:105) It follows from the lemma that the associated complex ( CW g ( L, T ∗ ( M \ K )) , m )has a subcomplex( CW ≥− (cid:15) ( L, T ∗ ( M \ K ); H g ) , m ) ∼ = ( C ∗ ( T ) , d )where d is the differential on C ∗ ( T ). We denote the associated homology by HW g ( L, T ∗ ( M \ K )) := HW ( L, T ∗ ( M \ K ); H g ) . WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 45
Not to further lengthen the paper and since the detailed construction is notexplicitly used in the present paper, we omit the details of this Morse-Bott con-struction till [BKO] where we provide them and describe its cohomology in termsof the Morse cohomology model of C ∗ ( T ). With this mentioned, we will pretendin the discussion below that ( ν ∗ T, H g ) is a nondegenerate pair with ν ∗ T implicitlyreplaced by a C -small perturbation L thereof. Let N ( K ) , N (cid:48) ( K ) be a tubular neighborhood of K such that N ( K ) ⊂ Int N (cid:48) ( K ) . We denote M \ N ( K ) = N cpt and similarly for N (cid:48) ( K ). We denote T = N ( K ) and L = ν ∗ T .We define the cylindrical adjustments g of the metric g on M with respect tothe exhaustion (4.1) by g = (cid:40) g on N (cid:48) , cpt da ⊕ g | ∂N cpt on N cpt \ K which is suitably interpolated on N cpt \ N (cid:48) , cpt and fixed. Theorem 10.2.
The quasi-isomorphism class of A ∞ algebra ( CW ( ν ∗ T, T ∗ ( M \ K ; H g )) , m ) , m = { m k } ≤ k< ∞ does not depend on the various choices involved such as tubular neighborhood N ( K ) and the metric g on M .Proof. The metric independence can be proved in the same as in the proof givenin the categorical level.Therefore we focus on the choice of tubular neighborhood. Let N , N be twodifferent tubular neighborhoods of K and denote by T , T be the boundaries T = ∂N and T = ∂N . Choose any diffeomorphism φ : M → M such that φ ( T ) = T (cid:48) and that it is isotopic to the identity fixing K . Then the symplecto-morphism ( dφ − ) ∗ , which is fiberwise linear (over the map φ ), induces a naturalquasi-isomorphism CW g ( ν ∗ T , T ∗ ( M \ K )) ∼ = CW φ ∗ g ( ν ∗ T , T ∗ ( M \ K )) . On the other hand, the latter is quasi-isomorphic to CW g ( ν ∗ T , T ∗ ( M \ K )) by themetric independence. Invariance under other changes can be proved similarly andso omitted.Denote by the resulting A ∞ algebra by CW ( ν ∗ T, T ∗ ( M \ K )) := CW ( ν ∗ T, T ∗ ( M \ K ); H g )whose quasi-isomorphism class is independent of the metric g . Finally we provethe invariance thereof under the isotopy of K . The proof is almost the same asthat of the categorical context given in the proof of Theorem 9.3. For readers’convenience, we duplicate the proof here with necessary changes made. Suppose K , K be isotopic to each other and let φ t : M → M be an isotopy such that K = φ ( K ) as before. Denote K t = φ t ( K ). We fix a metric g on M andconsider the family of metric g t := φ t ∗ g . We also fix a pair of precompact domains W ⊂ W (cid:48) ⊂ M \ K with smooth boundary and fix a cylindrical adjustment g (cid:48) of g outside W (cid:48) so that g (cid:48) = g | ∂W ⊕ dr on M \ W (cid:48) which is interpolated in W (cid:48) \ W to g . Then we consider the isotopy of pairs ( M \ K t , g t ). We denote W t = φ t ( W ) and W (cid:48) t = φ t ( W (cid:48) ). Then choose a smooth family of cylindrical adjustments g (cid:48) t for g t outside W (cid:48) t ⊂ M \ K t with g t = φ t ∗ ( g ) on W t which are interpolated in between.Then we construct an A ∞ mapΦ : CW ( ν ∗ T , T ∗ ( M \ K ); H g ) → CW ( ν ∗ T , T ∗ ( M \ K ); H g (cid:48) , g (cid:48) = φ t ∗ ( g )is defined by the same construction performed in the previous construction of A ∞ functor. By considering the inverse isotopy, we also haveΨ : CW ( ν ∗ T , T ∗ ( M \ K ); H g (cid:48) ) → CW ( ν ∗ T , T ∗ ( M \ K ); H g ) . Then we consider the isotopy which is a concatenation of φ t and its inverse isotopywhich is homotopic to the constant isotopy. Then the same construction of thehomotopy as the one in Section 7 can be applied to produce a A ∞ homotopybetween Ψ ◦ Φ and the identity map. This finishes the proof. (cid:3)
Definition 10.3 (Knot Floer algebra) . We denote by HW ∗ ( ∂ ∞ ( M \ K )) = ∞ (cid:77) d =0 HW d ( ∂ ∞ ( M \ K ))the resulting (isomorphism class of the) graded group and call it the knot Floeralgebra of K in M .The same argument also proves that the isomorphism class of the algebra dependsonly on the isotopy class of the knot K . Part C -estimates for the moduli spaces The compactifications of the moduli spaces such as M w ( x ; x ) = M w ( x ; H , J , η )(and also for N w ( x ; x )) are essential in the definition of the wrapped Floer coho-mology and its algebraic properties, the A ∞ structure. The main purpose of thepresent section is to establish the two C estimates, Proposition 6.2, 6.1, 7.14, and7.15 postponed in the previous sections.For this purpose, we observe that thanks to the choice we made for J to sat-isfy (5.4) in Remark 5.2, u satisfies (5.5) if and only if the composition v ( z ) = ψ − η ( z ) ( u ( z )) satisfies the autonomous equation ( dv − X H ⊗ β ) (0 , J g = 0 ,v ( z ) ∈ L i , for z ∈ ∂ Σ between z i and z i +1 where i ∈ Z k +1 . v ◦ (cid:15) j ( −∞ , t ) = x j ( t ) , for j = 1 , . . . , k.v ◦ (cid:15) (+ ∞ , t ) = x ( t ) . (10.1)Therefore it is enough to prove the relevant C -estimates for the map v which wewill do in the rest of this part.11. Horizontal C estimates for m -components Proof of Proposition 6.1.
Recall that on N end , we have the product metric g = da ⊕ h . Because of this we have the Riemannian splittings N end ∼ = [0 , ∞ ) × T with T = ∂N end and T ∗ N end ∼ = T ∗ ( ∂N end ) ⊕ T ∗ [0 , ∞ ) . WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 47
Furthermore the Sasakian almost complex structure has the splitting J g = i ⊕ J h where J h is the Sasakian almost complex structure on ∂N end and i is the standardcomplex structure on T ∗ [0 , ∞ ) ⊂ T ∗ R ∼ = C .Denote q = ( a, q T ) ∈ [0 , ∞ ) × T . Then we have the orthogonal decomposition p = ( p a , p T ) and hence | p | g = | p T | h + | p a | . Therefore the ( a, p a )-components of X H and JX H are given by π T ∗ [0 , ∞ ) ( X H ( q, p )) = p a ∂∂a , π T ∗ [0 , ∞ ) ( JX H ) = p a ∂∂p a . (11.1)Let z = x + √− y be a complex coordinate of (Σ , j ) such that β = dt e away from the singular points of the minimal area metric. If we write β = β x dx + β y dy , then (5.6) is separable.Another straightforward calculation shows that the ( a, p a )-component of theequation becomes ∂a ( v ) ∂x − ∂p a ( v ) ∂y − β x p a ( v ) = 0 ∂p a ( v ) ∂x + ∂a ( v ) ∂y − β y p a ( v ) = 0 (11.2)We note d ( β ◦ j ) = − (cid:18) ∂β x ∂x + ∂β y ∂y (cid:19) dx ∧ dy. In particular, if d ( β ◦ j ) = 0, this vanishes.For any given one-form β , a straightforward calculation using these identitiesleads to the following formula for the (classical) Laplacian∆( a ( v )) = p a ( v ) (cid:18) ∂β x ∂x + ∂β y ∂y (cid:19) − β x ∂ ( a ( v )) ∂y + β y ∂ ( a ( v )) ∂x (11.3)for any solution v of (5.6). Lemma 11.1.
Then for any one-form β on Σ , a ( v ) satisfies ∆( a ( v )) = − β x ∂ ( a ( v )) ∂y + β y ∂ ( a ( v )) ∂x (11.4) for any solution u of (5.6) .Proof. This immediately follows from (11.3) since β satisfies 0 = d ( β ◦ j ) = ( ∂β x ∂x + ∂β y ∂y ) dx ∧ dy . (cid:3) Therefore we can apply the maximum principle for v . Now let L , . . . , L k beLagrangian submanifolds contained in Int N cpt . Then we have a ( z ) ≤ a for a > z ∈ ∂ Σ = D \ { z , . . . , z k } . In particular, the end points of x i arecontained in Int N cpt . The maximum principle applied to the function t (cid:55)→ a ◦ x i ( t )on [0 , w i ] prevents the image of x i from entering in the cylindrical region N end . This proves that the whole image of x i is also contained in W . This finishes theproof of the proposition by applying the maximum principle to v based on Lemma11.1. (cid:3) Remark 11.2.
We comment that the conformal rescaling of u to v defined by v ( z ) = ψ − η ( z ) ( u ( z )) does not change the horizontal part, i.e., π ◦ v = π ◦ u .12. Vertical C estimates for m -components We next examine the C -bound in the fiber direction of T ∗ N .Writing ρ = e s ◦ v , a straightforward calculation (see [Se2, (3.20)]) derives∆ ρ = | dv − β ⊗ X H | − ρH (cid:48)(cid:48) ( ρ ) dρ ∧ βdx ∧ dy − ρH (cid:48) ( ρ ) dβdx ∧ dy (12.1)for any complex coordinates z = x + iy for (Σ , j ). We note that from this equation,the (interior) maximum principle applies.When ∂ Σ (cid:54) = ∅ , we also need to examine applicability of strong maximum principleon the boundary ∂ Σ. In [AS], Abouzaid-Seidel used certain integral estimates tocontrol C -bound instead of the strong maximum principle. Here we prefer to usethe strong maximum principle and so provide the full details of this application ofstrong maximum principle, especially for the moving boundary case.In this section we consider the case of fixed Lagrangian boundaries.For this purpose, the following C -bound is an essential step in the case of non-compact Lagrangian such as the conormal bundles L i = ν ∗ ( ∂N cpt ). For given x =( x , . . . , x k ) where x j ∈ X ( w j H ; L j − , L j ) with j = 1 , . . . , k and x ∈ X ( w j H ; L , L k ),we define ht ( x ; H, { L i } ) := max ≤ j ≤ k (cid:107) p ◦ x j (cid:107) C (12.2)Proof of the following proposition is a consequence of the strong maximum prin-ciple based on the combination of the following(1) ρ = r ◦ v = e s ◦ v with r = | p | g satisfies (12.1),(2) the conormal bundle property of L i and(3) the special form of the Hamiltonian H = r , which is a radial function.(See [EHS], [Oh2] for a similar argument in a simpler context of unperturbed J -holomorphic equation.) Proof of Proposition 6.2.
Since the interior maximum principle is easier and totallystandard for the type of equation (12.1), we focus on the boundary case.Due to the asymptotic convergence condition and (cid:98)
Σ is compact, the maximumof the function z (cid:55)→ p ( v ( z )) is achieved. If it happens on one of ∞ ’s in the strip-likeend, we are done.So it remains to examine that case where the maximum is achieved at z ∈ ∂ Σ.We will apply a strong maximum principle to prove that the maximum cannotbe achieved beyond the height of the asymptotic chords in the fiber direction of T ∗ N . However to be able to apply the strong maximum principle, we should dosome massaging the equation (12.1) into a more favorable form. Here the condition i ∗ β = 0 and dβ = 0 near the boundary enters in a crucial way.Choose a complex coordinate z = s + it on a neighborhood U of z so that z ( ∂ Σ ∩ U ) ⊂ R ⊂ C . WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 49
First the last term of (12.1) drops out by dβ = 0 near ∂ Σ. For the second term,we note dρ ∧ β = (cid:18) ∂ρ∂s ds + ∂ρ∂t dt (cid:19) ∧ ( β s ds + β t dt ) = (cid:18) ∂ρ∂s β t − ∂ρ∂t β s (cid:19) ds ∧ dt and so the second term becomes − ρH (cid:48)(cid:48) ( ρ ) (cid:18) ∂ρ∂s β t − ∂ρ∂t β s (cid:19) . On the other hand, β s = 0 since we imposed i ∗ β = 0 (3.5) and hence (12.1) isreduced to ∆ ρ = | dv − X H ( v ) ⊗ β | − ∂ρ∂s β t (12.3)on ∂ Σ.Since v ( ∂ Σ) ⊂ ν ∗ T i and z ∈ z i z i +1 (cid:55)→ v ( z ) defines a curve on L i and thefunction τ (cid:55)→ | p ( v ( s + 0 √− | achieves a maximum at z where z = s + 0 √− ∂r ◦ v∂s ( s ) = dr (cid:18) ∂v∂s ( z ) (cid:19) . But we note ∂v∂s ( z ) ∈ T v ( z ) L i ∩ ker dr v ( z ) and L i ∩ ker dr v ( z ) is a Legendrian subspace of T r − ( R ) with R = | p ( v ( z )) | .Therefore J ∂v∂s ( z ) ∈ ξ v ( z ) ⊂ ker dr and so dr ( − J ∂v∂s ( z )) = 0 . (12.4)Substituting − J ∂v∂s ( z ) = ∂v∂t ( z ) − β t X H ( v ( z ))into (12.4), we have obtained dr (cid:18) ∂v∂t ( z ) (cid:19) = dr ( β t X H ( v ( z ))) = 0which in turn implies ∂ρ∂s ( z ) = 0 in (12.3) and so(∆ ρ )( z ) ≥ . This contradicts to the strong maximum principle, unless Im v ⊂ r − ( R ). Thelatter is possible only when r ( x j ) ≡ R for all j = 0 , · · · , k for some R ≥
0. Butif that holds, the proposition already holds and there is nothing to prove.This completes the proof of the proposition. (cid:3) C estimates for moving Lagrangian boundary In this section, we first provide a direct proof of the uniform C estimate for themoving boundary condition without decomposing the isotopy in two stages hopingthat such an estimate may be useful in the future, even though this C -estimateis not used in the present paper. Then we will briefly indicate how the proofs ofeasier propositions, Propositions 7.14 and 7.15 can be obtained from the scheme ofthe proof with minor modifications.We consider a map v : Σ → T ∗ N satisfying a perturbed Cauchy-Riemann equa-tion ( dv − X H ⊗ β ) (0 , J = 0 ,v ◦ (cid:15) j ( −∞ , t ) = x j ( t ) , for j = 1 , . . . , k.v ◦ (cid:15) (+ ∞ , t ) = x ( t ) . (13.1)where x ∈ X ( w H ; L , X ) and x d ∈ X ( w d H ; L , X d ) and x d ∈ X ( w j H ; X j , X j +1 )for j = 1 , . . . , d −
1, and with moving boundary { L t } , a Hamiltonian isotopy from L to L : v ( z ) ∈ X i , for z ∈ z i z i +1 ⊂ ∂ Σ with 1 ≤ i ≤ d − L , for z ∈ z z ⊂ ∂ Σ L , for z ∈ z d z d +1 ⊂ ∂ Σ (13.2)
Proposition 13.1.
Let L := L i = ν ∗ ( a − ( a i )) , L := L j = ν ∗ ( a − ( a j )) with a i < a j and { x j } ≤ j ≤ k be given as above. Define the constant ht (cid:0) H ; L i , L i +1 ; { X k } dk =1 ; { x j } ≤ j ≤ k (cid:1) = max ≤ j ≤ k (cid:107) p ◦ x j (cid:107) C . Then max z ∈ Σ | p ( v ( z )) | ≤ ht (cid:0) H ; L i , L i +1 ; { X k } dk =1 ; { x j } ≤ j ≤ k (cid:1) for any solution u of (5.5) . We consider the case of moving the Lagrangian L i = ν ∗ T i to L i +1 = ν ∗ T i +1 for a given exhaustion sequence (2.4) where T i = a − ( i ) , T i +1 = a − ( i + 1). Themain point of this case is that the Lagrangian L i moves outward with respect tothe a -direction in the cylindrical region [0 , ∞ ) × , ∞ ). Remark 13.2.
It is very interesting to see that subclosedness of the one-form β also enters in our proof of the vertical C -bound. See the proof of Proposition 13.1below. Proof.
In order to produce a Hamiltonian flow which send L i to L i +1 , we start witha Hamiltonian function F : T ∗ N → R whose restriction to T ∗ N end agrees with thecoordinate function p a : T ∗ N end → R . Then we deduce X F | T ∗ N end = ∂∂a . Since the metric g is cylindrical on N end , the induced Sasakian metric (cid:98) g on T ∗ N end ∼ = T ∗ [0 , + ∞ ) ⊕ T ∗ T can be written as (cid:98) g = (cid:98) g a ⊕ (cid:98) g T .Let φ tF be a time t flow of X F on T ∗ N then its restriction to T ∗ N end becomesΦ tF (( a, p a ) , ( q T , p T )) = (( a + t, p a ) , ( q T , p T )) . WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 51
The local coordinate expression of the Lagrangian L i = ν ∗ ( b − ( k i )), where k i < { (( − k i , p a ) , ( q T , p a ∈ R , q T ∈ T } . For s ∈ [0 , L i + s := { (( − k i + s ( k − k i +1 ) , p a ) , ( q T , p a ∈ R , q T ∈ T } , a [0 , L i and L i +1 . By an easyobservation we have Lemma 13.3.
The flow Φ s ( k − k i +1 ) F of X F sends L i to L i + s for s ∈ [0 , . For each given test Lagrangians X , . . . , X k , we consider a moduli space N w ( x ; x ) = N w ( x ; x ; D m ) with N w ( x ; x ) := (cid:91) D∈FD w N w (( x ; x ); D )which consists of a map u : Σ → T ∗ N satisfying a perturbed Cauchy-Riemannequation (5.5) with moving boundary condition (13.2). Remark 13.4.
We would like to alert readers that as our proof clearly shows ingeneral construction of such a morphism is possible only in a certain direction thatfavors application of strong maximum principle.
For each given quadruple ( H ; L i , L i +1 , { x j } ≤ j ≤ k ), where x ∈ X ( w H ; L i +1 )and x j ∈ X ( w j H ; L i ) for j = 1 , . . . , k , we define a constant ht ( H ; L i , L i +1 , { x j } ≤ j ≤ k ) = max ≤ j ≤ k (cid:107) p ◦ x j (cid:107) C . Recall that our test Lagrangians X j are either compact or with cylindrical end ifnoncompact. Since the boundary condition is the fixed one on z j z j +1 , u ( z j z j +1 ) ⊂ X j , for 1 ≤ j ≤ k −
1, the strong maximum principle for (10.1) applies thereto.It remains to check on the strip-like end near z where one of the boundaryinvolves moving boundary { L s } from L i to L i +1 . The only place where this is notclear is the part where χ (cid:48) ( τ ) (cid:54) = 0, i.e., τ ∈ ( − , − ⊂ ( −∞ ,
0] in the strip-likeregion.In this part of the region, ∆ ρ is given by the same formula as (12.1) but withthe moving boundary condition. Recall the [0 , L i + s := { (( − k i + s ( k − k i +1 ) , p a ) , ( q T , p a ∈ R , q T ∈ T } . As before let z := (cid:15) ( τ ,
0) or (cid:15) ( τ ,
1) be a point in ∂ Σ ∩ (cid:15) ( Z − ) where amaximum of the radial function r ( z ) = | p ( v ( z )) | g is achieved. Without loss of anygenerality, we may assume that the point is z = (cid:15) ( τ ,
1) with − < τ < − U of z with coordinates ϕ : U ∩ Z − → H = { x + iy ∈ C : y ≥ } such that ϕ ( z ) = x + 0 i for some x and ϕ ( U ∩ ∂Z − ) ⊂ R ⊂ H . We note thatthe outward normal of Z − along t = 1 is ∂∂t and that of H along ∂ H = { y = 0 } isgiven by − ∂∂y . Therefore for the holomorphic map ϕ , we have ∂τ∂x <
0. This in turnimplies ∂ ( a ◦ v ) ∂x = ∂ ( a ◦ v ) ∂τ ∂τ∂x ≥ along { y = 0 } since ∂ ( a ◦ v ) ∂τ ≤ ∂Z − by the direction of the moving boundarycondition in (13.2).We also have dr (cid:0) ∂v∂x ( z ) (cid:1) = 0 and so 0 = θ (cid:0) J ∂v∂x ( z ) (cid:1) for r = | p | = 2 H g . Remark 13.5.
This time ∂v∂x ( z ) (cid:54)∈ T v ( z ) L ρ ( x ) and hence the argument used inthe later half of the fixed boundary case cannot be applied to the current situation.This is the precisely the reason why a direct construction of homotopy for themoving (noncompact) Lagrangian boundary has not been able to be made butother measures such as the cobordism approach [Oh4] or Nadler’s approach [N] aretaken. As we shall see below, we have found a way of overcoming this obstacle bya two different applications of strong maximum principle after combining all thegiven geometric circumstances and a novel sequence of logics.The condition v ◦ (cid:15) ( τ, ∈ L i + ρ ( τ ) at z and Lemma 13.3 imply (cid:16) φ ρ ( x )( k i − k i +1 ) F (cid:17) − ( v ◦ (cid:15) ( x, ∈ L i (13.4)where ρ ( x ) = ρ ( τ ( x )). By differentiating the equation for x and recalling z = (cid:15) ( x , ∂τ∂x ρ (cid:48) ( τ ( x ))( k i − k i +1 ) X F (cid:18)(cid:16) φ ρ ( x )( k i − k i +1 ) F (cid:17) − ( v ( z )) (cid:19) + d (cid:16) φ ρ ( x )( k i − k i +1 ) F (cid:17) − (cid:18) ∂v∂x ( z ) (cid:19) is a tangent vector of L i at (cid:16) φ ρ ( x )( k i − k i +1 ) F (cid:17) − ( v ( z )). Here F = − F ◦ φ tF is theinverse Hamiltonian which generates the inverse flow ( φ tF ) − . This implies − ∂τ∂x ρ (cid:48) ( τ ( x ))( k i − k i +1 ) X F ( v ( z )) + ∂v∂x ( z ) ∈ T v ( z ) L ρ ( x ) . (See [Oh1] for the same kind of computations used for the Fredholm theory for theperturbation of boundary condition.) Recall θ ≡ L ρ ( x ) . Therefore we obtain0 = θ (cid:18) − ∂τ∂x ρ (cid:48) ( τ ( x ))( k i − k i +1 ) X F ( v ( z )) + ∂v∂x ( z ) (cid:19) . By evaluating the equation in (13.2) against ∂∂x , we get ∂v∂x ( z ) + J (cid:18) ∂v∂y ( z ) − β t X H ( v ( z )) (cid:19) = 0 . Therefore the equation θ ◦ J = − dH g gives rise to12 ∂ | p ◦ v | ∂y ( z ) = ∂ ( H ◦ v ) ∂y ( z ) = − θ ◦ J ( ∂v∂y ) = θ (cid:18) ∂v∂x ( z ) (cid:19) = ∂τ∂x ρ (cid:48) ( τ ( x ))( k i − k i +1 ) θ ( X F ( v ( z )))= ∂τ∂x ρ (cid:48) ( τ ( x ))( k i − k i +1 ) p a ( v ( z )) . (13.5) Lemma 13.6.
We have p a ( v ( z )) ≥ and equality holds only when p a ◦ v ≡ ,i.e., when Image v is entirely contained in the zero section. WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 53
Proof.
In order to estimate p a ( v ( z )), by a direct computation from (11.2), wederive ∆( p a ◦ v ) = p a ◦ v (cid:18) ∂β x ∂y − ∂β y ∂x (cid:19) + (cid:18) β x ∂ ( p a ◦ v ) ∂y − β y ∂ ( p a ◦ v ) ∂x (cid:19) , where β = β x dx + β y dy near z . The sub-closedness assumption of β impliesthat ∂β x ∂y − ∂β y ∂x ≥ p a ( v ) and the (interior) minimum principle for the negative values,respectively.Now we consider the function p a ◦ v on ∂ Σ. Recalling the property i ∗ β = 0 on ∂ Σ which implies β x = 0, the above equation for ∆( p a ◦ v ) becomes∆( p a ◦ v ) = p a ◦ v (cid:18) ∂β x ∂y − ∂β y ∂x (cid:19) − β y ∂ ( p a ◦ v ) ∂x (13.6)on ∂ Σ. The inequality (13.3) on ∂Z − ∩ U implies ∂ ( p a ◦ v ) ∂y ( z ) ≥ a + ip a of x + iy . If we denote f ( z ) = − p a ◦ v ( z ), this inequality becomes − ∂f∂y ( z ) ≤ . (13.7)We would like to show f ( z ) = − p a ( v ( z )) ≤
0. Suppose to the contrary f ( z ) ≥ f ( z ) ≥ ∂ ( p a ◦ v ) ∂x ( z ) = 0. Therefore since − ∂∂y is the outwardnormal to { y = 0 } , (13.7) contradicts to the (strong) maximum principle. Therefore p a ( v ( z )) ≥ p a ◦ v ≡
0. This finishes the proof. (cid:3)
Substituting this lemma back into (13.5), we obtained − ∂ | p ◦ v | ∂y ( z ) ≤ p ◦ v ( z ) = 0. This contradicts to the strong maximumprinciple applied to the radial function ( r ◦ v ) = | p ◦ v | by the same way as for thevertical C estimates given in Section 12, unless the whole image of v is containedin the level set r − (0) = 0 N .But if the latter holds, all the Hamiltonian chords are constant chords and thesolution v must be constant maps. This is impossible because of the asymptoticcondition v ( (cid:15) − (( −∞ , ⊂ ν ∗ T i , i = 1 , · · · , k and v ( (cid:15) + ([0 , ∞ )) ⊂ ν ∗ T i +1 , i = 0 . This finishes the proof of Proposition 13.1. (cid:3)
Now we briefly indicate the main points of the proofs of Propositions 7.14 and7.15].For these cases, the boundaries are fixed as in the proofs of Proposition 6.1, 6.2is that the Hamiltonian involved is not τ -independent due to the appearance ofthe elongation function χ in H χ . We have only to consider the equation on thestrip-like region. Writing ρ = e s ◦ v , the calculation (see [Se2, (3.20)]) derives∆ ρ = | dv − β ⊗ X H χλ | − ρ ( H χλ ) (cid:48)(cid:48) ( ρ ) dρ ∧ βdτ ∧ dt + χ (cid:48) ( τ ) ∂H s ∂s (cid:12)(cid:12)(cid:12) s = χ ( τ ) (13.8)We note that from this equation, both the maximum principle and strong maximumprinciple apply for this non-autonomous case because χ (cid:48) ( τ ) ∂H s ∂s (cid:12)(cid:12)(cid:12) s = χ ( τ ) ≥ λ . We refer to the proof of Proposition 13.1 tosee how the strong maximum principle applies. Appendix A. Energy identity for Floer’s continuation equation
In this appendix, we recall the energy identity for the continuation equationgiven in [Oh5] and given its derivation which was originally given in [Oh2] in thecotangent bundle context.
Proposition A.1.
Suppose that u is a finite energy solution of (7.5) . Then (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂τ (cid:12)(cid:12)(cid:12)(cid:12) J χ ( τ ) dt dτ = A H + ( z + ) − A H − ( z − ) − (cid:90) ∞−∞ χ (cid:48) ( τ ) (cid:18)(cid:90) ∂H s ∂s (cid:12)(cid:12)(cid:12) s = χ ( τ ) ( u ( τ, t )) dt (cid:19) dτ. (A.1) Proof.
We derive (cid:90) (cid:90) (cid:12)(cid:12)(cid:12)(cid:12) ∂u∂τ (cid:12)(cid:12)(cid:12)(cid:12) J χ ( τ ) dt dτ = (cid:90) (cid:90) ω (cid:18) ∂u∂τ , J χ ( τ ) ∂u∂τ (cid:19) dt dτ = (cid:90) (cid:90) ω (cid:18) ∂u∂τ , ∂u∂t − X H χ ( τ ) ( u ) (cid:19) dt dτ = (cid:90) u ∗ ω + (cid:90) ∞−∞ (cid:90) dH χ ( τ ) (cid:18) ∂u∂τ (cid:19) dt dτ. The finite energy condition and nondegeneracy of z ± imply that the first improperintegral (cid:82) u ∗ ω converges and becomes (cid:90) u ∗ ω = (cid:90) u ∗ ( − dθ ) = − (cid:90) ( z + ) ∗ θ + (cid:90) ( z − ) ∗ θ. (A.2) WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 55
On the other hand, the second summand becomes (cid:90) ∞−∞ (cid:90) dH χ ( τ ) (cid:18) ∂u∂τ (cid:19) dt dτ = (cid:90) ∞−∞ (cid:90) ∂∂τ (cid:16) H χ ( τ ) ( u ( τ, t )) (cid:17) dt dτ − (cid:90) ∞−∞ χ (cid:48) ( τ ) (cid:18)(cid:90) ∂H s ∂s (cid:12)(cid:12)(cid:12) s = χ ( τ ) ( u ( τ, t )) dt (cid:19) dτ = (cid:90) H χ ( τ ) ( u ( ∞ , t )) dt − (cid:90) H χ ( τ ) ( u ( −∞ , t )) dt − (cid:90) ∞−∞ χ (cid:48) ( τ ) (cid:18)(cid:90) ∂H s ∂s (cid:12)(cid:12)(cid:12) s = χ ( τ ) ( u ( τ, t )) dt (cid:19) dτ. (A.3)By recalling u ( ± τ, t ) = z ± ( t ) and adding (A.2), (A.3), we have obtained (A.1). (cid:3) Appendix B. Gradings and signs for the moduli spaces
B.1.
Lagrangian branes.
Let us choose a quadratic complex volume form η M onthe symplectic manifold ( W = T ∗ ( M \ K ) , ω std ) with respect to the almost complexstructure J g introduced in Definition 2.6. The associated (squared) phase map is α M : Gr( T M ) → S , α M ( T L x ) = η M ( v ∧ v ∧ v ) | η M ( v ∧ v ∧ v ) | , where v , v , v is a basis of T L x and Gr( T M ) is the Lagrangian Grassmannian. Thevanishing condition of the relative first Chern class in Definition 4.1 implies thatthe Maslov class µ L ∈ H ( L ) vanishes on H ( L ). So we have a lifting α L : L → R of α M | T L satisfying exp(2 πiα L ( x )) = α M ( T L x ), which we call a grading on L .For the orientation of various types of moduli spaces, we need to consider a relative spin structure of each Lagrangian L . Since we have H ( M ; Z ) = 0, thesituation is rather simple. The relative spin structure P is consist of the choice oforientation of L and a Pin structure on T L , see [FOOO2, Section8],[Se1, Section11]for more general cases. Note that the vanishing condition of the second Stiefel-Whitney class w ( L ) in Definition 4.1 implies the existence of the Pin structures.We call a triple ( L, α , P ) a
Lagrangian brane .B.2.
Dimensions and orientations of moduli spaces.
Let ( L , L ) be apair of exact Lagrangian branes with a choice of Floer datum ( H, J ). Consider aHamiltonian chord x ∈ X ( H ; L , L ) and use a linearization dφ tH of the Hamiltonianflow of H to transport the linear Lagrangian brane L x (0) = (( T L ) x (0) , α ( x (0)) , ( P ) x (0) )to T M x (1) . Then the index and the orientation space are defined by comparing twolinear Lagrangian branes L x (0) and L x (1) , see [Se1, 11h] for the construction. Let usdenote them by µ ( x ) = µ ( H ; L x (0) , L x (1) ) and o ( x ) = o ( H ; L x (0) , L x (1) ), respectively.We have a more direct description of the grading in the cotangent bundle setupfrom [Oh2]. Note that the sequence of Riemannian metrics { g } i ∈ N on the knotcomplement M \ K give the canonical splittings of two Lagrangian subbundles T x ( t ) M = H x ( t ) ⊕ V x ( t ) , where the vertical tangent bundle V x ( t ) is canonically isomorphic to T ∗ π ( x ( t )) ( M \ K ),and the horizontal subbundle H x ( t ) with respect to the Levi-Civita connection of g is isomorphic to T π ( x ( t )) ( M \ K ) under T π : T M → T ( M \ K ).We now consider a canonical class of symplectic trivializationΦ : x ∗ T M → [0 , × C , satisfying Φ( H x ( t ) ) ≡ R and Φ( V x ( t ) ) ≡ i R for all t ∈ [0 , ,
1] iscontractible, such a trivialization exists. For example, if x ∈ X ( H g ; ν ∗ T i , ν ∗ T i ),then we have Φ( T x (0) ν ∗ T i ) = U Φ ⊕ U ⊥ Φ , Φ( T x (1) ν ∗ T i ) = W Φ ⊕ W ⊥ Φ , where U Φ , W Φ are 2-dimensional subspaces and U ⊥ Φ , W ⊥ Φ are the correspondingannihilators. Then the index µ ( x ) is defined by following the definition of theMaslov index in [RS] µ ( B Φ ( U Φ ⊕ U ⊥ Φ ) , W Φ ⊕ W ⊥ Φ ) , where B Φ : [0 , → Sp(6) is a symplectic path given by B Φ := Φ ◦ T φ tH g ◦ Φ − .Let Σ be a ( d + 1)-pointed disk equipped with a fixed conformal structure,with strip-like ends, and with brane structures ( L , . . . , L d ) on each boundarycomponent. Consider Floer data D in Definition 5.1 for x ∈ X ( L d , L ) , x = ( x , . . . , x d ) , x k ∈ X ( L k , L k − ) , k = 1 , . . . , d. (B.1)Let u ∈ M ( x , x ; D ) be a regular point, then the linearization of ( dv − X H ⊗ β ) (0 , J become D D ,u : ( T B D ) u → ( E D ) u , where B D is a Banach manifold of maps in W ,ploc (Σ , M ) satisfying W ,p -convergenceon each strip-like ends, and E D → B D is a Banach vector bundle whose fiber at u is L p (Σ , Ω , ⊗ u ∗ T M ).Let Σ be a ( d + 1)-pointed disk equipped with a fixed conformal structure, withstrip-like ends, and with a brane structure on each boundary component, set say L , . . . , L d . Consider Floer data D in Definition 5.1 for x ∈ X ( H ; L d , L ) and x k ∈ X ( H ; L k , L k − ), k = 1 , . . . , d . Let u ∈ M ( x ; x ; D ) be a regular point, where x = ( x , . . . , x d ), then we havedim u M ( x ; x ; D ) = µ ( x ) − µ ( x ) − · · · − µ ( x d );(Λ top T M ( x ; x ; D )) u ∼ = Λ top ker( D D ,u ) ⊗ Λ top Coker( D D ,u ) ∼ = o ( x ) ⊗ o ( x ) ∨ ⊗ · · · ⊗ o ( x d ) ∨ . Here, D D ,u is the linearized operator of ( dv − X H ⊗ β ) (0 , J , where ( H , J , β ) comesfrom the data D , and ∨ denotes the dual vector space.B.2.1. Sign convention for A ∞ structure map. We briefly explain the sign conven-tion adopted in Proposition 6.5. Let Σ be a ( d + 2 − m )-pointed disk equippedwith Floer data D and with brane structures (cid:126)L := ( L , . . . , L n , L n + m , . . . , L d ) , and Σ be an ( m + 1)-pointed disk with D and with branes (cid:126)L := ( L n , . . . , L n + m ) . WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 57
Now consider the gluing of (Σ , D , (cid:126)L ) and (Σ , D , (cid:126)L ) near the outgoing endof Σ and the ( n + 1)-th incoming end to obtain a ( d + 1)-pointed disk Σ with Floerdata D := D n +1 D and with the combined brane data (cid:126)L n +1 (cid:126)L = ( L , . . . , L d ) . For given regular points u ∈ M ( x ; x , . . . , x n , ¯ x, x n + m +1 , . . . , x d ; D ); u ∈ M (¯ x ; x n +1 , . . . , x n + m ; D ) , the standard gluing procedure gives rise to a regular point u ∈ M ( x ; . . . , x d ; D ) . This gluing process extends to a local diffeomorphism, let say φ , between neigh-borhood of ( u , u ) to that of u . Then its linearization Dφ fits into the followingcommutative diagram:(Λ top T M ( x , . . . , x d ; D )) u o ( x ) ⊗ o ( x ) ∨ ⊗ · · · ⊗ o ( x d ) ∨ (Λ top T M ( x ; D )) u ⊗ (Λ top T M ( x ; D )) u o ( x ) ⊗ o ( x ) ∼ = ∼ =(Λ top Dφ ) ( u ,u ( − ∆ The above diagram commutes when∆ = (cid:32) (cid:88) k>n + m µ ( x k ) (cid:33) · dim M ( x ; D ) u . Here x = ( x , . . . , x n , ¯ x, x n + m +1 , . . . , x d ); x = (¯ x, x n +1 , . . . , x n + m ) . and hence o ( x ) = o ( x ) ⊗ o ( x ) ∨ ⊗ · · · ⊗ o ( x n ) ∨ ⊗ o (¯ x ) ∨ ⊗ o ( x n + m +1 ) ∨ ⊗ · · · ⊗ o ( x d ) ∨ ; o ( x ) = o (¯ x ) ⊗ o ( x n +1 ) ∨ ⊗ · · · ⊗ o ( x n + m ) ∨ . Now we consider the M -parameterized Floer data used in Definition 5.3 D = D m = (cid:71) r ∈M D ( r ) , on ( d + 1)-pointed disks, with branes structures ( L , . . . , L d ) on the ends. Letus denote the corresponding parameterized moduli space by M D ( x ; x ) := (cid:71) r ∈M { r } × M ( x ; x ; D ( r )) , where ( x ; x ) are the same as in (B.1). For a regular point ( r, u ) of the parameterizedmoduli space, the extended linearized operator is given by D D ( r ) ,u : ( T M ) r × ( T B D ( r ) ) u → ( E D ( r ) ) u . We then havedim ( r,u ) M D ( x ; x ) = dim r M + µ ( x ) − µ ( x ) − · · · − µ ( x d );(Λ top M D ( x , x )) ( r,u ) ∼ = (Λ top T M ) r ⊗ Λ top ker( D D ( r ) ,u ) ∼ = (Λ top T M ) r ⊗ o ( x ) ⊗ o ( x ) ∨ ⊗ · · · ⊗ o ( x d ) ∨ . Now consider two Floer data D j parameterized over M j for j = 1 ,
2. The branedata and asymptotic data are given as before by ( (cid:126)L , (cid:126)L ) and ( x , x ). Thenthe gluing induces a Floer data D parameterized over M = R + × M × M , where R + plays a role of gluing parameter. Let us choose regular points( r , u ) ∈ M D ( x , . . . , x n , ¯ x, x n + m +1 , . . . , x d );( r , u ) ∈ M D (¯ x, x n +1 , . . . , x n + m ) , then for each gluing parameter (cid:96) ∈ R + we have a regular point( (cid:96), u ) ∈ M D ( x , . . . , x d ) . Note that we obtain(Λ top T M ) r ∼ = R ⊗ (Λ top T M ) r ⊗ (Λ top T M ) r , when (cid:96) is sufficiently large. By the same argument as above we have the followingdiagram: (Λ top T M D ( x , . . . , x d )) ( r,u ) (Λ top T M ) r ⊗ o ( x ) ⊗ o ( x ) ∨ ⊗ · · · ⊗ o ( x d ) ∨ (Λ top T M D ( x )) ( r ,u ) ⊗ (Λ top T M D ( x )) ( r ,u ) R ⊗ (Λ top T M ) r ⊗ o ( x ) ⊗ (Λ top T M ) r ⊗ o ( x ) ∼ = ∼ =(Λ top Dφ ) (( r ,u , ( r ,u ( − ∗ Here o ( x ) and o ( x ) are the same as before. The sign in the right vertical arrowcomes from the Koszul convention (cid:4) m = dim r M · index( D D ( r ) ,u ) + (cid:32) (cid:88) k>n + m µ ( x k ) (cid:33) · index( D D ( r ) ,u ) (B.2) ≡ m ( d − m −
1) + m (cid:32) (cid:88) k>n + m µ ( x k ) (cid:33) (B.3)and the sign difference between parameter spaces M and M × M (cid:78) ( r ,r ) = m ( d − n ) + m + n. (B.4)Recall from (6.6) that the structure map m k already has the sign † = (cid:80) ki =1 i · µ ( x i ).By considering all the above sign effects we have † u + † u + (cid:4) m + (cid:78) ( r ,r ) ≡ ‡ u + d (cid:88) i =1 ( i + 1) µ ( x i ) , where ‡ = (cid:80) ni =1 µ ( x i ) − n . This explains the sign convention in Proposition 6.5. WRAPPED FUKAYA CATEGORY OF KNOT COMPLEMENT 59
B.2.2.
Sign convection for A ∞ functor. For the sign convention in (7.13) and (7.14),we need to consider the previous argument with the parameter space N k +1 inDefinition 7.4. The notations in this section are the same as in Section 7.Firstly, let us consider a regular point( r, u ) ∈ N d +1 ( y ; x , . . . , x d ) (B.5)and its boundary strata( r , u ) ∈ N d − m +2 ( y ; x , . . . , x n , ˜ x, x n + m +1 , · · · , x d );( r , u ) ∈ M m +1 (˜ x ; x n +1 , . . . , x n + m )satisfying µ ( y ) = µ ( x ) + · · · + µ ( x n ) + µ (˜ x ) + µ ( x n + m +1 ) + · · · + µ ( x d ) + d − m ; µ (˜ x ) = µ ( x n +1 ) + · · · + µ ( y n + m ) + m − . As in the previous section we compare the sign difference between(Λ top N d +1 ) r ⊗ o ( y ) ⊗ o ( x ) ∨ ⊗ · · · ⊗ o ( x d ) ∨ and (Λ top N d − m +2 ) r ⊗ o ( y ) ⊗ o ( x ) ∨ ⊗ · · · ⊗ o (˜ x ) ⊗ · · · ⊗ o ( x d ) ∨ ⊗ (Λ top M m +1 ) r ⊗ o (˜ x ) ⊗ o ( x n +1 ) ∨ · · · o ( x n + m ) ∨ . The Koszul convention gives (cid:4) f ≡ m ( d − m ) + m (cid:32) (cid:88) k>n + m µ ( x k ) (cid:33) and a similar computation (cid:4) f + (cid:78) ( r ,r ) + † u + ♠ u ≡ ( ‡ u + 1) + (cid:32) d (cid:88) i =1 ( i + 1) µ ( x i ) + d (cid:33) verifies (7.13).Now we consider another type of boundary strata of (B.5) consisting of( r , u ) ∈ M (cid:96) +1 ( y ; y , . . . , y (cid:96) );( r , u ) ∈ N s +1 ( y ; x , . . . , x s );...( r (cid:96) , u (cid:96) ) ∈ N s λ +1 ( y (cid:96) ; x s + ··· + s (cid:96) − +1 , . . . , x d );satisfying degree conditions µ ( y ) = µ ( y ) + · · · + µ ( y (cid:96) ) + r − µ ( y ) = µ ( x ) + · · · + µ ( x s ) + s − µ ( y (cid:96) ) = µ ( x s + ··· + s (cid:96) − +1 ) + · · · + µ ( x d ) + s λ − top N d +1 ) r ⊗ o ( y ) ⊗ o ( x ) ∨ ⊗ · · · ⊗ o ( x d ) ∨ and (Λ top M (cid:96) +1 ) r ⊗ o ( y ) ⊗ o ( y ) ∨ ⊗ · · · ⊗ o ( y (cid:96) ) ∨ ⊗ (Λ top N s +1 ) r ⊗ o ( y ) ⊗ o ( x ) ∨ ⊗ · · · ⊗ o ( x s ) ∨ ⊗· · · ⊗ (Λ top N s λ +1 ) r (cid:96) ⊗ o ( y (cid:96) ) ⊗ o ( x s + ··· + s (cid:96) − +1 ) ∨ ⊗ · · · ⊗ o ( x d ) ∨ . Note that the Koszul sign convention induces (cid:4) f (cid:48) ≡ ( s (cid:48) + · · · + s (cid:48) λ ) (cid:96) + (cid:88) ≤ i ≤ j ≤ (cid:96) s (cid:48) i s (cid:48) j + (cid:96) − (cid:88) i =1 ( s (cid:48) + · · · + s (cid:48) i ) µ ( y i +1 ) , where s (cid:48) r = s r −
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Youngjin Bae, Research Institute for Mathematical Sciences, Kyoto University,Kyoto Prefecture, Kyoto, Sakyo Ward, Kitashirakawa Oiwakecho, Japan 606-8317
E-mail address : [email protected] Seonhwa Kim, Center for Geometry and Physics, Institute for Basic Sciences (IBS),Pohang, Korea
E-mail address : [email protected] Yong-Geun Oh, Center for Geometry and Physics, Institute for Basic Sciences (IBS),Pohang, Korea & Department of Mathematics, POSTECH, Pohang, Korea
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