Legendrian contact homology for attaching links in higher dimensional subcritical Weinstein manifolds
LLEGENDRIAN CONTACT HOMOLOGY FOR ATTACHING LINKSIN HIGHER DIMENSIONAL SUBCRITICAL WEINSTEINMANIFOLDS
CECILIA KARLSSON
Abstract.
Let Λ be a link of Legendrian spheres in the boundary of a subcriticalWeinstein manifold X . We show that the computation of the Legendrian contacthomology of Λ can be reduced to a computation of Legendrian contact homology in1–jet spaces. Since the Legendrian contact homology in 1–jet spaces is well studied,this gives a simplified way to compute the Legendrian contact homology of Λ.We restrict to the case when the attaching spheres of the subcritical handles of X do not interact with each other, and we will only consider mod 2 coefficients for now.The more general situation will be addressed in a forthcoming paper.As an application we compute the homology of the free loop space of CP . Introduction A Weinstein manifold is the symplectic counterpart of a Stein manifold in complexgeometry. More precisely, any Weinstein manifold X of dimension 2 n can be givena handle decomposition into symplectic handles of index at most n . The handlesare attached along isotropic spheres in the contact boundary of X , and after havingattached the handles of index less than n we get a subcritical Weinstein manifold . Thetop index handles are then attached along Legendrian spheres in the contact boundaryof the subcritical part of X . It has been shown that much of the symplectic topologyof X is encoded in the Legendrian attaching spheres. Indeed, the symplectic homologyof any subcritical Weinstein manifold vanishes, and by [BEE12] it follows that thesymplectic homology of X is isomorphic to the Hochschild homology of the Chekanov-Eliashberg DGA of the attaching link.The Chekanov-Eliashberg DGA A ( V, Λ) of a Legendrian Λ in a contact manifold(
V, λ ) is freely generated by the
Reeb chords of Λ. These are solution curves to theReeb vector field R λ associated to λ , defined by λ ( R λ ) = 1 , dλ ( R λ , · ) = 0, and the Reebchords should have start and end point on Λ. The grading is given by a Maslov-typeindex, and the differential counts certain pseudo-holomorphic curves. In the case when( V, λ ) is the contact boundary of a Weinstein manifold X , this is given by a count ofpseudo-holomorphic disks in R × V , capped off with pseudo-holomorphic planes in X .Such disks are called anchored in X . See Section 2.3. Another special case is when V isthe 1–jet space J ( M ) of a smooth manifold M . Then one counts pseudo-holomorphicdisks either in the symplectization of J ( M ) or in the Lagrangian projection T ∗ M . Thiscase is rather well-studied, and there are a number of computational tools available,even in higher dimensions. See e.g. [EES05, EES07, Ekh07, Kar16]. a r X i v : . [ m a t h . S G ] J u l CECILIA KARLSSON
In this paper we describe a setup where the Chekanov-Eliashberg DGA of the attach-ing spheres in the boundary V of a subcritical Weinstein manifold X with c ( X ) = 0can be computed from Legendrians in 1–jet spaces. In this way we don’t have toconsider pseudo-holomorphic disks anchored in X .This is a generalization of the work in [EN15], where the Chekanov-Eliashberg DGAis computed in the boundary of subcritical Weinstein 4-manifolds. We will assumethat n >
2, and we focus on a simplified situation where the attaching spheres of thesubcritical handles do not interact with each other. We also restrict to Z -coefficients.The more general situation will be dealt with in a forthcoming paper, together with acareful treatment of signs so that we can compute the Legendrian contact homologyof the attaching link over Z .To obtain our result, we need Λ to satisfy some assumptions when passing throughthe subcritical handles. Namely, if Λ passes through a handle of index k < n we assumeit to be of the form D k × Λ sub in the handle. Here D k is the core of the subcriticalhandle and Λ sub ⊂ S n − k − is a Legendrian submanifold with respect to the standardcontact structure. We also assume that Λ is contained in a 1–jet neighborhood ofΛ st = D k × Λ st , sub when passing through the handle, where Λ st , sub ⊂ S n − k − = { z ∈ C n − k ; | z | = 1 } is the standard Legendrian unknot, given by the real part of S n − k − .In addition we assume that the part of Λ outside the sub-critical handles is containedin a Darboux ball D a ⊂ S n − , which we then identify with a ball in J ( R n − ).In this way we cover Λ with charts of Legendrians in 1–jet spaces. This is in generalnot enough to be able to compute the Chekanov-Eliashberg DGA of Λ in 1–jet spaces,since we might have pseudo-holomorphic disks that leave the Darboux ball and the1–jet neighborhood of Λ st . To remedy this problem, we Legendrian isotope Λ in aneighborhood of the attaching regions for the subcritical handles, by performing ahigh-dimensional analogue of the dipping procedure in [Sab06]. As a result, we areable to to split the Chekanov-Eliashberg DGA of Λ in different parts:(1) For each handle of index k < n − A ( J ( R n − k − ) , Λ sub ).(2) For each handle of index k = n − J (Λ st ) and where the Legendrian is a dipped version of D k × Λ sub .(4) Finally, we get a sub-DGA A ( J ( R n − ) , Λ ∩ D a ).We will prove the following in the case when there is only one sub-critical handle of X . Theorem 1.1. [Proposition 3.6 and Proposition 3.7] The sub-DGAs (1) – (4) togethergive a DGA which is quasi-isomorphic to A ( V, Λ) . As an application we describe a Weinstein handle decomposition of T ∗ CP and com-pute the Chekanov-Eliashberg DGA of the index 4 attaching sphere. Using the relationbetween the Legendrian contact homology of the attaching spheres and the symplec-tic homology of the resulting Weinstein manifold [BEE12] together with the results of[AS06, Vit18, SW06], which relate the symplectic homology of T ∗ M with the singularhomology of the free loop space of M , this gives a description of the singular homologyof the free loop space of CP . CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 3
From [BEE12] it also follows that there is a relation between the Chekanov-EliashbergDGA of the Legendrian attaching spheres and the wrapped Fukaya category of the co-cores of the critical handles. By recent results in [GPS17, CDGG17] these cocoresgenerate the wrapped Fukaya category of the resulting Weinstein manifold. In [CM19]the authors use this together with the formula in [EN15] to give examples of mir-ror manifolds in homological mirror symmetry. Similar calculations are performedin [ACG + Outline.
In Section 2 we fix notation, give a brief introduction to Weinstein manifoldsand define Legendrian contact homology in contact manifolds which are Weinsteinfillable. We also describe the easier case when the contact manifold is a 1–jet spaceof a smooth manifold. In Section 3 we explain the assumptions needed for us toshow that Legendrian contact homology in the boundary of a subcritical Weinsteinmanifold reduces to a computation of Legendrian contact homology in some different1–jet spaces. We also describe the dipping procedure and give a more careful statementof Theorem 1.1. In Section 4 we give proofs of the results in Section 3. In Section 5we use a Weinstein handle decomposition of T ∗ CP to compute the singular homologyof the free loop space of CP . Acknowledgments.
We thank Tobias Ekholm for insightful discussions.2.
Background A Weinstein manifold is a symplectic manifold (
X, ω ) equipped with a Liouville vec-tor field Z and a Morse function which is gradient-like for Z . Along the boundary V of X we get an induced contact structure with contact form λ = ι Z ω . The Morse functionallows us to give a handle decomposition of X into Weinstein handles, defined below,and where the handles are attached along isotropic spheres in the contact boundary.If dim X = 2 n , these handles are of index at most n , and if X only have handles ofindex less than n we say that X is subcritical . A contact manifold ( V, λ ) that occursas the boundary of some Weinstein manifold as above is called
Weinstein fillable .2.1.
Notation.
For u = ( u , . . . , u l ) ∈ R l we write du = ( du , . . . , du l ) ∈ T ∗ R l ∂ u = ( ∂ u , . . . , ∂ u l ) ∈ T R l CECILIA KARLSSON and if du, dv ∈ T ∗ R l we write vdu = l X i =1 v i du i du ∧ dv = l X i =1 du i ∧ dv i . If M is a k -dimensional smooth manifold we write ( u, v, r ) = ( u , . . . , u k , v , . . . , v k , r )for the coordinates of the 1–jet space J ( M ) = T ∗ M × R of M , where u are thecoordinates on M , v are the cotangent coordinates and r is the coordinate in the R -direction.2.2. Geometry.
Let X be a subcritical Weinstein manifold of dimension 2 n , let V beits contact boundary. Assume that X admits a Weinstein handle decomposition X = B n ∪ n − [ k =1 k l [ i =1 H ki , where H k , . . . , H kk l are Weinstein handles of index k . We will use the following modelof such a handle: H k = { ( x, y, p, q ) ∈ R n − k × R k : − δ ≤ n − k X j =1
12 ( x j + y j ) + k X j =1 p j − q j ≤ δ } , where δ > Z = n − k X j =1
12 ( x j ∂ x j + y j ∂ y j ) + k X j =1 p j ∂ p j − q j ∂ q j with respect to the standard symplectic form ω st = dx ∧ dy + dp ∧ dq , and this vectorfield is transverse to the boundary H k ± = { ( x, y, p, q ) ∈ H k ; n − k X j =1
12 ( x j + y j ) + k X j =1 p j − q j = ± δ } , and points out of H k along H k + and into H k along H k − . If now X is a Weinsteinmanifold with contact boundary V , and if Υ ⊂ V is an isotropic sphere of dimension k − X along a neighborhood of Υ in V and the Liouville vector field of H k along H k − to attach H k to X to get a newWeinstein manifold X = X ∪ H k . See [Wei91]. The boundary of X will again be acontact manifold, where the contact form on H k + is given by(2.2) α + = ω st ( Z, • ) H k + = n − k X j =1
12 ( x j dy j − y j dx j ) + k X j =1 p j dq j + q j dp j . CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 5
Remark 2.1.
If (
V, λ ) is the contact boundary of a subcritical Weinstein manifoldthen we will identify it with standard contact S n − with handles attached, so that V = S n − ∪ n − [ k =1 k l [ i =1 H ki, + . LCH in fillable contact manifolds.
Here we give a brief overview of the defi-nition of Legendrian contact homology in the boundary of a Weinstein manifold. Werefer to [EGH00, BEE12, Ekh19] for a more thorough treatment. Let (
V, λ ) be the con-tact boundary of a Weinstein manifold X , and let Λ ⊂ V be a Legendrian submanifold.Assume that c ( X ) = 0. Then we define the Legendrian contact homology (LCH) of Λto be the homology of the differential graded algebra (DGA) A ( V, Λ) which is definedas follows.The algebra is freely generated over Z [ H ( X, Λ)] by the
Reeb chords of
Λ, whichare solution curves of the Reeb vector field R λ having start and end point on Λ. Thechords are graded by a Maslov type index, called the Conley-Zehnder index µ cz . Thatis, if c is a Reeb chord of Λ then the grading of c is given by(2.3) | c | = µ cz ( γ c ) − , where γ c is a closed path in V from the end point c + of c in Λ, going through Λ to thestarting point c − of c and then follows c to the end point c + in the case when c − and c + belong to the same component of Λ. In the case when the start and end point belongto different components of Λ the path γ c also contains a path from the componentof Λ containing c + to the component of Λ containing c − . The Conley-Zehnder indexmeasures how much the contact distribution ξ = Ker λ rotates along this path.The differential is defined by a count of anchored pseudo-holomorphic curves in X ,as follows. Let J be an almost complex structure on X which is compatible with thesymplectic form and which is cylindrical in a neighborhood R t × V of the boundary V , meaning that it is invariant under translations in the R –factor, gives a complexstructure on Ker λ and satisfies that J ( ∂ t ) = R λ .An anchored pseudo-holomorphic disk is a two-level J -holomorphic building, wherethe top level is given by a J -holomorphic map u : ( D m , ∂D m ) → ( R × V, R × Λ) , where D m is the unit disk in C with m punctures p , p , . . . , p m − along its boundary,where p is distinguished and located at 1. This puncture is called positive and thepunctures p , . . . , p m − are called negative . Near the puncture p the map u is requiredto be asymptotic to a Reeb chord a at + ∞ , and near the negative puncture p i it shouldbe asymptotic to a Reeb chord b i at −∞ for i = 1 , . . . , m −
1. The disk D m is alsoallowed to have interior punctures z , . . . , z l , so that near z i the map u is asymptoticto a cylinder over a Reeb orbit γ i at −∞ , i = 1 , . . . , l .The lower level consists of J -holomorphic maps v i : C → X , i = 1 , . . . , l , from thepunctured sphere and where v i maps a neighborhood of the puncture asymptoticallyto the Reeb orbit cylinder over the Reeb orbit γ i at + ∞ . CECILIA KARLSSON
Let M R × V ; XA ( a, b ), b = b · · · b m , denote the moduli space of such buildings, where A denotes the homology class of the building. Then the differential of A ( V, Λ) is definedon generators by(2.4) ∂ ( a ) = X dim M R × V ; XA ( a, b )=1 |M R × V ; XA ( a, b ) | A b where |M R × V ; XA ( a, b ) | is the mod 2 count of R -components in the moduli space, andthe differential is extended to the whole of A ( V, Λ) by the Leibniz rule. For proofs thatthe homology of this DGA gives a Legendrian invariant we refer to [EGH00, BEE12,Ekh19].2.4.
Legendrian contact homology in 1–jet spaces.
Let M be a smooth manifoldof dimension n . Then the 1 -jet space of M , J ( M ) = T ∗ M × R , is a contact manifoldwith contact form α = dz − vdu , where u are local coordinates on M , v are cotangentcoordinates and z is the coordinate in the R -direction. The Reeb vector field is givenby ∂ z .In this case one can use the Lagrangian projection Π C : J ( M ) → T ∗ M to study Legendrian submanifolds. The Legendrians are projected to exact, immersedLagrangians of T ∗ M under this projection, and the double points of Π C (Λ) correspondto Reeb chords of Λ. Moreover, after a small Legendrian isotopy of Λ we may assumethat it is chord generic , meaning that Π C (Λ) is an immersion with transverse doublepoints as the only intersections. This implies that if Λ is closed, then the number ofReeb chords of Λ is finite.2.4.1. Grading.
The grading of a Reeb chord of Λ can be explicitly described as follows.Consider the front projection Π F : J ( M ) → M × R , and the base projection Π : J ( M ) → M. We will assume that Λ is front generic . We refer to [[EES05], Section 3.2] for a definitionof this, but briefly this means that Π | Λ is an immersion outside a co-dimension 1singular set Σ ⊂ Λ, and that there is a subset Σ ⊂ Σ of codimension 1 so that thepoints in Σ \ Σ belong to a standard cusp singularity of the front projection. Thepoints in Σ \ Σ will be called the cusp edge points of the front of Λ.Let Λ , . . . , Λ s be the connected components of Λ. For each component Λ j fix a point q j ∈ Λ j so that q j does not project to a singularity under the front projection and sothat it does not coincide with a Reeb chord start or end point.For each pair Λ i , Λ j such that there is a Reeb chord between them, pick one suchchord c ij , which we will call a connecting chord . Let c ij, ± be the start and end pointof c ij so that z ( c ij, + ) > z ( c ij, − ). Suppose that c ij, + ∈ Λ j , c ij, − ∈ Λ i . Then there are CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 7 locally defined functions f i , f j : U → R , where U ⊂ M such that Π( c ij ) ∈ U , so that aneighborhood of c ij, − in Λ i and of c ij, + in Λ j is given by { ( u, df i ( u ) , f i ( u )); u ∈ U } and { ( u, df j ( u ) , f j ( u )); u ∈ U } , respectively, and c ij corresponds to a non-degenerate critical point of f j − f i . Let I ( c ij ) = ind c ij ( f j − f i )be the Morse index of f j − f i at this critical point. Definition 2.2.
Assume that Λ ⊂ J ( M ) is a front generic Legendrian submanifold.A path γ ⊂ Λ is admissible if it intersects the singularities of the front projection of Λ transversely at cusp edges. Definition 2.3. If γ ⊂ Λ is an oriented admissible path, let D ( γ ) ( U ( γ ) ) be thenumber of cusp edges of Λ that γ transverses downwards (upwards) with respect to the z -coordinate. Now pick admissible paths ˜ γ c ij,j ⊂ Λ j from p j to c ij, + and ˜ γ c ij,i ⊂ Λ i from p i to c ij, − .Let(2.5) I ij = D (˜ γ c ij,j ) − U (˜ γ c ij,j ) − D (˜ γ c ij,i ) + U (˜ γ c ij,i ) − I ( c ij ) , for i = j and c ij going from Λ i to Λ j , and let(2.6) I ji = − I ij , I ii = 0 . If now c is a Reeb chord of Λ with c ± ∈ Λ i ± , pick admissible paths γ c, ± ⊂ Λ i ± from c ± to q i ± . These paths are called capping paths for c . Definition 2.4.
The grading of a Reeb chord c of Λ is given by | c | = D ( γ c, + ) − U ( γ c, + ) − D ( γ c, − ) + U ( γ c, − ) + I ( c ) + I i − i + − . Differential.
The differential of A ( J ( M ) , Λ) is defined by counting pseudo-holomorphic disks of the Legendrian Λ. This can be done in two different ways, eitherby counting disks in the cotangent bundle T ∗ M , with the disks having boundary onthe Lagrangian Π C (Λ), or by counting disks in the symplectization of J ( M ), with thedisks having boundary on the Lagrangian R × Λ. In [DR16] it is proven that for certainchoices of almost complex structures these two different set-ups give the same count ofelements mod 2, and in [Kar20] this is proven to hold also with Z -coefficients.We give the definition of the count in the cotangent bundle, and refer to [Ekh08,DR16, Kar20] for the definition of the count in the symplectization.Let J be an almost complex structure of T ∗ M , compatible with the standard sym-plectic structure. Let D m +1 denote the punctured unit disk in C with m + 1 punctures p , . . . , p m cyclically ordered along the boundary in the counterclockwise direction,starting at p = 1. Let a, b , . . . , b m be Lagrangian projections of Reeb chords. Definition 2.5.
We say that u : ( D m +1 , ∂D m +1 ) → ( T ∗ M, Π C (Λ)) is a J -holomorphic disk of Λ with positive puncture a and negative punctures b = b · · · b m if CECILIA KARLSSON • ¯ ∂ J u := du + J du ◦ i = 0 , • u | ∂D m +1 \{ p ,...,p m } has a continuous lift ˜ u : ∂D m +1 \ { p , . . . , p m } → Λ , • u ( p ) = a , and ˜ u makes a jump from lower to higher z -coordinate when passingthrough p in the counterclockwise direction, • u ( p i ) = b i , i = 1 , . . . , m , and ˜ u makes a jump from higher to lower z -coordinatewhen passing through p i in the counterclockwise direction. We let M ( a, b ) = M Π C (Λ) ( a, b ) denote the moduli space of J -holomorphic disks ofΛ with positive puncture a and negative punctures b . We consider two disks in themoduli space to be equal if they only differ by a biholomorphic reparametrization ofthe domain.We define the differential of A ( J ( M ) , Λ) to be given by(2.7) ∂ ( a ) = X dim M ( a, b )=0 |M ( a, b ) | R b on generators a , and extend it to the whole of A (Λ) by the Leibniz rule. Here |M ( a, b ) | R ∈ R is the algebraic count of elements in the moduli space, which in thecase of R = Z is given by the modulo 2 count.In [EES07] it is proven that the homology of A ( V, Λ) is a well-defined Legendrianinvariant, that is, ∂ = 0 and the homology is invariant under Legendrian isotopies.2.4.3. Morse flow trees.
Instead of using pseudo-holomorphic disks to define the dif-ferential, one can as well use
Morse flow trees . These are defined as follows.Let Λ ⊂ J ( M ) be a chord generic Legendrian submanifold with simple front singu-larities , meaning that the codimension 2 subset Σ ⊂ Λ where the singularities of thebase projection does not consist of cusp singularities is empty. If dim Λ = 2 we mayalso allow swallow tail singularities, see [[Ekh07], Section 2.2.A].Away from the singular set Σ, the pre-image of an open set U ⊂ M under the baseprojection Π is given by the multi-1-jet lift of locally defined functions f , . . . , f k : M → R , that is, Π − ( U ) | Λ = k [ i =1 { ( x, df i ( x ) , f i ( x )); x ∈ U } . These locally defining functions of Λ are used to build the Morse flow trees. Moreprecisely, these trees are defined as follows.
Definition 2.6. A Morse flow tree is an immersed tree Γ in M satisfying the followingconditions. • The tree is rooted and oriented away from the root. The root is - or -valent. • Each edge γ of Γ is a solution curve of some local function difference: ˙ γ = ˙ γ ij ( t ) = −∇ ( f i − f j )( γ ij ( t )) , where f i > f j are locally defining functions of Λ . • The edge γ ij is given the orientation of −∇ ( f i − f j ) . CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 9 • The cotangent lift of Γ gives an oriented closed curve in Π C (Λ) , in the followingway. Each edge γ ij has two cotangent lifts ˆ γ ij,k = { ( x, df k ( x )); x ∈ γ ij } , k = i, j. If we give ˆ γ ij,i the orientation of γ ij , and ˆ γ ij,j the negative orientation of γ ij ,then the union of all the lifted edges of Γ are required to patch together to givea closed curve in Π C (Λ) ⊂ T ∗ M . • The vertices of Γ have valence at most , and are of the following form. – -valent punctures , which are critical points of the corresponding local func-tion difference, – -valent punctures , which are critical points of the corresponding local func-tion difference, – -valent Y -vertices , where flow lines γ ij , γ jk , γ ik meet, – -valent Y -vertices , similar to Y -vertices but contained in Π(Σ) , – -valent switch-vertices , contained in Π(Σ) , with corresponding flow lineswhich are tangent to
Π(Σ) at the vertex, – -valent end-vertices , contained in Π(Σ) , with corresponding flow lineswhich are transverse to
Π(Σ) at the vertex. • The root of the tree Γ is required to be a puncture, and is called the positivepuncture of the tree. All other punctures are called negative. Since a puncture p is a critical point of a local function difference, we have stable andunstable manifolds associated to p . These we denote by W s ( p ) and W u ( p ), respectively.The dimension of a Morse flow tree Γ with positive puncture a and negative punctures b , . . . , b m can be computed using data from the tree, and is given bydim(Γ) = 2 + dim W u ( a ) + m X j =1 (dim W s ( b j ) − n + 1) + e (Γ) − s (Γ) − Y (Γ) , where e (Γ), s (Γ), Y (Γ) is the number of end-, switch- and Y -vertices of Γ. Definition 2.7. A rigid Morse flow tree of Λ is a Morse flow tree of dimension whichis transversely cut out from the space of flow trees. In [Ekh07] it is proven that one can define the differential by counting rigid Morseflow trees of Λ instead of counting rigid pseudo-holomorphic disks. In some situationsthis gives an easier way to understand A ( J ( M ) , Λ), since this avoids solving the ¯ ∂ -equation which is a non-linear PDE.3. The Chekanov-Eliashberg DGA in a subcritical Weinstein manifold
Let Λ be a Legendrian submanifold of a contact manifold V which is the boundaryof a subcritical Weinstein manifold X of dimension 2 n , n >
2. Assume that c ( X ) = 0.In this section we describe the Chekanov-Eliashberg DGA of Λ, A ( V, Λ), in terms ofsub-DGAs which can be computed from Legendrians in 1–jet spaces. To do this weneed to some additional assumptions on V and Λ. Preliminary assumptions.
To simplify notation we will assume that X onlyhas one subcritical handle H k attached, and that this handle is attached along anisotropic sphere Υ ⊂ S n − = ∂B n . This easily generalizes to the case of havingseveral subcritical handles attached along isotropic spheres where no attaching spheresof the subcritical handles passes through any other subcritical handle.We will assume that there is a Darboux ball B A ⊂ S n − of radius A containing theattaching region N (Υ) of the handle. This means that we have a contactomorphism φ : B A → D A = { ( x, y, z ) ∈ J ( R n − ); | x | + | y | + z ≤ A } , φ ∗ ( dz − ydx ) = α S n − , where α S n − is the standard contact structure on S n − . Thus we can consider thehandle attachment as being performed in D A ⊂ J ( R n − ) instead. Let D HA denote theresulting surgered disk, and let B HA = φ − ( D HA ), where φ is extended by the identityover the handle. Lemma 3.1.
Let n ≥ and let Λ n − ⊂ V n − be a Legendrian submanifold such that Λ ⊂ B HA and assume that B A does not intersect any coordinate subspaces { z i = 0 } of S n − = {| z | = 1 } ⊂ C n . If A is sufficiently small, then A ( V, Λ) ’ quasi A ( D HA , φ (Λ)) .Proof. This follows similarly to [[EN15], Lemma 5.10]. (cid:3)
Assuming that V and Λ satisfy the requirements of this lemma we will consider Λas a subset of D HA from now on, dropping the map φ to simplify notation.We will need some further assumptions on Λ to be able to describe A ( D HA , Λ) interms of sub-DGAs of Legendrians in 1–jet spaces. First, we need to assume that thereis an a < A such that the attaching sphere of the handle is contained in D A \ D a andthat(3.1) Λ ∩ (cid:16) D HA \ (cid:16) H k + ∪ N (Υ) (cid:17)(cid:17) ⊂ D a . Let T ∗ ρ ( S k − ) = { ( u, v ) ∈ R k ; | u | = 1 , u · v = 0 , | v | < ρ } ,I ρ = { r ∈ R ; | r | < ρ } ,D n − kρ = { ( s, t ) ⊂ R n − k ; | s | + | t | < ρ } , and consider T ∗ ρ ( S k − ) × D n − kρ × I ρ with contact form α N = dr − vdu − tds . Thenthere is a contactomorphism ψ : N (Υ) → T ∗ ρ ( S k − ) × D n − kρ × I ρ for ρ , ρ , ρ sufficiently small, which maps Υ to the zero-section of T ∗ ρ ( S k − ).Let D k = { ( x, y, p, q ) ∈ H k ; x = y = p = 0 } be the core of the handle and let C n − k = { ( x, y, p, q ) ∈ H k ; q = 0 } be the cocore. Assume thatΛ ∩ ∂C n − k = { ( x, y, p, q ) ∈ H k + ; ( x, y ) ∈ Λ sub , p = q = 0 } CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 11 where Λ sub ⊂ S n − k − is a Legendrian submanifold with respect to the standardcontact structure. We also assume that Λ ∩ H k + is of the form(3.2) { ( r ( q ) x, r ( q ) y, p, q ) ∈ H k + ; ( x, y ) ∈ Λ sub , p = 0 } , where r : D k → R ≥ is a Morse function with exactly one critical point, located at theorigin and of index 0. We also assume that Λ ∩ H k + is contained in a 1–jet neighborhoodof the standard Legendrian cylinder in H k + , given byΛ st = { ( x, y, p, q ) ∈ H k + ; y = p = 0 } = { ( x, y, p, q ) ∈ H k + ; n − k X i =1 x i = 2( δ + k X i =1 q i ) }’ S n − k − × D k . By identifying H k + with H k − using the Liouville flow and then identifying a regionof H k − with the attaching region, we see that we might assume that the projection of ψ (Λ ∩ N (Υ)) to T ∗ ρ ( S k − ) × I ρ coincides with the zero section of T ∗ ρ ( S k − ) and thatthe projection of ψ (Λ ∩ N (Υ Ki )) to D n − kρ \ D n − kρ ’ S n − k − × [ ρ , ρ ] coincides with(3.3) ( ρ Λ sub , ρ ) , ρ ∈ [ ρ , ρ ]for some ρ >
0. See Section 4.From these assumptions it follows that we can cover Λ by charts given by Legendriansin D a ⊂ J ( R n − ) and in J ( S n − k − × D k ). In Section 3.2.1 we will describe an isotopyof Λ allowing us to describe A ( D HA , Λ) in terms of subalgebras, where each subalgebracan be computed in one of the 1–jet spaces just described.3.2.
The differential of A ( V, Λ) . The differential of A ( V, Λ) is a priori given by acount of pseudo-holomorphic curves anchored in X as in Section 2.3. However, bysimilar arguments as in [EN15] and also by the work in [Ekh19] it follows that it isenough to consider pseudo-holomorphic disks in the symplectization of V . In thissubsection we will investigate these disks further.Recall that the Chekanov-Eliashberg algebra A ( V, Λ) is generated by the Reeb cordsof Λ. By Lemma 3.1 and Lemma 4.2, 4.5, 4.7 it is enough to consider the following Reebchords of Λ, where we get different cases depending on the index k of the subcriticalhandle. Diagram chords:
The Reeb chords a , . . . , a ˜ m of Λ ∩ D a ⊂ ( J ( R n − ) , dz − ydx ). Handle chords, k < n − : Let b , . . . , b m be the Reeb chords of Λ sub seen as aLegendrian submanifold of J ( R n − k − ), using Lemma 3.1. Let b [ h ] , . . . , b m [ h ]be the copies of these chords located at Λ sub × { , } ⊂ H k + . Handle chords, k = n − : Note that Λ sub is a collection of s points for someinteger s . In this case we cannot make use of Lemma 3.1, but we also have totake long chords of Λ sub into account. Therefore, we will have infinitely manychords of Λ sub × { , } ⊂ H k + , labeled by c ij , ≤ i < j ≤ s , (3.4) c pij , ≤ i, j ≤ s , p ≥ . (3.5) We would like to be able to make a similar partition of the pseudo–holomorphiccurves which contribute to the differential. To be able to do this, we isotope Λ inthe attaching region, to introduce a high-dimensional counterpart of the dippings from[Sab06].Recall that we assume Λ to be of the form (3.2) and (3.3) in H k + and N (Υ), respec-tively. In Section 4 we prove that we have a sub-algebra A ( J ( R n − k − ) , Λ sub ) at theminimum q = 0 in the handle in the case when k < n −
1, and a subalgebra generatedby the chords in (3.4) and (3.5) and with differential explicitly described in Lemma4.7 in the case when k = n −
1. However, we might have pseudo-holomorphic diskswith positive punctures at diagram chords traveling into the handles. The dippingprocedure will help us to get control over these disks.3.2.1.
Dippings.
Let f : [ ρ , ρ ] → R be a positive Morse function which coincides with ρ for ρ < (cid:15) and ρ > (cid:15) for some ρ < (cid:15) < (cid:15) < ρ , and which has one maximumat ρ = p , one minimum at ρ = p for some (cid:15) < p < p < (cid:15) , and no other criticalpoints. Assume that f ( p ) = p + δ , f ( p ) = p − δ , where (cid:15) , (cid:15) , p , p , δ , δ are the dipping parameters and are to be chosen.Now we Legendrian isotopeΛ ∩ N (Υ) ’ { ( σ, ρ u, ρ v, ρ ); σ ∈ Υ , ( u, v ) ∈ Λ sub , ρ ∈ [ ρ , ρ ] } to the Legendrian { ( σ, f ( ρ ) u, f ( ρ ) v, ρ ); σ ∈ Υ , ( u, v ) ∈ Λ sub , ρ ∈ [ ρ , ρ ] } . To simplify notation we continue to denote the isotoped Legendrian by Λ.If k > sub get one S k − -family of Reeb chords for ρ = p and another S k − -family for ρ = p . Toavoid this situation let g : S k − → R be a positive Morse function with one maximumat σ ∈ S k − , one minimum at σ ∈ S k − and no other critical points. Legendrianisotope Λ ∩ N (Υ) to the Legendrian { ( σ, (1 + χ ( ρ ) g ( σ )) f ( ρ ) u, (1 + χ ( ρ ) g ( σ )) f ( ρ ) v, ρ ); σ ∈ Υ , ( u, v ) ∈ Λ sub , ρ ∈ [ ρ , ρ ] } , where χ : [ ρ , ρ ] → R is a bump function as in Figure 1. We continue to denote theisotoped Legendrian by Λ.By choosing the height h of the bump function χ small enough we can ensure thatwe get exactly four critical points for the function (1 + χ ( ρ ) g ( σ )) f ( ρ ) on S k − × [ ρ , ρ ],as in Figure 2. That is, we get critical points m = ( σ , p ) of index ks = ( σ , p ) of index 1 s = ( σ , p ) of index k − m = ( σ , p ) of index 0 . Let A d = { ( σ, u, v, ρ ) ∈ N (Υ); (cid:15) < ρ < (cid:15) } CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 13 ρ (cid:15) p p (cid:15) ρ h Figure 1.
The bump function χ . m m s s ρ S k − S k − Figure 2.
The critical points of the dipping function.be the region where the dipping is performed and consider A ( A d , Λ ∩ A d ). If 1
1, and the chords a , . . . , a ˜ m , , c ij , c ij [ m ] , c ij [ s ] , c ij [ s ] , c ij [ m ], 1 ≤ i < j ≤ s , c pij , 1 ≤ i, j ≤ s , p >
0, when k = n −
1, we proceed as follows. Let Λ , . . . , Λ s bethe connected components of Λ. For each component Λ j fix a point q j ∈ Λ j ∩ D a sothat q j does not project to a singularity under the front projection and so that it doesnot coincide with a Reeb chord start or end point. Also, for each pair Λ i , Λ j such thatthere is a Reeb chord between them, pick one such chord c ij ∈ D a as connecting chord.(Note that this is possible since we assume the dipping region to intersect D a .) Definition 3.2.
We say that a path γ ⊂ Λ is handle admissible if γ ∩ D a is admissibleand γ has constant projection to Λ sub in Λ \ (Λ ∩ D a ) ’ D k × Λ sub . Definition 3.3. If γ ⊂ Λ is handle admissible we let D ( γ ) = D ( γ ∩ D a ) , U ( γ ) = U ( γ ∩ D a ) , where D ( γ ∩ D a ) , U ( γ ∩ D a ) is computed as in Definition 2.3. Now choose handle admissible paths as follows. • For each connecting chord c ij choose paths ˜ γ c ij,j ⊂ Λ j from p j to c ij, + and˜ γ c ij,i ⊂ Λ i from p i to c ij, − . • For each diagram chord a = a , . . . , a ˜ m with a ± ∈ Λ l ± choose capping paths γ a ± ⊂ Λ l ± from a ± to q l ± .With these choices is it now possible to define gradings of a , . . . , a ˜ m as in Definition2.4.To define gradings of the handle and dipping chords we use the following results. Lemma 3.4.
Assume that k < n − . Then there is a function K : { , . . . , s } ×{ , . . . , s } → Z and a choice of capping paths for b [ p ] , . . . , b m [ p ] , p = m , s , m , s , h CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 15 so that | b i [ m ] | = | b i | sub + K ( i − , i +) + k | b i [ s ] | = | b i | sub + K ( i − , i +) + 1 | b i [ m ] | = | b i | sub + K ( i − , i +) | b i [ s ] | = | b i | sub + K ( i − , i +) + k − | b i [ h ] | = | b i | sub + K ( i − , i +) ,i = 1 , . . . , m , and where i − , i + ∈ { , . . . , s } satisfies b i, ± ∈ Λ i ± .Proof. To choose the capping paths, we first pick handle admissible paths as follows. • For each component Λ j of Λ sub let l be such that Λ j ⊂ Λ l and let ˜ γ jl ⊂ Λ l be apath from { m } × { q j } ∈ A d ∩ D a to q l . • For each connecting Reeb chord c ij of Λ sub , let l ± be such that Λ i ⊂ Λ l − , Λ j ⊂ Λ l + , and choose paths ˜ γ il − , con ⊂ Λ l − , ˜ γ jl + , con ⊂ Λ l + from { m } × { c ij, − } to q l − and from { m } × { c ij, + } to q l + , respectively.Let γ ∗ γ be the concatenation of the paths γ and γ . To simplify notation, if γ ⊂ Λ sub is a path we continue to write γ for the copy { m } × γ of γ in { m } × Λ sub .To define the capping path γ b ± for the Reeb chord b = b i [ m ], i = 1 , . . . , m , assumethat b i, ± ∈ Λ i ± ⊂ Λ l ± . We get the following cases, see Figures 4, 5 and 6. γ b ± = γ b i ± ∗ ˜ γ j ± l ± , i − = i + ,γ b i ± ∗ ˜ γ c i − i + ,i ± ∗ ˜ γ i ± l ± , con , i − 6 = i + , the connecting chord of Λ i − , Λ i + equals c i − i + ,γ b i ± ∗ ˜ γ c i + i − ,i ± ∗ ˜ γ i ± l ± , con , i − 6 = i + , the connecting chord of Λ i − , Λ i + equals c i + i − . To define capping paths for the chords b i [ p ], p = s , m , s , h we take the cappingpath of b i [ m ] and just extend it in a handle admissible way for i = 1 , . . . , m so thatthe extended parts do not intersect any cusps.Write I ( c ij ) for the Morse index of c ij regarded as a Reeb chord of Λ sub , and I ( c ij )for the Morse index of c ij regarded as a Reeb chord of Λ.If we now let K : { , . . . , s } × { , . . . , s } → Z be given by K ( i − , i +) = 0 , i − = i + ,K ( i − , i +) = D (˜ γ i + l + , con ) − U (˜ γ i + l + , con ) − D (˜ γ i − l − , con ) + U (˜ γ i − l − , con ) + I ( c i − i + ) + I l − l + ,i − 6 = i + , the connecting chord of Λ i − , Λ i + equals c i − i + ,K ( i − , i +) = D (˜ γ i + l + , con ) − U (˜ γ i + l + , con ) − D (˜ γ i − l − , con ) + U (˜ γ i − l − , con ) − I ( c i + i − ) + I l − l + ,i − 6 = i + , the connecting chord of Λ i − , Λ i + equals c i + i − , where I l − l + is given by (2.5) and (2.6) and computed with respect to the connectingchords in Λ.Comparing with Figures 4, 5 and 6 it is clear that the lemma follows. c i + i − ˜ γ c i + i − ,i − γ b + b γ b − ˜ γ c i + i − ,i + Λ i + Λ i − Figure 3.
The choice of admissible paths for Λ sub . γ b + γ b − Λ i + Λ i − b ˜ γ i + l q l q i Λ l Figure 4.
The choice of capping paths for b [ m ] in Λ in the case whenwhen i + = i − . (cid:3) Lemma 3.5.
Assume that k = n − . Then there is a function ˜ K : { , . . . , s } ×{ , . . . , s } → Z and a choice of capping paths for c ij [ p ] , ≤ i < j ≤ s , p = m , s , m , s , c ij , ≤ i < j ≤ s , c wij , ≤ i, j ≤ s , w > so that | c ij [ m ] | = ˜ K ( i, j ) + n − | c ij [ s ] | = ˜ K ( i, j ) + 1 | c ij [ m ] | = ˜ K ( i, j ) | c ij [ s ] | = ˜ K ( i, j ) + n − | c ij | = ˜ K ( i, j ) , | c wij | = ˜ K ( i, j ) − w ( n − . .Proof. For c ij and its copies this is similar to the proof of Lemma 3.4. For c pij , p > | c ij | together with Lemma 4.6 and the fact that the CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 17 γ b + γ b − Λ i + Λ i − b ˜ γ c i − i + ,i + q l Λ l ˜ γ i + l, con ˜ γ i − l, con ˜ γ c i − i + ,i − c i − i + Figure 5.
The choice of capping paths for b [ m ] in Λ in the case whenwhen i + = i − , l + = l − . γ b + γ b − Λ i + Λ i − b ˜ γ c i − i + ,i + q l + Λ l + ˜ γ i − l − , c o n ˜ γ c i − i + ,i − c i − i + q l − c l − l + ˜ γ i + l + , con ˜ γ c l − l + , l − ˜ γ c l − l + ,l + Λ l − Figure 6.
The choice of capping paths for b [ m ] in Λ in the case when i + = i − , l + = l − .grading of a Reeb chord can be computed in terms of the Conley-Zehnder index as in(2.3). (cid:3) Almost complex structure.
To choose an almost complex structure for R t × D HA we proceed as in [[EN15], Section 5.2.A]. That is, in D a ⊂ J ( R n − ) ’ C n − × R z we choose the standard complex structure on C n − and then extend it to the whole spaceby requiring that J ( ∂ t ) = ∂ z .In the handle we choose an almost complex structure which is standard in the 1–jetneighborhood of Λ st where we assume that Λ ∩ H k is contained. This means thatfor J (Λ st ) ’ T ∗ Λ st × R z we assume it to be given as in [[Ekh07], Section 4.3] in T ∗ Λ st , mapping the vertical subbundle of T ∗ Λ st to the horizontal subbundle, wherethese subbundles are defined using some metric connection. Again we extend it to acylindrical almost complex structure by setting J ( ∂ t ) = ∂ z . Extend this to a cylindricalalmost complex structure in the rest of the handle.Assuming that Λ st ∩ D a ⊂ N (Υ) is contained in a real plane ( R n − ×{ y }×{ z } ) ∩ D a ,the almost complex structure defined in the handle will coincide with the almost com-plex structure in R × D a in the 1–jet neighborhood of Λ st , assuming this is small enough,and we can interpolate the almost complex structures outside this neighborhood to givea cylindrical almost complex structure defined over the whole of R × D HA .3.2.4. Description of the differential.
We are now ready to state the main result of thispaper. Recall that Λ now represents the dipped version of the Legendrian attachingspheres.
Proposition 3.6.
Assume that k < n − . Then the DGA A ( V, Λ) is quasi-isomorphicto a DGA that splits into subalgebras A ( J ( R n − ) , Λ ∩ D a ) , A ( J (Λ st ) , Λ ∩ H k + ) , and A ( J ( R n − k − ) , Λ sub ) . Proposition 3.7.
Assume that k = n − . Then the DGA A ( V, Λ) is quasi-isomorphicto a DGA that splits into subalgebras A ( J ( R n − ) , Λ ∩ D a ) , A ( J (Λ st ) , Λ ∩ H k + ) , andthe DGA with generators c ij , ≤ i < j ≤ s , c pij , ≤ i, j ≤ s , p ≥ , graded by Lemma3.5, and with differential given by ∂c ij = s X l =1 c lj c il ∂c ij = δ ij + s X l =1 c lj c il + s X l =1 c lj c il ∂c pij = p X m =1 s X l =1 c mlj c p − lil , where p ≥ , c ij = 0 if i ≥ j , δ ij is the Kronecker delta and where we extend it to therest of the algebra by the Leibniz rule.Proof of Proposition 3.6 and 3.7. Similar to [EN15] the differential can be given by acount of pseudo-holomorphic disks in R × D Ha with boundary on R × Λ. After havingintroduced the dipping region this count reduces to the following.
Diagram disks:
Disks that have positive puncture at the diagram chords a , . . . , a ˜ m or at the dipping chords b i [ p ], i = 1 , . . . , m ( respectively c ij [ p ], 1 ≤ i < j ≤ s if k = n − p = s , m . These disks cannot leave R × D a because of the CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 19 dipping (they cannot pass through S k − × Λ sub × { p } ), which is easily seen byswitching to Morse flow trees instead of pseudo-holomorphic disks. Handle disks:
These are disks with positive punctures at the handle chords. InSection 4 we prove that we only have to consider disks with negative puncturesat handle chords and that we get subalgebras as in the statements above.
Dipping disks:
These are pseudo-holomorphic disks of Λ having positive punc-ture at some b i [ m ] or b i [ s ], i = 1 , . . . , m (respectively c ij [ p ], 1 ≤ i < j ≤ s , p = s , m if k = n − f is chosen sufficiently smallit follows by action reasons that these disks cannot leave a 1–jet neighborhoodof Λ st ⊂ H k + and hence we can use the techniques from Section 2.4 to find thepseudo-holomorphic disks. Note that we may identify the whole dipping areawith a subset of a 1–jet neighborhood of Λ st ⊂ H k + if the dipping parametersare small enough. (cid:3) Remark 3.8.
It is possible to give a more explicit description of the algebra in thedipping region using the techniques of broken Morse flow trees from [EK08].It follows that the Legendrian contact homology of Λ in V can be computed using1–jet space techniques together with the explicit description of the differential in thehandle when k = n −
1. It also follows that the coefficients reduce form Z [ H ( X, Λ)]to Z since no disk passes through a handle.4. The sub-DGA in the handle
In this section we prove that we get a sub-DGA of A ( D HA , Λ) generated by the Reebchords in the cocore of the handle. To do this, we will modify the model of the handlesfrom Section 2.2 slightly, to simplify the Reeb dynamics.4.1.
Geometry of a n -dimensional symplectic handle of index k. Let a , . . . , a n − k ∈ R be some positive constants that are linearly independent over Q and define H kδ = { ( x, y, p, q ) ∈ R n − k × R k : − δ ≤ n − k X j =1 a j x j + y j ) + k X j =1 p j − q j ≤ δ } . The handle still has Liouville vector field(4.1) Z = n − k X j =1
12 ( x j ∂ x j + y j ∂ y j ) + k X j =1 p j ∂ p j − q j ∂ q j which is transverse to the boundary H k ± = { ( x, y, p, q ) ∈ R n − k × R k : n − k X j =1 a j x j + y j ) + k X j =1 p j − q j = ± δ } , and points out of H kδ along H k + and into H kδ along H k − . Note that, topologically westill have H k − ’ R n − k × S k − , H k + ’ S n − k − × R k .The Liouville vector field induces contact forms α ± δ on H k ± :(4.2) α ± δ = ω st ( Z, • ) H k ± = n − k X j =1 a j x j dy j − y j dx j ) + k X j =1 p j dq j + q j dp j , with Reeb vector field given by N ˜ R , where(4.3) ˜ R = n − k X j =1 a j ( x j ∂ y j − y j ∂ x j ) + k X j =1 p j ∂ q j + q j ∂ p j | H k ± and N = n − k X j =1
14 ( x j + y j ) + k X j =1 p j + q j − . It follows that the differential equation for the Reeb flow is given by˙ x j = − N a j y j ˙ y j = N a j x j j = 1 , . . . , n − k, ˙ p j = N q j ˙ q j = 2 N p j j = 1 , . . . , k. Hence we get that the time t Reeb flow Φ tR = ( x ( t ) , y ( t ) , p ( t ) , q ( t )) is given by x j ( t ) = x j (0) cos N a j t ! − y j (0) sin N a j t ! , j = 1 , . . . , n − k, (4.4) y j ( t ) = x j (0) sin N a j t ! + y j (0) cos N a j t ! , j = 1 , . . . , n − k, (4.5) p j ( t ) = p j (0) cosh( N √ t ) + 1 √ q j (0) sinh( N √ t ) , j = 1 , . . . , k, (4.6) q j ( t ) = √ p j (0) sinh( N √ t ) + q j (0) cosh( N √ t ) , j = 1 , . . . , k. (4.7)Let us now consider some special cases.4.2. Index 1 handles.
In this case the attaching region is given by two disjoint ballsof dimension 2 n −
1. We describe models for the attaching of the handle H = H alongthese balls.Let B n − ρ = { ( u , v , . . . , , u n − , v n − , z ) ∈ R n − ; n − X i =1 u i + v i + z = ρ } equipped with the contact structure α b = dz + ( udv − vdu ). We will often omit thedimension and only write B ρ . Identify this with a ball in ( R n − , dz − ydx ), centeredat a = ( x, y, z ) via the contact embedding F a : B ρ → R n − , F a ( u, v, z ) = ( x + u, y + v, z + z + yu + 12 uv ) . Denote the image F a ( B ρ ) the standard contact ball of radius ρ centered at a . CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 21
For suitable ρ and a ± these balls will be our attaching locus, as follows. Fix twopoints a ± ∈ R n − of distance d >> ρ and a δ << ρ , let A ± ρ ( δ ) ⊂ H − be defined by A + ρ ( δ ) = { ( x, y, p, q ) ∈ H − ; p = 0 , q > , n − X i =1 a i ( x i + y i ) ≤ ρ } ,A − ρ ( δ ) = { ( x, y, p, q ) ∈ H − ; p = 0 , q < , n − X i =1 a i ( x i + y i ) ≤ ρ } . Define the map G : A ± ρ ( δ ) → B ρ , G ( x, y, , q ) = ( √ a x , √ a y , . . . , √ a n − x n − , √ a n − y n − , . Then G ∗ α b = α − δ | A ± ρ ( δ ) and since the Reeb vector field is transverse to A ± ρ ( δ ) and G ( A ± ρ ( δ )), respectively, its flow can be used to extend G to a contactomorphism froma neighborhood of A ± ρ ( δ ) to B ρ . Denote this neighborhood by B ± ρ ( δ ) ⊂ H − , and notethat this neighborhood is identified with a neighborhood of a ± via the composition F a ± ◦ G .The next step is to identify B ± ρ ( δ ) − B ± ρ ( δ ) ⊂ H − with a region in H + . To do thatwe use the Liouville flow in the handle, given by φ tZ (( x (0) , y (0) , p (0) , q (0)) =( e t x (0) , e t y (0) , . . . , e t x n − (0) , e t y n − (0) , e t p (0) , e − t q (0)) . Recall the standard Legendrian cylinder defined in Section 3.1. Perturb it to begiven byΛ st = { ( x, y, p, q ) ∈ H + ; y = p = 0 } = { ( x, y, p, q ) ∈ H + ; n − X i =1 a i x i = 2( δ + q ) } = s δ + q ) a ˆ x , , . . . , vuut δ + q ) a n − ˆ x n − , , , q where ˆ x , . . . , ˆ x n − are coordinates for S n − ⊂ R n − .Let T ( x, y, p, q ) be the time of the Liouville flow from H − to ( x, y, p, q ) ∈ H + . Lemma 4.1. If a = (cid:18)q δ + q ) a ˆ x , , . . . , r δ + q ) a n − ˆ x n − , , , q (cid:19) ∈ Λ st then T ( a ) = T ( q ) and T ( q ) decreases when q increases ( e − T ( q ) increases with q ).Proof. We should find points x i , q ∈ H − , i = 1 , . . . , n −
1, so that(4.8) x i = e − T ( a ) s δ + q ) a i ˆ x i , i = 1 , . . . , n − , q = e T ( a ) q. and such that(4.9) n − X j =1 a j x j − q = − δ . That means that T ( a ) should satisfy(4.10) n − X j =1 a j e − T ( a ) δ + q ) a j ˆ x j = e T ( a ) q − δ , which simplifies to(4.11) e − T ( a ) ( δ + q ) = e T ( a ) q − δ . From this we see that T ( a ) = T ( q ). Write u = e T ( a ) to simplify notation. Then (4.11)can be written as(4.12) δ + q = u q − uδ . When q = δ this equation has solution close to u = 3 /
2. Moreover, assuming that u > q δ = u + 1 u − , and since the function u +1 u − is strictly increasing to ∞ as u decreases from 2 to 1 , thesecond statement follows. (cid:3) This means that the image of Λ st in H − under the negative Liouville flow is givenby e − T ( q ) s δ + q ) a ˆ x , , . . . , e − T ( q ) vuut δ + q ) a n − ˆ x n − , , , e T ( q ) q , and if we view this in B ρ using the map G we get thatΛ st = (cid:18) e − T ( q ) q δ + q )ˆ x , , . . . , e − T ( q ) q δ + q )ˆ x n − , , , (cid:19) ⊂ B ρ , that is, this is nothing but a cone on S n − with radius decreasing into B ρ .Let us examine this in more detail. We have that ∂B n − ρ = S n − ⊃ S n − = { u + v = ρ } , which is a contact submanifold of B n − ρ . Moreover, after scaling if necessarily, we getthat Λ st ∩ S n − = { u = ρ, v = 0 } = S n − which is a Legendrian submanifold of S n − , denote it Λ st , sub . Let U ν ⊂ S n − be a1-jet neighborhood of Λ st , sub .Assume that Λ ∩ ∂B ρ = Λ ∩ S n − = Λ ∩ U ν which then is a Legendrian in S n − , which we denote by Λ sub . Assume that it does notintersect any coordinate subspaces { ( u i , v i ) = 0 } for i = 1 , . . . , n −
1. Further assumethat Λ ∩ B ρ is a cone on Λ sub , meaning that ifΛ sub = (˜ u, ˜ v ) ⊂ S n − ⊂ ∂B ρ , CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 23 then there is some positive parameter r which is strictly increasing with the radius of B n − ρ so that Λ ∩ ( B n − ρ \ B n − ρ ) = ( r ˜ u, r ˜ v, . Hence G − (Λ sub ) = ( a ˜ u , a ˜ v , . . . , a n − ˜ u n − , a n − ˜ v n − , , ⊂ H − and similar to the proof of Lemma 4.1 we see that we might assume that Λ is a coneon Λ sub in the handle, that isΛ ∩ H + = (˜ ra ˜ u , ˜ ra ˜ v , . . . , ˜ ra n − ˜ u n − , ˜ ra n − ˜ v n − , , q ) , where ˜ r : D → R ≥ is a Morse function with exactly one critical point, located at q = 0 and of index 0. .Let ˜ ν << δ and let D ˜ ν be a Darboux ball contained in { x + y + p + q ≤ ˜ ν } ,containing (Λ ∩ H + ∩ { q = 0 } ) ’ G − (Λ sub ) × { , } . Choose D ˜ ν so that it does notintersect any coordinate subspaces { ( x i , y i ) = 0 } for i = 1 , . . . , n − E δ ( a , . . . , a m ) = { P mi =1 a i | z i | = 2 δ } ⊂ C m . Lemma 4.2.
The Reeb chords of Λ ∩ H + ⊂ H + can be identified with the Reeb chordsof G − (Λ sub ) ⊂ E δ ( a , . . . , a n − ) , with degree shift given as in Lemma 3.4.Proof. Since p = 0 along Λ ∩ H + we must have p = q = 0 along Reeb chords of Λ ∩ H + ,since the Reeb flow is hyperbolic in the ( p, q )-plane. Thus the Reeb dynamics restrictto the one on E δ ( a , . . . , a n − ). (cid:3) We divide the Reeb chords of Λ ∩ H + into two types • Long chords which leave D ˜ ν , • Short chords which are completely contained in D ˜ ν . Lemma 4.3.
If we have a J -holomorphic disk of Λ with a positive puncture in H thenit cannot have a negative puncture outside H .Proof. This follows similarly as in the proof of [[EN15], Lemma 5.6]. Note that the m :th iterate of a Reeb orbit in the handle has grading of order m and action of order δm , which is what is needed for the proof to work out. (cid:3) Lemma 4.4.
Assume that n > . Then the DGA A ( H + , Λ ∩ H + ) is a sub-DGA of A ( D HA , Λ) , and this sub-DGA is quasi-isomorphic to A ( J ( R n − ) , Λ sub ) with a degreeshift given as in Lemma 3.4.Proof. The first statement follows from Lemma 4.3, and since Λ is conic in H + with aminimum at q = 0 it follows that A ( H + , Λ ∩ H + ) is quasi-isomorphic to A ( E δ ( a , . . . , a n − ) , G − (Λ sub )) with a degree shift given as in Lemma 3.4, which inturn is quasi-isomorphic to A ( J ( R n − ) , Λ sub ) by Lemma 3.1. (cid:3) Index k handles, k ∈ { , . . . , n − } . This is similar to the case of index 1handles, so we just describe the main differences.To describe the attaching map in the case when n >
2, recall the contact identifica-tion ψ : N (Υ) → T ∗ ρ ( S k − ) × D n − kρ × I ρ from Section 3.1. Let ρ = ( ρ , ρ , ρ ) and let(4.14) A ρ ( δ ) = { ( x, y, p, q ) ∈ H − ; p · q = 0 , n − k X i =1 a i ( x i + y i ) < ρ , | p | < ρ } and let G : A ρ ( δ ) → T ∗ ρ ( S k − ) × D n − kρ × I ρ be defined by G ( x, y, p, q ) =( q , . . . , q k , ( p · q ) q − p , . . . , ( p · q ) q k − p k , √ a x , . . . , √ a n − k x n − k , √ a y , . . . , √ a n − k y n − k , p · q + n − k X i =1 a i x i y i . Then G ∗ α N = α − δ | A ρ ( δ ) and since the Reeb vector fields are transverse to A ρ ( δ ) and G ( A ρ ( δ )), respectively, we can extend G to a contactomorphism from a neighborhoodof A ρ ( δ ) in H − to a neighborhood of G ( A ρ ( δ )).We now consider two different cases, k < n − k = n − k < n − . As in the case of index 1 handles we might assume that Λ is of theform(4.15) Λ ∩ H + = (˜ r ( q ) G − (Λ sub ) , , q ) ’ Λ sub × D k ⊂ H + where Λ sub ⊂ S n − k − is a Legendrian submanifold and ˜ r : D k → R ≥ is a Morsefunction with exactly one critical point, located at q = 0 and of index 0. Moreover,we assume that Λ ∩ H + is contained in a small 1–jet neighborhood of the perturbedstandard Legendrian cylinder in H Λ st = { ( x, y, p, q ) ∈ H k + ; y = p = 0 } = { ( x, y, p, q ) ∈ H k + ; n − k X i =1 a i x i = δ + k X i =1 q i }’ S n − k − × D k . Lemma 4.5.
The Reeb chords of Λ ∩ H + are contained in the slice p = q = 0 , and A ( H + , Λ ∩ H + ) is a sub-DGA of A ( V, Λ) isomorphic to A ( J ( R n − k − ) , Λ sub ) with adegree shift as in Lemma 3.4.Proof. This is similar to the case of index 1 handles. (cid:3)
Index n − handles. First of all let us assume that a = 1 and let H = H n +1 tosimplify notation.From the Reeb vector field formulas (4.4) – (4.7) we see that there is exactly onegeometric Reeb orbit γ in H + , given by γ = { p = q = 0 , x + y = 2 δ } , CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 25 and this orbit intersectsΛ st = { ( x, y, p, q ) ∈ H k + ; y = p = 0 , x > } = { ( x, y, p, q ) ∈ H k + ; x = δ + n − X i =1 q i } ’ D n − in the point x = √ δ, y = p = q = 0.Pick a trivialization of the contact structure ker α + given by ( v , iv , . . . , v n − , iv n − ), v j = 2 p j ∂ x + q j ∂ y −
12 ( x ∂ p j − y ∂ q j ) + R ( j ) , where R ( j ) is a linear combination of vectors ∂ q i , ∂ p i with coefficients given by constantstimes exactly one of the coordinate functions q l , p l , i, l = j . This trivialization corre-sponds to the trivialization of the contact planes in D a ⊂ J ( R n − ) ’ C n − × R inducedby the Lagrangian projection form the choice of trivialization ∂ t , i∂ t , . . . , ∂ t n − , i∂ t n − on C n − = { ( z = s + it ) } . Lemma 4.6. If γ m denotes the m:th iterate of γ , then the Conley-Zehnder index of γ m with respect to the trivialization ( v , iv , . . . , v n − , iv n − ) is given by CZ ( γ m ) = − m ( n − . Proof.
First we compute the Conley-Zehnder index with respect to the trivialization( ∂ p , ∂ q , . . . , ∂ q n − , ∂ p n − ). Since the linearized flow is hyperbolic in these directions weget that the index equals 0. Next we need to calculate the Maslov index of the pathof matrices that express ( ∂ p , ∂ q , . . . , ∂ q n − , ∂ p n − ) in the basis ( v , iv , . . . , v n − , iv n − ).We notice that ∂ p j = − x v j − y iv j x + y ∂ q j = 2 y v j − x iv j x + y . After renormalizing we might assume that x + y = 1, that γ has period 2 π and thatwe start at the point x = 1 , y = 0. Then we get ∂ p j = − cos( t ) v j − sin( t ) iv j ∂ q j = sin( t ) v j − cos( t ) iv j . Since the matrix ψ ( t ) = " − cos t − sin t sin t − cos t has crossings only when t = π for t ∈ [0 , π ], and˙ ψ ( t ) | t = π = " sin t − cos t cos t sin t t = π = " − we get that the crossing form has signature − ∂ p j , ∂ q j )-plane.Since the problem splits into these n − − n −
1) foreach iterate of γ . (cid:3) Now let Λ , Λ , . . . , Λ s be the connected components of Λ ∩ H + . Let ( u, v, s ) becoordinates in a 1-jet neighborhood of Λ st . Then we might assume that Λ j is given bythe 1-jet lift of the Morse function f j : R n − → R , f j ( u ) = j(cid:15) + jκ | u | , j = 1 , . . . , s . Here 0 < κ << (cid:15) are two small constants.Then the Reeb chords from Λ i to Λ j is given bya) i < j : For each integer w ≥ c wij of length 2 πwδ +( j − i ) (cid:15) ,b) i ≥ j : For each integer w > c wij of length 2 πwδ − ( i − j ) (cid:15) .In all cases the Reeb chords follow the orbit γ . Lemma 4.7.
The DGA A ( H n − , Λ ∩ H n − ) is a sub-algebra of A ( D HA , Λ) with gener-ators c ij , ≤ i < j ≤ s , c pij , ≤ i, j ≤ s , p ≥ , graded by Lemma 3.5. If we set c ij = 0 , i ≥ j , then the differential is given by ∂c ij = s X l =1 c lj c il ∂c ij = δ ij + s X l =1 c lj c il + s X l =1 c lj c il ∂c pij = p X m =1 s X l =1 c mlj c p − lil , where p ≥ , δ ij is the Kronecker delta and where we extend it to the rest of the algebraby the Leibniz rule.Proof. That the Reeb chords c ij , 1 ≤ i < j ≤ s , c pij , 1 ≤ i, j ≤ s , p ≥ M ( c mij ; c m lj , c m il ) with m = m + m , and that thesemoduli spaces consist of one point each, up to translation and reparametrization. (cid:3) The singular homology of the free loop space of CP In this section we give a Weinstein handle decomposition of T ∗ CP and computethe Chekanov-Eliashberg DGA of the Legendrian attaching sphere. By the results of[AS06, Vit18, SW06] and [BEE12] this gives a description of the singular homology ofthe free loop space of CP .5.1. Weinstein handle decomposition of T ∗ CP . Recall that CP is obtained from B by attaching a 2-handle along a knot ˜Υ ⊂ S with framing 1, and then attachinga 4-handle along the boundary of B ∪ ˜Υ H . Thus, to give a Weinstein handle decom-position of T ∗ CP we should attach one subcritical handle to ∂B along an isotropic S with the correct framing and then attach the critical handle along a Legendrian S which goes through the subcritical handle along the standard Legendrian cylinder.Another way of seeing this is to consider the Legendrian˜Λ = { x + x + x + x = 1 } ⊂ S = { z = x + iy ∈ C ; | z | = 1 } CH FOR ATTACHING LINKS IN HIGHER DIMENSIONS 27 with the isotropic attaching sphere Υ of the subcritical handle as being a subset of ˜Λ,and then let Λ be the Legendrian submanifold we get by replacing a neighborhood ofΥ with the standard Legendrian cylinder in H .If we view ˜Λ as a Legendrian submanifold of J ( R ) ’ ∂B \ { pt } , then it can begiven by the 1–jet lift of the locally defined functions f ± : R → R , f ± ( u , u , u ) =˜ f ± ( | u | ) with ˜ f ± as in Figure 7. (Recall that the 1–jet lift of f : R n → R is given by { ( u, df ( u ) , f ( u )); u ∈ R n } ⊂ J ( R n ).) ˜ f + ˜ f − − − Figure 7.
The locally defined functions ˜ f ± .We see that ˜Λ has exactly one Reeb chord a of grading 3, located at the origin, andthat there is an S –family of Morse flow trees going from a to the cusp edge ’ S .Identifying ˜Λ \ { (0 , , } with R via stereographic projection (mapping (0 , , −
1) tothe origin) we get that the lifts of the flow trees of ˜Λ go radially from ∞ to the origin.Assume that the unit sphere S ⊂ R represents the cusp edge singularity of ˜Λ.Now we describe our choice of attaching sphere Υ for the subcritical handle in thispicture. To that end, pick a point p ∈ S ⊂ R . Then a neighborhood of p in R canbe identified with J ( I ) ’ T ∗ I × R where I ⊂ S is some interval and T ∗ I ’ D is adisk contained in S , and the R –direction corresponds to the radial direction in R .Let Υ be given by the standard Legendrian unknot as in Figure 8.It follows that the lifted flow trees of ˜Λ intersect Υ either 0,1 or 2 times, in 2–dimensional, 1–dimensional and 0–dimensional families, respectively. Moreover, thereis exactly one tree that intersects Υ in 2 points, namely the tree which goes throughthe point p . This tree gives rise to a rigid flow tree Γ of Λ, and this will be the onlyrigid flow tree. Since Λ coincides with Λ st in the subcritical handle we have that Λ sub p R I p T ∗ I Figure 8.
The front and Lagrangian projection of the standard Legen-drian unknot in J ( I ).is given by the standard Legendrian unknot in Figure 8, which has exactly one Reebchord b , and | b | sub = | b | = 1. The tree Γ has positive puncture at a , one Y -vertex overthe point p and 2 negative punctures at b [ h ]. See Figure 9. Υ ΥΛ sub Λ sub Figure 9.
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