Chekanov-Eliashberg dg-algebras for singular Legendrians
CCHEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULARLEGENDRIANS
JOHAN ASPLUND AND TOBIAS EKHOLM
Abstract.
The Chekanov–Eliashberg dg-algebra is a holomorphic curve invariant associ-ated to Legendrian submanifolds of a contact manifold. We extend the definition to Leg-endrian embeddings of skeleta of Weinstein manifolds. Via Legendrian surgery, the newdefinition gives direct proofs of wrapped Floer cohomology push-out diagrams [21]. It alsoleads to a proof of a conjectured isomorphism [16, 23] between partially wrapped Floercohomology and Chekanov–Eliashberg dg-algebras with coefficients in chains on the basedloop space. Introduction
We introduce holomorphic curve invariants of Legendrian embeddings of a Weinstein do-main V of dimension (2 n −
2) with a given handle decomposition h into the (2 n − ∂W of a Weinstein 2 n -manifold W . Here, a Legendrian em-bedding f : V → ∂W is an embedding that extends to a contact embedding F : ( − (cid:15), (cid:15) ) × V → ∂W , f = F | V ×{ } , where R × V is the contactization of V , for some (cid:15) > h are isotropic disks in ∂W . Inparticular, the ( n − l are Legendrian and we think of the wholeskeleton of V , i.e., the union l ∪ l of subcritical core disks l and l , as a singular Legendrianin ∂W .To consider ordinary smooth Legendrian ( n − ⊂ ∂W from this point ofview, take V to be a small neighborhood of the zero-section in the cotangent bundle T ∗ Λ.Then the invariant of V agrees with the Chekanov–Eliashberg dg-algebra of the Legendriansubmanifold Λ with coefficients in chains on the based loop space, see Theorem 1.2.The generalization of dg-algebras of Legendrian submanifolds to embeddings of generalWeinstein domains V ⊂ ∂W is useful from many points of view. For example, it givesLegendrian surgery formulas for wrapped Floer cohomology in W stopped at V , see Theorem1.1, dg-algebras for non-closed Legendrian submanifolds Λ ⊂ ∂X with Legendrian boundaryin ∂V , and cut-and-paste formulas for holomorphic curve invariants that parallels results in[21], see Remark 1.4.The remainder of Section 1 is organized as follows. In Section 1.1 we describe our con-struction in the basic example of the cotangent bundle T ∗ R n of n -space. In Section 1.2 wedefine Chekanov–Eliashberg dg-algebras in general and state our main results. In Section1.3 we discuss generalizations and computations.1.1. A basic example.
We give a Legendrian surgery description of T ∗ R n and assume forsimplicity that n > R n \ { } is simply connected. (We consider n = 2 in Example JA is supported by the Knut and Alice Wallenberg Foundation.TE is supported by the Knut and Alice Wallenberg Foundation and the Swedish Research Council. a r X i v : . [ m a t h . S G ] F e b JOHAN ASPLUND AND TOBIAS EKHOLM T ∗ S n is more basic than T ∗ R n and we start from there.Consider the standard symplectic 2 n -space B = ( R n , (cid:80) nj =1 dx j ∧ dy j ) with ideal contactboundary the standard contact sphere ∂B = S n − . Let Λ n − ⊂ S n − denote the Leg-endrian unknot, a Legendrian ( n − n -plane through the origin in R n . The Chekanov–Eliashberg dg-algebra ofΛ n − , CE ∗ (Λ n − ; B ) is the unital algebra C [ a n − ] on one generator a n − , corresponding tothe unique Reeb chord of the standard representative of Λ n − in a Darboux ball (Figure 1.1left), of degree | a n − | = − ( n − n -handle to B (viewed as the 2 n -ball with a half-infinite symplecti-zation collar attached) along Λ n − using the standard attaching map, the resulting Weinsteinmanifold is the cotangent bundle T ∗ S n and the co-core disk C in the handle becomes thecotangent fiber in T ∗ S n . Legendrian surgery [7, 13, 16] gives a geometrically induced quasi-isomorphism of A ∞ -algebras from the wrapped Floer cohomology of C , CW ∗ ( C ; T ∗ S n ), to CE ∗ (Λ n − ; B ), where CE ∗ (Λ n − ; B ) is viewed as a chain complex generated by monomialsin Reeb chords with differential induced by the dg-algebra differential, with product given byconcatenation of monomials, and with all higher A ∞ -operations trivial. In this simple exam-ple, CE ∗ (Λ n − ; B ) can be directly compared to the description of CW ∗ ( C ; T ∗ S n ) as chainson the based loop space C −∗ (Ω S n ) of S n with the Pontryagin product, see [1]. Concretely, weuse Adams’ construction [2, 16] to represent C −∗ (Ω S n ) as the reduced cobar construction onthe Morse complex of S n . Taking a Morse function with only two critical points then givesa quasi-isomorphism of algebras C −∗ (Ω S n ) ≈ C [ y n − ] where the generator y n − has degree − ( n −
1) and corresponds to the maximum of the Morse function. The surgery isomorphismthen maps y n − to a n − .Next, consider T ∗ R n . Here we view Λ n − instead as a Legendrian stop [23, 21] in the idealcontact boundary of B . This can be thought of, see [18, 16, 6] as attaching a punctured ver-sion of the Weinstein handle, T ∗ (Λ n − × [0 , ∞ )), to Λ n − . In the case under consideration, itis clear that the result is T ∗ R n . According to [16, Conjecture 3] the wrapped Floer cohomol-ogy CW ∗ ( C ; T ∗ R n ) is isomorphic to CE ∗ (Λ n − , C −∗ (ΩΛ n − ); B ), the Chekanov–Eliashbergdg-algebra with loop space coefficients. Concretely (recall n > C (cid:104) a n − , y n − (cid:105) , with differential ∂a n − = y n − induced by a countof rigid disks with a point constraint, see Figure 1.1. It is then quasi-isomorphic to theground field C in agreement with the Floer cohomology of the fiber in T ∗ R n .We now instead consider T ∗ R n from the point of view in the current paper, where we takea Weinstein handle perspective and represent T ∗ R n as a Weinstein cobordism with negativeend the contactization of T ∗ Λ n − and one critical handle with co-core disk C at the criticalpoint, C then corresponds to the fiber in T ∗ R n . More precisely, let V ⊂ ∂B be a smallneighborhood of the zero section in T ∗ Λ n − , equip it with a handle decomposition h witha zero handle with core disk l and an ( n − l . Let V denote thesubcritical part of V , i.e., a neighborhood of l . We construct a handle decomposition on theWeinstein cobordism representing T ∗ R n as follows. Start from the 2 n -ball and first attachthe handle V × D ∗ (cid:15) [ − , D ∗ (cid:15) [ − ,
1] denotes an (cid:15) -disk subbundle of the cotangentbundle T ∗ [ − , − (cid:15), (cid:15) ) × V to ∂B (cid:116) ( R × T ∗ Λ n − ) viewed as the positive end of B (cid:116) ( R × ( R × T ∗ Λ n − )). Denote the resulting manifold B V . HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 3
To connect to chains on the based loop space, take l to be the constraining point in Λ n − corresponding to the generator y n − (the maximum of the Morse function). Then the bound-ary ∂l of the core disk of the top handle l intersects the boundary ∂V of the neighborhoodin a standard Legendrian ( n − n − . The full cobordism corresponding to T ∗ R n isobtained by attaching a standard Weinstein handle to the Legendrian ( n − h )which consists of two copies of l joined across the handle via ∂l × [ − ,
1] = Λ n − × [ − , h ) then has two Reeb chords: a n − and a n − , the latter in-side the handle. The differential in CE ∗ (Σ( h ); B V ) satisfies ∂a n − = a n − , see Figure 1.1and CE ∗ (Σ( h ); B V ) is isomorphic to CW ∗ ( C ; T ∗ R n ). Our approach here is to define theChekanov–Eliashberg dg-algebra of ( V, h ), CE ∗ (( V, h ); B ) as CE ∗ (Σ( h ); B V ), the usual dg-algebra of the Legendrian sphere Σ( h ). For smooth Legendrians ( V a small neighborhood ofthe zero section T ∗ Λ) the new definition will correspond to the usual Chekanov–Eliashbergdg-algebra with loop space coefficients, see Theorem 1.2, but the definition makes sense moregenerally for any Legendrian admitting a Weinstein thickening, see [5] for one-dimensionalexamples.If we compare the usual chains on the loop space dg-algebra with the one defined here, wefind that in the former approach the algebra contains generators of both topological (chainson the based loop space) and dynamical (Reeb chords) nature, whereas in the latter allgenerators are of dynamical nature. a n − Λ n − a n − Λ n − V Figure 1.
Front projection of Legendrian unknot Λ n − with a point con-strained holomorphic curve that gives ∂a n − = a n − .1.2. General definition and main results.
Our definition of dg-algebras for general We-instein domains is the following generalization of the cobordism method described for T ∗ R n in Section 1.1. Consider a Weinstein (2 n − V with handle decomposition h , where h encodes both handles and attaching maps. We write V for the subcritical part of V , ∂V for its contact boundary, l j , j = 1 , . . . , m for the core-disks of the critical handles and ∂l j fortheir boundaries, i.e., the attaching spheres for the corresponding handle, and ∂l = (cid:83) mj =1 ∂l j for their union. Assume that ( V, h ) ⊂ ∂W is a Legendrian embedding into the contactboundary of a Weinstein 2 n -manifold W . For simplicity, we will assume throughout that c ( W ) = c ( V ) = 0.We build a cobordism W V (which is our version of W stopped at V , see [23, 21]) withnegative end R × V (in the negative end we think of V as the Weinstein manifold whichis the completion of the embedded Weinstein domain) in two steps: First we construct W V by attaching V × D ∗ (cid:15) [ − , D ∗ (cid:15) [ − ,
1] denotes an (cid:15) -disk subbundle of the cotangentbundle T ∗ [ − , ∂W (cid:116) ( R × V ) along V × D ∗ (cid:15) [ − , | {− , } . Then the full cobordism W V is JOHAN ASPLUND AND TOBIAS EKHOLM obtained from W V by attaching critical Weinstein n -handles to a link of Legendrian spheresΣ( h ) ⊂ ∂W V defined as follows. There is one sphere component Σ( h j ) ≈ S n − of Σ( h ) foreach top-dimensional handle h j of V . Here Σ( h j ) consists of two copies of l j ≈ D n − , oneembedded in ∂W \ (( − (cid:15), (cid:15) ) × V ) with boundary on { }× ∂V and one in ( R × V ) \ (( − (cid:15), (cid:15) ) × V ).The copies are joined across the handle V × D ∗ (cid:15) [ − ,
1] by ∂l j × [ − , ≈ S n − × [ − , Wl j R × ( R × V ) l j V × D ∗ ϵ [ − , ∂l j × [ − , ( − ϵ, ϵ ) × V ( − ϵ, ϵ ) × V ∂W Figure 2.
The Weinstein cobordism W V is constructed by attaching V × D ∗ (cid:15) [ − ,
1] along V × D ∗ (cid:15) [ − , | {− , } .We then define the dg-algebra of ( V, h ), CE ∗ (( V, h ); W ) as the usual dg-algebra of itsattaching spheres,(1.1) CE ∗ (( V, h ); W ) := CE ∗ (Σ( h ); W V ) , where Σ( h ) ⊂ ∂W V is the Legendrian link of attaching spheres described above and where W V in CE ∗ (Σ( h ); W V ) indicates that the differential counts holomorphic disks anchored in W V , see [7, 13, 16]. It is easy to see that Legendrian isotopies of ( V, h ) induce Legendrianisotopies of Σ( h ) and it follows that CE ∗ (( V, h ); W ) is a Legendrian isotopy invariant.The dg-algebra in (1.1) naturally contains the dg-algebra of the attaching spheres of V as a subalgebra. Indeed, as explained above, the attaching spheres ∂l ⊂ ∂V of the tophandles of V in ∂V appears as ‘equators’ of the spheres in Σ( h ) and, shrinking the handle,a straightforward action argument shows that equipping V with what we call an indexdefinite Weinstein structure, see Section 2.1, the chords of ∂l generate a dg-subalgebra of CE ∗ (Σ( h ); W V ) canonically identified with CE ∗ ( ∂l ; V ).It follows by Legendrian surgery [7, 13, 16] that the dg-algebra CE ∗ (( V, h ); W ) containsthe wrapped Floer cohomology algebra CW ∗ ( c ; V ) ≈ CE ∗ ( ∂l ; V ) of the union c = (cid:83) j c j ofthe co-core ( n − l j in the critical handles h j of V , as a subalgebra. Fromthe point of view of Floer cohomology with coefficients it is then clear that our dg-algebrashave the most general coefficients: the endomorphism algebra of generators of the Fukayacategory of the embedded domain itself. In this sense our construction generalizes Floercohomology with coefficients in chains on the based loop space to objects in the Fukayacategory represented by skeleta of Weinstein manifolds, no matter how singular their skeletamay be, see Section 6.2.1. HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 5
With CE ∗ (( V, h ); W ) defined we consider its properties. Let W V denote the Weinsteinmanifold W stopped at V , see [16, 23, 21]. Let C ( h ) = (cid:83) mj =1 C ( h j ) denote the union ofLagrangian co-core n -disks C ( h j ) of W V in the Weinstein n -handles attached to Σ( h j ) ⊂ ∂W V . Theorem 1.1.
There is a natural surgery isomorphism
Φ : CW ∗ ( C ( h ); W V ) −→ CE ∗ (( V, h ); W ) of A ∞ -algebras, where the right hand side is viewed as a complex generated by Reeb chordmonomials with differential induced by the dg-algebra differential, product given by concate-nation, and all higher operations trivial. Given the Legendrian surgery isomorphism from [7, 13, 16], Theorem 1.1 follows immedi-ately from the definition of CE ∗ (( V, h ); W ) and is one of its main motivations.We next relate our new definition of the dg-algebra to the standard version. Let Λ ⊂ ∂W be a smooth Legendrian and consider its Chekanov–Eliashberg dg-algebra with coefficientsin chains on the based loop space CE ∗ (Λ , C −∗ (ΩΛ); W ), as defined in [16]. Let ( V (Λ) , h (Λ))denote a small disk sub-bundle of the cotangent bundle T ∗ Λ of Λ with a handle decompositionwith a single top-dimensional handle.
Theorem 1.2.
There is a natural quasi-isomorphism
Ψ : CE ∗ (( V (Λ) , h (Λ)); W ) −→ CE ∗ (Λ , C −∗ (ΩΛ); W ) . Theorem 1.2 is proved in Section 4. We point out that Theorems 1.1 and 1.2 togetherestablish [16, Conjecture 3].Consider next two Weinstein manifolds W and W (cid:48) that both contain ( V, h ) as a Legen-drian embedding in their boundary. Then we can join W and W (cid:48) by attaching a V -handle, V × D ∗ (cid:15) [ − , − (cid:15), (cid:15) ) × V to build a new Weinstein manifold W V W (cid:48) . As in the con-struction of the handle decomposition of X above, the V -handle gives a link of Legendrianattaching spheres Σ ( h ) in ∂ ( W V W (cid:48) ), where V is the subcritical part of V and where acomponent Σ ( h j ) consists of two copies of l j , inside ∂W and ∂W (cid:48) , joined by ∂l j × [ − , Theorem 1.3.
There is a natural push-out diagram CE ∗ ( ∂l ; V ) −−−→ CE ∗ (( V, h ); W (cid:48) ) (cid:121) (cid:121) CE ∗ (( V, h ); W ) −−−→ CE ∗ (Σ ( h ); W V W (cid:48) ) , where, for a suitable contact form and almost complex structure on the V -handle, all mapsare defined by inclusion on Reeb chord generators.Remark . Combining Theorems 1.1 and 1.3 and the Legendrian surgery isomorphism CW ∗ ( C ( h ); W V W (cid:48) ) ≈ CE ∗ (Σ ( h ); W V W (cid:48) ), where C ( h j ) is the co-core disk in theWeinstein handle attached to Σ ( h j ) and C ( h ) = (cid:83) j C ( h j ), we obtain the followingpush-out diagram of [21]: CW ∗ ( c ; V ) −−−→ CW ∗ ( C (cid:48) ( h ); W (cid:48) V ) (cid:121) (cid:121) CW ∗ ( C ( h ); W V ) −−−→ CW ∗ ( C ( h ); W V W (cid:48) ) , JOHAN ASPLUND AND TOBIAS EKHOLM where W V and W (cid:48) V denote W and W (cid:48) stopped at V and C ( h ) and C (cid:48) ( h ) denote the unionof Lagrangian disks dual to the top handles h of V in W and W (cid:48) , respectively.1.3. Generalizations, computations, and examples.
We also describe natural general-izations of the basic dg-algebras introduced above and illustrate how to work with them bydescribing schemes for computation and studying examples.In Section 6.1 we discuss relative versions of the dg-algebra. More precisely, with V ⊂ ∂W as above if Λ ⊂ ∂W is a Legendrian submanifold with boundary ∂ Λ in ∂V then there isa Chekanov–Eliashberg dg-algebra CE ∗ (Λ; V ; W ) associated to Λ. We show that relativeversions of the theorems above hold. In Section 6.2 we discuss analogues of exact Lagrangianfillings and cobordisms for our singular Legendrians ( V, h ) and sketch constructions of theirFloer cohomology and of induced cobordism maps.In Section 7 we describe how to find generators and compute differentials in the case whenthe contact manifold ∂W is a contactization. We also study several concrete examples, e.g.,we exhibit Legendrian tri-valent graphs that do not admit Lagrangian fillings with restrictedsingularities. Acknowledgements.
TE thanks Sylvain Courte and Vivek Shende for valuable discussions.2.
Weinstein handles and decompositions of V -handles In this section we discuss handle decompositions of the basic building block of our con-struction. Let (
V, h ) be a (2 n − λ ,Liouville vector field z , handle decomposition h , and exhausting Morse function φ : V → [0 , ∞ ) with a single non-degenerate minimum at value 0. Then there is ρ > φ − ([ ρ, ∞ )) ≈ [0 , ∞ ) × ∂V is the positive half of the symplectization of the contact bound-ary ∂V of V . By scaling, we may take ρ > F z ( v, t ) denote the flowof z with initial condition v ∈ V . The skeleton of V is the set of points v ∈ V such thatlim t →∞ φ ( F z ( v, t )) < ∞ . The skeleton is then a union of core disks in the handles of thedecomposition h . We will often think of the exhausting function φ as being approximatelyzero on the skeleton so that φ − ( δ ) for δ > δ → V -handles.2.1. Weinstein handles.
Consider R n with the standard symplectic form ω = (cid:80) nj =1 dx j ∧ dy j . For 0 ≤ k ≤ n , the vector field Z k = k (cid:88) j =1 (2 x j ∂ x j − y j ∂ y j ) + n (cid:88) j = k +1
12 ( x j ∂ x j + y j ∂ y j )is a Liouville vector field, L Z k ω = ω . We consider for δ ≥ ∂ ± δ H k = (cid:40) ( x, y ) : k (cid:88) j =1 ( x j − y j ) + n (cid:88) j = k +1 12 ( x j + y j ) = ± δ (cid:41) HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 7 and R n = H k with the Liouville field Z k as a symplectic cobordism with negative andpositive ends ∂ − δ H k ≈ S k − × D k +2( n − k ) ,∂ δ H k ≈ D k × S n − k − . We say that D k × { } ⊂ H k is the core disk of the handle with attaching sphere S k − × { } ⊂ ∂ − δ H k , and that { } × D n − k ⊂ H k is the co-core disk and { } × S n − k − ⊂ ∂ δ H k the co-coresphere .The induced contact form on ∂ ± δ H k is the restriction of the form α k = k (cid:88) j =1 (2 x j dy j − y j dx j ) + n (cid:88) j = k +1 ( x j dy j + y j dx j ) . The corresponding Reeb vector field R k ; ± δ along ∂ ± δ H k is R k ; ± δ = 1 N ( x, y ) (cid:32) k (cid:88) j =1 (2 x j ∂ y j − y j ∂ x j ) + n (cid:88) j = k +1 ( x j ∂ y j + y j ∂ x j ) (cid:33) , where N ( x, y ) = k (cid:88) j =1 (4 x j + y j ) + n (cid:88) j = k +1 ( x j + y j ) . It follows in particular that any Reeb orbit in ∂ δ H k lies in the middle of the handle, i.e., at( x j , y j ) = (0 ,
0) for j = 1 , . . . , k , where the Reeb flow agrees with the standard Reeb flow onthe (2( n − k ) − J on R n , J ∂ x j = ∂ y j , in a neighbor-hood of ( x, y ) = (0 ,
0) takes Z k to R k and that it extends as an almost complex structure J over H k that leaves the contact planes in ∂ ± δ H k invariant for all δ > ∂ ± δ H k forany δ >
0. We call such an almost complex structure handle adapted .An arbitrary Weinstein 2 n -manifold W admits a handle decomposition H , which can bedescribed as follows in terms of an exhausting function φ : W → [0 , ∞ ). The function φ has d ,1 ≤ d ≤ n critical levels 0 < δ < · · · < δ d < ∞ . For 0 < δ < δ , φ − ([0 , δ )) is isomorphic tothe 2 n -ball, H in the notation above. Furthermore, for δ k − < δ (cid:48) < δ k < δ < δ k +1 , φ − ([0 , δ ])is obtained from φ − ([0 , δ (cid:48) )) by attaching a finite number of k -handles H k along an isotropiclink ∂L k of ( k − φ − ( δ (cid:48) ).If d < n then we say that W is subcritical.Our next result controls Reeb orbits in the boundary of a subcritical manifold. We considera parameter η that controls the size of the attaching regions of the handles in W . The linkof isotropic attaching spheres ∂L k − has a neighborhood of the form ∂L k − × D n + k . Welet η > ∂W η for the boundary of theWeinstein domain with η -sized handles. Using the Liouville flow in local models it is clearthat if η (cid:48) < η then there is a topologically trivial symplectic cobordism with positive end ∂W η and negative end ∂W η (cid:48) which can be taken to be standard in the middle of each handle. Lemma 2.1.
For generic attaching spheres the following holds. For any a > there exists η > such that for all η < η , any Reeb chord of ∂W η of action < a lies in the middle ofsome handle. Furthermore, for all η < η (cid:48) < η rigid holomorphic cylinders in the cobordism JOHAN ASPLUND AND TOBIAS EKHOLM gives a natural one to one correspondence between Reeb orbits of action a < in ∂W η (cid:48) and ∂W η . Finally, there exists a contact form on ∂W η , arbitrarily close to the standard contactform in the middle of each handle such that the minimal Conley–Zehnder index of an orbitin a k -handle is ( n − k ) + 1 .Proof. A subcritical isotropic sphere generically has no Reeb chords. Therefore for any a > η such that any Reeb flow line starting in an η -neighborhood of the sphere returnsonly if it has action > a . It follows from this that all Reeb orbits must lie inside the handles.The claim then follows from well-know properties of the standard contact sphere, which sitsin the middle of each handle. (cid:3) We call contact forms on subcritical manifolds of the form above index definite . Note thatif Λ ⊂ ∂W is a Legendrian sphere in the index definite boundary of a subcritical Weinsteinmanifold of dimension 2 n >
2, then its dg-algebra can be computed without any anchoring:punctured spheres in W have minimal dimension n − ( n −
1) + 1 + ( n −
3) = n − >
0, seee.g., [10, Appendix A] for the well-known dimension formula.2.2.
Reeb dynamics and attaching spheres of V -handles. Let V be a Weinstein (2 n − h . Consider the contactization of V , i.e., the contactmanifold R × V with contact form dζ + λ , where ζ is a linear coordinate on R . We willconstruct a cobordism V × T ∗ [ − ,
1] with negative end ( R × V ) × {− , } and positive endthe contact manifold obtained by removing (( − (cid:15), (cid:15) ) × V ) × {− , } from ( R × V ) × {− , } and joining the boundaries by ( V × S ∗ (cid:15) [ − , ∪ (cid:0) ∂V × D ∗ (cid:15) I | {− , } (cid:1) , where S ∗ (cid:15) [ − ,
1] and D ∗ (cid:15) [ − ,
1] denotes the (cid:15) -sphere and (cid:15) -disk cotangent bundles, respectively. The cobordismwill furthermore come with a natural handle decomposition induced by h . In order to get amore precise description we will use an explicit model of the handle that we describe next.Consider symplectic R with coordinates ( x, y ) and symplectic form dx ∧ dy . Consider theproduct V × R with symplectic form ω = dλ + dx ∧ dy . Then Z = z + 2 x∂ x − y∂ y is a Liouville vector field for ω and we consider H δ ( V ) ≈ V × R as an exact symplecticcobordism with positive and negative ends the contact hypersurfaces(2.1) G ± δ ( V ) = { ( x, y, v ) ∈ R × V : x − y + φ ( v ) = ± δ } . The induced contact form on G δ ( V ) is α ± δ = (2 xdy + ydx + λ ) | G ± δ and the corresponding Reeb vector field(2.2) R ± δ = β ( x, y, v )(2 x∂ y + y∂ x ) + r λ ( x, y, v ) , where β ( x, y, v ) is a smooth function, non-zero if ( x, y ) (cid:54) = (0 ,
0) and r λ is a smooth vectorfield in the kernel of dλ | { φ ( v )= − x + y ± δ } at points where z (cid:54) = 0 and equal to zero where z = 0. Lemma 2.2.
The Reeb vector field R δ has the following properties. The subset of G δ givenby ( x, y ) = (0 , and φ ( v ) = δ is invariant under the flow of R δ and along this subset R δ agrees with the Reeb vector field of ∂V .Proof. Clear from (2.2). (cid:3)
HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 9
Now take V = V , the subcritical part of V . Write l = (cid:83) j l j for the core-disks of the tophandles h j in h and ∂l ⊂ ∂V for their boundaries which are the attaching ( n − h j . Consider ∂l = (cid:8) ( x, y, v ) ∈ R × V : x = 0 , v ∈ ∂l × [0 , ∞ ) , (0 , y, v ) ∈ G δ ( V ) (cid:9) . It has the following properties (below the (contact homology) grading of a Reeb orbit is thedimension of the moduli space of holomorphic planes with positive asymptotic at the orbit).
Lemma 2.3.
The subset ∂l ⊂ G δ is a smooth Legendrian submanifold of topology ∂l × R .The Reeb chords of ∂l all lie over ( x, y ) = 0 and are in natural 1-1 grading preservingcorrespondence with the Reeb chords of ∂l ⊂ ∂V . Furthermore, the Reeb orbits in G δ are innatural 1-1 correspondence with Reeb orbits in ∂V , where the grading of an orbit in G δ isone above the grading of the corresponding orbit in ∂V .Proof. With Reeb vector field R δ as in (2.2), the ( x, y )-coordinate of any Reeb flow line notstarting at ( x, y ) = 0, moves along the curves { x − y = const } with non-zero speed, andthe ( x, y )-coordinate of flow lines starting at ( x, y ) = (0 ,
0) is fixed. Since the restriction of R δ to ( x, y ) = 0 agrees with the Reeb vector field in ∂V it follows that Reeb chords are in 1-1correspondence as claimed. It remains to show that this correspondence is index preserving.Consider a Reeb chord c of ∂l ⊂ ∂V . The degree | c | of c is −| c | = µ ( c ) −
1, where µ ( c )is the Maslov index of the tangent planes to the Legendrian along a path from the top endpoint of c to the bottom endpoint, followed by the linearized Reeb flow along the chord andfinally closed up by a positive rotation along the K¨ahler angle. Including the chord as a chordof ∂l , the only change comes from the linearized Reeb flow in the additional R -direction.The Reeb vector field is y∂ x which means that y -axis at the bottom endpoint arrives rotatedslightly in the negative and closed up by a small positive rotation this contributes zero tothe Maslov index. It follows that the gradings of c as a chord of ∂l and ∂l agree.The statement on Reeb orbits is similar. The difference in grading arises as follows: aswith the Maslov index of the chord, the Conley–Zehnder index CZ of an orbit does notchange from ∂V to G δ . Since the grading of an orbit is CZ + ( n − n −
1, the grading then increases by one. (cid:3)
Consider now adding V × D ∗ (cid:15) [ − ,
1] as above to V × R × ∂ [ − ,
1] along V × ( − (cid:15), (cid:15) ).Smoothing the union of the two copies of l in V × R × ∂I and ∂l , we get a collection ofLegendrian spheres Σ( h ) in the upper contact boundary. Lemma 2.4.
The Legendrian spheres Σ( h ) are attaching spheres for V × R with Liouvillefield as in the cobordism with positive end G δ and negative end G − δ , see (2.1) .Proof. Consider an exhausting function φ for V such that V = φ − ([0 , δ ]) with critical levelfor critical points of maximal index at δ . Then the level set δ gives the contact manifoldwith attaching spheres Σ( h ). The critical points in the core disks lie at the level δ andsub-level sets of levels > δ give V × R . (cid:3) Holomorphic curves in V -handles. We next consider almost complex structures on H δ ( V ). Let J V be a handle adapted almost complex structure on V , see Section 2.1. Write z for the Liouville vector field of V and let r = J V z . We think of H δ ( V ) \ ( { x = 0 }× skeleton( V ))as the symplectization of ∂H δ . We will define an almost complex structure on G δ thattakes the Liouville vector field to the Reeb vector field and then extend it to all H δ ( V ) bytranslation along the Liouville vector field. Consider the tangent space to G δ ( V ). • At ( x, y ) = 0, the tangent space is spanned by ker dφ , ∂ x , and ∂ y . We define J as J V on T V and
J ∂ x = ∂ y . • At points ( x, y, v ) ∈ G δ ( V ), where v is not a critical point of φ and ( x, y ) (cid:54) = 0, thetangent space is the sum of the (2 n − dφ ) and the one-dimensional subspaces generated by vector field ( (cid:112) x + y ) − ( y∂ x + 2 x∂ y ) and thevector field − (4 x + y ) z + dφ ( z )(2 x∂ x − y∂ y ). Here we define J as the restriction of J V on the contact hyperplane in ker φ and J z = 1 ω ( z, r ) + 4 x + y r, J ∂ x = 1 ω ( z, r ) + 4 x + y ∂ y . • At points ( x, y, v ) where v is critical for φ we define J to equal J V on T V and J (2 x∂ x − y∂ y ) = x + y (2 x∂ y + y∂ x ). (Note that, v critical and ( x, y, v ) ∈ G δ imply( x, y ) (cid:54) = 0.)Since J V is handle adapted it follows that J is smooth. We finally extend J by translationalong the Liouville vector field. Let π : H δ → R be the projection to the ( x, y )-plane. Lemma 2.5.
The hypersurfaces V ( x,y ) = π − ( x, y ) are J -complex. In particular, any holo-morphic curve in the symplectization of G δ not contained in V ( x,y ) intersects V ( x,y ) positivelyand the intersection number is locally constant in ( x, y ) .Proof. The splitting
T H δ ( V ) = T V ⊕ T R is preserved along the flow lines of the Liouvillefield Z = z + 2 x∂ x − y∂ y and J takes T V ( x,y ) to T V ( x,y ) by definition. (cid:3) We next consider holomorphic curves in the V -handle. Recall that we have an indexdefinite Weinstein structure on V . For dim( V ) = 2 n − > ∂l ⊂ ∂V is defined without anchor-ing. We start with this case, where there will be a canonical identification of dg-algebras. Inthe lowest dimensional case, dim( V ) = 2 the basic orbit has grading 0 and its effects mustbe taken into account, see Remarks 2.7 and 2.8. Lemma 2.6.
Assume n > . For a > , any J -holomorphic curve in the symplectization R × G δ with boundary on R × ∂l and with positive puncture at a Reeb chord over y = 0 corresponding to a Reeb chord of ∂l of action < a , lies entirely over ( x, y ) = 0 . Also, there isa natural 1-1 correspondence between such rigid holomorphic disks and rigid J V -holomorphicdisks in R × ∂V with boundary on R × ∂l .Proof. If a curve intersects V ( x,y ) but is not contained in it then by Lemma 2.5 it is unboundedoutside ( x, y ) = 0 which shows it would have some positive asymptote not at ( x, y ) = 0 incontradiction to our assumption. Consequently, all rigid curves actually are contained in( x, y ) = 0. The correspondence between curves is obvious, it remains only to show that therigid disks are transversely cut out. This is straightforward, the linearization corresponds to astabilization with small angles at the punctures which is easily seen to be an isomorphism. (cid:3) Remark . We consider the counterpart of Lemma 2.6 when n = 1. In this case V is a2-disk and ∂l is a collection of points in its S -boundary. This type of handle was studiedin [17]. It was shown that the resulting holomorphic curves stay inside the handle and thedg-algebra was computed and was called the internal algebra [17, Section 2.3]. This algebrais canonically isomorphic to CE ∗ ( ∂l, V ), see [17, Sections 4 and 5]. HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 11
Remark . We give a brief discussion of geometric aspects of the case n = 1. Disks con-tributing to the differential in CE ∗ ( ∂l ; D ) are anchored at the basic Reeb orbit of index0. Including D in the middle of the 4-dimensional handle, the grading of the Reeb orbitincreases to 1, there is no anchoring disks anymore, and the upper levels (in the symplectiza-tion) in the anchored curves are now of dimension ≤
0. In order to achieve transversality onemust break the symmetry of the Lagrangian and the Reeb chords are no longer containedin ( x, y ) = 0. In particular, the trivial holomorphic strips over the Reeb chords that extendin the Liouville direction now appear in the handle model H δ as thin strips around { y = 0 } stretching to infinity. The perturbation breaks the symmetry and the strips go to infinity inonly one direction. The disk family with a positive asymptotic at the Reeb chord going oncearound lie arbitrarily close to the broken disk of the augmented curve in the middle and thebasic disk over the half axis chosen by the perturbation. Counts of other anchored disks canbe concluded formally from properties of the differential. Geometrically, they all lie close tothe anchored curve in the middle with half lines of standard disks attached. Remark . Curves in fibers V ( x,y ) , ( x, y ) (cid:54) = (0 ,
0) as in Lemma 2.6 are not fixed by theLiouville flow. To find their asymptotics at the negative end one can consider the intersectionwith the level surface G δ (cid:48) as we translate the curve to infinity in the positive Liouvilledirection.Consider now adding V × D ∗ (cid:15) [ − ,
1] as above to V × R × ∂ [ − ,
1] along V × ( − (cid:15), (cid:15) ).Smoothing the union of the two copies of l in V × R × ∂I and ∂l , we get a collection ofLegendrian spheres Σ( h ) in the upper contact boundary. Corollary 2.10.
The Chekanov–Eliashberg dg-algebra of Σ( h ) is canonically isomorphic tothe dg-algebra of ∂l ⊂ ∂V , using the identification of Reeb chords and holomorphic disks inLemma 2.6 for n > and the corresponding identification from Remark 2.7 for n = 1 . (cid:3) Chekanov–Eliashberg dg-algebras and Legendrian surgery formulas
In this section we define the Chekanov–Eliashberg dg-algebra of a Weinstein (2 n − V, h ) ⊂ ∂W with handle decomposition h . It follows from the definition and theLegendrian surgery formula that the dg-algebra is isomorphic to the endomorphism algebraof co-core disks dual to the top-dimensional core-disks of V in the partially wrapped Fukayacategory with a stop at V . We also show that the dg-algebra can be decomposed into anexterior piece generated by Reeb chords in ∂W connecting the ( n − l of top-dimensional handles in h and the dg-algebra of their attaching ( n − ∂l in ∂V , the subcritical part of V , which here appears as a sub-algebra, compare Corollary 2.10.3.1. Definition of Chekanov–Eliashberg dg-algebras.
Let W be a Weinstein 2 n -manifoldand let ( V, h ) ⊂ ∂W be a Legendrian embedding of the Weinstein (2 n − V withhandle decomposition h . Then a small neighborhood of V in ∂W can be identified with V × ( − (cid:15), (cid:15) ), where the second factor is along the Reeb flow. We define W V , W stopped at V ,to be the Weinstein cobordism with negative end V × R obtained by attaching a V -handle,i.e., an (cid:15) -neighborhood V × D ∗ (cid:15) [ − ,
1] of V × [ − ,
1] in V × T ∗ [ − , V × ( − (cid:15), (cid:15) ) to ∂W (cid:116) ( R × V ). (As mentioned in Section 1, we think of V in the negative end R × V as theWeinstein manifold which is the completion of the embedded Weinstein domain.) Lemma 3.1.
The negative end of the cobordism W V have no closed Reeb orbits. If W comeswith a handle decomposition H then h induces a handle decomposition of W V with handles H ∪ h [1] , where h [1] is the set of handles of h with dimension shifted up by and geometricallydescribed in Lemma 2.4.Proof. The first statement holds by construction since the Reeb flow on R × V has no orbits.The second statement follows from Lemma 2.4. (cid:3) Let ( V , h ) be the subcritical part of ( V, h ). Consider the subset W V ⊂ W V whichis obtained by attaching a V -handle to W (cid:116) ( R × V ). Lemma 2.4 then gives a link ofLegendrian attaching ( n − h ) ⊂ ∂ + W V , where ∂ + W V denotes the positiveboundary of W V . Here there is one component Σ( h j ) for each top-dimensional handle h j of V and attaching Weinstein n -handles to W V along Σ( h ) we obtain W V . As in Lemma3.1, there are no Reeb orbits in the negative end of the cobordism W V . We can thereforeuse it for anchoring holomorphic disks and the ordinary Chekanov–Eliashberg dg-algebra CE ∗ (Σ( h ); W V ) is defined as in [7, 13, 16]. We define the Chekanov–Eliashberg dg-algebraof ( V, h ) ⊂ W to be the dg-algebra of the attaching link Σ( h ). More precisely, we have thefollowing. Definition 3.2.
Let (
V, h ) ⊂ ∂W be a Legendrian embedding. The Chekanov–Eliashbergdg-algebra of (
V, h ), CE ∗ (( V, h ); W ) is CE ∗ (( V, h ); W ) := CE ∗ (Σ( h ); W V ) , where the link of Legendrian spheres Σ( h ) ⊂ ∂ + W V is as described above and where theright hand side is the standard Legendrian invariant as defined in [7, 13, 16]. Proof of Theorem 1.1.
Theorem 1.1 follows from the Legendrian surgery formula [7, 13, 16]and the definition of CE ∗ (( V, h ); W ) as the Legendrian dg-algebra of the link of attachingspheres Σ( h ) of core disks dual to C ( h ). (cid:3) Generators.
Definition 3.2 gives CE ∗ (( V, h ); W ) as the usual Legendrian dg-algebraof a link Σ( h ) and as such, it is generated by Reeb chords. The Reeb chord generators can bedescribed as follows. Recall that we write l for the union of the core ( n − V and ∂l ⊂ ∂V for their Legendrian attaching spheres in the contact boundary ∂V of the subcritical part of V . Lemma 3.3.
For any given action level a > , for all sufficiently small handles (i.e.,sufficiently small (cid:15) > in the handle V × D ∗ (cid:15) I ), after arbitrarily small perturbation of ( V, h ) , Reeb chords of Σ( h ) of action < a in ∂W V are in natural 1-1 correspondence withReeb chords connecting core-disks in the top handles l ⊂ ∂W and Reeb chords of ∂l ⊂ ∂V .Furthermore, all chords can be assumed transverse.Proof. The subcritical part of the skeleton of V has dimension n −
2. Hence for any a > V of the subcritical part of the skeleton such that no Reebflow line starting on l × { (cid:15) } ⊂ V × ( − (cid:15), (cid:15) ) of action < a hits V × ( − (cid:15), (cid:15) ). The result thenfollows from the definition of Σ( h ) and Lemma 2.4. (cid:3) A basic property of the differential.
We next consider the differential of the dg-algebra. The differential is defined in the standard way in terms of counts of holomorphicdisks in W V with boundary condition Σ( h ).Assume that ( V, h ) is in general position so that Lemma 3.3 holds. Lemma 2.6 showsthat CE ∗ (( V, h ); W ) has a subalgebra canonically identified with the dg-algebra of the top-dimensional attaching spheres of V . To compute the differential it remains to describe its HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 13 action on Reeb chords outside the V × D ∗ (cid:15) [ − , ∂W is a contactization in Section 7.1. Here we establish auseful general property of curves contributing to the differential, such curves cannot ‘cross’the handle. Lemma 3.4.
No holomorphic curve in R × ∂ + W V with positive puncture at a Reeb chord of Σ( h ) can cross the handle. In other words, such a curve lies entirely in W ∪ ( V × D ∗ (cid:15) [ − , .Proof. To see this note that inside the handle such a curve must intersect the hypersurface { y = 0 } which is foliated by the J -complex submanifolds V ( x, . It follows by positivityof intersections that the curve is either contained in a J -complex submanifold V ( x, or itintersects every fiber over ( x,
0) non-trivially. This means the curve must have a positiveasymptotic insider the handle not over the origin, which is incompatible with having positivepuncture only at a Reeb chord of Σ( h ) by Lemma 3.1. It follows that the curve cannot crossthe handle. (cid:3) Remark . The proof of Lemma 3.4 shows that any holomorphic curve that passes throughthe handle must have positive asymptotics at a Reeb chord or orbit that goes through thehandle. In case V have critical handles there are such chords and orbits. In the subcriticalindex definite case they are ruled out by Lemma 3.1.4. Non-singular Legendrians treated as singular Legendrians
In this section we show that the new definition of Chekanov–Eliashberg dg-algebras agreeswith the standard definition for smooth Legendrians. As a consequence we conclude that[16, Conjecture 3] holds.Let Λ ⊂ ∂W be a smooth connected Legendrian submanifold with a handle decomposition h with a single top handle and let V = D ∗ (cid:15) Λ denote a small neighborhood of Λ in T ∗ Λ. Bythe Darboux theorem for Legendrians there is a Legendrian embedding V → W canonicallyassociated to Λ for all sufficiently small (cid:15) > CE ∗ (( V, h ); W ): Σ( h ) ⊂ ∂W V denotes the attaching sphere ofthe top handle of W V . It consists of two copies of the core disk l of the top handle joinedby [ − , × ∂l across the handle. We next note that the cobordism W V contains a naturalLagrangian cobordism Q (cid:48) with topology (Λ × [ − , \ D n , where D n is a disk with boundaryΣ( h ). The positive boundary of Q is Σ( h ) and the negative boundary is Λ × {− , } .We will consider a subset Q ⊂ Q (cid:48) with the topology of Λ × [ − , δ ]. More precisely, we take Q = Q (cid:48) ∩ V × D ∗ (cid:15) [ − , δ ], where δ >
0. (Here we think of the critical points of the cobordismas sitting in V × CE ∗ (Λ , C −∗ (ΩΛ)) = CE ∗ (Λ , C −∗ (Ω Q )) , where the map on coefficients is induced by inclusion. We then note that there is a naturalcobordism map Φ : CE ∗ (Σ( h ); W V ) −→ CE ∗ (Λ , C −∗ (Ω Q )) , defined by moduli spaces of holomorphic disks with positive puncture in the positive endand negative punctures in the negative and ‘loop space coefficients’ in the cobordism Q .More formally, let a be a Reeb chord of Σ( h ) and let M ( a ) denote the moduli space ofholomorphic disks in the cobordism W V with boundary on Q (cid:48) and negative punctures atReeb chords of Λ. Any component M ( a ; b . . . b m ) of such a moduli space where the negative Wl j R × ( R × V ) l j V × D ∗ ϵ [ − , Q ′ ∂W L Λ L Λ L Figure 3.
The Lagrangian cobordism Q (cid:48) with positive boundary Σ( h ) andnegative boundary Λ × {− , } is shaded in gray. The handles of Λ belowmaximal dimension gives a Lagrangian filling L of ∂l in V .punctures are at Reeb chords b , . . . , b m defines a sum of monomials in CE ∗ (Λ , C −∗ (Ω Q (cid:48) ))as follows. The fundamental chain of the moduli space gives a chain of based loops in theproduct, with one factor for each boundary segment. Applying the Alexander–Whitneydiagonal approximation as in [16] we get a sum of monomials σ b b . . . b m + b σ b . . . b m + · · · + b . . . b m σ m , in the dg-algebra, where σ j are chains of based loops in Q (cid:48) , see [16]. We write [ M ( a )] forthe sum of all such elements over all components in the moduli space with positive punctureat a and define Φ( a ) = [ M ( a )] ∈ CE ∗ (Λ , C −∗ (Ω Q )) . Lemma 4.1.
The map Φ is a chain map.Proof. The proof of the chain map property follows as in [16, Proposition 21] once we knowthat all holomorphic disks with positive puncture at a Reeb chord of Σ( h ) has its boundaryin Q ⊂ Q (cid:48) . This follows from Lemma 3.4. (cid:3) We will next show that Φ is a quasi-isomorphism. To this end we use a particular repre-sentation of the loop space of Λ. More precisely, we replace Λ by a space ¯Λ in which a diskcontaining the core of the top cell l is identified to a point. We first consider the restrictionof the map Φ to the short Reeb chords inside the handle. It follows from Lemma 2.6 thatthis map is the natural curve counting map from CE ∗ ( ∂l ; ∂V ) that associates to a Reebchord the chain of loops in ¯Λ carried by the moduli space of disks with a positive punctureat that Reeb chord. We call this map ψ . Lemma 4.2.
The map ψ : CE ∗ ( ∂l ; ∂V ) −→ C −∗ (Ω ¯Λ) is a quasi-isomorphism. HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 15
Proof.
Consider the wrapped Floer cohomology of the fiber F in the middle of the top handle.Note that V = T ∗ Λ. Recall the standard quasi-isomorphism α : CW ∗ ( F ; V ) −→ C −∗ (Ω ¯Λ) , see e.g., [1, 6], and the Legendrian surgery isomorphism, [7, 13, 16], β : CW ∗ ( F ; V ) −→ CE ∗ ( ∂l ; ∂V ) . By SFT-stretching, α = ψ ◦ β. Since α and β are both quasi-isomorphisms so is ψ . (cid:3) Proof of Theorem 1.2.
After Lemma 4.2, the standard action argument applies: Lemma 4.2gives the desired small action isomorphism at the action level of the chords in the middle ofthe handle, as we increase the action, trivial strips over the Reeb chords of l shows that thechain map Φ is triangular with respect to the action filtration with ± (cid:3) Theorem 1.2 can be specialized in various ways. Here we consider the case when there isan augmentation of the dg-algebra corresponding to the loop space. If (cid:15) : CE ∗ ( ∂l ; ∂V ) → C is an augmentation then we can form the (cid:15) -partially linearized Chekanov–Eliashberg dg-algebra CE ∗ (( V, h ); W ; (cid:15) ) generated by chords of l only and with differential given by thedifferential in CE ∗ (( V, h ); W ) followed by (cid:15) on short chords in the subalgebra CE ∗ ( ∂l ; V ),compare [7, Section 4.6]. Corollary 4.3. If (cid:15) = (cid:15) L , where L is a Lagrangian filling of ∂l in V then CE ∗ (( V, h ); W ; (cid:15) ) is isomorphic to the usual Legendrian dg-algebra of Λ , where Λ is obtained by capping l offby L .Proof. The chain map relating the algebras is the composition of Φ with the map that takesdegree zero chains to 1 and other chains to 0. The quasi-isomorphism statement follows asin the proof of Theorem 1.2, by restricting attention to long chords only. (cid:3) Cut and paste
In this section we prove Theorem 1.3 and consider its consequences for stop removal.5.1.
A natural push-out diagram.
Assume (
V, h ) is Legendrian embedded in the bound-ary of two Weinstein manifolds W and W (cid:48) , respectively. Consider W V and W (cid:48) V as defined inSection 3.1. Consider the Weinstein manifold W V W (cid:48) obtained by connecting W to W (cid:48) bya V -handle. More precisely, we construct W V W (cid:48) by attaching V × D ∗ (cid:15) [ − ,
1] exactly as inthe construction of W V from W and R × ( R × V ) in Section 3.1. Proof of Theorem 1.3.
Consider the manifold W V W (cid:48) and note that the core disks l ofthe top handles of h in W and W (cid:48) joined by ∂l × [ − ,
1] form Legendrian attaching spheresΣ ( h ) in ∂W V W (cid:48) for W V W (cid:48) . In other words, we obtain W V W (cid:48) by attaching Weinstein n -handles to W V W (cid:48) along Σ( h ) ⊂ ∂W V W (cid:48) .Lemma 3.4 shows that holomorphic disks that contribute to the differential in CE ∗ (Σ( h ); W V W (cid:48) )of a Reeb chords of Σ( h ) in ∂W or ∂W (cid:48) stays in W ∪ ( D ∗ (cid:15) [ − , × V ) and W (cid:48) ∪ ( D ∗ (cid:15) [0 , × V ),respectively, and that disks with positive puncture at a Reeb chord inside D ∗ (cid:15) [ − , × ∂V stays over (0 , × V if n > { y = 0 } if n = 2, see Remark2.7. The theorem follows. (cid:3) Stop removal.
In this section we consider the operation of removing a stop. Consideras usual a Legendrian embedding (
V, h ) ⊂ ∂W with m top-dimensional handles h j with coredisks l j , j = 1 , . . . , m . Definition 5.1.
The Legendrian embedding of (
V, h ) is loose if the core disks l j of all thetop handles h j , j = 1 , . . . , m admit disjoint loose charts in ∂W , see [19].As expected the Chekanov–Eliashberg dg-algebra of a loose embedding of ( V, h ) is trivial:
Lemma 5.2.
Let ( V, h ) ⊂ ∂W be a loose Legendrian embedding then CE ∗ (( V, h ); W ) istrivial.Proof. The dg-algebra CE ∗ (( V, h ); W ) is defined as the dg-algebra of the link Σ( h ) ⊂ ∂W V where each component Σ( h j ) contains the core disk l j of h j . The loose charts for l j giveloose charts for Σ( h ) which then is loose and hence its dg-algebra is quasi-isomorphic to thetrivial algebra. (cid:3) Let V be a Weinstein (2 n − X = V × D ∗ [ − , z + x∂ x + y∂ y where z is a Liouville vectorfield on V , see Figure 4. Then V × { ( − , } is a Legendrian embedding of V in X . y x Figure 4.
The projection of the Liouville vector field on the Weinstein man-ifold X to the factor D ∗ [ − , Lemma 5.3.
For any handle decomposition h of V , the Legendrian embedding of ( V, h ) corresponding to V × { ( − , } is loose in X .Proof. We round corners as in [3, Section 2.5.1], see Figure 4 for a comparison with the usualhandle. Then applying [8, Proposition 2.8] (compare [17, Remark 2.23]), we get a loose chartfor each top handle. (cid:3)
We use the lemmas above to describe the effect of adding X to a Weinstein manifoldstopped at V . Geometrically, it is clear, by canceling critical points that this correspondsto removing the stop. Here we state two results that are simple consequences of that butwhich are sometimes useful in calculations. Corollary 5.4.
Let X be as in Lemma 5.3 and let ( V, h ) ⊂ ∂W then CW ∗ ( C ( h ); W V X ) is quasi-isomorphic to the trivial algebra. HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 17
Proof.
As in the proof of Lemma 5.2 the link of attaching spheres Σ( h ) of the core disks dualto the co-cores in C ( h ) is loose. The result then follows from Lemma 5.2 combined withTheorem 1.1 and Theorem 1.3, see also Remark 1.4. (cid:3) Assume now that W is a Weinstein manifold with co-core disks C (cid:48) dual to its criticalhandles with Legendrian boundary ∂C (cid:48) ⊂ ∂W . Let ( V, h ) ⊂ ∂W be a Legendrian embeddingsuch that V ∩ ∂C (cid:48) = ∅ . Let W V denote W stopped at V , then we can remove the stop byforming W = W V X . Corollary 5.5.
Let X be as in Lemma 5.3. Then there is a quasi-isomorphism CW ∗ ( C (cid:48) ∪ C ( h ); W V X ) ≈ CW ∗ ( C (cid:48) ; W ) . Proof.
Follows from Corollary 5.4, Theorem 1.3 and Theorem 1.1, see also Remark 1.4. (cid:3) Generalizations
In this section we discuss some natural generalizations of the results in earlier sections.6.1.
Legendrian submanifolds with boundary.
Consider a Legendrian embedding (
V, h ) ⊂ ∂W as above. Let Λ be a ( n − ∂ Λ. Consider a Legendrian em-bedding Λ → ∂W such that ∂ Λ ⊂ ∂V is a Legendrian embedding as well. We construct anon-compact Legendrian Λ ⊂ ∂W V with ideal boundary ∂ Λ ⊂ × ∂V ⊂ R × V by adding ∂ Λ × [ − ,
1] and the Legendrians lift of the positive cone on ∂ Λ in R × R × ∂V . We definethe Chekanov–Eliashberg dg-algebra CE ∗ (Λ; V ; W )as the ordinary Chekanov–Eliashberg dg-algebra of Λ ⊂ ∂W V . Lemma 6.1.
The dg-algebra CE ∗ (Λ; V ; W ) contains a subalgebra canonically isomorphic to CE ∗ ( ∂ Λ; V ) .Proof. As before, the sub-algebra is generated by Reeb chords in the middle of the handle.The proof is a repetition of the proof of Lemma 2.6. (cid:3)
We next give a surgery description of CE ∗ (Λ; V ; W ). By general position, we may assumethat ∂ Λ ⊂ ∂V is disjoint from the boundary ∂c of co-core disks of the top handles in h . Thislast condition means that we can consider ∂ Λ ⊂ ∂V , disjoint from the attaching spheres ∂l of the top handles in V . This in turn means that we can compute the dg-algebra byLegendrian surgery. We write Λ → l to denote either a Reeb chord from Λ to l in ∂W or aReeb chord from ∂ Λ to ∂l in ∂V and use the notation l → l , l → Λ, and Λ → Λ similarly.
Proposition 6.2.
In terms of data in W and V , the dg-algebra CE ∗ (Λ; V ; W ) admitsthe following description. It is generated by composable words of Reeb chords of the form Λ → l → l → · · · → l → Λ . The differential is induced by the Legendrian dg-algebradifferential of Λ ∪ Σ( h ) ⊂ ∂ + W V and the subalgebra of Lemma 6.1 is generated by composablewords of Reeb chords ∂ Λ → ∂l → · · · → ∂l → ∂ Λ in ∂V .Proof. This follows from [7, Theorem 5.10] by an argument directly analogous to the proofof Theorem 1.1. (cid:3)
As in the absolute case we also have push-out diagrams. More precisely, in the situationof Theorem 1.3, if W and W (cid:48) contains Legendrians Λ ⊂ ∂W and Λ (cid:48) ⊂ ∂W (cid:48) with commonboundary ∂ Λ = ∂ Λ (cid:48) ⊂ ∂V . Then we can join Λ and Λ (cid:48) via ∂ Λ × [ − ,
1] across the V -handle D ∗ (cid:15) V to form the closed Legendrian submanifold Λ ∂V Λ (cid:48) . We next give a surgery descriptionof the dg-algebra CE ∗ (Λ ∂V Λ (cid:48) , W V W (cid:48) ). As above, by general position ∂ Λ ⊂ ∂V andΛ V Λ is obtained from the Legendrian Λ V Λ ⊂ ∂ + W V after surgery on Σ ( h ). We usenotation analogous to that in Proposition 6.2. Proposition 6.3.
In terms of data in W , W (cid:48) , and V , the dg-algebra CE ∗ (Λ ∂V Λ (cid:48) ; W V W (cid:48) ) admits the following description. It is generated by composable words of Reeb chords of theform Λ → l → l → · · · → l → Λ . The differential is induced by the Legendrian dg-algebradifferential of Λ ∂V ∪ Σ( h ) ⊂ ∂ + W V . Furthermore, the subalgebra generated by composablewords of Reeb chords ∂ Λ → ∂l → · · · → ∂l → ∂ Λ in ∂V is isomorphic to CE ∗ ( ∂ Λ; V ) .Proof. This again follows from [7, Theorem 5.10]. (cid:3)
Proposition 6.4.
There is a push-out diagram CE ∗ ( ∂ Λ; ∂V ) −−−→ CE ∗ (Λ (cid:48) ; V ; W (cid:48) ) (cid:121) (cid:121) CE ∗ (Λ , V ; W ) −−−→ CE ∗ (Λ ∂V Λ (cid:48) ; W V W (cid:48) ) , where all maps are defined by inclusion on Reeb chord generators in the Legendrian surgerypresentations of the algebras, see Propositions 6.2 and 6.3.Proof. Directly analogous to the proof of Theorem 1.3. (cid:3)
Floer cohomology of singular exact Lagrangian fillings and cobordisms.
Inthis section we outline a construction of the counterpart of ‘cobordism maps’ for usualdg-algebras in the setting of singular Legendrians, induced by corresponding singular La-grangians. A more complete treatment of the subject will appear elsewhere. We refer toSection 7.2.2 for illustrations of the construction with detailed calculations in concrete ex-amples.Consider a Weinstein manifold X with contactization R × X . We say that an embeddingof a Weinstein domain V with handle decomposition h , ( V, h ) → X is exact if it lifts to aLegendrian embedding into the contactization. Note that such a lift is determined up to atranslation in R , one for each connected component of V .6.2.1. Floer cohomology.
To motivate our next definition we consider smooth exact La-grangians L , . . . , L k in X . Then the operation m k on the Floer cohomology complex CF ∗ ( L ∪ · · · ∪ L k ; X ) that counts holomorphic disks with k inputs and one output, canbe defined as the dual of the differential in the dg-algebra of a Legendrian link ˜ L . Here ˜ L isobtained by shifting the Legendrian lifts of L j so that L j sits above L k for all k ≤ j . TheFloer cohomology complex CF ∗ ( L ; X ) of any Lagrangian L ⊂ X carries the structure of an A ∞ -algebra. We use parallel copies of L as in [16] to define operations and in this way theLegendrian lift approach works in general for exact Lagrangian submanifolds.We consider exact embeddings ( V, h ) ⊂ X as singular Lagrangian embeddings of theskeleton of ( V, h ) and define the Floer cohomology CF ∗ (( V, h ); X ) as the A ∞ -algebra withdifferential and operations given by the duals of the differential in CE ∗ (( ˜ V , ˜ h ) , R × ( R × X ))for lifts ( ˜ V , ˜ h ) into R × X , where the lifts are related exactly as for smooth Lagrangians. HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 19
This then leads to new ways of representing objects in the Fukaya category of X , see [12]for immersed Lagrangian 2-spheres treated in this way.6.2.2. Energy filtration, fillings, and cobordisms.
We next consider the case when the singularLagrangian is allowed to have boundary at infinity. Again we start in the smooth case. Let L ⊂ X be an exact Lagrangian submanifold with Legendrian boundary ∂L . Then there is anatural map CE ∗ ( ∂L ; X ) −→ ( CF ∗ ( L ; X )) (cid:48) , where right hand side is the (completed) co-algebra dual to the Floer cohomology algebra,see [16], which is defined by counting curves with one positive puncture at a Reeb chord andseveral negative punctures at intersection points of a system of parallel copies of L . Here itis natural to replace the dual in the right hand side by the Chekanov–Eliashberg dg-algebraof a Legendrian lift of L into R × L , as in Section 6.2.1, and we get the corresponding map(6.1) CE ∗ ( ∂L ; X ) −→ CE ∗ ( L ; R × ( R × X )) . The projection of this chain map to the subalgebra generated by the unit in L is called an augmentation .In case ∂L is a sphere we may think of the augmentation in terms of Legendrian surgeryas follows. Let the co-core of the handle attached be C and let (cid:98) L be the Lagrangian L with the core disk attached. Then the Floer complex CF ∗ ( C, L ) has only one generator,the unique intersection point between the core and the co-core. However, the Floer complex CF ∗ ( C, (cid:98) L ) is a module over CW ∗ ( C ), where the module structure is obtained by countingdisks with several input (positive) punctures at generators of CW ∗ ( C ), one input genera-tor of CF ∗ ( C, (cid:98) L ), and one output generator of CF ∗ ( C, (cid:98) L ). By SFT-stretching it followsthat the surgery isomorphism CW ∗ ( C ) → CE ∗ ( ∂L ; X ) takes the module structure to theaugmentation.The cobordism map on the form (6.1), generalizes to the singular case. Consider a Wein-stein domain K ⊂ X such that in the ideal boundary of X , K agrees with V + × (( − (cid:15), (cid:15) ) × R ),or in terms of a compact model of K and X , there is some region near the boundary of X where K agrees with V × D ∗ (cid:15) ( − η, V × D ∗ (cid:15) ( − η, | { } lies in ∂X .We consider a handle decomposition H of K with, except for standard handles, also hashandles with boundary in the handle decomposition h + of V + . Here, the k -dimensional core∆ of a handle with boundary from H has boundary given by ( k − δ + in handles from h . In the compact model above it means that there is a neighborhood ofthe boundary of ( η,
0] where ∆ is a product, δ + × ( − η, K to a Legendrian embedding into R × X ,and add a K -handle K × D ∗ (cid:15) [ − ,
1] to X × R . We need to explain what this looks like at theboundary. To this end we consider two intervals [ − , (cid:48) and [ − , (cid:48)(cid:48) . At the boundary we addthe V -handle V × D ∗ (cid:15) [ − , (cid:48) , where the fiber at − ∈ [ − ,
1] is attached at 0 ∈ ( − η,
0] andwhere we use the standard Liouville vector field in the handle with a saddle point at (0 , K -handle now looks like ( V × D ∗ (cid:15) [ − , (cid:48) ) × D ∗ (cid:15) [ − , (cid:48)(cid:48) , where D ∗ (cid:15) [ − , (cid:48)(cid:48) correspond tothe Reeb and symplectization directions in X × R × R . We start from the Liouville vectorfield that is radial along fibers in D ∗ (cid:15) [ − , (cid:48)(cid:48) . This gives a Bott situation with a family ofReeb chords and orbits along the 0-section. We impose the boundary condition that these areperturbed out by a Liouville vector field that point into the handle at the point − ∈ [ − , X, K ). Note that using this model, the attaching locus for the top-dimensional handle cores L in H is a Legendrian cobordism ∂L which near the boundary looks like the product of the0-section in D ∗ (cid:15) [ − , (cid:48)(cid:48) and ∂l + , and hence after the perturbation has Legendrian ends inthe attaching spheres ∂l + , where l + are the top-dimensional core disks in V and ∂l + theirattaching spheres in ∂V .We define the dg-algebra CE ∗ (( K, H ); R × ( R × X )) in parallel with the dg-algebra above.It is generated by Reeb chords of the top handles L in H inside R × X and Reeb chordsof the Legendrian attaching cobordisms ∂L of these handles. In this setting we think ofthe Chekanov–Eliashberg dg-algebra CE ∗ (( V + , h + ); X ) as generated by Reeb chords in ∂X and of Reeb chords inside the V -handle as sitting at the ideal boundary of the positivesymplectization end of X and of ∂K , respectively. Counting disks with one positive punctureat these chords at infinity then gives a chain map CE ∗ (( V + , h + ); X ) −→ CE ∗ (( K, H ); R × ( R × X )) . By shrinking the handle we find that the sub-algebra CE ∗ ( ∂l + , ∂V ) maps to the sub-algebra CE ∗ ( ∂L ; ∂K ).Also the discussion about augmentations have counterparts in this set up. Here we considerthe Floer cohomology with the fiber of the singular Lagrangian. Consider the case thatthere are no Reeb chords of L in X × R then the counterpart of the augmentation is thedg-algebra map above followed by the projection to the subalgebra CE ∗ ( ∂L ; ∂K ) whichgives the module structure of the Floer cohomology CF ∗ ( C, K ) that must now be viewed asa Floer cohomology ’with coefficients’ in the dg-algebra of the link of the singularity of K .As the Floer cohomology itself is again very simple, the augmentation naturally takes valuesin the dg-algebra of this link of singularities, see Section 7 for examples. l + V +0 l − V − ∂ + X∂ − X K ∂L Figure 5.
The subcritical part of the Weinstein domain K in the Weinsteincobordism X . The cores of the top handles L are shaded in gray, and itsboundary in ∂K is drawn in blue.One can also extend the discussion here and allow the ‘cobordism’ ( K, H ) to have anegative end as well. We first take X to have a negative end ∂ − X that we assume is filledby a Weinstein manifold X − and in ∂ − X we have a Legendrian embedding ( V − , h − ). We HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 21 then require that (
K, H ) agrees with V − × (( − (cid:15), (cid:15) ) × R ) near the negative end and thatthe handle decomposition H of K is allowed to have handles with boundary. With such ahandle structure we can then again lift K to a Legendrian embedding with a cylindrical endin R × X . The only difference from the treatment above is that we perturb the Bott familyover D ∗ (cid:15) [ − , (cid:48)(cid:48) with a Liouville vector field that points out of the cobordism in the negativeend.The attaching locus for a top-dimensional handle is then a Legendrian cobordism ∂L withLegendrian ends in the attaching spheres ∂l + at the positive end and in ∂l − in the negativeend.Here we define the dg-algebra CE ∗ (( K, H ); R × ( R × X )) in parallel with the above asgenerated by Reeb chords of the top handles L in R × X , Reeb chords of the Legendrianattaching cobordisms ∂L , as well as Reeb chords between core disks l − and their attachingspheres ∂l − in the negative end. Here the two types of Reeb chords in the negative end forma subalgebra isomorphic to CE ∗ (( V − , h − ); X − ).We again think of the Chekanov–Eliashberg dg-algebra of ( V + , h + ) as generated by Reebchords of l + in ∂X and Reeb chords of ∂l + as sitting at the ideal boundary of the positivesymplectization end of X and of ∂K , respectively. Counting disks with one positive punctureat these chords at infinity then again gives a chain map CE ∗ (( V + , h + ); X ) −→ CE ∗ (( K, H ); R × ( R × X )) , where there are now chords and disks at the negative end in the right hand side.7. Computations, examples, and applications
In this section we first describe a method for computing the differential in Chekanov–Eliashberg dg-algebras when there is a global Reeb-projection. We then study a number ofexamples including non-existence results for Lagrangian fillings with restricted singularitiesand illustrations of how dg-algebras of singular Legendrians contain information of nearbysmooth Legendrians.7.1.
The differential for contactizations.
In this section we consider a useful way tocompute CE ∗ (( V, h ); W ) in the case that ∂W is a contactization. Consider thus the casewhen W = R × ( R × P ), where P is an exact symplectic manifold, R × P its contactizationand R × ( R × P ) the further symplectization. Note that there are no Reeb orbits in R × P which allows us to work without anchoring.Assume that V ⊂ R × P is a generic Legendrian embedding so that π | V , where π isthe projection projecting out R , is an embedding. Consider then a new contact manifold R × P ◦ , where P ◦ is obtained from P by removing V leaving the negative contact boundary ∂V and inserting in its place the negative end [0 , −∞ ) × ∂V . Another way to think aboutthis contact manifold is the manifold that results from R × P if all Reeb flow lines throughthe skeleton on V is removed.The projection of the core ( n − l of the top handles in h to P ◦ is then an immersedexact Lagrangian ¯ l with negative end ∂l in ∂V . By construction, the Legendrian lift l ⊂ R × P ◦ of the exact Lagrangian ¯ l has constant z -coordinate of equal value on all the componentsof ∂l in the negative end. We can then define the Legendrian dg-algebra CE ∗ ( l, R × P ◦ ) inthe standard way, with generators Reeb chords of double points of ¯ l and Reeb chords of ∂l in the negative end, compare [9]. Recall that we use an index definite Weinstein structureon V . This means anchoring is trivial if dim( V ) ≥ V ) = 2 we anchor at Reeb orbits in the negative end ∂V using punctured spheres in the filling V ≈ R , see Remarks2.7 and 2.8. Lemma 7.1.
The natural identification of Reeb chord generators gives a chain isomorphism CE ∗ ( l ; R × P ◦ ) ≈ −→ CE ∗ (( V, h ); R × ( R × P )) . Proof.
Using a complex structure on the V -handle as in Section 2.3 it is straightforward tocheck that curves contributing to the differential in the left and right hand sides above canbe identified:First, it follows from Lemma 2.5 that holomorphic disks with negative punctures at thechords in the handle must lie in V ,y ) -fibers, i.e., fibers over (0 , y ) in the model of the sym-plectization of G δ , which have the form ∂V × R . (Here the R -translation corresponds simplyto moving the curve in the fiber along the y -axis.) Second, curves in the symplectizationof R × P ◦ are determined up to translation by their projection to P ◦ . Consequently, the J V -biholomorphic map from ∂V × [0 , ∞ ) ⊂ P ◦ to the V -fibers over the y -axis relates thecurves in question. The lemma follows. (cid:3) Examples.
In this section we study Chekanov–Eliashberg dg-algebras for singular Leg-endrians in several examples.7.2.1.
The n -point algebra and T ∗ R from the point of view of singular Legendrians. Westudy Chekanov–Eliashberg dg-algebras in dimension 1 and consider the case left out inSection 1.1.
Example 7.2.
We discuss the relation between wrapped Floer cohomology and Chekanov–Eliashberg dg-algebras for Weinstein handles attached to the 2-disk, following [20]. Thedg-algebra involved will appear as ’singularity link dg-algebra’ in several examples below.Consider ( R , dx ∧ dy ) with ideal contact boundary ( S , ( xdy − ydx )). Let Λ be n distinctpoints in the ideal contact boundary and let V = T ∗ Λ ⊂ S . Since V is zero-dimensional,the only generators of CE ∗ (( V, h ); R ) are Reeb chords in S of the core disks of the tophandles l = Λ, and these are the following: • c ij for 1 ≤ i < j ≤ n , • c pij for 1 ≤ i, j ≤ n and p ≥ c pij is the Reeb chord starting at the i th point, ending at the j th , andpassing through the reference point ∗ p times. ∗ Figure 6.
Generator Reeb chords c , c , and c of CE ∗ (( V, h ); R ) whenΛ consists of three distinct points.Choosing Maslov potentials ( m (1) , . . . , m ( n )) for the points in Λ we get the followinggrading of the generators:(7.1) | c pij | = 1 − p − m ( j ) + m ( i ) . HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 23
The differential ∂ is given by ∂ ( c ij ) = n (cid:88) k =1 ( − m ( i )+ m ( k ) c kj c ik (7.2) ∂ ( c ij ) = δ ij + n (cid:88) k =1 ( − m ( i )+ m ( k ) c kj c ik + n (cid:88) k =1 ( − m ( i )+ m ( k ) c kj c ik (7.3) ∂ ( c pij ) = p (cid:88) (cid:96) =0 n (cid:88) k =1 ( − m ( i )+ m ( k ) c p − (cid:96)kj c (cid:96)ik , p ≥ , (7.4)where δ ij = e i = e j when i = j and δ ij = 0 otherwise. (The idempotents come from disksanchored in the R -filling.) In these formulas we use the convention c ij = 0 for i ≥ j . Asusual, the differential extends to all of CE ∗ (( V, h ); C ) by Leibniz rule. This means that CE ∗ (( V, h ); R ) is the internal algebra I n of [20, Definition 8]. C C C Figure 7. R V with co-core disks C = C ∪ C ∪ C when Λ is three distinct points.Add a stop at V and let C = C ∪ · · · ∪ C n be the union of the co-core disks dual to thetop handles in V . Then CW ∗ ( C ; R V ) is generated by n Reeb chords { c , . . . , c ( n − n , c n } in ∂ R V , where c ij denotes the unique Reeb chord starting at ∂C i and ending at ∂C j . Aftera choice of Maslov potentials ( m (1) , . . . , m ( n )) of the components of C we get the followinggrading: | c i ( i +1) | = 1 + m ( i + 1) − m ( i ) , ≤ i ≤ n − , | c n | = − m (1) − m ( n ) . Non-vanishing A ∞ -operations have as input any cyclic permutation of the cyclic sequence ofReeb chords ( c n , c ( n − n , . . . , c ). That is, for any 1 ≤ i ≤ n − m n ( c ( i − i , c ( i − i − , . . . , c n , c ( n − n , . . . , c ( i +1)( i +2) , c i ( i +1) ) = e i . Then [20, Corollary 11], shows that there is a quasi-isomorphism CW ∗ ( C ; R V ) ≈ I n = CE ∗ (( V, h ); R ). Example 7.3.
We study the remaining n = 2 case of the example Section 1.1 . Consider( R , dx ∧ dy + dx ∧ dy ), with ideal contact boundary ( S , ( x dy − y dx + x dy − y dx )).Consider the unknot Λ = ∂ R ⊂ S , and let V ⊂ S be a small neighborhood of the zero-section in T ∗ Λ. The handle decomposition of V is given by h = { N ( t ) } , and the 1-handle h is a cotangent neighborhood of the knot itself after removing the point t as in Figure 8. a t ∗ t Figure 8.
Lagrangian projection of the unknot in a Darboux chart and thehandle decomposition of its cotangent neighborhood.The right hand part of Figure 8 shows V = h with ∂l , which consists of the points denoted1 and 2 in ∂V . Let l denote the core disk of the top handle h .The Chekanov–Eliashberg dg-algebra of ∂l ⊂ ∂V is computed as in Example 7.2 andis generated by t and t pij for p ≥ ≤ i, j ≤
2. After choosing Maslov potential( m (1) , m (2)) = (1 ,
0) their gradings are | t pij | = 1 − p + m ( j ) − m ( i ).The Chekanov–Eliashberg dg-algebra CE ∗ (( V, h ); R ) is generated by the generators of CE ∗ ( ∂l ; V ) as described above and one additional generator a of degree −
1. By Lemma7.1, the differential ∂ is computed via Figure 8 as ∂a = 1 − t (7.5) ∂t = ∂t = 0(7.6) ∂t ij = δ ij − (cid:88) k =1 t kj t ik − (cid:88) k =1 t kj t ik i ≤ j (7.7) ∂t pij = p (cid:88) (cid:96) =1 2 (cid:88) k =1 ( − m ( i )+ m ( k ) t p − (cid:96)kj t (cid:96)ik , p ≥ . (Here δ ij is the Kronecker delta.) Note that our ground field in this case is C , sinceΛ is connected. As in [20, Theorem 12] we can consider the quasi-isomorphic model of CE ∗ (( V, h ); R ) with generators { a, t , t , t , t } and differential as in (7.5), (7.6) and(7.7). One can show that the homology is concentrated in degree 0 (see [20, Proposition 14])and we compute(7.8) CE (( V, h ); R ) ≈ C [ t , t ] / (cid:104) − t , − t t , − t t (cid:105) ≈ C , As in Section 1.1, R V ≈ T ∗ R and we get the desired quasi-isomorphisms: CE (( V, h ); R ) ≈ CW ∗ ( C ; R V ) = CW ∗ ( T ∗ ξ R ; T ∗ R ) ≈ C −∗ (Ω ξ R ) ≈ C . Singular Lagrangian fillings in R . We study dg-algebra maps of singular Lagrangianfillings in R and use them to show non-existence results for Lagrangians with restrictedsingularities. Throughout we write R for the standard symplectic 4-space with contactform dx ∧ dy + dx ∧ dy . We will think of R in two ways, as the symplectization of R with contact form dz − ydx and as the Weinstein manifold with a single 0-handle. Wewill often present front pictures of Legendrians (projections to the xz -plane) and also forLagrangians in R when viewed the symplectization of R where Lagrangians appear asasymptotically conical fronts, see [14, Section 2] for details. HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 25
Example 7.4.
Let Γ be the singular exact Lagrangian cobordism in R with front as inFigure 9. Let Λ + and Λ − denote the positive and negative boundaries of Γ respectively. Let( V ± , h ± ) be Weinstein thickenings of Λ ± . Let K ⊂ R be a Weinstein domain which agreeswith ( − (cid:15), (cid:15) ) × V + and ( − (cid:15), (cid:15) ) × V − in the positive and negative ideal contact boundaries of R , see Section 6.2.2. Let H be a handle decomposition of K induced by Γ, and denote theunion of the core disks of the critical handles of H by L . a − a − x y ˆ xa +1 b a +2 Λ + Λ − Figure 9.
The front of the singular saddle cobordism Γ.Consider CE ∗ (( K, H ); R × ( R × R )) as described in Section 6.2. It contains the dg-subalgebra CE ∗ ( ∂L ; ∂K ) ⊂ CE ∗ (( K, H ); R × ( R × R )) . Recall that the boundary condition at the negative end of the K -handle means that theLiouville field points out of the handle which means there is a maximum along core of theattaching 1-handle with boundary in ∂K . We indicate the location of this maximum byˆ x in Figure 9 and denote the corresponding dg-algebra generators by (cid:8)(cid:98) x pij (cid:9) . Together withthe generators of the three point algebras in the negative end these form the Chekanov–Eliashberg dg-algebra (cid:99) I of 3 distinct points in S , multiplied by the 0-section in T ∗ R witha maximum and two minima in it. Apply [15, Corollary 5.6] to Example 7.2 to see thedifferential d :(7.9) d (cid:98) x pij = x pij − y pij + G ( ∂ (cid:98) x pij ) , where ∂ (cid:98) x pij is the differential of (cid:98) x pij regarded as a generator of the 3-point algebra I and G is the following operator on monomials: G ( (cid:98) x p i j (cid:98) x p i j . . . (cid:98) x p m i m j m ) = (cid:98) x p i j x p i j . . . x p m i m j m + ( − | x p i j | y p i j (cid:98) x p i j . . . x p m i m j m + · · · + ( − | x p i j | + ··· + | x pm − im − jm − | y p i j y p i j . . . (cid:98) x p m i m j m . Holomorphic curves in R × ( R × R ) with boundary on R × K can be understood throughMorse flow trees, which here since there are only two sheets locally are in fact flow lines. They give the cobordism dg-algebra map defined on generators as followsΦ : CE ∗ (( V + , h + ); R ) −→ CE ∗ (( K, H ); R × R × R ) ,a +1 (cid:55)−→ a − + (cid:98) x ,a +2 (cid:55)−→ a − ,b (cid:55)−→ y . The disks giving the first terms in the first two equations correspond to straight flow lines.The last equation comes from the flow line starting at b and hitting the singular locus andthen continuing as a flow line in the singular locus to negative infinity. The second termin the first equation can be understood as coming from the 1-parameter family of disksobtained by gluing the strip at positive infinity with positive puncture at a and negativepuncture at b to the disk from b to the singular locus. This one parameter family travelsinto the cobordism and gets rigidified when it hits (cid:98) x . The rigidified disk then has a negativepuncture at (cid:98) x . We verify algebraically that it indeed is a dg-algebra map:( d ◦ Φ)( a +1 ) = d ( (cid:98) x + a − ) = 1 + y = Φ(1 + b ) = (Φ ◦ ∂ )( a +1 )( d ◦ Φ)( a +2 ) = d ( a − ) = 1 + y = Φ(1 + b ) = (Φ ◦ ∂ )( a +2 )( d ◦ Φ)( b ) = d ( y ) = (0 ,
0) = (Φ ◦ ∂ )( b ) . Example 7.5.
Let Λ ⊂ R be the singular Legendrian whose front projection is in Figure10. Let ( V, h ) be a Weinstein thickening of Λ. We show that Λ does not admit any singularLagrangian filling with only ’Y-singularities’, i.e., Γ is a union of smooth 2-dimensionalLagrangian strata that meets along S × Y , where Y is a trivalent graph and S is a 1-manifold with at least one component which has boundary on the trivalent Y graphs aroundthe singularities of Λ.To see this, assume that Γ is such a filling and let K ⊂ R be its Weinstein thickening. Let H be a handle decomposition of K with critical core disks L and with boundary on the coredisks of Λ. We compute CE ∗ (( K, H ); R × ( R × R )) and show that there is no dg-algebra x ya a Figure 10.
The front projection of the singular Legendrian Λ.map CE ∗ (( V, h ); R ) → CE ∗ (( K, H ); R × ( R × R )) to conclude that no such singular exactLagrangian filling exists.Since there are no double points of L , CE ∗ (( K, H ); R × ( R × R )) ≈ CE ∗ ( ∂L ; ∂K ). Since ∂L is the product of the 3-point Legendrian in S and a compact 1-manifold S we findthat CE ∗ ( ∂L ; ∂K ) is a free product of algebras one for each component S over the ringof idempotents of the 2-cells. The algebra corresponding to a closed component is (cid:99) I , see(7.9) with generators x pij and y pij identified. Here we can think of the generators as sittingat a maximum and a minimum in the circle component. The algebra corresponding to thecomponent with boundary at infinity is I itself, see Example 7.2, with generators sittingat a minimum, recall the inwards boundary condition at positive infinity . We consider the HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 27 map CE ∗ (( V, h ); R ) → CE ∗ (( K, H ); R × ( R × R )) → I , where the last map is the projection to the sub-algebra corresponding to the component of S with boundary. We write { x pij } and { y pij } for the generators of the two copies of I at thesingularities of Λ and { q x pij } for the generators of I corresponding to the minimum in thenon-closed component of S . The differential in CE ∗ (( V, h ); R ) is given on the generators a and a by ∂a = e − x ∂a = e − y . The differential on the generators { x pij } and { y pij } is given by (7.2), (7.3) and (7.4) and hasthe following property. For any generator c ∈ I , either ∂c or ∂c − e i , for some i ∈ { , , } , isa sum of word length 2 generator monomials. It follows that for any monomial of generators w of word length k , ∂w = s k − + s k +1 where s k ± are sums of monomials of word length k ± s k +1 is not the empty word.The dg-algebra map φ induced by the proposed singular exact Lagrangian filling Γ actsas follows on { x pij } and { y pij } : φ ( x pij ) = φ ( y pij ) = q x pij , and therefore(7.10) ∂ ( φ ( a )) = φ ( ∂a ) = e + q x . Equation (7.10) contradicts I being non-trivial as follows. Write φ ( a ) = t e + t o , where t e and t o are linear combinations of monomials of even and odd word length, respectively.Since the differential changes word length mod 2 it follows that ∂t o = e which is not truein I .One can also show the non-existence of Γ from a geometrical point of view. Concatenatingthe singular exact Lagrangian cobordism in Example 7.4, which is depicted in Figure 9,with Γ gives a singular exact Lagrangian filling of the unknot. Removing top dimensionalcomponents that have no boundary at infinity, one would construct an embedded exactLagrangian filling of the unknot of genus ≥
1, but such a filling does not exist.
Example 7.6.
Consider R , as the completion of the symplectic ball with ideal contactboundary standard contact S . Consider the singular exact LagrangianΓ = { ( x , x ) } ∪ { ( y , x ) | y ≥ } ⊂ R . We point out that if we identify R with T ∗ R then Γ ≈ R ∪ L + { x =0 } ⊂ T ∗ R where L + { x =0 } is the positive conormal bundle of the hypersurface { x = 0 } ⊂ R . Thus, Γ is an arborealsingularity, see [22, 4, 3].Let Λ = Γ ∩ S be the singular Legendrian boundary of Γ. After Legendrian isotopy, Λ liesin a Darboux ball identified with an open subset of standard contact R , and Λ appears as inFigure 11. Consider V = T ∗ Λ ⊂ S with handle decomposition h as indicated in Figure 11.Let K ⊂ R be a Weinstein domain which agrees with ( − (cid:15), (cid:15) ) × V in the ideal contactboundary of R . Let H be a handle decomposition of K induced by Γ, and denote the unionof the core disks of the critical handles of H by L . The Chekanov–Eliashberg dg-algebra of( V, h ) is generated by the long Reeb chords a , b and by the collection (cid:8) x pij (cid:9) ∪ (cid:8) y pij (cid:9) which ba xy l l l
31 2 ∗
31 2 ∗ xy Figure 11.
Lagrangian projection of Λ in a Darboux ball (left) with magni-fied singularity links (right).generates the dg-subalgebra CE ∗ ( ∂l ; ∂V ), and is equal to two copies of I , see Example7.2. The differential ∂ in CE ∗ (( V, h ); R ) of the generators a and b is given by ∂a = e + y bx + y x − y x ∂b = x − y , and on the generators x pij and y pij by (7.2), (7.3) and (7.4).The Chekanov–Eliashberg dg-algebra of ( K, H ) is CE ∗ ( ∂L ; ∂K ) = I , see Example 7.2.Denote the generators of CE ∗ ( ∂L ; ∂K ) by (cid:8) q x pij (cid:9) . The induced dg-algebra map is then ε : CE ∗ (( V, h ); R ) −→ CE ∗ (( K, H ); R × ( R × R )) a (cid:55)−→ q x x pij , y pij (cid:55)−→ q x pij b (cid:55)−→ . The second equation comes from flow lines along the interval of singularities starting atinfinity and ending at the minimum q x . To see the first equation, follow the family of holo-morphic disks at infinity with boundary on l into the core disk with boundary L . At someinstance the boundary hits the singularity link, projecting to a complex line perpendicularto the complexification of the line of singularities, one finds that this disk is asymptotic tothe chord that goes once around. The contributing configuration is then this disk with aflow line in the manifold of once around chords that ends at the minimum q x attached. Example 7.7.
As in Example 7.6 we consider the singularity link of an arboreal singularityin R . Let L = { ( x , x ) } ∪ { ( y , x ) | y ≥ } ∪ { ( x , iy ) | y ≤ } ⊂ R . Then using R ≈ T ∗ R , we have Γ ≈ R ∪ L + { x =0 }∪{ x =0 } ⊂ T ∗ R where L + { x =0 }∪{ x =0 } is thepositive conormal bundle of the hypersurface { x = 0 } ∪ { x = 0 } ⊂ R , where we assumethat { x = 0 } and { x = 0 } have been equipped with opposite co-orientations, and Γ is anarboreal singularity.Let Λ = Γ ∩ S and consider V = T ∗ Λ ⊂ S with handle decomposition h as indicatedin Figure 12. Let l denote the union of the core disks of the top handles. Let K ⊂ R be a HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 29
Weinstein domain which agrees with ( − (cid:15), (cid:15) ) × V in the ideal contact boundary of R . Let H be a handle decomposition of K with boundary h . Denote the union of the core disksof the critical handles of H by L . The Chekanov–Eliashberg dg-algebra CE ∗ (( V, h ); R ) is b a a yx wv Figure 12.
Lagrangian projection of the singularity link Λ.generated by Reeb chords of l ⊂ S , denoted a , a and b in Figure 12. Other generators areReeb chords of ∂l ⊂ ∂V , i.e., of the singularity links at x, y, z, w : (cid:8) x pij (cid:9) ∪ (cid:8) y pij (cid:9) ∪ (cid:8) v pij (cid:9) ∪ (cid:8) w pij (cid:9) , where the notation is as usual. This collection generates the dg-subalgebra CE ∗ ( ∂l ; V )which contains four copies of I , see Example 7.2.The differential ∂ on a , a , b is ∂a = e + v bx − v w x + v y x ∂a = e − y bw − y x w + y v w ∂b = y x − v w . As in Example 7.5, we show that Λ does not admit a singular exact Lagrangian fillingΓ with only Y-singularities. Assume that such a filling Γ exists. Then since Λ has fourY-singularities, Γ is a union of smooth 2-dimensional strata meeting along S × Y where Y is a trivalent graph and S is a 1-manifold with at least two components with boundaryon the vertices of Λ. The two components with boundary subdivides the four collection (cid:8) x pij (cid:9) , (cid:8) y pij (cid:9) , (cid:8) v pij (cid:9) , and (cid:8) w pij (cid:9) into two pairs connected by the components, and Γ inducesa dg-algebra map ε : CE ∗ (( V, h ); R ) −→ CE ∗ (( K, H ); R × ( R × R )) −→ I ∗ I , where I ∗ I denotes the free algebra of two copies of I over the ring of idempotents. Wefind that ∂ ( ε ( b )) = ε ( ∂b ) = ε ( y ) ε ( x ) − ε ( v ) ε ( w )(7.11) ∂ ( ε ( a )) = ε ( ∂a ) = e + ε ( v ) ε ( b ) ε ( x ) − ε ( v ) ε ( w ) ε ( x ) + ε ( v ) ε ( y ) ε ( x )(7.12)As in Example 7.5, the differential on I changes word length mod 2 and the same is truefor the differential on I ∗ I . We conclude first from (7.11) that ε ( b ) is a sum of monomialsof generators of odd word length. Hence we have from (7.12) that ∂ ( ε ( a )) = e + r o , where r o is a sum of monomials of generators of odd word length. Then as in Example 7.5, write ε ( a ) = t o + t e , with t e and t o linear combinations of monomials of even and odd wordlength and conclude ∂t o = e . Since no such equation holds in I ∗ I , the singular exactLagrangian filling Γ with only Y-singularities cannot exist. Example 7.8.
As in Examples 7.6 and 7.7 we consider the singularity link of an arborealsingularity in R . Let Γ := R ∪ L + { x =0 } { x ≤ , x = x } ⊂ T ∗ R where L + { x =0 } { x ≤ , x = x } is the positive conormal bundle of { x = 0 } ∪ { x ≤ , x = x } ⊂ R . Then Γ is the arboreal A -Lagrangian, [22, 4, 3]. The Lagrangian projection of Λ isshown in Figure 13. As above we let ( V, h ) be a fattening of Λ and Let (
K, H ) ⊂ R be afilling. The Chekanov–Eliashberg dg-algebra CE ∗ (( V, h ); R ) is generated by the long Reeb ba xyw va Figure 13.
The figure shows the Lagrangian projection of Λ.chords a , a , b of l ⊂ S . The differential ∂ is given on the generators a , a , b by ∂a = w + y bx − y x + y v x ∂a = e − w x v − w ( y v + ( a x − y x + y bx + y v x ) v ) ∂b = v x − y . The differential on generators x pij , y pij , v pij , w pij is as usual.As in Example 7.7, we show that Λ does not admit any singular exact Lagrangian fillingΓ with only Y-singularities: such Γ would induce a dg-algebra map ε : CE ∗ (( V, h ); R ) −→ CE ∗ (( K, H ); R × ( R × R )) −→ I ∗ I and we would have ∂ ( ε ( b )) = ε ( ∂b ) = ε ( v ) ε ( x ) − ε ( y ) . Recall from Example 7.7 that the differential in I ∗ I changes word length mod 2. Wewrite ε ( b ) = t o + t e , where t e and t o are linear combinations of monomials of even and oddword length, respectively. Then ∂ ( t e ) = ε ( y ), but this is not true in I ∗ I , since ε ( y )equals a short chord in CE ∗ ( ∂L ; ∂K ) and such a chord is not homologous to 0 in I . Example 7.9.
Consider R with its standard symplectic form and ideal contact boundarystandard contact S . As in the R -examples above, we draw Legendrians in a Darboux chart HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 31 of S that we think of as standard contact R . We will consider a singular Legendrian Λ ⊂ S which has one Reeb chord of length zero. Resolving this double point we get embeddedLegendrian tori. We show here the dg-algebra of the singular Legendrian Λ when equippedwith suitable augmentations of its singularity link subalgebra is dg-algebra equivalent to anynearby smooth Legendrian torus, compare Corollary 5.4. The smooth nearby Legendriantori were studied in [11].Consider the singular Legendrian submanifold Λ with front as in Figure 14. The singularityof Λ is one immersed point (which can be though of as a Reeb chord of length zero). Let V = T ∗ Λ ⊂ S be a Weinstein thickening of Λ with handle decomposition h . Here h has one0-handle centered at the singular double points of Λ and one 1-handle which together form S × D . Finally there is one 2-handle. We denote its core disk l . The attaching sphere ∂l for l can then be described as follows. The intersection of the boundary of the 0-handles withΛ is a Legendrian Hopf link in S . The 1-handle is attached on this Hopf link, connectingits components by the standard two strand Legendrian through the handle, see [17]. a b ah h Figure 14.
The front of the singular Legendrian Λ with an immersed point.The 0-handle is a neighborhood of the immersed point and the 1-handle is aneighborhood of a curve connecting the singular point to itself. The family ofexternal Reeb chords is indicated in green.The dg-algebra of (
V, h ) is then the following. At the minimum of the 1-handle sits thedg-algebra I , we denote its generators c pij . The differential is as in Example 7.2. Werepresent the singularity link as the boundary of two transverse planes. This dg-algebra isquasi-isomorphic to the standard Hopf link dg-algebra of [12, Section 3]. Reeb chords ofthe Hopf link in S come in pairs of S -families of length kπ , k >
0. We will use only theshortest chord families which after Morsification give rise to two chords each: p and (cid:98) p , and q and (cid:98) q . Since the 1-handle connects the Hopf link components the differential is as follows,see Figure 15: ∂p = ∂q = 0 , ∂ (cid:98) p = p − c pc , ∂ (cid:98) q = q − c qc . Also the exterior Reeb chords come in an S -family. After Morsification we get two chords (cid:98) a and a . The differential is as follows: ∂ (cid:98) a = a − c ac + (cid:98) p − c ,∂a = e − p, see Figure 15.The left hand picture shows the differential of (cid:98) p : both flow lines of Reeb chord endpointshit the attaching locus of h , enter the handle and hit the short chords in the middle of thehandle. p b p c l l b a + b a − a + a − b p + b p − Figure 15.
The differential.The right hand picture shows the boundary of the curves that contribute to the differentialof the exterior chords on a cylinder parameterizing the singular Legendrian, the boundarymaps to the core disk of the 0-handle, l . The differential of a is the vertical line connecting a + to a − and the vertical lines connecting to the boundary. The disks for ∂ (cid:98) a are the linewhich is tangent to the attaching locus of the 1-handle. It gives c , a flow line to theminimum in the 1-handle connects the tangency point to c . The pair lines intersectingthe 1-handle and ending at a ± gives c ac , and the pair ending at a ± not intersecting the1-handle gives a , flow lines to the short chords are split off at the intersections. Finally thelines to (cid:98) p ± gives the contribution (cid:98) p .The Legendrian tori in [11] are obtained by resolving the double point of Λ in two dif-ferent ways, as a Lagrangian cone and as a cusp edge. This in turn correspond to distinctLagrangian fillings of the Hopf link which induces different augmentations of its dg-algebra,see [12, Equations (4.2) and (4.4)]. The resulting augmentations here are (cid:15) and (cid:15) (cid:48) : (cid:15) ( c ) = (cid:15) (cid:48) ( c ) = λ, (cid:15) ( c ) = (cid:15) (cid:48) ( c ) = λ − ,(cid:15) ( p ) = µ, (cid:15) (cid:48) ( p ) = µ − µλ. References [1] Mohammed Abouzaid. On the wrapped Fukaya category and based loops.
J. Symplectic Geom. ,10(1):27–79, 2012.[2] J. F. Adams. On the cobar construction.
Proc. Nat. Acad. Sci. U.S.A. , 42:409–412, 1956.[3] Daniel ´Alvarez-Gavela, Yakov Eliashberg, and David Nadler. Arborealization III: Positive arborealiza-tion of polarized Weinstein manifolds. preprint, arXiv:2011.08962 , 2020.[4] Daniel ´Alvarez-Gavela, Yakov Eliashberg, and David Nadler. Arborealization I: Stability of arborealmodels. preprint, arXiv:2101.04272 , 2021.[5] Byung Hee An and Youngjin Bae. A Chekanov–Eliashberg algebra for Legendrian graphs.
Journal ofTopology , 13(2):777–869, 2020.[6] Johan Asplund. Fiber Floer cohomology and conormal stops. to appear in J. Symplectic Geom.,arXiv:1912.02547 , 2019.[7] Fr´ed´eric Bourgeois, Tobias Ekholm, and Yasha Eliashberg. Effect of Legendrian surgery.
Geom. Topol. ,16(1):301–389, 2012. With an appendix by Sheel Ganatra and Maksim Maydanskiy.[8] Roger Casals and Emmy Murphy. Legendrian fronts for affine varieties.
Duke Math. J. , 168(2):225–323,2019.[9] Baptiste Chantraine, Georgios Dimitroglou Rizell, Paolo Ghiggini, and Roman Golovko. Floer theoryfor Lagrangian cobordisms.
J. Differential Geom. , 114(3):393–465, 2020.
HEKANOV–ELIASHBERG DG-ALGEBRAS FOR SINGULAR LEGENDRIANS 33 [10] K. Cieliebak, T. Ekholm, and J. Latschev. Compactness for holomorphic curves with switching La-grangian boundary conditions.
J. Symplectic Geom. , 8(3):267–298, 2010.[11] Georgios Dimitroglou Rizell. Knotted Legendrian surfaces with few Reeb chords.
Algebr. Geom. Topol. ,11(5):2903–2936, 2011.[12] Georgios Dimitroglou Rizell, Tobias Ekholm, and Dmitry Tonkonog. Refined disk potentials for im-mersed Lagrangian surfaces. to appear in J. Differential Geom., arXiv:1806.03722 , 2018.[13] Tobias Ekholm. Holomorphic curves for Legendrian surgery. preprint, arXiv:1906.07228 , 2019.[14] Tobias Ekholm, Ko Honda, and Tam´as K´alm´an. Legendrian knots and exact Lagrangian cobordisms.
J. Eur. Math. Soc. (JEMS) , 18(11):2627–2689, 2016.[15] Tobias Ekholm and Tam´as K´alm´an. Isotopies of Legendrian 1-knots and Legendrian 2-tori.
J. SymplecticGeom. , 6(4):407–460, 2008.[16] Tobias Ekholm and Yanki Lekili. Duality between Lagrangian and Legendrian invariants. preprint,arXiv:1701.01284 , 2017.[17] Tobias Ekholm and Lenhard Ng. Legendrian contact homology in the boundary of a subcritical Wein-stein 4-manifold.
J. Differential Geom. , 101(1):67–157, 2015.[18] Tobias Ekholm, Lenhard Ng, and Vivek Shende. A complete knot invariant from contact homology.
Invent. Math. , 211(3):1149–1200, 2018.[19] Yakov Eliashberg and Emmy Murphy. Lagrangian caps.
Geom. Funct. Anal. , 23(5):1483–1514, 2013.[20] Tolga Etg¨u and Yankı Lekili. Fukaya categories of plumbings and multiplicative preprojective algebras.
Quantum Topol. , 10(4):777–813, 2019.[21] Sheel Ganatra, John Pardon, and Vivek Shende. Covariantly functorial wrapped Floer theory on Liou-ville sectors.
Publ. Math. Inst. Hautes ´Etudes Sci. , 131:73–200, 2020.[22] David Nadler. Arboreal singularities.
Geom. Topol. , 21(2):1231–1274, 2017.[23] Zachary Sylvan. On partially wrapped Fukaya categories.
J. Topol. , 12(2):372–441, 2019.
Department of mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden
Email address : [email protected] Department of mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden andInstitut Mittag-Leffler, Aurav 17, 182 60 Djursholm, Sweden
Email address ::