Floer homology and Lagrangian concordance
We derive constraints on Lagrangian concordances from Legendrian submanifolds of the standard contact sphere admitting exact Lagrangian fillings. More precisely, we show that such a concordance induces an isomorphism on the level of bilinearised Legendrian contact cohomology. This is used to prove the existence of non-invertible exact Lagrangian concordances in all dimensions. In addition, using a result of Eliashberg-Polterovich, we completely classify exact Lagrangian concordances from the Legendrian unknot to itself in the tight contact-three sphere: every such concordance is the trace of a Legendrian isotopy. We also discuss a high dimensional topological result related to this classification.
Key words and phrases:Lagrangian concordance, Floer homology, Legendrian unknot, non-symmetry
2010 Mathematics Subject Classification:Primary 53D12; Secondary 53D42
In this paper we are interested in exhibiting various rigidity phenomena for Lagrangian concordances between Legendrian submanifolds of the standard contact sphere (or equivalently of the standard contact space ). Recall that the standard contact structure on is given by , where is a one-form on . A submanifold is Legendrian if it is -dimensional and tangent to . A Lagrangian concordance is a special case of an exact Lagrangian cobordism which is diffeomorphic to ; see Definition 2.2. In particular, Lagrangian concordant Legendrian submanifolds are diffeomorphic.
The classical (complete) obstruction for the existence of immersed Lagrangian cobordisms is the formal Lagrangian class [Gro2, Lee]. For example, when , an immersed Lagrangian concordance between two knots exists if and only if the two knots have the same rotation numbers. The classical obstruction to the existence of an embedded (not necessarily exact) oriented Lagrangian cobordism comes from the Thurston-Bennequin invariant, which is defined as the linking number , where is obtained by pushing slightly in the direction of the Reeb vector field. If is a (not necessarily exact) oriented Lagrangian cobordism from to , the corresponding Thurston-Bennequin invariants are related by
In high dimension, recent results of Eliashberg-Murphy in [EM] implies that exact Lagrangian cobordisms satisfy an -principle similar to the one in [Lee, Gro2] when the negative end is loose as defined in [Mur]. Thus, in order to expect rigidity phenomena, in the present paper we will study Lagrangian cobordisms whose negative end admits an exact Lagrangian filling, i.e. an exact Lagrangian cobordism from to .
In fact, it is well-known that a Legendrian submanifold of which admits an exact filling cannot be loose. In the forthcoming paper [CDRGG], we will study exact Lagrangian cobordisms between Legendrian submanifolds admitting augmentations. This condition, in particular, implies that these Legendrian submanifolds are not loose. The geometrical situation is however more involved due to the fact that Legendrian submanifolds admitting augmentations are not necessarily fillable.
Observe that there are very few known examples and constructions of cobordisms in the case when admits an exact Lagrangian filling. It seems like the only ones are so-called decomposable cobordisms, i.e. exact Lagrangian cobordisms built by concatenating cobordisms of the following two types:
Our goal is to extract obstructions to the existence of a Lagrangian concordance from a Legendrian submanifold that admits an exact Lagrangian filling from its Legendrian contact homology [EGH], [Che], [EHK]. One of the first results in this direction was in [Cha1], where it is shown that the Legendrian representative of the knot described in Figure 2 (with maximal Thurston-Bennequin invariant) is not Lagrangian concordant to the unknot with . On the other hand is Lagrangian fillable by a disc which can be seen as the concatenation of the standard Lagrangian disc that fills and a Lagrangian concordance from to . In other words, this result shows that the relation of being Lagrangian concordant is not symmetric, in particular there are Lagrangian concordances that cannot be inverted (unlike those arising from the trace of a Legendrian isotopy).
In dimension three, this result was later generalised in [BS] and [CNS]. The latter article is based upon the technique of rulings, which is a combinatorial invariant related to Legendrian contact homology which can be defined in the case . It is shown there that a Lagrangian concordance from to imposes restrictions on the possible rulings of . Namely, such a Legendrian knot cannot have two normal rulings.
Our main result is similar in spirit, but concerns the bilinearised Legendrian contact cohomology induced by a pair of augmentations. This invariant can be defined in all dimensions.
Our main rigidity result for Lagrangian concordances is the following:
Suppose that is a Lagrangian concordance from to in . If are augmentations of the Chekanov-Eliashberg algebra of that are induced by exact Lagrangian fillings, it follows that the induced bilinearised map
in cohomology is an isomorphism.
The previous theorem applies in the more general case when the ambient contact manifold is a contactisation of a Liouville domain as defined in Section 2.1.
of linearised cohomologies in the case when is induced by a filling. Namely, both the left and the right-hand side are isomorphic to the singular homology of the filling (with a shift in grading).
In general, we will use to denote the standard Legendrian -dimensional sphere, which can be obtained as the intersection . The standard representative of this Legendrian submanifold admits a unique augmentation of its Legendrian contact homology algebra. The following important corollary follows from the fact that is one-dimensional and concentrated in degree (see [EES2, Example 4.2]) for this augmentation.
If the Legendrian submanifold admits two fillings inducing augmentations , , for which is not one-dimensional and concentrated in degree , it follows that there is no exact Lagrangian concordance to from to .
In the case when a result due to Eliashberg and Polterovich [EP] shows that, up to compactly supported Hamiltonian isotopy, the unknot has a unique filling by a disc. Then a fillable Legendrian knot which is exact Lagrangian cobordant to the unknot is automatically doubly slice, as observed in [Cha1, Theorem 1.2]. A smooth knot in is called slice if it bounds an embedded disc in , and it is called doubly slice if can be obtained as the transverse intersection of an unknotted embedding of with the unit sphere . In Theorem 4.4 we elaborate on this idea to give a complete classification of exact Lagrangian concordances from to itself up to compactly supported Hamiltonian isotopy.
A similar classification result in high dimensions is out of reach of current technology. However, there are strong topological constraints in the case when has a single generic Reeb chord and admits an exact Lagrangian filling (e.g. this is the case for the standard representative of ). Restrictions on the topology and smooth structure of such a Legendrian submanifold have previously been established in [ES1, ES2] by Ekholm-Smith. Also, in the case when , a result due to Abouzaid, Fukaya-Seidel-Smith, Nadler, and Kragh (see Remark 4.10) implies that any exact Lagrangian is contractible. We generalise this result in Theorem 4.9.
We end with the following corollary. Using the result of Eliashberg-Polterovich it was shown in [Cha1, Theorem 6.1] that every exact Lagrangian cobordism from the standard representative of the one-dimensional Legendrian unknot to itself is a concordance which, moreover, induces the identity automorphism of its Chekanov-Eliashberg algebra. Combining Theorem 1.1, Theorem 4.9, together with an algebraic consideration, we obtain an analogous result in high dimensions as well:
Suppose that and let be a Legendrian submanifold having a single generic Reeb chord (e.g. ), or suppose that and that . Any exact Lagrangian cobordism from a Legendrian submanifold to , where moreover admits an exact Lagrangian filling, induces a unital DGA morphism
which is a monomorphism admitting a left-inverse. In the case when satisfies the same assumptions as , is moreover a concordance and the above map is an isomorphism.
We now give an outline of the paper. In Section 2, we review all preliminary definitions and results in order to prove the main results of the paper. In particular, we recall the definitions of a Lagrangian cobordisms and concordances in Section 2.1. We recall the front spinning construction in Section 2.2. In sections 2.3 to 2.5 we recall the definition of bilinearised Legendrian contact homology and its relation with the Floer homology of Lagrangian fillings. In sections 2.6 to 2.9 we define the moduli spaces of pseudo-holomorphic curves. Notably in Section 2.7 we use the results by Lazzarini in [Laz1] to prove a useful transversality property for pseudo-holomorphic curves with one positive puncture. Theorem 1.1 is proved in Section 3 by a computation using wrapped Floer homology. Section 4 is devoted to the study of Lagrangian concordances from to inside , which we classify up to Hamiltonian isotopy. We also give a proof of Theorem 4.9 using wrapped Floer homology with local coefficients. We conclude in Section 5 with non-symmetry results for high-dimensional Lagrangian concordances in the above rigid settings.
The first and third authors would like to thank the organisers of the 21st Gökova Topology Conference, May 2014.
2.1. Geometric set-up
A contact manifold is a pair consisting of an odd-dimensional manifold together with a smooth maximally non-integrable field of tangent hyperplanes . Given the choice of a contact one-form , i.e. a one-form such that , the associated Reeb vector field is uniquely determined by
The symplectisation of is the exact symplectic manifold , where denotes the standard coordinate of the -factor. In general, an exact symplectic manifold is a pair consisting of an even-dimensional smooth manifold together with an exact non-degenerate two-form .
An -dimensional submanifold is called Legendrian if . A half-dimensional submanifold, or immersed submanifold, is called exact Lagrangian if the pull-back of to is exact. We are interested in the following relation between Legendrian submanifolds.
A properly embedded exact Lagrangian submanifold of the symplectisation without boundary is called an exact Lagrangian cobordism from to if it is of the form
Observe that it follows from the definition that necessarily are Legendrian submanifolds.
The Legendrian submanifold is called the positive end of , while is called the negative end of . In the special case when we say that is an exact Lagrangian filling of .
Lagrangian concordances are particular examples of exact Lagrangian cobordisms.
A Lagrangian concordance from a Legendrian submanifold to a Legendrian submanifold in a symplectisation is a Lagrangian cobordism from to whose compact part is diffeomorphic to .
Note that if there exists a Lagrangian concordance from to then, in particular, is diffeomorphic to . The exactness of a Lagrangian concordance is immediate as all the topology is concentrated in the cylindrical ends, where the form vanishes.
The Maslov number of a Lagrangian cobordism is defined to be the generator of the image of the Maslov class (see e.g. [EES2] for more details). Its relevance comes from the fact that gradings in Floer homology are defined modulo . In the special case of a Lagrangian concordance, grading consideration are vastly simplified as the Maslov number of a concordance is twice the rotation number of the Legendrian (positive or negative) end.
The contactisation of an exact symplectic manifold is the contact manifold together with a natural choice of contact form, where denotes a coordinate on the -factor. The Reeb vector field with respect to this contact form is given by . The natural projection is called the Lagrangian projection. Given a closed Legendrian submanifold , it follows that is an exact Lagrangian immersion whose double points correspond to integral curves of the Reeb vector field having endpoints on ; such integral curves are called the Reeb chords on . The set of all Reeb chords on will be denoted by . We say that is chord generic if the double points of are transverse, which in particular implies that .
We will here mainly be interested in the following two contact manifolds: the standard and the standard . First we define the standard . The contactisation of the standard symplectic vector space , where is the Liouville form, is the standard contact -space , . Note that Gromov’s theorem [Gro1] implies that a closed exact Lagrangian immersion in must have at least one double-point or, equivalently, that any closed Legendrian submanifold of the standard contact vector space must have a Reeb chord.
Then we define the standard . Considering the primitive of the standard symplectic form on , the standard contact sphere is given by , where is induced by the standard embedding as the unit sphere. Observe that the complement of a point of the standard contact sphere is contactomorphic to the standard contact vector space [Gei, Proposition 2.1.8]. The exact symplectic manifold can be identified with the symplectisation of .
2.2. The front spinning construction
Given a Legendrian submanifold , the so called front spinning construction produces a Legendrian embedding of inside , as described by Ekholm, Etnyre and Sullivan [EES2]. In [Gol2] this construction was extended to the -front spinning, which produces a Legendrian embedding of inside . It was also shown that this construction extends to exact Lagrangian cobordisms. Below we recall these constructions.
induces an embedding
which, in turn, has a canonical extension to an embedding
preserving the Liouville forms. Using to denote the zero-section, the fact that is a Legendrian submanifold of the contactisation of thus provides an embedding of into the contactisation of .
Suppose that we are given a Legendrian submanifold . The above Legendrian embedding of is called the -spin of and will be denoted by .
Observe that the symplectisation of the standard contact vector space is symplectomorphic to . For an exact Lagrangian cobordism from to , the image of the exact Lagrangian submanifold under the above embedding can be seen as an exact symplectic cobordism from to inside the symplectisation of .
Suppose that we are given an exact Lagrangian cobordism from to inside the symplectisation of . The above exact Lagrangian cobordism from to diffeomorphic to is called the -spin of and will be denoted by .
2.3. Bilinearised Legendrian contact cohomology
Legendrian contact homology (LCH) is a modern Legendrian isotopy invariant defined for Legendrian submanifolds of by Chekanov [Che] in a combinatorial way, and later generalised to the contactisation of a Liouville domain by Ekholm, Etnyre and Sullivan in [EES3] using holomorphic curves. This invariant can be seen as a part of the symplectic field theory program, which was proposed by Eliashberg, Givental and Hofer in [EGH].
Given a chord-generic Legendrian submanifold , we associate to it a unital non-commutative differential graded algebra over freely generated by the set of Reeb chords . This algebra is sometimes called the Chekanov-Eliashberg algebra of . The differential is defined on the generators by the count
of pseudo-holomorphic discs with one positive asymptotics to the Reeb chord and several negative asymptotics to the Reeb chords . Here the sum is taken over the one-dimensional components of the moduli spaces, where the -action is induced by translation, for some generic choice of cylindrical almost complex structure. See Sections 2.6, 2.7 below for the definitions of these moduli spaces. The differential is then extended to all of via the Leibniz rule and linearity. For the details of this construction we refer to [EES3].
In order to extract finite-dimensional linear information out of this DGA, Chekanov considered augmentations as bounding cochains for Legendrian contact homology.
Let be a DGA over a unital commutative ring . An augmentation of is a unital DGA map
where all elements of are concentrated in degree .
In other words, an augmentation is a unital algebra map such that
Given two augmentations of the Chekanov-Eliashberg algebra , we can define the bilinearised Legendrian contact cohomology complex, which is the finite-dimensional -vector space with basis , whose boundary is given by the count
of pseudo-holomorphic discs, where the sum is taken over the rigid components (modulo translation) of the moduli spaces for some generic choice of cylindrical almost complex structure. The homology of this complex will be denoted by .
This cohomology theory is a generalisation of the original Chekanov’s linearised Legendrian contact cohomology, see [Che]. If , then these two theories coincide. The set of isomorphism classes of bilinearised Legendrian cohomologies is a Legendrian isotopy invariant [BC, Theorem 1.1].
In the spirit of symplectic field theory, Ekholm has shown [Ekh1] that an exact Lagrangian cobordism from to together with the choice of a generic almost complex structure induces a DGA morphism
defined on generators by
where the sum is taken over the rigid components of the moduli spaces for some generic choice of compatible almost complex structure. In particular, an exact Lagrangian filling induces an augmentation because the Chekanov-Eliashberg algebra associated to the empty set is the ground ring. Moreover, given two augmentations , , of there is an induced chain map
which is defined by
We now consider the case , and when is an augmentation of . It follows from the fact that counts pseudo-holomorphic discs (which in particular are connected) that there is an induced augmentation of which on the Reeb chord takes the value and vanishes on the Reeb chords from to and from to . For the same reason, the subset
consisting of Reeb chords starting on and ending on spans a subcomplex of . We will denote this subcomplex by
If are sufficiently -close, it follows from the invariance theorem in [EES3] that the canonical identification of the generators induces an isomorphism between the Chekanov-Eliashberg algebras and . In particular, there is a canonical bijective correspondence between the augmentations of and in this case.
Use to denote the translation , i.e. the time- Reeb flow for the standard contact form.
Consider the cylindrical lift of a fixed regular almost complex structure on (see Section 2.7). If the Legendrian submanifold is sufficiently -close to , for each there is a canonical isomorphism
of complexes, where we have identified the augmentations of the Chekanov-Eliashberg algebras of and by Remark 2.6, and used in the definition of the differentials.
2.4. Wrapped Floer homology
Wrapped Floer homology is a version of Lagrangian intersection Floer homology for certain non-compact Lagrangian submanifolds. It first appeared in [AS1], and different versions were later developed in [AS2], [Abo2], [FSS1], and [Ekh2]. We will be using the set-up of the latter version, which is useful for establishing a connection between wrapped Floer homology and bilinearised Legendrian contact cohomology.
In the following we will let be exact Lagrangian fillings of , respectively, which are assumed to intersect transversely, and hence in a finite set of double-points. We use to denote the augmentation induced by , . The wrapped Floer homology complex is defined to be
with respect to the above decomposition, where
Here the above sums are taken over the rigid components of the moduli spaces, and , while .
It immediately follows that is a subcomplex, whose corresponding quotient complex can be identified with the complex . For obvious reasons, we will denote the latter quotient complex by
since the differential of this complex counts ordinary Floer strips.
In the current setting, we have the following invariance result
Proposition 2.8 (Proposition 5.12 in [Dr2]).
If are exact Lagrangian fillings inside the symplectisation of a contactisation, it follows that is an acyclic complex or, equivalently, that
is a quasi-isomorphism, where denotes the augmentation induced by , .
2.5. Relations between wrapped Floer homology and bilinearised LCH
In the case when is an exact Lagrangian filling of , the DGA morphism described in Equation 2 is an augmentation of the Chekanov-Eliashberg algebra of defined by counting elements in the moduli space for each Reeb chord on .
In [Ekh2], Ekholm outlined an isomorphism, first conjectured by Seidel, relating the linearised Legendrian contact cohomology and the singular homology of a filling. The details of this isomorphism were later worked out in [DR2].
Let be a Legendrian submanifold admitting an exact Lagrangian filling . There is an isomorphism
where is the augmentation induced by . Here all the gradings are taken modulo the Maslov number of .
The above theorem follows from the following basic result, together with a standard computation. Recall that the Hamiltonian flow is simply a translation of the -coordinate by .
Theorem 2.10 (Theorem 4.2 in [Bc]).
Let be a Legendrian submanifold admitting exact Lagrangian fillings inducing the augmentations of , . For each sufficiently small and an appropriate choice of compatible almost complex structure there is an isomorphism
The bilinearised Legendrian contact homology of the -spin induced by a pair , of -spins of fillings can be computed using the following Künneth-type formula.
Let be the augmentation of induced by an exact Lagrangian filling and let be the augmentation of induced by the exact Lagrangian filling , . There is an isomorphism of graded -vector spaces
Use the Künneth-type formula for Lagrangian Floer homology, see e.g. [Li] or [HLS, Section 2.6], together with the isomorphism in Theorem 2.10. Namely, has a neighbourhood symplectomorphic to , where is a neighbourhood of the zero section and moreover is identified with , . The Künneth-type formula now gives
where the latter factor is isomorphic to by a standard computation. ∎
Finally, observe that this Künneth-type formula is analogous to the version for generating family homology proved in [SS, Proposition 5.4] for spins of Legendrian manifolds admitting generating families.
2.6. Pseudo-holomorphic discs with boundary on a Lagrangian cobordism
In this section we describe the moduli spaces of pseudo-holomorphic discs involved it the construction of Legendrian contact homology. Recall that a compatible almost complex structure on the symplectisation of is cylindrical if
is invariant under translations of the -coordinate;
In the following we let be an exact Lagrangian cobordism from to inside the symplectisation of , where is assumed to be cylindrical outside of . Also, we let be a compatible almost complex structure on which is cylindrical outside of a compact subset of .
The so-called Hofer-Energy of a map from a Riemann surface with boundary is defined as
where is the set consisting of smooth functions satisfying for , and . Observe that the Hofer-Energy is non-negative whenever is -holomorphic, i.e. satisfies , for an almost complex structure of the above form.
Consider the piecewise smooth function which satisfies for , for , while for . We define the -energy of by
We will study -holomorphic discs , where the map is defined outside of a finite set of boundary points, usually called the (boundary) punctures, and required to have finite Hofer energy. At a puncture we require that either , in which case we call the puncture positive, or , in which case we call the puncture negative. The finiteness of the Hofer energy implies that is asymptotic to cylinders over Reeb chords at its boundary punctures. This fact follows by the same arguments as the analogous statement in the case when the boundary is empty and all punctures are internal, which was proven in [HWZ].
Let a Reeb chord on and a word of Reeb chords on . We use
to denote the moduli space consisting of -holomorphic discs as above that moreover satisfy the properties that:
has a unique positive puncture , at which it is asymptotic to a cylinder over ; and
has negative punctures, and at the -th negative puncture on the oriented boundary arc it is asymptotic to a cylinder over .
Choose a primitive of which is constant on the negative end. Such a primitive exists by the definition, since is an exact Lagrangian cobordism of the form , and because vanishes on ,where is cylindrical. For a Reeb chord , we define
For and , we now write
where denote the starting and the ending points of the Reeb chord , respectively.
A standard computation utilising Stoke’s theorem and the assumption that is exact shows that
The -energy of , where satisfies the above properties, is given by
where, in the case , if and only if is constant. In particular, must have a positive puncture unless it is constant.
We will also be interested in the case when , where each component is embedded, but where we allow intersections inside . Suppose that and both start on and end on , has both endpoints on , and has both endpoints on . Letting and , we write
The reason for this notation is that, using an appropriate conformal identification of the domain, we will consider such a disc to be a -holomorphic strip
having one boundary component on and one boundary component on (although possibly having additional negative punctures on each boundary arc). Here is asymptotic to a cylinder over and as and , respectively, using the coordinates on .
We will also consider the moduli spaces , where we allow any of or to be double-points in , in which case a strip is required to converge and as and , respectively. For a double-point , we define
where are potentials of which are required to coincide on the negative ends. Similarly to Formula (3) one can compute
Let , where and are allowed to be either Reeb chords or double points, and suppose that is an almost complex structure satisfying the above properties. It follows that
where, in the case , if and only if is constant.
2.7. Transversality results
In the case when is a trivial cylinder and is a contactisation, we will chose to be the uniquely defined cylindrical lift of a compatible almost complex structure on , i.e. the cylindrical almost complex structure for which the canonical projection is -holomorphic. Observe that the moduli spaces of -holomorphic discs with boundary on the exact Lagrangian immersion having one positive puncture is transversely cut-out for a suitable generic choice of by [EES3]. Finally, the latter moduli spaces being transversely cut out implies that the moduli spaces are transversely cut out as well [DR2, Theorem 2.1].
In the setting when is not cylindrical, the following technical result will be crucial for achieving transversality. Recall that that a pseudo-holomorphic map from a punctured Riemann surface is called simple if the subset
is open and dense. Standard techniques [MS2] show that the simple pseudo-holomorphic curves are transversely cut out solutions for a generic almost complex structure.
We will prove that a disc with boundary on an exact Lagrangian cobordism in having exactly one positive puncture asymptotic to a cylinder over the Reeb chord is simple. Otherwise the techniques in [Laz1], [Laz2] could be used to extract a non-constant pseudo-holomorphic disc with boundary on without any positive puncture, thus leading to a contradiction. Apart from exactness, the following property is crucial here: the Reeb chord is an embedded integral curve and hence that is an embedding onto its image inside the subset for sufficiently large.
Let be an exact Lagrangian cobordism. Then is simple.
We start by observing that is an embedding near each of its punctures because of the asymptotic properties. Moreover, for sufficiently large we may assume that is arbitrarily close to a parametrisation of in the -topology while is arbitrarily close to a parametrisation of in the -topology.
Let denote the positive puncture, and , , the negative punctures. We define
where each is a connected punctured neighbourhood of satisfying for . As a consequence of Carleman’s similarity principle, see e.g. [Laz1, Lemma 4.2], and either intersect in a discrete set, or coincide after a holomorphic identification of the domains. In particular, the domain is contained in a closed domain diffeomorphic to a disc with smooth boundary and moreover satisfying the following property: the restriction of to two arcs in are embeddings which either are disjoint or coincide.
For a sufficiently small neighbourhood