Lagrangian Cobordism Research Articles

Existence of generating families on Lagrangian cobordisms

For an embedded exact Lagrangian cobordism between Legendrian submanifolds in the 1-jet bundle, we prove that a generating family linear at infinity on the Legendrian at the negative end extends to a generating family linear at infinity on the Lagrangian cobordism after stabilization if and only if the formal obstructions vanish. In particular, a Lagrangian filling with trivial stable Lagrangian Gauss map admits a generating family linear at infinity.

Lagrangian cobordisms and Lagrangian surgery

Lagrangian k -surgery modifies an immersed Lagrangian submanifold by topological k -surgery while removing a self-intersection. Associated to a k -surgery is a Lagrangian surgery trace cobordism. We prove that every Lagrangian cobordism is exactly homotopic to a concatenation of suspension cobordisms and Lagrangian surgery traces. This exact homotopy can be chosen with as small Hofer norm as desired. Furthermore, we show that each Lagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain associated with the handle attachment to the Floer cochain generated by the self-intersection. We give a sample computation for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds. In an appendix, we describe a 2-ended embedded monotone Lagrangian cobordism which is not the suspension of a Hamiltonian isotopy following a suggestion of Abouzaid and Auroux.

Lagrangian cobordism functor in microlocal sheaf theory I

AbstractLet be Legendrian submanifolds in the cosphere bundle . Given a Lagrangian cobordism of Legendrians from to , we construct a functor between sheaf categories of compact objects with singular support on and its right adjoint on sheaf categories of proper objects, using Nadler–Shende's work. This gives a sheaf theory description analogous to the Lagrangian cobordism map on Legendrian contact homologies and the right adjoint on their unital augmentation categories. We also deduce some long exact sequences and new obstructions to Lagrangian cobordisms between high‐dimensional Legendrian submanifolds.

Legendrian satellites and decomposable cobordisms

In this paper, we investigate the interactions between the Legendrian satellite construction and the existence of exact, orientable Lagrangian cobordisms between Legendrian knots. Given Lagrangian cobordisms between two Legendrian knots and between two Legendrian tangles, we construct a Lagrangian cobordism between Legendrian satellites of the knots by the closures of the tangles, with extra twists on both the top and the bottom satellite to compensate for the genus of the cobordism. If the original cobordisms were decomposable, then a decomposable cobordism between satellites exists as well, again with extra twists.

Lagrangian cobordisms in Liouville manifolds

Floer theory for Lagrangian cobordisms was developed by Biran and Cornea in a series of papers [Lagrangian cobordism. I, J. Amer. Math. Soc. 26 (2013) 295–340; Lagrangian cobordism and Fukaya categories, Geom. Funct. Anal. 24 (2014) 1731–1830; Cone-decompositions of Lagrangian cobordisms in Lefschetz fibrations, Selecta Math. 23 (2017) 2635–2704] to study the triangulated structure of the derived Fukaya category of monotone symplectic manifolds. This paper explains how to use the language of stops to study Lagrangian cobordisms in Liouville manifolds and the associated exact triangles in the derived wrapped Fukaya category. Furthermore, we compute the cobordism groups of non-compact Riemann surfaces of finite type.

A Lagrangian pictionary

We describe a dictionary of geometry ⟷ algebra in Lagrangian topology. As a by-product, we obtain a tautological (in a particular sense which we explain) proof of a folklore conjecture (sometimes attributed to Kontsevich) claiming that the objects and structure of the derived Fukaya category can be represented through immersed Lagrangians. Our construction is based on certain Lagrangian cobordism categories endowed with a structure called surgery models.

Lagrangian cobordism and tropical curves

Abstract We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.

Rational equivalence and Lagrangian tori on K3 surfaces

Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L \subset X to the skyscraper sheaf of a point y \in Y . We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the “Beauville–Voisin subring” in the Chow groups of Y , and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.

Genus One Cobordisms Between Torus Knots

Abstract We determine the pairs of torus knots that have a genus one cobordism between them, with one notable exception. This is done by combining obstructions using $\nu ^+$ from the Heegaard Floer knot complex and explicit constructions of cobordisms. As an application, we determine the pairs of torus knots related by a single crossing change. Also, we determine the pairs of Thurston–Bennequin number maximizing Legendrian torus knots that have a genus one exact Lagrangian cobordism, with one exception.

The relative Gromov width of Lagrangian cobordisms between Legendrians

We obtain upper and lower bounds for the relative Gromov width of Lagrangian cobordisms between Legendrian submanifolds. Upper bounds arise from the existence of $J$-holomorphic disks with boundary on the Lagrangian cobordism that pass through the center of any given symplectically embedded ball. The areas of these disks --- and hence the sizes of these balls --- are controlled by a real-valued fundamental capacity, a quantity derived from the algebraic structure of filtered linearized Legendrian Contact Homology of the Legendrian at the top of the cobordism. Lower bounds come from explicit constructions that use neighborhoods of Reeb chords in the Legendrian ends. We also study relationships between the relative Gromov width and another quantitative measurement, the length of a Lagrangian cobordism.

A note on coherent orientations for exact Lagrangian cobordisms

Let L \subset \R \times J^1(M) be a spin, exact Lagrangian cobordism in the symplectization of the 1-jet space of a smooth manifold M . Assume that L has cylindrical Legendrian ends \Lambda_\pm \subset J^1(M) . It is well known that the Legendrian contact homology of \Lambda_\pm can be defined with integer coefficients, via a signed count of pseudo-holomorphic disks in the cotangent bundle of M . It is also known that this count can be lifted to a mod 2 count of pseudo-holomorphic disks in the symplectization \R \times J^1(M) , and that L induces a morphism between the \Z_2 -valued DGA:s of the ends \Lambda_\pm in a functorial way. We prove that this hold with integer coefficients as well. The proofs are built on the technique of orienting the moduli spaces of pseudo-holomorphic disks using capping operators at the Reeb chords. We give an expression for how the DGA:s change if we change the capping operators.

In simply connected cotangent bundles, exact Lagrangian cobordisms are h-cobordisms

Abstract Let Q be a simply connected manifold. We show that every exact Lagrangian cobordism between compact, exact Lagrangians in T * Q is an h-cobordism. This is a corollary of the Abouzaid–Kragh Theorem.

Topology of (small) Lagrangian cobordisms

We study the following quantitative phenomenon in symplectic topology: In many situations, if a Lagrangian cobordism is sufficiently small (in a sense specified below) then its topology is to a large extend determined by its boundary. This principle allows us to derive several homological uniqueness results for small Lagrangian cobordisms. In particular, under the smallness assumption, we prove homological uniqueness of the class of Lagrangian cobordisms which, by Biran-Cornea's Lagrangian cobordism theory, induces operations on a version of the derived Fukaya category. We also establish a link between our results and Vassilyev's theory of Lagrange characteristic classes. Most currently known constructions of Lagrangian cobordisms yield small Lagrangian cobordisms in many examples.

Invariants of Lagrangian cobordisms via spectral numbers

We extend parts of the Lagrangian spectral invariants package recently developed by Leclercq and Zapolsky to the theory of Lagrangian cobordism developed by Biran and Cornea. This yields a nondegenerate Lagrangian “spectral metric” which bounds the Lagrangian “cobordism metric” (recently introduced by Cornea and Shelukhin) from below. It also yields a new numerical Lagrangian cobordism invariant as well as new ways of computing certain asymptotic Lagrangian spectral invariants explicitly.

The fundamental group of a rigid Lagrangian cobordism

In this article we extend the construction of the Floer fundamental group to the monotone Lagrangian setting (for weakly exact or monotone Lagrangians with large minimal Maslov number) and use it to study the fundamental group of a Lagrangian cobordism $$W\subset (\mathbb {C}\times M, \omega _{st}\oplus \omega )$$ between two Lagrangian submanifolds $$L, L'\subset ( M, \omega )$$ . We show that under natural conditions the inclusions $$L,L'\hookrightarrow W$$ induce surjective maps $$\pi _{1}(L)\twoheadrightarrow \pi _{1}(W)$$ , $$\pi _{1}(L')\twoheadrightarrow \pi _{1}(W)$$ and when the previous maps are injective then W is an h-cobordism. In particular, in dimension at least 6, W is topologically trivial in this case.

Dehn twist exact sequences through Lagrangian cobordism

This paper introduces a new Lagrangian surgery construction that generalizes Lalonde–Sikorav and Polterovich’s well-known construction, and combines this with Biran and Cornea’s Lagrangian cobordism formalism. With these techniques, we build a framework which both recovers several known long exact sequences (Seidel’s exact sequence, including the fixed point version and Wehrheim and Woodward’s family version) in symplectic geometry in a uniform way, and yields a partial answer to a long-term open conjecture due to Huybrechts and Thomas; this also involved a new observation which relates projective twists with surgeries.

Stability conditions and Lagrangian cobordisms

In this paper we study the interplay between Lagrangian cobordisms and stability conditions. We show that any stability condition on the derived Fukaya category $D\mathcal{F}uk(M)$ of a symplectic manifold $(M,\omega)$ induces a stability condition on the derived Fukaya category of Lagrangian cobordisms $D\mathcal{F}uk(\mathbb{C} \times M)$. We also discuss a relation between stability conditions and the Lagrangian cobordisms group. This provides a general framework in which Haug's result, that the Lagrangian cobordism group of $T^2$ is isomorphic to $K_0(D\mathcal{F}uk(T^2))$, can be understood.

The augmentation category map induced by exact Lagrangian cobordisms

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

Exact Lagrangian cobordism and pseudo-isotopy

We show that under some topological assumptions, an exact Lagrangian cobordism [Formula: see text] of dimension [Formula: see text] is a Lagrangian pseudo-isotopy. This result is a weaker form of a conjecture proposed by Biran and Cornea, which states that any exact Lagrangian cobordism is Hamiltonian isotopic to a Lagrangian suspension.

Nonorientable Lagrangian cobordisms between Legendrian knots

In the symplectization of standard contact $3$-space, $\mathbb R \times \mathbb R^3$, it is known that an orientable Lagrangian cobordism between a Legendrian knot and itself, also known as an orientable Lagrangian endocobordism for the Legendrian knot, must have genus $0$. We show that any Legendrian knot has a non-orientable Lagrangian endocobordism, and that the crosscap genus of such a non-orientable Lagrangian endocobordism must be a positive multiple of $4$. The more restrictive exact, non-orientable Lagrangian endocobordisms do not exist for any exactly fillable Legendrian knot but do exist for any stabilized Legendrian knot. Moreover, the relation defined by exact, non-orientable Lagrangian cobordism on the set of stabilized Legendrian knots is symmetric and defines an equivalence relation, a contrast to the non-symmetric relation defined by orientable Lagrangian cobordisms.