PPRIME-LOCALIZED WEINSTEIN SUBDOMAINS
OLEG LAZAREV AND ZACHARY SYLVAN
Abstract.
For any high-dimensional Weinstein domain and finite collection of primes, weconstruct a Weinstein subdomain whose wrapped Fukaya category is a localization of theoriginal wrapped Fukaya category away from the given primes. When the original domainis a cotangent bundle, these subdomains form a decreasing lattice whose order cannot bereversed.Furthermore, we classify the possible wrapped Fukaya categories of Weinstein subdomainsof a cotangent bundle of a simply connected, spin manifold, showing that they all coincidewith one of these prime localizations. In the process, we describe which twisted complexesin the wrapped Fukaya category of a cotangent bundle of a sphere are isomorphic to genuineLagrangians. Introduction
Main results.
One of the main problems in symplectic topology is to understand theset of Lagrangians L in a symplectic manifold X . For example, Arnold’s nearby Lagrangianconjecture states that any closed exact Lagrangians L in T ∗ M nstd is Hamiltonian isotopic tothe zero-section M ⊂ T ∗ M std ; by work [14, 3, 20] on this conjecture, all such Lagrangiansare homotopy equivalent to M n . Each closed exact Lagrangian L ⊂ X gives a Liouvillesubdomain T ∗ L of X and the skeleton of T ∗ L , the stable set of its Liouville vector field,is precisely L . More generally, any Weinstein domain V deformation retracts to a possiblysingular Lagrangian skeleton. Therefore a Weinstein subdomain V ⊂ X can be considereda singular Lagrangian in X . In this paper, we consider the problem of constructing andclassifying Weinstein subdomains of a fixed Weinstein domain, as well as the wrapped Fukayacategories W ( V ; R ) of such subdomains (here, R is a commutative coefficient ring). We willonly consider Weinstein subdomains V ⊂ X with the stronger property that X \ V is also aWeinstein cobordism, i.e. V is the sublevel set of an ambient Weinstein Morse function on X ; see [10] for background on the geometry of Weinstein domains.There is a (cohomologically) fully faithful embedding of W ( X ; R ) into Tw W ( X ; R ), thecategory of twisted complexes on W ( X ; R ). Since Tw W ( X ; R ) is a formal algebraic enlarge-ment of a geometric category, this functor is usually not a quasi-equivalence. To understandwhich A ∞ -categories actually arise from Weinstein subdomains, it turns out we will haveto understand which twisted complexes come from actual geometric Lagrangians. In otherwords, we will largely be concerned with understanding the image of this embedding. Wegive examples when this functor is a quasi-equivalence (Proposition 2.2) and describe itsimage when X = T ∗ S nstd (Example 1.10); see Section 1.2. This type of question about thegeometricity of twisted complexes has previously been studied by [18, 8]. a r X i v : . [ m a t h . S G ] S e p OLEG LAZAREV AND ZACHARY SYLVAN
Given a small A ∞ -category C over Z and set of objects A of C , one can form the quotient A ∞ -category C / A , which comes with a localization functor C → C / A ; see [25, 26]. Inparticular, given a collection of prime numbers P ⊂ Z , one can form C (cid:20) P (cid:21) := C / { cone( p · Id L ) | p ∈ P, L ∈ C} (1.1)the localization of C away from the primes P . Quotienting by cone( p · Id L ) kills the objectcone( p · Id L ), which has the effect of making the morphism p · Id L a quasi-isomorphism,i.e. inverting p . Hence if hom ∗C ( L, K ) is a cochain complex of free Abelian groups, thenhom ∗C [1 /P ] ( L, K ) is quasi-isomorphic to hom ∗C ( L, K ) ⊗ Z Z [ P ], which explains our notation C [ P ]. We will also allow P to be empty or contain 0, in which case C [ P ] is the originalcategory C or the trivial category, respectively.Our first result is that any high-dimensional Weinstein domain has Weinstein subdomainswhose Fukaya categories are localizations away from any finite collection of primes P . Fur-themore, these subdomains are almost symplectomorphic, i.e. their symplectic forms arehomotopic through non-degenerate 2-forms, and hence indistinguishable from the point ofview of classical smooth topology. We note that by Gromov’s h-principle [17] for opensymplectic manifolds, any two almost symplectomorphic Weinstein domains are actuallyhomotopic through symplectic structures (but may not be symplectomorphic). Theorem 1.1.
For any Weinstein domain X n with n ≥ and finite collection of primenumbers P , that is possibly empty or contains , there is a Weinstein subdomain X P ⊂ X such that Tw W ( X P ; Z ) ∼ = Tw W ( X ; Z )[ P ] and the Viterbo transfer functor V : Tw W ( X ; Z ) → Tw W ( X P ; Z ) is localization away from P . In particular, Tw W ( X P ; F p ) = 0 if p ∈ P or ∈ P , and Tw W ( X P ; F p ) ∼ = Tw W ( X ; F p ) otherwise. Furthermore, we can arrange that(1) The Weinstein cobordism X \ X P is smoothly trivial and hence X P is almost symplec-tomorphic to X .(2) If Q ⊂ P or ∈ P , we can exhibit a Weinstein embedding ϕ P,Q : X P (cid:44) → X Q with theproperty that if R ⊂ Q ⊂ P , then ϕ P,Q ◦ ϕ Q,R is Weinstein homotopic to ϕ Q,R .(3) If P is empty, then X P is X . If ∈ P , then X P is the flexibilization X flex of X andthe Weinstein embedding X P ⊂ X is unique up to Weinstein homotopy.Remark . For us, the objects of W ( X ; R ) are graded exact spin Lagrangian submanifolds(branes) in X that are closed or have conical Legendrian boundary in a collar of ∂X . Wewill usually not specify what type of grading data our Lagrangian should have, except when X is a cotangent bundle and we will use the canonical Z -grading.In Section 3.1 we will briefly allow some branes to be equipped with rank 1 local systems.We will generally not treat these as honest members of W ( X ; R ), but they are certainly(isomorphic to) members of Tw W ( X ; R ).More precisely, there is a Weinstein homotopy of the Weinstein structure on X to adifferent structure X (cid:48) so that X P is a sublevel set of the Weinstein Morse function on X (cid:48) . RIME-LOCALIZED WEINSTEIN SUBDOMAINS 3
That is, X P is itself a Weinstein domain and X (cid:48) \ X P is a Weinstein cobordism. We also notethat Theorem 1.1 holds for any grading of X (and the induced grading on its subdomains).Our construction is related to a result of Abouzaid and Seidel [5], who also showed thatany Weinstein domain X n , n ≥ , can be modified to a produce a new Weinstein domain X (cid:48) P , almost symplectomorphic to X , with the property that SH ∗ ( X P ; F q ) ∼ = SH ∗ ( X ; F q ) if q (cid:54)∈ P and SH ( X P ; F q ) = 0 otherwise. Theorem 1.1 proves this property on the level ofFukaya categories, which implies the result on the level of symplectic cohomology [15]. Theother main difference between our domain X P and the domain X (cid:48) P produced by Abouzaidand Seidel [5] is that X P is manifestly a subdomain of X while X (cid:48) P is an abstract Weinsteindomain. The construction of Abouzaid and Seidel involves modifying a Lefschetz fibrationfor X by enlarging the fiber and adding new vanishing cycles, and there is no obvious mapbetween X and X (cid:48) P . Our construction involves removing a certain regular Lagrangian disk(which also appears in Abouzaid-Seidel’s work) so that X P is automatically a subdomain of X ; constructing these regular disks requires n ≥
5, hence the restriction on n in Theorem 1.1.Both our construction and that of Abouzaid-Seidel require many choices, but we conjecturethat one can make these choices so that the resulting Weinstein domains X P , X (cid:48) P agree. Remark . We expect that Theorem 1.1 also holds for an infinite collection of primes P ifwe allow X P to be a symplectic manifold that is the intersection of infinitely many Weinsteindomains. Namely, if P = { p , p , . . . } and P i := { p , . . . , p i } , then Theorem 1.1 provides adecreasing collection of Weinstein subdomains X ⊃ X P ⊃ X P ⊃ X P ⊃ · · · and we can set X P := (cid:84) i ≥ X P i . Then we expect that Tw W ( X P ) ∼ = Tw W ( X )[ P ] (or can take this as adefinition). However, we do not know whether X P is a Weinstein manifold in the sense of[10], i.e. an increasing union of finite type Weinstein domains. If P is the set of all primes,we consider X P to be a symplectic ‘rationalization’ of X , analogous to the rationalization ofclassical spaces. Remark . An analog of Theorem 1.1 is true for Weinstein domains with Weinstein stops.For example, in Theorem 2.3 we prove that there is a Legendrian sphere Λ P ⊂ ∂B nstd so thatTw W ( B nstd , Λ P ) ∼ = Tw W ( B nstd , Λ ∅ )[ P ] ∼ = Tw Z [ P ], where Λ ∅ is the Legendrian unknot, andthere is a smoothly trivial Lagrangian cobordism L ⊂ ∂B nstd × [0 ,
1] whose positive, negativeends ∂ ± L coincide with Λ ∅ , Λ P respectively. Note that ( B nstd , Λ ∅ ) is the standard Weinsteinhandle of index n ; we call ( B nstd , Λ P ) a Weinstein P -handle of index n . The constructionof the Weinstein subdomain X P in Theorem 1.1 can be viewed as replacing all standardWeinstein handles of index n with Weinstein P -handles. This is similar to the classicalrationalization of a CW complex, in which all standard cells are replaced with ‘rational’cells.Next we consider Weinstein subdomains of the cotangent bundle T ∗ M std of a smoothmanifold M . Using Theorem 1.1 and the additional fact that Tw W ( T ∗ M std ; F p ) is non-trivial for any p , we show that T ∗ M std has many infinitely different Weinstein subdomains. Corollary 1.5. If n ≥ , then for any finite collection P of primes numbers, possibly emptyor containing zero, there is a Weinstein subdomain T ∗ M nP ⊂ T ∗ M nstd almost symplectomor-phic to T ∗ M std so that Tw W ( T ∗ M P ; Z ) ∼ = Tw W ( T ∗ M ; Z )[ P ] . Furthermore, we can arrange OLEG LAZAREV AND ZACHARY SYLVAN for T ∗ M P to be a Weinstein subdomain of T ∗ M Q if and only if Q ⊂ P or ∈ P , i.e. theproduct of primes in P divides the product of those in Q . The claim here is stronger than in Theorem 1.1: here T ∗ M P is a Weinstein subdomain of T ∗ M Q if and only if Q ⊂ P , or 0 ∈ P (in fact, ‘Weinstein’ subdomain can be replaced with‘Liouville’ subdomain). The proof of Corollary 1.5 carries over to any Weinstein domain X for which W ( X ; F p ) is non-trivial for all p , e.g. if X has a closed exact Lagrangian.Furthermore, by the ‘only if’ part of the claim, our subdomains form a decreasing latticewhose order cannot be reversed. For example, there is an infinite decreasing sequence T ∗ M std (cid:41) T ∗ M (cid:41) T ∗ M , (cid:41) T ∗ M , , (cid:41) · · · (cid:41) T ∗ M P k (cid:41) · · · (cid:41) T ∗ M = T ∗ M flex where P k is the set of the first k primes; the other subdomains T ∗ M P where P (cid:54) = P k , e.g. T ∗ M , , contain T ∗ M P k for sufficiently large k . In particular, T ∗ M std has many singular Lagrangians given by the skeleta of T ∗ M P . These skeleta are not Hamiltonian isotopicsince otherwise we could find a Liouville embedding of T ∗ M Q into T ∗ M P for P ⊃ Q . Wecontrast this with the nearby Lagrangian conjecture which claims that all closed exact smooth Lagrangians of T ∗ M std are Hamiltonian isotopic. Finally, we note that T ∗ M P has no closedexact smooth Lagrangians if P is non-empty since its Fukaya category over F p vanishes.Our second main result about subdomains of T ∗ M std is a converse to Corollary 1.5: theFukaya category of any Weinstein subdomain of T ∗ M std is a localization of Tw W ( T ∗ M std ; Z )away from some finite collection of primes. Here we use the Z -grading on T ∗ M std and itssubdomains induced by the Lagrangian fibration by cotangent fibers. Theorem 1.6. If M n is a closed, simply connected, spin manifold and i : X (cid:44) → T ∗ M std isa Weinstein subdomain, then Tw W ( X ; Z ) ∼ = Tw W ( T ∗ M std ; Z )[ P ] for some finite collec-tion of primes P , that is possibly empty or contains , and is unique (unless P contains ). Under this equivalence, the Viterbo transfer functor Tw W ( T ∗ M ; Z ) → Tw W ( X ; Z ) islocalization away from P . Furthermore, either the restriction i ∗ : H n ( T ∗ M n ; Z ) → H n ( X ; Z ) is an isomorphism or W ( X ; Z ) ∼ = 0 (or both). For n ≥
5, Theorem 1.6 combined with Corollary 1.5 completely classify which categoriesappear as Fukaya categories (with integer coefficients) of Weinstein subdomains of cotangentbundles of closed, simply connected, spin manifolds. For n ≤
4, the question remainsopen whether the categories Tw W ( T ∗ M nstd ; Z )[ P ] actually appear as Fukaya categories ofsubdomains. Indeed, in the n = 1 case, the only subdomains of T ∗ S std are T ∗ S std or B std ,which algebraically correspond to the cases P = ∅ and P = 0. We note that the conditionon the map i ∗ shows that any Weinstein ball Σ ⊂ T ∗ M std has trivial W (Σ). There are norestrictions on i ∗ in degrees less than n , as in the case of T ∗ M std ∪ H n − ⊂ T ∗ M std . Finally,we note that the ‘both’ case does occur in the case of T ∗ M nflex ⊂ T ∗ M nstd .We emphasize that Theorem 1.6 classifies Weinstein subdomains of X ⊂ T ∗ M std ; namely, X is itself a Weinstein domain and T ∗ M std \ X is a Weinstein cobordism (after Weinsteinhomotopy of T ∗ M std ). We do not know if our result holds for more general Liouville sub-domains X ⊂ T ∗ M std , for which either X is not a Weinstein domain or T ∗ M std \ X is nota Weinstein cobordism. However, in the only known examples of subdomains X ⊂ T ∗ M std for which T ∗ M std \ X is not a Weinstein cobordism, X is a flexible domain [13] and hence RIME-LOCALIZED WEINSTEIN SUBDOMAINS 5 has trivial Fukaya category. Furthermore, our classification is quite special to cotangentbundles: for a general Weinstein domain X , there are subdomains X for which Tw W ( X )is different from Tw W ( X )[ P ] for any collection of primes P . For example, the boundaryconnected sum T ∗ M std (cid:92)T ∗ N std of two cotangent bundles T ∗ M, T ∗ N has a natural collectionsubdomains indexed by pairs of collections of primes P, Q , namely T ∗ M P (cid:92)T ∗ N Q .1.2. Outline of proofs.
We now outline the proofs of our two main results: Theorem 1.1and Theorem 1.6. We focus primarily on the latter result, whose proof involves describingwhich twisted complexes in Tw W ( T ∗ M std ) are quasi-isomorphic to actual Lagrangians, i.e.the image of the functor W ( T ∗ M std ) (cid:44) → Tw W ( T ∗ M std ).To see the connection, consider a Weinstein subdomain X n ⊂ X n . The Weinsteincobordism X \ X has index n Lagrangian co-core disks D , . . . , D k which are objects of W ( X ). Ganatra, Pardon, and Shende [16] proved thatTw W ( X ) ∼ = Tw W ( X ) / ( D , . . . , D k )and the localization functor Tw W ( X ) → Tw W ( X )has a geometric interpretation and is called the Viterbo transfer functor . See [32] for resultswhen
X, X are both Weinstein but X \ X is not necessarily a Weinstein cobordism. So todescribe Tw W ( X ), it suffices to describe the quasi-isomorphism classes of the Lagrangiandisks D , . . . , D k in Tw W ( X ). To prove Theorem 1.1, we construct a disjoint collection ofdisks D , . . . , D k ⊂ X n , n ≥ , so that Tw W ( X ; Z ) / ( D , . . . , D k ) ∼ = Tw W ( X ; Z )[ P ]. Byremoving the Weinstein handles associated to these disks, we get the subdomain X P withthe desired property Tw W ( X P ; Z ) ∼ = Tw W ( X ; Z ) / ( D , . . . , D k ) ∼ = Tw W ( X ; Z )[ P ]. Remark . In fact, the localization C / A by some objects A ⊂ C depends only on the split-closure of A in C [16], which is the kernel of the localization C → C / A . A subcategory C (cid:48) ⊂ C is split-closed if for any two objects A, B of C for which A ⊕ B is an object of C (cid:48) , then A, B are also objects of C (cid:48) . More generally, there is a correspondence between localizing functors C → D and split-closed subcategories of C .Since any Weinstein domain X ⊂ T ∗ M std has Tw W ( X ) ∼ = Tw W ( T ∗ M std ) / ( D , . . . , D k )for some collection of Lagrangian disks D , . . . , D k in T ∗ M std , to prove Theorem 1.6 we needto classify the objects of Tw W ( T ∗ M std ) that are quasi-isomorphic to embedded Lagrangiandisks. By work of Abouzaid [1], any object of Tw W ( T ∗ M std ) is quasi-isomorphic to atwisted complex of the cotangent fibers T ∗ q M ; after taking boundary connected sums ofthese cotangent fibers along isotropic arcs, we can replace this twisted complex with a singleembedded Lagrangian disk equipped with a bounding cochain . However for Theorem 1.6, weneed to consider Lagrangian disks without bounding cochains and as we will see in Theorem1.8 below, not every twisted complex in Tw W ( T ∗ M std ) is quasi-isomorphic to such a disk.In the following key result, we characterize those twisted complexes in Tw W ( T ∗ M std ) thatare quasi-isomorphic to Lagrangian disks. To make this precise, we fix some notation. Let A be an object of some pre-triangulated A ∞ -category C over Z . A homotopy unit e ∈ end C ( A ) of OLEG LAZAREV AND ZACHARY SYLVAN A gives an A ∞ -homomorphism Z → end C ( A ), which induces a functor Tw Z → Tw end C ( A ).Applying this to C = Tw W ( T ∗ M std ; Z ), and A = T ∗ q M , we get the composition of functors ⊗ T ∗ q M : Tw Z → Tw end( T ∗ q M ) ∼ −→ Tw W ( T ∗ M std ; Z ) (1.2)Note here that Tw Z is the category of finite cochain complexes, i.e. those Z -cochain com-plexes whose underlying graded Abelian group is free and finitely generated. The functor ⊗ T ∗ q M sends such a twisted complex on Z to the corresponding twisted complex on T ∗ q M .In particular, the differential consists entirely of morphisms that are all integer multiples ofthe unit. By Abouzaid’s theorems [4, 1], the second functor is actually a quasi-equivalence,meaning that every object of Tw W ( T ∗ M std ; Z ) is a twisted complex of T ∗ q M with differentialgiven by arbitrary elements of end( T ∗ q M ). As we will see, the composite functor ⊗ T ∗ q M isnot essentially surjective, but for nice M every Lagrangian disk is contained in its essentialimage. More generally, we have the following result. Theorem 1.8.
Let M n be a closed, simply-connected, spin manifold and let i : L n (cid:44) → T ∗ M nstd be an exact Lagrangian brane. If i : L n (cid:44) → T ∗ M nstd is null-homotopic as a continuous map,then L is in the image of ⊗ T ∗ q M . More precisely, L is quasi-isomorphic to CW ∗ ( M, L ; Z ) ⊗ T ∗ q M in Tw W ( T ∗ M std ; Z ) , where the cochain complex CW ∗ ( M, L ; Z ) is considered an object of Tw Z . Combining this result with the construction of the Lagrangian disks in the Theorem 1.1,we have the following description of the image of ⊗ T ∗ q M . Corollary 1.9. If M n is a closed, simply-connected, spin manifold and L ⊂ T ∗ M std is aLagrangian disk, then L is in the essential image of ⊗ T ∗ q M . If n ≥ , then every object of Tw W ( T ∗ M std ) in the image of ⊗ T ∗ q M is quasi-isomorphic to a Lagrangian disk. Theorem 1.8 translates the purely topological condition that the Lagrangian is null-homotopic into the Floer-theoretic condition on its quasi-isomorphism class in the Fukayacategory. The proof of Theorem 1.8 actually shows that this topological condition can beweakened to the algebraic condition that the restriction homomorphism i ∗ : C ∗ ( T ∗ M ; Z ) → C ∗ ( L ; Z ) on singular cochain algebras is homotopic as an A ∞ -homomorphism to a map thatfactors through Z . In Proposition 3.2, we prove a generalization of Theorem 1.8 for arbi-trary Lagrangians i : L (cid:44) → T ∗ M std that are not nessarily null-homotopic: we prove that the CW ∗ ( M, M )-module CW ∗ ( M, L ) is in the image of the compositionMod C ∗ ( L ) i ∨ −→ Mod C ∗ ( T ∗ M ) ∼ = Mod CW ∗ ( M,M ) , where i ∨ is the pullback functor on modules induced by the restriction homomorphism i ∗ : C ∗ ( T ∗ M ) → C ∗ ( L ).The proof of Corollary 1.9 uses the Koszul duality between the wrapped Floer cochainsof a cotangent fiber and those of the zero-section of a cotangent bundle, the fact that thezero-section M is homotopy equivalent to the ambient manifold T ∗ M std , and a certain com-mutativity property of the closed-open map that holds for arbitrary Liouville domains (see RIME-LOCALIZED WEINSTEIN SUBDOMAINS 7
Proposition 3.2 and Remark 3.4). Consequently, Corollary 1.9 is quite special to cotangentbundles and analogous results do not hold for general Weinstein domains. Even if X n hasa single index n handle with co-core D n , then it is not true that any Lagrangian disk L ⊂ X is isomorphic to C ∗ ⊗ D for some cochain complex C ∗ over Z (but since D is a generatorof Tw W ( X ), L is isomorphic to a twisted complex of D whose differential has arbitrarymorphisms). For example, this is the case if X n is one of the exotic cotangent bundlesconstructed in [21] that have many closed regular Lagrangians with different topology.In the following example, we illustrate the above results when M = S n . We describe theimage of the functor W ( T ∗ S nstd ) (cid:44) → Tw W ( T ∗ S nstd ) and give examples of Lagrangians thatare not in image of the functor ⊗ T ∗ q S n : Tw Z → Tw W ( T ∗ S nstd ). Example 1.10.
We first Floer-theoretically classify all exact Lagrangian branes in T ∗ S nstd .If L ⊂ T ∗ S n is closed, then it is quasi-isomorphic to the zero-section S n ⊂ T ∗ S n by [14]; if L n ⊂ T ∗ S n has non-empty boundary, then any embedding i : L n (cid:44) → T ∗ S n is automaticallynull-homotopic and hence in the image of ⊗ T ∗ q M : Tw Z → Tw W ( T ∗ S nstd ; Z ); this impliesthat L is quasi-isomorphic to a disk if n ≥
5. However, there are many exact Lagrangians L ⊂ T ∗ S nstd that are not homotopy-equivalent to a disk or n -sphere: any smooth n -manifold L with non-empty boundary and trivial complexified tangent bundle has an exact Lagrangianembedding into T ∗ S n for n ≥
3; see [12, 23]. Using the above classification, one can check thatfor any Lagrangian L ⊂ T ∗ S nstd with non-empty boundary, the wrapped Floer cohomology HW ∗ ( L, L ) is either trivial or infinite-dimensional (over some field F p ). This implies thefollowing new case of the Arnold chord conjecture: any Legendrian Λ ⊂ ST ∗ S nstd that boundsan exact Lagrangian brane (so graded, spin) in T ∗ S nstd has at least one Reeb chord for anycontact for any contact form; see [19, 30, 27] for existing results.Although all Lagrangians with non-empty boundary are in the image of the functor ⊗ T ∗ q S n ,we now show the zero-section S n ⊂ T ∗ S n is not; this is compatible with the fact that i : S n (cid:44) → T ∗ S n is not null-homotopic. Indeed, any Lagrangian L that is in the image of ⊗ T ∗ q M : Tw Z → Tw W ( T ∗ S nstd )represents something in the image of the pullback functor i ∨ : M od Z → M od C ∗ ( S ) and so hasthe property that the product CW ∗ ( S n , L ) ⊗ CW n ( S n , S n ) → CW ∗ + n ( S n , L )must vanish on cohomology (since CW ∗ ( S n , S n ) ∼ = C ∗ ( S n ) → Z vanishes in degree n ).Since this product does not vanish for L = S n , this Lagrangian is not in the image of ⊗ T ∗ q S n . However since T ∗ q S n generates Tw W ( T ∗ S nstd ), the zero-section S n is still sometwisted complex of T ∗ q S n . It turns out that S n is quasi-isomorphic to T ∗ q S n [ n ] γ → T ∗ q S n ,where γ is the generator of CW ( T ∗ q S n [ n ] , T ∗ q S n ) = CW − n ( T ∗ q S n , T ∗ q S n ) ∼ = C n − (Ω S n ) ∼ = Z .Note that γ is not a multiple of the unit.In all, we have shown that if n ≥
5, the image of the full and faithful embedding W ( T ∗ S nstd ) (cid:44) → Tw W ( T ∗ S nstd ) ∼ = Tw (cid:8) T ∗ q S n (cid:9) is quasi-isomorphic to the subcategory (cid:8) C ∗ ⊗ T ∗ q S n | C ∗ is a cochain complex over Z (cid:9) ∪ (cid:110) T ∗ q S n [ n ] γ → T ∗ q S n (cid:111) . OLEG LAZAREV AND ZACHARY SYLVAN
For more general manifolds M , W ( T ∗ M ) has other objects besides the zero-section andLagrangian disks, e.g. the surgery of the zero-section and a cotangent fiber.Finally, we use Corollary 1.9 to prove Theorem 1.6 classifying the wrapped Fukaya cate-gories of subdomains of T ∗ M std . Proof of Theorem 1.6.
Let X n ⊂ T ∗ M std be a Weinstein subdomain and C n := T ∗ M std \ X n the complementary Weinstein cobordism. Then C = C sub ∪ H n ∪ · · · ∪ H nk , where all han-dles of C sub are subcritical, i.e. have index less than n . The Viterbo restriction induces anequivalence Tw W ( X ∪ C sub ; Z ) ∼ = Tw W ( X ; Z ) on the subcritical cobordism [16]. Also by[16], Tw W ( X ∪ C sub ; Z ) ∼ = Tw W ( T ∗ M std \ ( D (cid:113) · · · (cid:113) D k ); Z ) ∼ = Tw W ( T ∗ M std ; Z ) / ( D , . . . , D k ) , where D , . . . , D k ⊂ T ∗ M std are the Lagrangian co-cores of H n , . . . , H nk ; recall that thisquotient category depends just on the subcategory split-generated by these disks by Remark1.7. Now by Corollary 1.9, D i ∼ = CW ∗ ( M, D i ) ⊗ T ∗ q M in Tw W ( T ∗ M ; Z ) where CW ∗ ( M, D i )is considered as an object of Tw Z , or equivalently a cochain complex over Z . Any cochaincomplex of free Abelian groups splits as a direct sum of twisted complexes of the form Z [1] m → Z for some integer m and free groups Z (and their shifts). If CW ∗ ( M, D i ) hasa Z -summand, then D i split-generates and hence Tw W ( T ∗ M std ; Z ) / ( D , . . . , D k ) is trivial.Otherwise, let p , . . . , p j be the collection of primes dividing m in the summand Z [1] m → Z .Then the split-closure of Z [1] m → Z coincides with that of the objects Z [1] p → Z , . . . , Z [1] p j → Z .So if P denotes the set of primes obtained this way over all D , . . . , D k , the split-closure of( D , . . . , D k ) coincides with that of T ∗ q M [1] p → T ∗ q M ∼ = cone( p · Id T ∗ q M ) where p ∈ P . Since T ∗ q M generates Tw W ( T ∗ M std ; Z ), the subcategory split-generated by ( D , . . . , D k ) coincideswith that split-generated by { cone( p · Id L ) | p ∈ P, L ∈ Tw W ( T ∗ M std ; Z ) } , and so Tw W ( X ; Z ) ∼ = Tw W ( T ∗ M std ; Z ) (cid:2) P (cid:3) as desired. Also, P is unique since W ( X ; F q )vanishes if q ∈ P and Tw W ( X ; F q ) ∼ = Tw W ( T ∗ M std ; F q ) is non-trivial if q (cid:54)∈ P and 0 (cid:54)∈ P .Finally, if i ∗ : H n ( T ∗ M ; Z ) → H n ( X ; Z ) is not an isomorphism, then [ D i ] ∈ H n ( T ∗ M ; Z ) ∼ = Z is non-zero for some D i and so the algebraic intersection number M · D i ∈ Z is non-zero.Since this intersection number is precisely the Euler characteristic χ ( CW ∗ ( M, D i )) of theFloer cochains CW ∗ ( M, D i ), the direct sum decomposition of CW ∗ ( M, D i ) discussed abovemust contain a free group Z , which implies that Tw W ( X ; Z ) is trivial. (cid:3) Remark . Abouzaid observed that Corollary 1.9, and hence Theorem 1.6, extends to thecase where M has finite fundamental group and spin universal cover. Indeed, in that caseany Lagrangian disk L ⊂ T ∗ M lifts to a disk ˜ L ⊂ T ∗ ˜ M . Applying Corollary 1.9 to ˜ L , weobtain an isomorphism ˜ L ∼ = K ∗ ⊗ T ∗ q ˜ M RIME-LOCALIZED WEINSTEIN SUBDOMAINS 9 for some complex K ∗ ∈ Tw Z . Presenting the upstairs category W ( T ∗ ˜ M ) using pulled-backFloer data, we can push this isomorphism back down to W ( T ∗ M ) to conclude L ∼ = K ∗ ⊗ T ∗ q M. The authors expect the same to hold if π ( M ) is infinite, but that requires extendingTheorem 1.8 to the non-compact case.As we have seen, any Weinstein subdomain X ⊂ T ∗ M std induces a localization (Viterbo)functor Tw W ( T ∗ M std ; Z ) → Tw W ( X ; Z ) and hence by Remark 1.7 is associated to a split-closed subcategory of Tw W ( T ∗ M std ; Z ). So Theorem 1.6 can be viewed as a classification ofthe split-closed subcategories of Tw W ( T ∗ M std ; Z ) coming from this geometric setting. Thefact that these correspond to subsets of prime integers stems from the corresponding fact forTw Z (and the crucial Corollary 1.9). More generally, Hopkins and Neeman [29] proved thatsplit-closed subcategories of D b Mod R correspond to certain subsets of Spec( R ); in the globalsetting, Thomason [33] proved that split-closed subcategories of D b Coh( X ) that are closedunder the tensor product correspond to certain closed subsets of X . Although the wrappedFukaya category does not generally have a monoidal structure, we pose the open problem ofclassifying Fukaya categories of Weinstein subdomains of arbitrary Weinstein domains as away of extending these results to the symplectic setting.1.3. Acknowledgements.
We would like to thank Mohammed Abouzaid and Paul Seidelfor helpful discussions, particularly concerning Proposition 3.2. The first author was par-tially supported by an NSF postdoctoral fellowship, award 1705128; the second author waspartially supported by the Simons Foundation through grant
Proof of results
Constructing Lagrangian disks.
Our construction of Weinstein subdomains of aWeinstein domain X n depends on the existence of certain Lagrangian disks near the index n co-cores of X n . Since a neighborhood of an index n co-core is T ∗ D n , it suffices to constructthese Lagrangians in T ∗ D n . In this section, we will exhibit these Lagrangian disks in T ∗ D n and study their isomorphism classes in the partially wrapped category Tw W ( T ∗ D n , ∂D n ; Z ).Recall that objects of Tw W ( T ∗ D n , ∂D n ; Z ) are twisted complexes of exact Lagrangiansin T ∗ D n whose boundary is disjoint from ∂D n . We use the canonical Z -grading of T ∗ D n viathe Lagrangian fibration by cotangent fibers. By [16, 9], the category Tw W ( T ∗ D n , ∂D n ) isgenerated by the cotangent fiber T ∗ D n ⊂ T ∗ D n at the origin 0 ∈ D n . end ∗ ( T ∗ D n , T ∗ D n ),the partially wrapped Floer cochains of T ∗ D n , is quasi-isomorphic to Z , hence there is acohomologically full and faithful A ∞ -functor CW ∗ ( T ∗ D n , ) : Tw W ( T ∗ D n , ∂D n ) → Mod Z Here Mod Z denotes the dg-category of right Z -modules. Since T ∗ D n generates the par-tially wrapped Fukaya category, this functor has image Tw Z , the category of cochain com-plexes whose underlying graded Abelian group is free and finitely generated. The equiv-alence between Tw W ( T ∗ D n , ∂D n ) and Tw Z takes an object L of Tw W ( T ∗ D n , ∂D n ) to CW ∗ ( T ∗ D n , L ), viewed as cofibrant cochain complex over Z .Let D n − ⊂ T ∗ D n be a negative perturbation of the zero-section D n , i.e. the result ofapplying the negative wrapping (cid:80) q i ∂ p i to D n so that ∂D n − is disjoint from the stop ∂D n .Note that D n − is Lagrangian isotopic to T ∗ D n in the complement of ∂D n by geodesic flow.Hence CW ∗ ( T ∗ D n , L ) is quasi-isomorphic to CW ∗ ( D n − , L ). Since all Reeb chords out of ∂D n − hit the stop in small time, CW ∗ ( D n − , L ) is quasi-isomorphic to CF ∗ ( D n − , L ), the unwrapped Floer cochains which can be explicitly computed.Next we review certain regular Lagrangian disks in T ∗ D n introduced by Abouzaid andSeidel in Section 3b of [5] and study their isomorphism class in Tw W ( T ∗ D n , ∂D n ). Let U ⊂ S n − be a compact codimension zero submanifold with smooth boundary. Let g : S n − → R be a C -small function so that g is strictly negative in the interior of U , zero on ∂U , strictly positive on S n − \ U , and has zero as a regular value. Next, extend g to a smoothfunction f : R n → R so that f is C -small in the unit disk and satisfies f ( tq ) = | t | f ( q ) for | q | ≥ / , t ≥
1. Let Γ( df ) be the graph of df in T ∗ R n and let D U = Γ( df ) ∩ T ∗ D n . Since f ishomogeneous for | q | ≥ / g , D U has Legendrian boundary whichis disjoint from ∂D n . Furthermore, there is a Lagrangian isotopy Γ( d ( sf )) from D U to thezero-section D ⊂ T ∗ D n (which intersects the stop ∂D precisely when s = 0). After fixing agrading on D , the isotopy Γ( d ( sf )) induces a preferred grading on D U . In particular, D U with this Z -grading is an object of Tw W ( T ∗ D n , ∂D n ).We now compute the isomorphism class of D U in Tw W ( T ∗ D n , ∂D n ), following [5]. Namely,as noted in Lemma 3.3 of [5], we can scale f so that the intersection points of D nU and D n − have small action and then by a classical computation of Floer, CW ( D − , D U ) is quasi-isomorphic to Morse cochains of f . Since R n is contractible, this is quasi-isomorphic to˜ C ∗− ( U ), reduced Morse cochains on U . Hence, under the equivalence CW ∗ ( T ∗ D n , ) be-tween Tw W ( T ∗ D n , ∂D ; Z ) and Tw Z , the image of the disk D U in Mod Z is quasi-isomorphicto ˜ C ∗− ( U ). Note that since CW ∗ ( T ∗ D n , T ∗ D n ) ∼ = Z , the image of the twisted complex˜ C ∗− ( U ) ⊗ T ∗ D n under the functor CW ∗ ( T ∗ D n , ) is also ˜ C ∗− ( U ). Since the CW ∗ ( T ∗ D n , )functor is cohomologically full and faithful, the disk D U is quasi-isomorphic to the twistedcomplex ˜ C ∗− ( U ) ⊗ D − ∼ = ˜ C ∗− ( U ) ⊗ T ∗ D n in Tw W ( T ∗ D n , ∂D ; Z ). Remark . Our definition of the disk D U agrees with that in Abouzaid-Seidel [5]. However,they make an inconsequential misidentification of the Floer complex with the Morse complexto obtain ˜ C ∗− ( U ) as CF ∗ ( D U , D ) instead of CF ∗ ( D, D U ).Using the disks D U , we now show that for sufficiently large n any Lagrangian in T ∗ D n (or twisted complex of Lagrangians) is quasi-isomorphic to a Lagrangian disk. Note thatthis is stronger than the statement that any Lagrangian is a twisted complex of disks, whichfollows from the fact that T ∗ D n generates Tw W ( T ∗ D n , ∂D n ; Z ). RIME-LOCALIZED WEINSTEIN SUBDOMAINS 11
Proposition 2.2. If n ≥ , every object of Tw W ( T ∗ D n , ∂D n ; Z ) is quasi-isomorphic toan exact Lagrangian disk. In particular, W ( T ∗ D n , ∂D n ; Z ) → Tw W ( T ∗ D n , ∂D n ; Z ) is aquasi-equivalence.Proof. An arbitrary object of Tw W ( T ∗ D n , ∂D n ; Z ) can be identified with some finite di-mensional cochain complex of free Abelian groups via the quasi-equivalence CW ∗ ( T ∗ D n , ).Every such cochain complex ( C ∗ , ∂ ) splits as a direct sum of twisted complexes of the form Z [ d + 1] m → Z [ d ] for some integer m or complexes Z [ d ] with no differential. To see this, weuse the fact that the short exact sequence 0 → ker ∂ n → C n → im ∂ n → ∂ n is free.Next, we recall that given two exact Lagrangians L, K ⊂ X and a framed isotropic arcbetween their Legendrian boundaries ∂L, ∂K ⊂ ∂X , one can form a new exact Lagrangian L(cid:92)K ⊂ X , the isotropic boundary connected sum of L, K . If
L, K are Z -graded Lagrangians,then there is a choice of framing for the isotropic arc (the space of such choices up tohomotopy is a Z -torsor) so that L(cid:92)K also has a Z -grading that restricts to the Z -grading of L and K and hence L(cid:92)K is quasi-isomorphic to L ⊕ K in Tw W ( X ); whenever we discuss theisotropic connected sum of two Lagrangians we mean the sum using any isotropic arc withthis framing. The actual geometric disk will depend on the homotopy class of the arc, butsince we are only concerned with the resulting object of W ( X ) we will ignore the distinction.Returning to X = T ∗ D n , note that we can assume that any two Lagrangians L, K ⊂ T ∗ D n are disjoint since we can view T ∗ D n as the result of gluing two copies of T ∗ D n together andplace L in one copy and K in the other copy. So in light of the above discussion andthe splitting from the previous paragraph, it suffices to prove that the twisted complexes Z [ d + 1] m → Z [ d ] and Z [ d ] are quasi-isomorphic to embedded Lagrangian disks. The lattercomplex is quasi-isomorphic to T ∗ D n with the appropriate grading so it suffices to provethat Z [ d + 1] m → Z [ d ] is quasi-isomorphic to a disk.As noted in Abouzaid-Seidel [6], for n ≥ m ≥
0, there is a codimension 0 Moorespace U m ⊂ S n − with ˜ C ∗ ( U m ) ∼ = Z [ − m → Z [ − V obtained by attaching D to S along a degree m map S → S ; then V embeds into S n − for n ≥ n = 5 by the explicit map D → C given by z → ((1 −| z | ) z, z m ). Let U m be a neighborhood of V in S n − . Then D U m is quasi-isomorphicto Z [ − m → Z [ − D U m by d + 3 and the resulting disk D U m [ d + 3] is quasi-isomorphic to Z [ d + 1] m → Z [ d ], as desired. (cid:3) We observe that not every object of Tw W ( T ∗ D n , ∂D n ; Z ) is quasi-isomorphic to a disk D U . This is because CW ∗ ( D − , D U ) is a cochain complex that is supported between degrees0 and n − U ⊂ S n − ) or a shift thereof (if we shift the grading on D nU ) while ageneral cochain complex can have arbitrarily wide support. However, Proposition 2.2 showsthat every object of Tw W ( T ∗ D n , ∂D n ; Z ) is quasi-isomorphic to the boundary connectedsum of possibly several different D U , with possibly different gradings.2.2. Constructing subdomains.
Now we use the Lagrangian disks from the previous sec-tion to construct Weinstein subdomains of a Weinstein domain X and prove Theorem 1.1. As stated in Remark 1.4, the construction of subdomains also holds when the ambient We-instein domains has stops. The most important case for us is when X = ( T ∗ D n , ∂D n ), thestopped domain considered in the previous section. As we will see, Theorem 1.1 for arbitraryWeinstein domains follows from this case.In the following, we say a stopped Weinstein domain ( X , Λ ) is a Weinstein subdomainof ( X, Λ) if X = X ∪ C for some Weinstein cobordism C which is trivial along Λ = Λ.In particular, there is a smoothly trivial regular Lagrangian cobordism between Λ and Λ in X \ X which allows to identify the linking disk of Λ in X with the linking disk of Λ in X . We say that this cobordism is flexible if the attaching spheres of the index n handlesare loose in the complement of Λ . We also say that two Weinstein subdomains X , X ⊂ X are Weinstein homotopic if the following holds: there is a homotopy of Weinstein Morsefunctions f t , ≤ t ≤
1, on X that have c as a regular level set for all t and X and X arethe c -sublevel sets of f and f , respectively. Theorem 2.3.
Let n ≥ . For any finite collection of prime numbers P , that is possiblyempty or contains , there is a Legendrian sphere Λ P ⊂ ∂B nstd formally isotopic to thestandard unknot Λ ∅ so that ( B nstd , Λ P ) embeds as a Weinstein subdomain of ( B nstd , Λ ∅ ) =( T ∗ D n , ∂D n ) with the following properties:(1) The Viterbo restriction functor Tw W ( B nstd , Λ ∅ ; Z ) → Tw W ( B nstd , Λ P ; Z ) induces an equivalence Tw W ( B nstd , Λ P ; Z ) ∼ = Tw W ( B nstd , Λ ∅ ; Z ) (cid:20) P (cid:21) ∼ = Tw Z (cid:20) P (cid:21) . (2) ( B nstd , Λ P ) embeds as a Weinstein subdomain of ( B nstd , Λ Q ) if and only if Q ⊂ P or ∈ P . In such cases, we can construct such an embedding with the property thatthe Weinstein cobordism between unstopped domains is trivial, i.e. ∂B n × [0 , , andif R ⊂ Q ⊂ P , the composition ( B nstd , Λ P ) ⊂ ( B nstd , Λ Q ) ⊂ ( B nstd , Λ R ) is Weinsteinhomotopic to ( B nstd , Λ P ) ⊂ ( B nstd , Λ R ) obtained by viewing P ⊂ R .(3) There is a smoothly trivial regular Lagrangian cobordism L ⊂ ∂B nstd × [0 , with ∂ − L = Λ P and ∂ + L = Λ Q if and only if Q ⊂ P or ∈ P . Furthermore, fortwo disjoint subsets of primes P , P , the Legendrian sphere Λ P (cid:113) P is the isotropicconnected sum Λ P (cid:92) Λ P of Λ P , Λ P embedded in disjoint Darboux balls in ∂B nstd .(4) If ∈ P , then Λ P ⊂ ∂B nstd is loose. In particular, we have a sequence of LegendriansΛ unknot = Λ ∅ , Λ , Λ , , Λ , , , Λ , , , , . . . , Λ = Λ loose in ∂B nstd and Lagrangian cobordisms in ∂B nstd × [0 ,
1] connecting consecutive Legendriansinterpolating between Λ unknot and Λ loose , analogous to the sequence of subdomains in The-orem 1.1. We note that such Legendrians do not exist for n = 2 as proven in [11]: if L is a decomposable Lagrangian cobordism (a condition similar to regularity) with negativeend Λ and positive end Λ ∅ , then either Λ = Λ ∅ or Λ is stabilized in the sense of [28], soTw W ( B std , Λ; Z ) ∼ = Tw Z or Tw W ( B std , Λ; Z ) ∼ = 0 are the only possibilities. RIME-LOCALIZED WEINSTEIN SUBDOMAINS 13
Remark . The construction of the isotropic connected sum Λ (cid:92) Λ of two LegendriansΛ , Λ in the statement of Theorem 2.3 above is similar to the boundary connected sumof two Lagrangians discussed in the proof of Proposition 2.2 (the former happens on theboundary of the latter) and also depends on a framed isotropic arc between Λ and Λ .However if Λ is contained in a Darboux chart of ( Y, ξ ) disjoint from Λ , then the isotropicconnected sum Λ (cid:92) Λ is actually independent of the isotropic arc and its framing; this isbecause we can isotope Λ to a small neighborhood of Λ via the original isotropic arc andthen isotope it back to its original position using a new isotropic arc.More precisely, we can identify the Darboux chart containing Λ with the cotangent bundleof a small piece of the framing-thickened arc and use this to produce a family of Darbouxcharts. If the two arcs have the same framing at the endpoints, the resulting family ofDarboux charts is a loop, which means that Λ returns to itself. Proof of Theorem 2.3.
We prove this theorem in several stages: first we construct Λ p when p is a single prime and prove that it has the claimed geometric properties, then we constructΛ P for a general set of primes P , and finally we prove our claims about the Fukaya categoryof ( B nstd , Λ P ).2.2.1. Λ p for a single prime p . We first consider the case when the collection of primes P consists of a single prime p . As discussed in the previous section, let U p ⊂ S n − be afixed p -Moore space. Then the Lagrangian disk D p := D U p ⊂ ( T ∗ D n , ∂D n ) is isomorphicto T ∗ D n [1] p → T ∗ D n in Tw W ( T ∗ D n , ∂D n ; Z ). Also, if p = 0, we set U = S n − (as afull subset of S n − ) and form D := D nS n − , which is Lagrangian isotopic in ( T ∗ D n , ∂D n )to the the cotangent fiber T ∗ D n . If p is the empty set, we set U = B n − ⊂ S n − andform D ∅ := D nB n − , which is a small Lagrangian disk that is disjoint from the zero-section D n ⊂ T ∗ D n ; note that any two such small Lagrangian disks are isotopic in ( T ∗ D n , ∂D n ). Inparticular, D ∅ is the zero object in Tw W ( T ∗ D n , ∂D n ; Z ). To construct ( B nstd , Λ P ), we will carve out these Lagrangian disks as we now explain.In general, given a Liouville domain X n and an exact Lagrangian disk D n ⊂ X n withLegendrian boundary, there is a Liouville subdomain X ⊂ X (which we say is obtainedby carving out D n from X ) and a Legendrian sphere Λ ⊂ ∂X so that X = X ∪ H n Λ andthe co-core of H n Λ is D n ; see [12] for details. If X is a Weinstein domain and D n ⊂ X is a regular Lagrangian, then X ⊂ X is a Weinstein subdomain. The disks D p ⊂ T ∗ D n we consider are indeed regular; in fact D p = Γ( df ) is isotopic through Lagrangians withLegendrian boundary ( D p ) s = Γ( sdf ) ∩ T ∗ D n to the zero-section D n ⊂ T ∗ D n . Therefore, T ∗ D n \ D p is homotopic to the Weinstein domain T ∗ D n \ D n , which is actually the subcriticaldomain T ∗ ( S n − × D ) = B nstd ∪ H n − . Since D p is disjoint from ∂D n , we can consider( T ∗ D n , ∂D n ) \ D p as T ∗ ( S n − × D ) with some stop, namely the image of ∂D n .Since the subdomain T ∗ D n \ D p is obtained by carving out D p , there is a Legendrian Λ ⊂ ∂ ( T ∗ D n \ D p ) disjoint from ∂D n so that ( T ∗ D n , ∂D n ) \ D p ∪ H n Λ = ( T ∗ D n , ∂D n ) and the co-coreof H n Λ is D p . Because n ≥
5, there is a unique loose Legendrian Λ loose ⊂ ∂ ( T ∗ D n \ D p ) thatis formally isotopic to Λ and is loose in the complement of ∂D n ; see [28]. Next we form thestopped domain ( T ∗ D n , ∂D n ) \ D p ∪ H nflex by attaching the handle H nflex along Λ loose . We notethat the ambient Weinstein domain T ∗ D n \ D p ∪ H nflex is flexible since T ∗ D n \ D p is subcritical and H nflex is attached a loose Legendrian. Furthermore, it is formally symplectomorphic tothe standard Weinstein ball since Λ loose is formally isotopic to Λ (and attaching a handleto Λ reproduces B nstd ). Therefore by the h-principle for flexible Weinstein domains [10], T ∗ D n \ D p ∪ H nflex is Weinstein homotopic to B nstd . Under this identification with B nstd , thestop ∂D n ⊂ T ∗ D n \ D p ∪ H nflex becomes some Legendrian in ∂B nstd which we call Λ p . That is,we set ( B nstd , Λ p ) := ( T ∗ D n , ∂D n ) \ D p ∪ H nflex We will show that ( B nstd , Λ p ) satisfies the claimed properties.First we show that ( B nstd , Λ p ) is a Weinstein subdomain of ( T ∗ D n , ∂D n ). Note that( B nstd , Λ ∅ ) is precisely ( T ∗ D n , ∂D n ). This is because ( T ∗ D n , ∂D n ) \ D ∅ = ( T ∗ D n , ∂D n ) ∪ H n − and the Legendrian Λ from the previous paragraph intersects the belt sphere of H n − exactlyonce; so H n − ∪ H nflex are cancelling handles and hence( T ∗ D n , ∂D n ) \ D ∅ ∪ H nflex = ( T ∗ D n , ∂D n ) ∪ H n − ∪ H nflex = ( T ∗ D n , ∂D n )Now we consider the case when p is a (non-zero) prime. It is clear that ( T ∗ D n , ∂D n ) \ D p is a subdomain of ( T ∗ D n , ∂D n ) by construction; we claim that it is still a subdomain evenafter attaching the flexible handle H nflex to ( T ∗ D n , ∂D n ) \ D p . To see this, let C be theWeinstein cobordism between ( T ∗ D n , ∂D n ) \ D p and ( T ∗ D n , ∂D n ) given by the handle H n Λ (whose co-core is D p ). By [24], we can Weinstein homotope C , in the complement of ∂D n ,to a Weinstein cobordism H nflex ∪ H n − ∪ H n Λ (cid:48) , where H nflex is attached along Λ loose and H n − ∪ H n Λ (cid:48) is a smoothly trivial Weinstein cobordism whose attaching spheres are disjointfrom ∂D n . So we have the following equalities, up to Weinstein homotopy:( B nstd , Λ p ) ∪ H n − ∪ H n Λ (cid:48) = ( T ∗ D n , ∂D n ) \ D p ∪ H flex ∪ H n − ∪ H n Λ (cid:48) (2.1)= ( T ∗ D n , ∂D n ) \ D p ∪ C = ( T ∗ D n , ∂D n ) (2.2)which show that ( B nstd , Λ p ) is a subdomain of ( B nstd , Λ ∅ ) = ( T ∗ D n , ∂D n ). Furthermore, theconstruction in [24] shows that Λ (cid:48) is loose (but not in the complement of Λ loose or ∂D n ) sinceΛ is loose (but not in the complement of ∂D n ). So the Weinstein cobordism H n − ∪ H n Λ (cid:48) is flexible (but not in the complement of the stop ∂D n ) and therefore is homotopic to ∂B nstd × [0 , H n − , H n Λ (cid:48) are disjoint from ∂D n , we can view ∂D n × [0 , ∂D n in T ∗ D n \ D p ∪ H nflex and ∂D n in T ∗ D n .Under our identifications, this produces a smoothly trivial regular Lagrangian cobordism(regular in that the Liouville vector field can be made tangent to it) between Λ p and Λ ∅ in ∂B nstd × [0 ,
1] as desired. We also observe that Λ p is formally Legendrian isotopic to Λ ∅ in ∂B nstd because the attaching spheres Λ and Λ loose are formally Legendrian isotopic in thecomplement of ∂D n . More precisely, note that ∂D n (cid:113) Λ and ∂D n (cid:113) Λ loose are formally isotopicLegendrian links. Furthermore, there is a genuine Legendrian isotopy from Λ to Λ loose (butnot in the complement of ∂D n ) and so this extends to a Legendrian isotopy from ∂D n (cid:113) Λ loose to ∂D n (cid:113) Λ, where ∂D n is some other Legendrian that becomes Λ p after handle attachmentto Λ. Since a genuine Legendrian isotopy preserves formal Legendrian isotopies, ∂D n (cid:113) Λand ∂D n (cid:113) Λ are also formally Legendrian isotopic links. So when we attach a handle to Λ
RIME-LOCALIZED WEINSTEIN SUBDOMAINS 15 to get B nstd , ∂D n and ∂D n are still formally Legendrian isotopic in ∂B nstd , which is preciselythe statement that Λ ∅ , Λ p are formally Legendrian isotopic.Next we consider the case when p = 0. Recall that in this case, D is the cotangentfiber T ∗ D n ⊂ T ∗ D n . Then ( T ∗ D n , ∂D n ) \ D p is ( T ∗ ( S n − × D ) , S n − × { } ). We note that S n − × { } ⊂ ∂T ∗ ( S n − × D ) is loose since this Legendrian crosses the belt sphere of theindex n − n − S n − ) exactlyonce; see [10] for this looseness criterion. To construct ( B nstd , Λ ) from ( T ∗ D n , ∂D n ) \ D p =( T ∗ ( S n − × D ) , S n − × { } ), we attach an index n handle H nflex along the Legendrian Λ loose that is loose in the complement of ∂D n = S n − × { } (and is formally isotopic to theLegendrian Λ). Since ∂D n = S n − × { } is loose and Λ loose is loose in the complementof ∂D n , then ∂D n = S n − × { } is in fact also loose in the complement of Λ loose , i.e. ∂D n = S n − × { } and Λ loose form a loose link; see [10] for an argument explaining thisfact. In particular, the loose chart of ∂D n persists under attaching the handle H nflex alongΛ loose and so ∂D n ⊂ ( T ∗ D n , ∂D n ) \ D p ∪ H nflex is still loose. By definition, this means thatthe stop Λ in ( B nstd , Λ ) is loose as desired. As in the previous paragraph, ( B nstd , Λ ) is aWeinstein subdomain of ( B nstd , Λ ∅ ). Hence there is a regular Lagrangian cobordism from aloose Legendrian to the Legendrian unknot, as originally proven in [13, 23].This proves all of claims (2), (3), (4) when P consists of a single element.2.2.2. Λ P for a collection of primes P . Now we construct Λ P when P = { p , . . . , p k } is acollection of primes with multiple elements. We consider disjoint Weinstein balls B nstd,i sothat Λ p i ⊂ ∂B nstd,i and do a simultaneous boundary connected sum to the B nstd,i and Λ i , asin the construction of regular Lagrangians [12]:( B nstd , Λ P ) := ( B n , Λ p ) (cid:92) · · · (cid:92) ( B nk , Λ p k )Namely, we attach index 1 Weinstein handles to the disjoint union of Weinstein balls B nstd, (cid:113) · · · (cid:113) B nstd,k so that the attaching spheres of these index 1 handles, i.e. two points, are on different Λ i ;we simultaneously do Legendrian surgery on the Λ i via isotropic arcs in the 1-handles. Theresulting Legendrian Λ P is connected and in fact coincides with the usual isotropic connectedsum of Legendrians Λ p , . . . , Λ p k embedded in disjoint Darboux balls in a single ∂B nstd . Thisalso shows that up to Legendrian isotopy, Λ P does not depend on the order of the set P .Next we show that ( B nstd , Λ P ) is a Weinstein subdomain of ( B nstd , Λ Q ) if Q ⊂ P . Via ourprevious identification, ( B nstd , Λ P ) is the same as(( T ∗ D n , ∂D n ) \ D p ∪ H nflex ) (cid:92) · · · (cid:92) (( T ∗ D n , ∂D n ) \ D p k ∪ H nflex ) (2.3)where we choose points on each ∂D n to do the simultaneous boundary connected sum. Soif Q ⊂ P , ( B nstd , Λ P ) differs from ( B nstd , Λ Q ) by a boundary connected sum with( T ∗ D n , ∂D n ) \ D p ∪ H nflex for all p ∈ P \ Q . We saw previously that ( T ∗ D n , ∂D n ) \ D p ∪ H nflex is a subdomain of( T ∗ D n , ∂D n ) and hence ( B nstd , Λ P ) is a subdomain of ( B nstd , Λ Q ) boundary connected sumwith several copies of ( T ∗ D n , ∂D n ), one for each p ∈ P \ Q . Since doing boundary connected sum with ( T ∗ D n , ∂D n ) does not change the Weinstein homotopy type, the latter domain isstill ( B nstd , Λ Q ) and so ( B nstd , Λ P ) is a subdomain of ( B nstd , Λ Q ) as desired. This also showsthat if R ⊂ Q ⊂ P , the Weinstein cobordism ( B nstd , Λ R ) \ ( B nstd , Λ P ) is homotopic to the con-catenation of Weinstein cobordisms ( B nstd , Λ Q ) \ ( B nstd , Λ P ) and ( B nstd , Λ R ) \ ( B nstd , Λ Q ). Since( B nstd , Λ P ) is a subdomain of ( B nstd , Λ Q ), we have a Lagrangian cobordism in ∂B nstd × [0 , P and positive boundary Λ Q by definition.If 0 ∈ P , then Λ P is loose since it is the isotropic connected sum of Λ P \ and Λ , whichwe already saw to be loose. Let Q be another set of primes. Then Λ P and Λ P ∪ Q are bothloose unknots (since P ∪ Q contains 0) and so Λ P and Λ P ∪ Q are Legendrian isotopic bythe h-principle for loose Legendrians [28]. By the previous discussion, this implies that( B nstd , Λ P ) = ( B nstd , Λ P ∪ Q ) is a subdomain of ( B nstd , Λ Q ) since now Q ⊂ P ∪ Q .This proves all of claims (2), (3), (4), except the ‘only if’ part of claim (2), (3).2.2.3. Fukaya category of ( B nstd , Λ P ) . Finally, we compute the partially wrapped Fukaya cat-egory of ( B nstd , Λ P ). By the description in Equation 2.3, ( B nstd , Λ P ) is the result of carvingout the disks D p , . . . , D p k from ( B nstd , Λ ∅ ) = ( T ∗ D n , ∂D n ) and then attaching some flexiblehandles; here the disks are embedded disjointly by viewing ( T ∗ D n , ∂D n ) as the boundaryconnected sum of several disjoint copies of ( T ∗ D n , ∂D n ). By [16, 31], there is a geometricallydefined Viterbo transfer functorTw W ( T ∗ D n , ∂D ) → Tw W (( T ∗ D n , ∂D ) \ D p )which is localization by D p . That is, Tw W (( T ∗ D n , ∂D ) \ D p ) ∼ = Tw W ( T ∗ D n , ∂D ) /D p andthe Viterbo functor is the algebraic localization by the object D p . By construction, theLagrangian D p of Tw W ( T ∗ D n , ∂D n ; Z ) is isomorphic to the twisted complex T ∗ D n [1] p → T ∗ D n = cone( p · Id T ∗ D n ) . So Tw W (( T ∗ D n , ∂D ) \ D p ) ∼ = Tw W ( T ∗ D n , ∂D ) / cone( p · Id T ∗ D n ) Furthermore, the localiza-tion by a collection of objects depends only the split-closure of that collection of objects.Since T ∗ D generates Tw W ( T ∗ D n , ∂D n ), we have the equivalenceTw W ( T ∗ D n , ∂D ) / cone( p · Id T ∗ D n ) ∼ = Tw W ( T ∗ D n , ∂D ) / { cone( p · Id L ) | L ∈ Tw W ( T ∗ D n , ∂D n ) } =: Tw W ( T ∗ D n , ∂D ; Z ) (cid:20) p (cid:21) Combining with the previous equivalence, we haveTw W (( T ∗ D n , ∂D ) \ D p ; Z ) ∼ = Tw W ( T ∗ D n , ∂D ; Z ) (cid:20) p (cid:21) (2.4)Similarly, when we carve out multiple disks D p , . . . , D p k , we invert p , . . . , p k in the Fukayacategory. Attaching flexible handles does not affect the Fukaya category and soTw W ( B nstd , Λ P ; Z ) ∼ = Tw W ( T ∗ D n , ∂D n ; Z ) (cid:20) P (cid:21) (2.5)as desired. If p is zero, then D p = T ∗ D n and Tw W ( T ∗ D n , ∂D ) /T ∗ D n ∼ = 0, which is indeedthe case for ( B nstd , Λ ) since Λ is loose. RIME-LOCALIZED WEINSTEIN SUBDOMAINS 17
Remark . We note that the above discussion does not automatically show that thatthe equivalence in Equation 2.5 is given by the Viterbo functor induced by the Weinsteinembedding of ( B nstd , Λ P ) into ( B nstd , Λ ∅ ) = ( T ∗ D n , ∂D n ) due to the presence of the extraflexible handles. However this is indeed the case. Recall that the Weinstein cobordismbetween these two domains is H n − ∪ H n Λ (cid:48) , which comes from a construction in [24, 22].The proof there shows that the co-core of H n Λ (cid:48) is D p (cid:92)D p ⊂ ( T ∗ D n , ∂D n ) and so the Viterbofunctor between these two domains is localization by D p (cid:92)D p . Now D p (cid:92)D p ∼ = D p ⊕ D p [1] and D p have the same split-closure, and so localization by D p (cid:92)D p is the same as localization by D p , as in Equation 2.5.Finally, we prove the ‘only if’ part of claims (2), (3). Suppose that ( B nstd , Λ P ) is Weinsteinsubdomain of ( B nstd , Λ Q ) but Q (cid:54)⊂ P and 0 (cid:54)∈ P . There would be a localization functor fromthe Fukaya category of ( B nstd , Λ Q ) to that of ( B nstd , Λ P ) over any coefficient ring R . However,if we take R = F q for any q ∈ Q \ P , we have D q ∼ = cone(0 T ∗ D n ) ∼ = T ∗ D n [1] ⊕ T ∗ D n inTw W ( B nstd , Λ ∅ ; F q ) since q ≡ F q . This object split-generates Tw W ( B nstd , Λ ∅ ; F q ) and soTw W ( B nstd , Λ Q ; F q ) ∼ = Tw W ( B nstd , Λ ∅ ; F q ) /D q ∼ = 0On the other hand, all p ∈ P are invertible in F q because q ∈ Q \ P by assumption and p (cid:54) = 0.Therefore D p ∼ = cone( p · Id T ∗ D n ) ∼ = 0 in Tw W ( B nstd , Λ ∅ ; F p ) for all p ∈ P and soTw W ( B nstd , Λ P ; F q ) ∼ = Tw W ( B nstd , Λ ∅ ; F q ) / ∼ = Tw W ( B nstd , Λ ∅ ; F q ) ∼ = Tw F q which is non-trivial. Since there cannot be a localization functor from the trivial categoryto Tw F q , ( B nstd , Λ P ) cannot be a Weinstein subdomain of ( B nstd , Λ Q ). This proves the ‘onlyif’ part of claim (2). If there is a smoothly trivial regular Lagrangian cobordism from Λ P toΛ Q in ∂B nstd × [0 , B nstd , Λ P ) is a Weinstein subdomain of ( B nstd , Λ Q ) and so the ‘onlyif’ part of claim (3) follows from that for claim (2). (cid:3) Now we show that Theorem 2.3 implies Theorem 1.1 concerning Weinstein subdomainsof an arbitrary
Weinstein domain. Recall that an index n Weinstein handle can be viewedas the stopped domain ( T ∗ D n , ∂D n ) = ( B nstd , Λ ∅ ). We will consider the stopped domains( B nstd , Λ P ) in Theorem 2.3 as generalized Weinstein handles. Definition 2.6. A P -Weinstein handle of index n is the stopped domain ( B nstd , Λ P ).Here our model for the P -Weinstein handle uses explicit embeddings of Moore spaces into S n − and hence is well-defined. When attaching Weinstein handles, one implicitly uses thecanonical parametrization of ∂D n ⊂ T ∗ D n . Via the construction in the proof of Theorem 2.3,this parametrization gives the Legendrians Λ P ⊂ ∂B n a parametrization as well. Therefore,given a parametrized Legendrian sphere Λ in a contact manifold ( Y, ξ ), we can attach a P -Weinstein handle ( B nstd , Λ P ) to it and produce a Weinstein cobordism, just like we do forusual Weinstein handles. To prove Theorem 1.1, we replace all standard Weinstein n -handles( B nstd , Λ ∅ ) with Weinstein P -handles ( B nstd , Λ P ). Proof of Theorem 1.1.
Let X n be a Weinstein domain with n ≥ C n , . . . , C nk ⊂ X n the Lagrangian co-core disks of its index n handles H n , . . . , H nk . Hence there is a subcriticalWeinstein domain X ⊂ X and Legendrian spheres Λ , . . . , Λ k ⊂ ∂X so that X = X ∪ H n Λ ∪ · · · ∪ H n Λ and the co-core of H n Λ i is C i ⊂ X . That is, X is obtained from X by carvingout the Lagrangian disks C , . . . , C k . This gives the following decomposition of X : X = ( X , Λ , . . . , Λ k ) ∪ Λ =Λ ∅ ( B nstd , Λ ∅ ) ∪ · · · ∪ Λ k =Λ ∅ ( B nstd , Λ ∅ ) (2.6)where the i th copy of ( B nstd , Λ ∅ ) is glued to X by identifying Λ ∅ with Λ i . Now we define X P to be the following Weinstein domain: X P := ( X , Λ , . . . , Λ k ) ∪ Λ =Λ P ( B nstd , Λ P ) ∪ · · · ∪ Λ k =Λ P ( B nstd , Λ P ) (2.7)Namely, we replace each standard Weinstein n -handle ( B nstd , Λ ∅ ) by a P -Weinstein handle( B nstd , Λ P ). Remark . We note that attaching P -Weinstein handles ( B nstd , Λ P ) to ( X , Λ , . . . , Λ k ) is thesame as attaching standard Weinstein handles ( B nstd , Λ ∅ ) to X with some modified attachingLegendrian Λ Pi ⊂ ∂X . In fact, Λ Pi is the isotropic connected sum Λ i (cid:92) Λ P of Λ i ⊂ ∂X andΛ P ⊂ ∂B nstd , which we place into a Darboux chart in ∂X disjoint from Λ i . To see this, notethat gluing ( B nstd , Λ P ) to ( X , Λ i ) by identifying Λ P with Λ i ⊂ ∂X is the same as gluing acylinder T ∗ ( S n − × D ) to ( X , Λ i ) (cid:113) ( B nstd , Λ P ) by identifying S n − × i and S n − × P . The cylinder can be decomposed into a standard Weinstein index 1 handle and astandard Weinstein index n handle. So we first do simultaneous index 1 handle attachmentto ( X , Λ i ) and ( B nstd , Λ P ), with attaching sphere a point in Λ i and a point in Λ P , to produce( X (cid:92)B nstd , Λ i (cid:92) Λ P ). If we identify X (cid:92)B n with X , then Λ P becomes a Legendrian in ∂X (ina Darboux chart disjoint from Λ i ) and Λ i (cid:92) Λ P is precisely the isotropic connected sum of Λ i and Λ P in ∂X . Then we attach the (standard) index n Weinstein handle of the cylinder T ∗ ( S n − × D ) along Λ i (cid:92) Λ P . Thus, the decomposition of X P in Equation 2.7 can alternativelybe described as( X , Λ (cid:92) Λ P , . . . , Λ k (cid:92) Λ P ) ∪ Λ (cid:92) Λ P =Λ ∅ ( B nstd , Λ ∅ ) ∪ · · · ∪ Λ k (cid:92) Λ P =Λ ∅ ( B nstd , Λ ∅ ) (2.8)In particular, the attaching spheres for the (standard) index n handles for X and X P differby a purely local modification, namely an isotropic connected sum with Λ P .Now Claims 1), 2), 3) in Theorem 1.1 follow from the analogous claims in Theorem 2.3.For example, X ∅ = X since ( B nstd , Λ ∅ ) is the standard Weinstein handle ( T ∗ D n , ∂D n ). Also,since ( B nstd , Λ P ) is a Weinstein subdomain of ( B nstd , Λ Q ) for Q ⊂ P , X P is a Weinsteinsubdomain of X Q and this Weinstein embedding is also functorial with respect to inclusionsof various subsets of primes. If 0 ∈ P , then X P is flexible. To see this, recall that Λ P ⊂ ∂B nstd is loose by Theorem 2.3; this implies that the attaching spheres Λ Pi ⊂ ∂X for X P are alsoloose since by Remark 2.7, Λ Pi is the isotropic connected sum of Λ i with Λ P , which is aloose Legendrian loosely embedded in a Darboux chart disjoint from Λ i . If 0 ∈ Q ⊂ P , thenthe cobordism between X P and X Q is flexible since the cobordism between ( B nstd , Λ P ) and( B nstd , Λ Q ) is also flexible (in the complement of Λ P ).Finally, we compute Tw W ( X P ; Z ). Since X P is a Weinstein subdomain of X , there is aViterbo transfer functor: Tw W ( X ; Z ) → Tw W ( X P ; Z )As in the proof of Theorem 2.3, this functor is localization by D p ⊂ ( T ∗ D n , ∂D n ) (or equiv-alently by D p (cid:92)D p ) and D p ∼ = cone( p · Id T ∗ D n ). On the other hand, T ∗ D n ⊂ ( T ∗ D n , ∂D n ) = RIME-LOCALIZED WEINSTEIN SUBDOMAINS 19 ( B nstd , Λ ∅ ) is precisely the co-core C ni of H n Λ i under the decomposition of X in Equation 2.6and so D p is isomorphic to cone( p · Id C ni ). By [16, 9], the co-cores C ni of all the H n Λ i generateTw W ( X ). So localizing by cone( p · Id C ni ) for all i is the same as localizing by cone( p · Id L )for all L ∈ Tw W ( X ; Z ). That is, Tw W ( X P ; Z ) ∼ = Tw W ( X ; Z )[ P ] as desired. (cid:3) We observe that our construction of X P depends on many choices. For example, it dependson the choice of initial Weinstein presentation for X . There are Weinstein homotopic presen-tations for X with different numbers of index n handles; hence in this case, our constructionwould involve carving out different numbers of Lagrangian disks (and then attaching theappropriate flexible cobordism). There are also choices to be made in constructing the P -handles ( B nstd , Λ P ). We fixed a p -Moore space U ⊂ S n − so that ˜ C ∗ ( U ) = Z [ − p → Z [ −
3] andused this to construct D p := D U and then form ( B nstd , Λ P ). In fact, we could have taken any U ⊂ S n − so that ˜ C ∗ ( U ) is quasi-isomorphic to (cid:76) i ( Z [ k i + 1] p → Z [ k i ]) for any k i . Repeatingthe construction for such U , we would also have Tw W ( B nstd , Λ P ; Z ) ∼ = Tw W ( B nstd , Λ ∅ ; Z )[ P ]as well.Now that we have described the subdomains X P of X , we can explain the differencebetween our construction and that of Abouzaid and Seidel [6] more precisely. Abouzaid andSeidel [6] starts with a Lefschetz fibration for X n whose fiber is a Weinstein domain F n − .They then embed the Lagrangian disks D n − p into F n − so that they are in a neighborhoodof the co-cores C n − i of the critical index n − H n − i of F n − ; using these disks,they build a larger fiber F (cid:48) (which has F as a Weinstein subdomain) and add new vanishingcycles to create a new Lefschetz fibration, which is their space X (cid:48) P . On the other hand, theconstruction in Theorem 1.1 embeds the disks D np into the total space X n so that they arein a neighborhood of the co-cores C ni of the critical index n handles H ni of X n ; we thencarve out these disks. The construction of Abouzaid-Seidel holds only for n ≥
6. Because wework near the index n handles instead of the index n − n ≥ T ∗ M std . Proof of Corollary 1.5.
The only extra feature of this result over Theorem 1.1 is the ‘onlyif’ part of the statement: T ∗ M P ⊂ T ∗ M Q if and only if Q ⊂ P or 0 ∈ P . To prove this, werepeat the proof in Theorem 2.3 that ( B nstd , Λ P ) is a subdomain of ( B nstd , Λ Q ) if and only if Q ⊂ P . Namely, suppose that T ∗ M P ⊂ T ∗ M Q is a Weinstein subdomain but Q (cid:54)⊂ P and0 (cid:54)∈ P . Then there is a Viterbo localization functor on Fukaya categories over F q for q ∈ Q \ P .However, Tw W ( T ∗ S nQ ; F q ) ∼ = 0 but Tw W ( T ∗ S nP ; F q ) ∼ = Tw W ( T ∗ S n ; F q ) ∼ = Tw C ∗ (Ω S n ; F q )is non-trivial and so there cannot be such a localization functor. (cid:3) Remark . A similar argument using the fact that the Viterbo map on symplectic coho-mology is a unital ring map shows that that T ∗ S nP cannot be a Liouville subdomain of T ∗ S nQ if Q (cid:54)⊂ P and 0 (cid:54)∈ P .2.2.4. Exotic presentations.
We now briefly explain the connection between the subdomainsof T ∗ S nstd constructed in Corollary 1.5 and certain ‘exotic’ Weinstein presentations of T ∗ S nstd studied by the first author in [22]; the reader can safely skip this section without interruptingthe flow of this paper. There are many different Legendrian spheres Λ k ⊂ ∂B nstd so that B nstd ∪ H n Λ k is Weinsteinhomotopic to the standard presentation B nstd ∪ H n Λ ∅ ; we call this an exotic presentationsince Λ k is different from Λ ∅ . Under the resulting identification, the co-core of H n Λ k is (cid:92) ki =1 T ∗ x i S n (cid:92) k − j =1 T ∗ y j S n , the boundary connected sum of several copies of the cotangent fiber T ∗ q S n , possibly with the opposite orientation.Recall that for any U ⊂ S n − , we consider a Lagrangian disk D U ⊂ ( T ∗ D n , ∂D n ) with D U ∼ = ˜ C ∗− ( U ) ⊗ T ∗ D n and in the proof of Theorem 2.3, we observed that T ∗ D n \ D U is thesubcritical domain T ∗ ( S n − × D ). There are two disjoint Legendrian spheres Λ ,U , Λ ,U ⊂ ∂ ( T ∗ ( S n − × D )); here Λ ,U is the image of the original Legendrian stop ∂D n ⊂ ∂T ∗ D n andΛ ,U is the Legendrian obtained by carving out D U , i.e. the co-core of a handle attachedalong Λ ,U is D U (this Legendrian is called Λ in the proof of Theorem 2.3). Starting from thestandard presentation ( B nstd , Λ ∅ ) ∪ Λ ∅ =Λ ∅ ( B nstd , Λ ∅ ) of T ∗ S nstd and taking U to be a p -Moorespace, the construction in Theorem 1.1 produces the Weinstein subdomain T ∗ S np ⊂ T ∗ S nstd as T ∗ S np := ( B nstd , Λ ∅ ) ∪ Λ ∅ =Λ ,U ( T ∗ ( S n − × D )) , Λ ,U , Λ ,U,loose ) ∪ Λ ,U,loose =Λ ∅ ( B nstd , Λ ∅ )where Λ ,U,loose is a loose version of Λ ,U . That is, T ∗ S np is obtained by attaching a flexiblehandle to the (unstopped) Weinstein domain( B nstd , Λ ∅ ) ∪ Λ ∅ =Λ ,U ( T ∗ ( S n − × D )) , Λ ,U )where U is a p -Moore space.Now, if U is a neighborhood of (cid:96) ki =1 B n − (cid:113) ∨ k − j =1 S ⊂ S n − , the disjoint union of k balls B n − and the wedge sum of k − S , then D U is Lagrangian isotopic in ( T ∗ D n , ∂D n )to (cid:92) ki =1 T ∗ x i D n (cid:92) k − j =1 T ∗ y j D n . Furthermore, Λ ,U ⊂ ∂ ( T ∗ S n − × D ) is loose. This is becausethe subdomain obtained by carving out the disjoint union (cid:96) ki =1 T ∗ x i D n (cid:96) k − j =1 T ∗ y j D n and thesubdomain obtained by carving out the boundary connected sum (cid:92) ki =1 T ∗ x i D n (cid:92) k − j =1 T ∗ y j D n arerelated by flexible cobordism (see [22] for the proof). In particular, the attaching spheresfor the cobordism are loose in the complement of the stop. Since the former domain has aloose stop, so does the latter by [10]. Hence( B nstd , Λ ∅ ) ∪ Λ ∅ =Λ ,U ( T ∗ ( S n − × D )) , Λ ,U )is a flexible Weinstein domain X , with no stop. In fact, X is the standard Weinstein ballbecause [ (cid:92) ki =1 T ∗ x i D n (cid:92) k − j =1 T ∗ y j D n ] = [ T ∗ x D n ] ∈ H n ( T ∗ D n ; Z ), which implies that it has trivialhomology and hence is a smooth ball by the h-cobordism theorem. In conclusion, the stoppeddomain ( B nstd , Λ ∅ ) ∪ Λ ∅ =Λ ,U ( T ∗ ( S n − × D )) , Λ ,U , Λ ,U )is precisely ( B nstd , Λ k ), where Λ k is the Legendrian from [22], since by construction the co-coreof a handle attached along Λ ,U is D U = (cid:92) ki =1 T ∗ x i D n (cid:92) k − j =1 T ∗ y j D n .We end with a discussion of which ingredients were necessessary in the construction ofthese exotic presentations. First and foremost, we need to realize (cid:92) ki =1 T ∗ x i D n (cid:92) k − j =1 T ∗ y j D n as D U and hence embed S into S n − as a proper subset. This requires n ≥ n = 2. Interestingly, these exotic presentations failto exist for the same reason that the existence h-principle for Legendrians fails when n = 2. RIME-LOCALIZED WEINSTEIN SUBDOMAINS 21
Indeed, consider a proper subdomain U ⊂ B n − ⊂ S n − and a Legendrian Λ n − ⊂ ∂B nstd .Then one can form the Legendrian U -stabilization St U (Λ) ⊂ ∂B nstd of Λ (see [10, 28]) whoseThurston-Bennequin invariant differs from that of Λ by the Euler characteristic χ ( U ). If n ≥
3, any integer can be realized as the Euler characteristic of U by taking U ⊂ B n − to bea neighborhood of (cid:96) j B n − (cid:113) ∨ k S ⊂ S n − , the space we previously considered. This allowsone to realize all formal Legendrian embeddings by actual Legendrians for n ≥
3. However,if n = 2, all proper subdomains U ⊂ S have χ ( U ) >
0. Indeed the Bennequin inequalityproves that the Thurston-Bennequin invariant of any smoothly trivial Legendrian in ∂B std is at most −
1; so the existence h-principle fails for n = 2. We expect that there is preciseconnection between the U -stabilized Legendrians St U (Λ) from [10, 28] and the Lagrangiandisks D U here, since both construction involving pushing a smooth subdomain U in oneLegendrian through another.3. Classifying Lagrangian disks
In this section, we prove Theorem 1.8: if M is simply connected and spin, and i : L (cid:44) → T ∗ M is null-homotopic, then L ∼ = CW ∗ ( M, L ) ⊗ T ∗ q M n in Tw W ( T ∗ M ; Z ). To accomplish this, wewill apply Koszul duality to characterize objects of Tw W ( T ∗ M ; Z ) as modules over the A ∞ -algebra CW ∗ ( M, M ) ∼ = C ∗ ( M ). Here it is crucial that we work with the Z -graded wrappedFukaya category, where the Z -grading comes from the Lagrangian fibration by cotangentfibers. Any Lagrangian disk, since it is contractible, can be Z -graded; the zero-section M ⊂ T ∗ M std can also be Z -graded for this grading. Hence these Lagrangians define objectsof the Z -graded Fukaya category.3.1. C ∗ ( X ) -modules. We begin with a general discussion of how to view Floer complexesas modules over Morse cochain algebras. The outcome is Proposition 3.2, which says thatthe module structures are unexpectedly topological. This is what will allow us to draw Floertheoretic conclusions from the topological assumption of null-homotopy.For now, we will work in a general Liouville domain X . Given two Lagrangian branes K, L ⊂ X , possibly equipped with rank 1 local systems, we can endow CW ∗ ( K, L ) withthe structure of a right C ∗ ( X )-module in a number of ways. In each case, we model the A ∞ structure on our cochain algebras C ∗ ( X ), C ∗ ( K ), and C ∗ ( L ) with Morse complexes andperturbed gradient flow trees [2] associated to exhausting Morse functions f X , f K , and f L .Let us fix some notation. The moduli space of domains controlling the A ∞ operations isthe space T d +1 R d +1 of metric ribbon trees with d + 1 infinite leaves and no finite leaves, labeled x , . . . , x d incounterclockwise order. More explicitly, a point p ∈ R d +1 is an isomorphism class [ T p ], where T p is a noncompact tree with • d + 1 ends and no mono- or bivalent vertices, • a ribbon structure, which for a tree is the same as a homotopy class of planar em-beddings, • an edge metric, meaning that we can measure the distance between any two pointsof T p (not necessarily vertices), and • a labeling of the ends by x , . . . , x d in counterclockwise order with respect to theribbon structure.The fibration T d +1 → R d +1 is the tautological one, which over each p is a representative T p .In what follows, we will imagine x as the bottom of T p and the other x i as the top, whichwill allow us to use the prepositions “below” or “above” to mean “closer to x ” or “closer tosome other x i ”, respectively.The restriction homomorphisms i ∗ K : C ∗ ( X ) → C ∗ ( K ) and i ∗ L : C ∗ ( X ) → C ∗ ( L ) are con-trolled by the space G d +1 S d +1 of grafted trees , which are metric ribbon trees T as above with the additional data of a(necessarily finite) subset D ⊂ T which separates x from the other leaves and whose el-ements are equidistant from x . For d ≥ S d +1 has a natural R -action which translates D , and the quotient is canonically identified with R d +1 (for d = 1, S is a single point).However, the natural compactification R d +1 models the associahedron, while S d +1 modelsthe multiplihedron. The restriction homomorphism { F d | d = 1 , . . . , ∞} : C ∗ ( X ) → [ C ∗ ( K ) or C ∗ ( L )]is then given by counting isolated perturbed gradient flow trees of shape T q for some q ∈ S d +1 ,where the portion of T q above (resp. below) D maps into X (resp. K or L ). Note that,because we work with a perturbed gradient flow, we do not need to require f X to restrict to f K or f L . Of course, if we wanted to we could arrange that f X restricts to one of these Morsefunctions, but generally it would impossible to achieve both. Fortunately, all the resultinghomomorphisms are homotopic.To make Floer complexes into C ∗ ( X )-modules, we need chain-level PSS-type structures,which are built from short trees or short grafted trees . A short tree with d inputs is a rootedmetric ribbon tree with d infinite leaves and no finite leaves (except possibly the root). Theroot is labeled y , while the leaves are labeled x , . . . , x d in counterclockwise order. A shortgrafted tree is a short tree equipped with the additional data of a dividing set D as aboveeither separating y from the x i or equal to { y } . We will denote the spaces of short trees andshort grafted trees by R d +1 s and S d +1 s , respectively. There are canonical piecewise smoothhomeomorphisms R d +1 s ∼ = R d +1 × R ≥ (3.1)for d ≥ S d +1 s ∼ = R d +1 s × R ≥ (3.2) RIME-LOCALIZED WEINSTEIN SUBDOMAINS 23 for d ≥
1, i.e. all d . In (3.1), the R ≥ factor measures the distance between the root y andthe first vertex, while in (3.2) it measures the distance between y and the dividing set.The PSS-type structures in question all come from moduli spaces of strips with somenumber of short Morse trees attached at marked points. Definition 3.1. A hedge comprises(1) a smooth function f : R → [0 , k points z , . . . , z k on the graph Γ( f ) ⊂ R × [0 ,
1] with strictly increasing R components, and(3) for each z i , a short tree T i .Identifying z i with the root y i of the tree T i induces a total lexicographic order of the leaves x ij of the trees T i , namely x i,j < x i (cid:48) j (cid:48) if either i < i (cid:48) or both i = i (cid:48) and j < j (cid:48) .Fix a number c ∈ (0 , H dc of hedges with d leaves x ij and f ( s ) = c comesa priori as a disjoint union of components indexed by partitions of the leaves into trees T i .However, there is a natural way to glue the various components to build a connected modulispace. To see this, note that the boundary strata (before compactification) come from oneor more roots y i becoming multivalent, or in horticultural terms from some tree T i becomingmaximally short. Such configurations can also be achieved by having multiple smaller shorttrees attached to distinct marked points collide. The result is that we can make H dc into a connected, smoothly stratified, topological manifold without boundary , see Figure 1. This isgood enough to construct operations in Floer theory. H dc has a natural compactification H d , where the codimension 1 boundary strata come intwo types. The first is associated with Morse breaking, where a single short tree will breakinto a short tree and a (long) tree. The second is a type of Floer breaking associated withthe marked points z i moving apart, so that the limiting configuration is made up of twohedges.An X -valued perturbation datum for a hedge H amounts to a perturbation datum foreach short tree T i , which is just an ε -parametrized family of vector fields on X for eachedge ε of T i which vanishes outside a compact subset of ε . Given a Morse-Smale pair on X and a Floer datum for the pair ( K, L ), we can define a hedge map out of H to be a tuple( u, τ , . . . , τ k ), where • u is a Floer trajectory with boundary on ( K, L ), • τ i is a perturbed gradient flow tree in X parametrized by T i , and • τ i ( y i ) = u ( z i ).If H ∈ H d , we can analogously define a K -valued perturbation datum for H to be a familyof vector fields on K , and a hedge map to involve gradient flow trees in K ; if H ∈ H d , wecan do the same with L .For generic Morse-Smale pairs, smooth translation-invariant families of perturbation dataon H dc , and Floer data on X , the spaces of d -leaved hedge maps are smoothly stratified topo-logical manifolds of the expected dimension. Counting such maps which are isolated up totranslation makes CW ∗ ( K, L ) into a right C ∗ ( X )-module, which we’ll denote CW ∗ ( K, L ) X,c . Figure 1.
The space H c ; the boundary consists of Floer breaking (lowerboundary in the diagram) and Morse breaking (upper boundary).Similarly, when c = 0 or 1, we can make CW ∗ ( K, L ) into a right C ∗ ( K )- or C ∗ ( L )-module CW ∗ ( K, L ) K, or CW ∗ ( K, L ) L, , respectively.The key holomorphic curve ingredient of our story is that these modules are all homotopic(and therefore quasi-isomorphic) when pulled back to C ∗ ( X ): Proposition 3.2.
For c , c ∈ [0 , , there is a homotopy CW ∗ ( K, L ) X,c (cid:39) CW ∗ ( K, L ) X,c (3.3) of right C ∗ ( X ; Z ) -modules.Similarly, for any c , there are homotopies CW ∗ ( K, L ) X,c (cid:39) i ∨ K CW ∗ ( K, L ) K, (3.4) CW ∗ ( K, L ) X,c (cid:39) i ∨ L CW ∗ ( K, L ) L, , (3.5) where i ∨ K : Mod C ∗ ( K ; Z ) → Mod C ∗ ( X ; Z ) is the pullback functor under the restriction homomorphism of cochains i ∗ K : C ∗ ( X ) → C ∗ ( K ) , and similarly for i ∨ L .Remark . The key takeaway of Proposition 3.2 is not just that CW ∗ ( K, L ) has a canoni-cally defined C ∗ ( X )-module structure, but that this module structure is determined by eitherthe C ∗ ( K )- or the C ∗ ( L )-module structure. RIME-LOCALIZED WEINSTEIN SUBDOMAINS 25
Proof.
Pick a smooth function f : R → [0 ,
1] interpolating between f ( s ) = c for s near + ∞ and f ( s ) = c for s near −∞ . Write H df for the space of hedges with marked points z i onthe graph of f . Counting isolated (no longer up to translation) hedge maps parametrized by H df defines the homotopy (3.3).For the second part, we prove (3.4), since the proof of (3.5) is identical. For this, wemay apply the first part to assume c = 0, so we need only produce a homotopy between CW ∗ ( K, L ) X, and CW ∗ ( K, L ) K, . We do this by generalizing the notion of a hedge to thatof a grafted hedge . This is the same as Definition 3.1, except f ≡ T i are replaced by short grafted trees. A (ordinary) hedge can thus be viewed as a special caseof a grafted hedge, where all the dividing points are at the root y i . Using this identification,we can extend the definition of the spaces H dc to negative values of c . Concretely, we declare H dc to be the space of d -leaved grafted hedges, where each tree is attached to the strip at t = 0 and has dividing set at distance | c | from the root. For negative c , H dc continues tohave a natural compactification H dc , and there is a canonical diffeomorphism H dc ∼ = H dc (cid:48) forany c, c (cid:48) ∈ ( −∞ , H a grafted hedge, a hedge map out of H is a tuple ( u, τ , . . . , τ k ), where • u is a Floer trajectory with boundary on ( K, L ). • τ i is a perturbed grafted gradient flow tree with leaves in X and root in K parametrizedby T i , and • τ i ( y i ) = u ( z i ).Now the diffeomorphism H dc ∼ = H dc (cid:48) is compatible with both the internal stratification andthe boundary decompositions, so follows that hedge maps parametrized by H dc continue todefine C ∗ ( X )-module structures CW ∗ ( K, L ) X,c for c <
0. Moreover, the same argument asfor nonnegative c shows that these module structures are homotopic – just interpolate thedividing sets rather than the attaching points.To conclude, observe that the pullback module i ∗ K CW ∗ ( K, L ) K, is what we get by sendingthe dividing set to infinity. While it is delicate to do that directly, it is enough to movethe dividing set close to infinity: below any given action bound, gluing theory establishes abijection of spaces of hedge maps. This ensures first that the module structure maps stabilizeto the pulled-back ones, and second that the homotopies eventually become trivial. (cid:3) Remark . A version of proposition 3.2 remains true with C ∗ ( X ) replaced by symplecticcochains SC ∗ ( X ), C ∗ ( K or L ) replaced by CW ∗ ( K or L ), and the restriction maps replacedby closed-open maps. In that case, one is forced to use left CW ∗ ( L )-modules. While weexpect all of the resulting homotopies to be intertwined by the relevant A ∞ algebra ho-momorphisms, sticking to Morse cochains allows us to avoid a good deal of combinatorialmessiness.Recall that for a Weinstein domain X , SC ∗ ( X ) is quasi-isomorphic to the Hochschildcochains CC ∗ ( W ( X )) of W ( X ) [15]. Using this quasi-isomorphism, we note that Proposition3.2 has a purely categorical analog. For any A ∞ category A , there is an A ∞ -homomorphism CC ∗ ( A ) → hom ∗ ( X, X ) and hence a pullback map on modules, i.e. π X : Mod end ∗ ( X ) → Mod CC ∗ ( A ) . Since CC ∗ ( A ) is an E -algebra, there is also an A ∞ -homomorphism CC ∗ ( A ) → hom ∗ ( X, X ) op and hence a similar pullback functor π X : Mod end ∗ ( X ) op → Mod CC ∗ ( A ) . For any two objects
X, Y ∈ A , composition of morphisms in A makes hom( X, Y ) an objectof Mod end( X ) and also of Mod end( Y ) op . Then the categorical analog of Proposition 3.2 is thatthe objects π X hom( X, Y ) and π Y hom( X, Y ) are quasi-isomorphic in Mod CC ∗ ( A ) .For the actual statement in Proposition 3.2, we work with C ∗ ( X ), the low-action partof CC ∗ ( W ( X )), and need to identify CC ∗ ( W ( X )) → hom ∗ ( L, L ) with the restriction map C ∗ ( X ) → C ∗ ( L ) on Morse cochains. Here it is essential that our Lagrangian L is not equipped with a bounding cochain, which destroys the action filtration on Floer cochainsand hence our access to the low-energy, topological subcomplex.While so far we have considered general A ∞ presentations of our Morse cochain complexes,the above constructions work just as well for their strict unitalizations C ∗ s ( − ). Indeed,suppose X is connected, and pick a positive exhausting Morse function f on X with aunique degree 0 critical point. Define C ∗ s ( X ) := CM ≥ ( f ) ⊕ Z · with the restricted A ∞ structure on CM ≥ ( f ) (which is well-defined because µ k increasesreduced degree, which is non-negative by assumption) and for which is a strict unit. Any A ∞ homomorphism C ∗ ( X ) → A for A a strictly unital A ∞ algebra induces a strictly unital homomorphism C ∗ s ( X ) → A . Because modules are just functors to the strictly unital dg-category Ch , we conclude Corollary 3.5. If X , K , and L are connected, then Proposition 3.2 continues to hold in therealm of strictly unital modules with C ∗ ( X ) replaced with C ∗ s ( X ) , and similarly with K and L . (cid:3) Corollary 3.6.
Let M be a closed connected manifold. If the restriction A ∞ -homomorphism i ∗ : C ∗ ( T ∗ M ; Z ) → C ∗ ( L ) factors up to homotopy through the canonical augmentation, C ∗ ( T ∗ M ; Z ) C ∗ ( L ; Z ) Z i ∗ ε can η then CW ∗ ( M, L ) M, is isomorphic to a module in the image of Tw Z ⊂ Mod Z ε can −−→ Mod C ∗ ( M ; Z ) . Proof.
Replacing C ∗ ( − ) by C ∗ s ( − ), we may assume all algebras and maps are strictly unital.In particular, the pullback functor η ∗ : Mod C ∗ ( L ; Z ) → Mod Z RIME-LOCALIZED WEINSTEIN SUBDOMAINS 27 preserves strict unitality of modules. Since a strictly unital Z -module is just a chain complex,the Z -module η ∗ ( CW ( M, L ) L, ) coincides with its underlying chain complex, which lies inTw Z because M is compact.The result now follows from Corollary 3.5 (on each connected component of L ), togetherwith the observation that the restriction C ∗ ( T ∗ M ) → C ∗ ( M ) is an isomorphism. (cid:3) Disks in cotangent bundles.
In the previous section, we studied properties of Floermodules CW ∗ ( K, L ) over various Morse cochain algebras. In this section, we restrict to thecase of T ∗ M , where M is a simply connected, spin manifold. We use Koszul duality to showthat the module structure over C ∗ ( M ) knows everything about the Fukaya category andprove Theorem 1.8.We first construct a presentation of the wrapped Fukaya category which is well-adapted totalking about modules over C ∗ ( M ). First, write C for the semiorthogonally glued category (cid:104) M Morse , W ( T ∗ M ) (cid:105) , where end ∗ ( M Morse ) = C ∗ s ( M ), and hom ∗C ( M Morse , L ) = CW ∗ ( M, L ). The mixed A ∞ opera-tions count generalized hedges, i.e. usual perturbed holomorphic disks whose first boundarylies geometrically on M , together with short perturbed gradient flow trees in M attached atboundary marked points. We will obtain our desired presentation by localizing C : Lemma 3.7.
Let e ∈ hom ( M Morse , M ) be a cocycle representing the unit in CW ∗ ( M, M ) .Define W Morse ( T ∗ M ) := C / cone( e ) , (3.6) so that we have tautological functors W ( T ∗ M ) W Morse ( T ∗ M ) end ∗C ( M Morse ) = C ∗ s ( M ) . i W i M Then i W is a quasi-equivalence and i M is fully faithful.Proof. For any object X ∈ W ( T ∗ M ), precomposition with e induces a quasi-isomorphismhom ∗C ( M, X ) ∼ = hom ∗C ( M Morse , X ) . This means that cone( e ) is left-orthogonal to every X ∈ W ( T ∗ M ), which implies that i W isfully faithful. Because i W ( M ) is isomorphic to M Morse in W Morse ( T ∗ M ), i W is also essentiallysurjective, which means it’s an equivalence.The proof for i M is identical, except cone( e ) is right-orthogonal to M Morse by the classicalLagrangian PSS isomorphism. (cid:3)
The benefit of W Morse ( T ∗ M ) is that it allows for direct Koszul duality between the Morsecochain algebra on the zero section and the wrapped Fukaya algebra of the fiber. In partic-ular, we do not have to transfer Corollary 3.6 through Floer’s isomorphism. Proposition 3.8. If M is a simply connected, spin manifold, then the restricted Yonedafunctor Y : W Morse ( T ∗ M ) Mod W Morse ( T ∗ M ) Mod C ∗ ( M )Yoneda i ∗ M is fully faithful. Remark . Note that we have used the full-faithfullness of i M from Lemma 3.7 to write C ∗ ( M ) rather than end ∗ ( M Morse ). At the level of objects, Y just sends L to CW ∗ ( M, L ) M, . Proof.
Lemma 3.7 and Abouzaid’s theorems [4, 1] give us a chain of quasi-equivalencesTw W Morse ( T ∗ M ) Tw W ( T ∗ M ) Tw (end ∗ ( T ∗ q M )) Tw ( C −∗ (Ω M )) . ∼ = F ∼ = ∼ = The resulting functor F sends the cotangent fiber T ∗ q M to the rank 1 free module.Let us study what happens to M Morse . We know CW ∗ ( M Morse , T ∗ q M ) ∼ = Z , since thezero section and fiber have just one intersection point. This means that F ( M Morse ) isan augmentation, and in fact it is the canonical augmentation of C −∗ (Ω M ). Indeed, all C −∗ (Ω M )-modules whose cohomology is Z are quasi-isomorphic. To see, use the homo-logical perturbation lemma to replace C −∗ (Ω M ) with its cohomology H −∗ (Ω M ). This issupported in non-positive degrees and, because M is simply connected, has H (Ω M ) ∼ = Z .Since the A ∞ -module operation µ k | : H −∗ (Ω M ) ⊗ k ⊗ Z → Z has degree 1 − k and H −∗ (Ω M ) is supported in non-positive degrees, the only non-trivial A ∞ -operation is the product µ | : H (Ω M ) ⊗ Z → Z ; this is the identity operation.By [7], the standard augmentation and the rank 1 free module of C −∗ (Ω M ) are Koszul dualif M is simply-connected, so M Morse is Koszul dual to T ∗ q M and the proposition follows. (cid:3) Remark . Simply-connectedness and Z -grading are standard essential ingredients forKoszul duality. The spin condition also seems essential in our proof, but we do not have anexample showing that Proposition 3.8 fails without it.We now have the necessary ingredients to prove Theorem 1.8. Proof of Theorem 1.8.
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