05.12.04 · symplectic / 12-singularities

Lagrangian and Legendrian cobordism

shipped3 tiersLean: none

Anchor (Master): Arnold 1980 *Funct. Anal. Pril.* 14; Eliashberg 1984 *Funct. Anal. Pril.* 18; Audin 1985 *Comm. Math. Helv.* 60; Eliashberg-Mishachev *Introduction to the h-Principle* AMS 2002; Biran-Cornea 2013 *Geom. Topol.* 17

Intuition Beginner

Two shapes count as the same in cobordism theory when one can be deformed into the other through a higher-dimensional surface that connects them. Picture a soap film: at one end the rim is a circle, at the other end the rim is a figure eight, and the film is smooth all the way through. The boundary of the film records both shapes at once, and the film itself is the cobordism.

A Lagrangian cobordism is the symplectic version of this picture. The two ends are Lagrangian submanifolds inside a symplectic manifold; the surface connecting them is a Lagrangian submanifold in a slightly bigger ambient space, with the extra two dimensions playing the role of the cobordism direction. The whole surface must satisfy the Lagrangian vanishing condition at every interior point, not just at the ends.

This idea matters because it gives a topological way to compare Lagrangians. Two Lagrangians related by a smooth path are cobordant, but cobordism is strictly weaker: some Lagrangians can be reached by surgery moves that are not isotopies. The classification of Lagrangian cobordism classes turns out to be governed by a short list of integer invariants, the Maslov number among them.

Visual Beginner

The diagram shows a strip-shaped Lagrangian cobordism inside $M \times R^{2}$ . At the far left the strip is asymptotic to a vertical cylinder over the circle $L_{0}$ ; at the far right it is asymptotic to a vertical cylinder over the figure eight $L_{1}$ . In the middle a single transverse double point is smoothed away by a Polterovich surgery, producing the topology change between the two ends.

The picture records two things at once: the cobordism is a single connected Lagrangian, and the topology of the two endpoint Lagrangians can differ because the cobordism is allowed to perform surgeries in the interior.

Worked example Beginner

Take the plane $R^{2}$ with coordinates $(q, p)$ and the standard area form $ω = d q \land d p$ . The unit circle $L_{0} = {q^{2} + p^{2} = 1}$ is a one-dimensional Lagrangian (any curve in a two-dimensional symplectic surface is Lagrangian, because the form vanishes for dimension reasons). The figure eight $L_{1}$ , parametrised by $t \mapsto (sin 2 t, sin t)$ for $t \in [0, 2 π]$ , is another one-dimensional Lagrangian, this time with a self-intersection at the origin.

A Lagrangian cobordism between these two curves lives in $R^{2} \times R^{2}$ with the product form $ω + d s \land d r$ , where $(s, r)$ are the extra two coordinates. Outside a compact region the cobordism is a product: on the left it equals $L_{0} \times {s = - 1} \times R$ , on the right $L_{1} \times {s = + 1} \times R$ . In the middle, a tube of dimension two interpolates by smoothing the double point of the figure eight into a small handle.

Counting invariants is direct. The rotation number of the unit circle (how many times the tangent vector turns under a full traversal) is $1$ . The rotation number of the figure eight, traversed once, is $0$ , because the two lobes turn in opposite directions. The Maslov number of the cylindrical cobordism turns out to equal the difference $1 - 0 = 1$ , which is exactly the integer class of the cobordism in Arnold's $Z \oplus Z$ classification.

What this tells us: the Lagrangian-cobordism class is detected by two integers, and the difference of rotation numbers at the two ends is one of them.

Check your understanding Beginner

Formal definition Intermediate+

Let $(M, ω)$ be a symplectic manifold of dimension $2 n$ , and equip $R^{2}$ with the standard area form $ω_{R^{2}} = d s \land d r$ in coordinates $(s, r)$ . The product $M \times R^{2}$ carries the symplectic form $ω \oplus ω_{R^{2}}$ . A submanifold $Λ \subset M \times R^{2}$ of dimension $n + 1$ is Lagrangian when $(ω \oplus ω_{R^{2}}) ∣_{Λ} = 0$ .

Two compact Lagrangian submanifolds $L_{0}, L_{1} \subset (M, ω)$ of dimension $n$ are Lagrangian-cobordant if there exists a Lagrangian submanifold $Λ \subset M \times R^{2}$ of dimension $n + 1$ and a constant $T > 0$ such that $$ \Lambda \cap \bigl( M \times (-\infty, -T] \times \mathbb{R} \bigr) = L_0 \times (-\infty, -T] \times {r_0}, $$ $$ \Lambda \cap \bigl( M \times [T, +\infty) \times \mathbb{R} \bigr) = L_1 \times [T, +\infty) \times {r_1}, $$ for some heights $r_{0}, r_{1} \in R$ . The two ends are cylindrical in the $s$ -direction. The convention follows Arnold 1980 ^{[Arnold 1980]} and Audin 1985 ^{[Audin 1985]}.

The pair $(L_{0}, L_{1})$ being cobordant defines an equivalence relation: reflexivity by the product cobordism $L \times R \times {r}$ ; symmetry by reflection in $s$ ; transitivity by truncating two cobordisms at their cylindrical ends and gluing along a common Lagrangian. The set of equivalence classes forms an abelian group $Ω_{n}^{L} (M, ω)$ under disjoint union, with inversion realised by the reflected cobordism.

Legendrian variant. Let $(N, α)$ be a contact manifold of dimension $2 n + 1$ in the sense of 05.12.03. The contactisation product $N \times R$ with coordinate $t$ on the $R$ -factor carries the contact form $α_{prod} = e^{t} α + d t$ in the standard construction. Two compact Legendrian submanifolds $Λ_{0}, Λ_{1} \subset (N, α)$ of dimension $n$ are Legendrian-cobordant when there exists a Legendrian $Σ \subset (N \times R, α_{prod})$ of dimension $n + 1$ with cylindrical ends $Λ_{0} \times (- \infty, - T]$ at $t = - \infty$ and $Λ_{1} \times [T, + \infty)$ at $t = + \infty$ . The Legendrian projection of $Σ$ to $N$ recovers a Lagrangian cobordism in the symplectisation $N \times R$ of $(N, α)$ via the symplectisation construction.

Cobordism invariants. The Maslov class $μ_{L_{i}} \in H^{1} (L_{i}; Z)$ pulls back to the Maslov class $μ_{Λ} \in H^{1} (Λ; Z)$ under inclusion, and the class on each end recovers $μ_{L_{i}}$ . The Maslov index $Maslov (L) \in Z$ of a Lagrangian $L$ with vanishing first cohomology (e.g. a Lagrangian sphere) is the integer obtained by pairing $μ_{L}$ with a generator of $H_{1} (L; Z)$ , after choosing one. For oriented Lagrangian curves in $R^{2}$ , the Maslov index coincides with the rotation number, the winding number of the unit tangent around the unit circle in $R^{2}$ .

Polterovich-surgery construction. Given two Lagrangians $L_{0}, L_{1} \subset (M, ω)$ intersecting transversally at a single point $p$ , the Lagrangian connect sum $L_{0} #_{p} L_{1}$ is a new Lagrangian obtained by replacing a small neighbourhood of $p$ in $L_{0} \cup L_{1}$ with a Lagrangian handle modelled on the standard Lagrangian $T^{*} S^{n - 1}$ near its zero-section, scaled so the handle fits the gluing data ^{[Polterovich 1991]}. The trace of the surgery move — interpolating between $L_{0} ⊔ L_{1}$ and $L_{0} #_{p} L_{1}$ as the handle scale shrinks to zero — is a Lagrangian cobordism in $M \times R^{2}$ with the disjoint union on one end and the surgery on the other. Polterovich's construction is the basic surgery move in the Lagrangian-cobordism category.

The sign convention follows 05.12.01: $ω = \sum d q_{i} \land d p_{i}$ on $M$ and $ω \oplus ω_{R^{2}}$ on $M \times R^{2}$ . The Maslov class $μ$ is normalised so that $μ ([γ]) \in Z$ for any Lagrangian loop $γ : S^{1} \to L$ .

Key theorem with proof Intermediate+

Theorem (Arnold 1980). The Legendrian-cobordism group of compact Legendrian curves in the contact three-manifold $J^{1} (R) = T^{*} R \times R$ , equivalently the wave-front cobordism group of immersed plane curves with cusps, is $$ \Omega^L(\mathbb{R}^2) ;\cong; \mathbb{Z} \oplus \mathbb{Z}. $$ The first $Z$ -factor is generated by the rotation index $r (γ)$ of the front projection $γ : S^{1} \to R^{2}$ , equal to the winding number of the unit tangent around the unit circle in $R^{2}$ , evaluated for the smooth approximation of $γ$ around its cusps. The second $Z$ -factor is generated by the Maslov index $μ (γ) \in Z$ , equal to the signed count of cusps of $γ$ . The pair $(r (γ), μ (γ))$ determines the Legendrian-cobordism class.

Proof. The argument has three steps: (i) both invariants are cobordism-invariant; (ii) the pair $(r, μ)$ takes every value in $Z^{2}$ ; (iii) no further invariant is required.

Step (i): Cobordism invariance. Let $Σ \subset J^{1} (R) \times R$ be a Legendrian cobordism between Legendrian curves $Λ_{0}$ and $Λ_{1}$ , with cylindrical ends at $t = \pm \infty$ . The Maslov class of a Legendrian curve is the pull-back of the universal Maslov class on $Λ (1)$ along the Gauss map, by 05.12.01. The Gauss map of $Σ$ extends both Gauss maps of $Λ_{0}, Λ_{1}$ to a map $Σ \to Λ (1)$ ; the pull-back $μ_{Σ} \in H^{1} (Σ; Z)$ pulls back further to $μ_{Λ_{i}}$ along the inclusions $Λ_{i} ↪ Σ$ . Since $Σ$ has cylindrical ends, $H^{1} (Σ; Z)$ surjects onto $H^{1} (Λ_{0}; Z) \oplus H^{1} (Λ_{1}; Z)$ via the restriction map, and the cobordism gives the relation $μ_{Λ_{0}} = μ_{Λ_{1}}$ as elements of $H^{1} (S^{1}; Z) = Z$ . The rotation-index invariance is analogous: the unit tangent of the front projection defines a map $Λ_{i} \to S^{1}$ , and the cobordism extends this to a map $Σ \to S^{1}$ — equality of rotation numbers across the cobordism follows from the same restriction-map argument applied to the unit tangent.

Step (ii): Surjectivity of $(r, μ) : Ω^{L} (R^{2}) \to Z^{2}$ . It suffices to exhibit Legendrian curves realising the generators $(1, 0)$ and $(0, 1)$ . The unit circle $γ_{1} (θ) = (cos θ, sin θ)$ , lifted to a Legendrian in $J^{1} (R)$ as the front projection, has rotation index $1$ and no cusps, so $μ = 0$ : it represents $(1, 0)$ . The Legendrian unknot whose front projection is the figure eight $γ_{2} (t) = (sin 2 t, sin t)$ for $t \in [0, 2 π]$ has rotation index $0$ (the two lobes turn in opposite directions) and one cusp at each lobe-tip after Legendrian smoothing, giving $μ = 1$ : it represents $(0, 1)$ . The pair $(γ_{1}, γ_{2})$ generates $Z \oplus Z$ under disjoint union.

Step (iii): No further invariant. Two Legendrian curves with the same $(r, μ)$ admit a Legendrian cobordism. The construction proceeds by reducing to a normal form via Reidemeister-style Legendrian moves and Polterovich surgery: each Legendrian curve is cobordant to a disjoint union of $r$ standard circles (each representing $(1, 0)$ ) plus $μ$ standard figure eights (each representing $(0, 1)$ ). The reduction uses the h-principle for Legendrian immersions of plane curves: any two Legendrian curves with the same total Maslov-and-rotation data are connected by a sequence of cusp-cancellations and rotation-equivalent reparametrisations, all of which are realised by Legendrian cobordisms in $J^{1} (R) \times R$ .

Combining the three steps, $(r, μ) : Ω^{L} (R^{2}) \to Z \oplus Z$ is a well-defined surjective group homomorphism with zero kernel, hence an isomorphism ^{[Arnold 1980]}. $□$

Bridge. This unit pulls together 05.12.01 (the universal Maslov class on $Λ (n)$ — used here as the source of the integer cobordism invariant) and 05.12.03 (Legendrian singularities and wave-front evolution — supplying the cusp-count interpretation of the Maslov index). The bordism-quotient construction is the symplectic specialisation of the unoriented Pontryagin-Thom framework of 03.06.12, with $Ω_{*}^{L}$ replacing $Ω_{*}^{O}$ .

Exercises Intermediate+

Exercise 4 (medium, short-answer).

Show that the trace of a smooth Lagrangian isotopy ${L_{t}}_{t \in [0, 1]} \subset (M, ω)$ is a Lagrangian cobordism between $L_{0}$ and $L_{1}$ . Identify the symplectic form on the ambient $M \times R^{2}$ .

Hint

The trace is the set ${(x, s, r) : x \in L_{f (s)}}$ for a suitable cutoff $f$ matching the cylindrical-ends condition.

Answer

Choose a smooth monotone cutoff $f : R \to [0, 1]$ with $f (s) = 0$ for $s \leq - 1$ and $f (s) = 1$ for $s \geq 1$ . Define $Λ = {(x, s, r) \in M \times R^{2} : x \in L_{f (s)}, r \in R}$ . Outside the strip $- 1 \leq s \leq 1$ , $Λ$ coincides with $L_{0} \times (- \infty, - 1] \times R$ on the left and $L_{1} \times [1, \infty) \times R$ on the right, satisfying the cylindrical-ends condition. The dimension count gives $dim Λ = n + 1$ . The Lagrangian condition on $Λ$ holds because at each interior point $(x, s, r)$ , the tangent space splits as $T_{x} L_{f (s)} \oplus R \cdot \partial_{s} \oplus R \cdot \partial_{r}$ , and the symplectic form $ω \oplus d s \land d r$ vanishes on this product: $ω ∣_{T_{x} L_{f (s)}} = 0$ because $L_{f (s)}$ is Lagrangian, and the $(s, r)$ -component is the symplectic complement of the diagonal direction $\partial_{s}$ paired against $\partial_{r}$ . The ambient form is $ω \oplus d s \land d r$ .

Rubric: full credit for the cutoff construction and the dimension-plus-Lagrangian-condition verification; partial credit for either alone.

Exercise 5 (medium, short-answer).

Describe the Polterovich Lagrangian-surgery move at a transverse double point of two Lagrangians $L_{0}, L_{1} \subset (M, ω)$ . What is the dimension of the cobordism produced, and what role does the local model $T^{*} S^{n - 1}$ play?

Hint

The handle interpolates between $L_{0} \cup L_{1}$ near the double point and the smoothed $L_{0} #_{p} L_{1}$ via a Lagrangian handle of fixed local form.

Answer

Near a transverse intersection point $p \in L_{0} \cap L_{1}$ , the Darboux theorem identifies a neighbourhood of $p$ in $(M, ω)$ with a neighbourhood of $0$ in $(R^{2 n}, ω_{0})$ such that $L_{0}$ becomes the position-axis Lagrangian $R^{n} \times {0}$ and $L_{1}$ becomes the momentum-axis Lagrangian ${0} \times R^{n}$ . The Lagrangian handle $H_{ϵ} = {(q, p) : q \cdot p = ϵ, ∣ q ∣^{2} = ∣ p ∣^{2}}$ for small $ϵ > 0$ is diffeomorphic to a tube neighbourhood of the zero-section in $T^{*} S^{n - 1}$ , with the standard symplectic structure on the cotangent bundle, and interpolates between $R^{n} \cup_{p} R^{n}$ (at $ϵ = 0$ , two transverse planes meeting at the origin) and the smoothed Lagrangian $L_{0} #_{p} L_{1}$ (for $ϵ > 0$ , a single Lagrangian with the handle filling the corner). The trace of the move as $ϵ$ varies from $0$ to a fixed positive value defines a Lagrangian cobordism inside $M \times R^{2}$ of dimension $n + 1$ , with $L_{0} ⊔ L_{1}$ on one end and $L_{0} #_{p} L_{1}$ on the other. The local model $T^{*} S^{n - 1}$ is the universal handle: every transverse-double-point smoothing is locally diffeomorphic to this picture, which is why the Polterovich move is well-defined.

Rubric: full credit for naming the Darboux normal form, the $T^{*} S^{n - 1}$ handle model, and the dimension $n + 1$ of the cobordism trace.

Exercise 6 (hard, short-answer).

State Eliashberg's 1984 result computing the Lagrangian-immersion cobordism group $Ω_{*}^{L, imm}$ of immersed Lagrangian submanifolds modulo cobordism, and explain the role played by the h-principle.

Hint

Eliashberg used the contact h-principle for Legendrian immersions; Audin's translation identifies the result with a Thom-spectrum-type homotopy group built from $U / O$ .

Answer

Eliashberg 1984 Funct. Anal. Appl. 18 ^{[Eliashberg 1984]} established the contact h-principle for Legendrian immersions: any formal Legendrian immersion (a smooth map together with a homotopy of its derivative through pointwise-Legendrian linear maps) is homotopic through formal Legendrian immersions to a genuine Legendrian immersion, with the homotopy class realised by an h-principle argument lifting Smale-Hirsch immersion theory to the contact setting. Audin 1985 Comm. Math. Helv. 60 ^{[Audin 1985]} translated this into the cobordism statement: the Lagrangian-immersion cobordism group $Ω_{n}^{L, imm} (pt)$ is isomorphic to the $n$ -th homotopy group of the Thom spectrum $M U / M O$ associated to the inclusion $O ↪ U$ , or equivalently to the $n$ -th stable homotopy group of the suspension of $U / O$ . The h-principle is the bridge between the geometric cobordism group (genuine Lagrangian immersions modulo cobordism) and the homotopy-theoretic group (formal Lagrangian immersions, classified by maps to $U / O$ ). The proof reduces the geometric problem to a topological problem, which is then computed by classical algebraic topology.

Rubric: full credit for naming the h-principle, the Thom-spectrum identification with $U / O$ , and the role of Audin in translating Eliashberg's result; partial credit for two of the three components.

Exercise 7 (hard, short-answer).

Biran and Cornea 2013 Geom. Topol. 17 lifted Lagrangian cobordism to a categorical statement: a Lagrangian cobordism induces an exact triangle in the derived Fukaya category. State the precise form of their result and explain why monotonicity is required.

Hint

A Lagrangian cobordism with three cylindrical ends $L_{0}, L_{1}, L_{2}$ produces a distinguished triangle $L_{0} \to L_{1} \to L_{2} \to L_{0} [1]$ in $D Fuk (M, ω)$ .

Answer

Biran-Cornea ^{[Biran-Cornea 2013]} show that a monotone Lagrangian cobordism $V \subset M \times R^{2}$ with three cylindrical ends $L_{0}, L_{1}, L_{2}$ — corresponding to a tree-shaped cobordism with one input and two outputs — induces an exact triangle $$ L_1 \to L_2 \to L_0 \to L_1[1] $$ in the derived Fukaya category $D Fuk (M, ω)$ of monotone Lagrangians. The monotonicity hypothesis $μ_{L} = λ \cdot [ω ∣_{L}]$ for a fixed $λ > 0$ is required to ensure: (i) the Floer differential on each $L_{i}$ is well-defined over $Z /2$ (no holomorphic-disc bubbling shifts the chain complex); (ii) the cobordism Floer theory inherits the same control on disc bubbling, so the connecting morphisms in the exact triangle are honest chain maps. Without monotonicity, the moduli spaces of holomorphic discs with cobordism boundary can be non-compact, and the categorical statement fails. The result lifts cobordism from a set-level equivalence relation to a triangle relation in a triangulated category, with the cobordism class encoded as the homotopy class of the connecting morphism. The Biran-Cornea theorem is the structural endpoint of the Arnold programme: cobordism is not just an invariant of Lagrangians but a piece of the homological algebra of the symplectic manifold.

Rubric: full credit for the exact-triangle statement, naming monotonicity as the disc-bubbling control, and locating the result inside the Fukaya category; partial credit for any two of the three.

Lean formalization Intermediate+

Lagrangian and Legendrian cobordism are not yet present in Mathlib at the time of writing; the unit ships with lean_status: none. The Mathlib-gap roadmap in the unit metadata describes the missing infrastructure (symplectic product, cylindrical-ends predicate, Polterovich surgery, cobordism quotient, Maslov-index homomorphism). A future Lean formalisation would proceed in three layers: (a) the symplectic-product layer $M \times R^{2}$ with its product symplectic form; (b) the cobordism predicate as a sigma-type combining a Lagrangian $Λ$ with cylindrical-ends data; (c) the quotient construction realising $Ω_{n}^{L} (M)$ as a quotient abelian group with the Maslov index as a homomorphism. Each layer factors through existing Mathlib machinery (manifolds, smooth submanifolds, quotient groups), but the connecting glue is absent.

Advanced results Master

Fix a symplectic manifold $(M, ω)$ of dimension $2 n$ . The product $M \times R^{2}$ with symplectic form $ω \oplus d s \land d r$ is the standard ambient space for Lagrangian cobordism. The Lagrangian-cobordism set $Ω_{n}^{L} (M, ω)$ is an abelian group under disjoint union, with the product cobordism $L \times R \times {0}$ as identity and reflection in $s$ providing inversion. The Maslov class $μ_{Λ} \in H^{1} (Λ; Z)$ , viewed as a homomorphism from $H_{1} (Λ; Z)$ to $Z$ , restricts to $μ_{L_{i}}$ on each end and so descends to a homomorphism $$ \mu_* : \Omega_n^{\mathcal L}(M, \omega) \to H^1_{\mathrm{cob}}(M; \mathbb{Z}) $$ to a cobordism-classified cohomology group ^{[Arnold 1980]}.

Arnold's plane-curve classification. For $M = pt$ and $n = 1$ the Lagrangian (equivalently Legendrian, via the contactisation) cobordism group is $$ \Omega^L(\mathbb{R}^2) \cong \mathbb{Z} \oplus \mathbb{Z}, $$ generated by the rotation index $r$ and the Maslov index $μ$ . The rotation-index generator is the standard circle in $R^{2}$ , lifted to a Legendrian curve in $J^{1} (R)$ by the front projection; the Maslov-index generator is the standard figure eight, lifted to a Legendrian with two cusps after smoothing. The classification proof reduces to an h-principle argument for Legendrian immersions of plane curves: any two Legendrian curves with the same rotation index and Maslov index are connected by a sequence of cusp-cancellations and Reidemeister-style moves, all of which are realised as Legendrian cobordisms ^{[Arnold 1980]}.

Eliashberg's Lagrangian-cobordism computation at a point. Eliashberg 1984 Funct. Anal. Appl. 18 ^{[Eliashberg 1984]} computed the Lagrangian-cobordism group of $pt$ for closed Lagrangians using the contact h-principle for Legendrian immersions in the contactisation. The result identifies $Ω_{n}^{L} (pt)$ with the $n$ -th homotopy of a Thom-spectrum-style object built from the inclusion $O (n) ↪ U (n)$ ; equivalently, after Audin's 1985 Comm. Math. Helv. 60 ^{[Audin 1985]} translation, with $π_{n}^{st} (U / O)$ shifted by the dimension. The h-principle is the structural bridge between the geometric definition (Lagrangian immersions modulo cobordism) and the homotopy-theoretic one (formal Lagrangian immersions classified by maps to $U / O$ ).

Polterovich Lagrangian surgery. Given two Lagrangians $L_{0}, L_{1} \subset (M, ω)$ intersecting transversally at $p$ , the Lagrangian surgery $L_{0} #_{p} L_{1}$ is the smoothing obtained by replacing a small neighbourhood of $p$ in $L_{0} \cup L_{1}$ with a Lagrangian handle modelled on the tube neighbourhood of the zero-section in $T^{*} S^{n - 1}$ . The trace of the surgery scaling parameter is a Lagrangian cobordism $V_{p} \subset M \times R^{2}$ with $L_{0} ⊔ L_{1}$ on one end and $L_{0} #_{p} L_{1}$ on the other. In symplectic dimension $\geq 4$ , Polterovich's construction is the building block of all Lagrangian cobordisms: every elementary cobordism is either an isotopy trace or a Polterovich surgery trace, by an h-principle argument adapted from Smale's handle calculus ^{[Polterovich 1991]}.

Generating-family viewpoint. A Lagrangian cobordism $Λ \subset M \times R^{2}$ between cotangent-bundle Lagrangians $L_{0}, L_{1} \subset T^{*} Q$ for a base $Q$ can be presented locally as the graph of a generating function $F : Q \times R \times R^{k} \to R$ — generating the cobordism Lagrangian via $Λ_{F} = {(q, \partial_{q} F, s, \partial_{s} F) : \partial_{u} F (q, s, u) = 0}$ . The cylindrical-ends condition translates to a tameness condition on $F$ at $s = \pm \infty$ . The Picard-Lefschetz monodromy of the fibration $Λ \to R_{(s, r)}^{2}$ , considered along a generic path between the two ends, encodes the cobordism class as a product of Dehn twists in the Lagrangian-vanishing-cycle calculus. This generating-family picture is the bridge to mirror symmetry: a Lagrangian cobordism in a symplectic Calabi-Yau corresponds under mirror symmetry to an exact triangle of coherent sheaves on the mirror, exactly the Biran-Cornea statement on the symplectic side.

Biran-Cornea categorical lift. For monotone $(M, ω)$ , Biran-Cornea 2013 Geom. Topol. 17 ^{[Biran-Cornea 2013]} show that a monotone Lagrangian cobordism with three cylindrical ends $L_{0}, L_{1}, L_{2}$ corresponding to a tree-shaped cobordism (one input, two outputs) induces an exact triangle $$ L_1 \to L_2 \to L_0 \to L_1[1] $$ in the derived Fukaya category $D Fuk (M, ω)$ . The connecting morphism is constructed from the Floer complex of the cobordism viewed as a Lagrangian in the product. Monotonicity is necessary to control holomorphic-disc bubbling, ensuring the Floer differential on each end and the connecting morphism are well-defined chain maps. The categorical statement lifts the Arnold-Eliashberg-Audin classification from a group-theoretic invariant to a structural feature of $D Fuk (M, ω)$ : the cobordism category is a piece of triangulated structure rather than a separate object.

Maslov-index and area homomorphisms. Two universal cobordism invariants are the Maslov class $μ_{Λ} \in H^{1} (Λ; Z)$ and the symplectic-area class. For a Lagrangian sphere $L ≅ S^{n}$ in a monotone symplectic manifold with $μ_{L} = k \cdot α$ for $α$ a generator of $H^{1} (S^{n}; Z) = 0$ (so $k = 0$ for $n \neq = 1$ ), the integer $k$ vanishes structurally and the cobordism group of Lagrangian spheres in a fixed homology class is dominated by the symplectic-area invariant. For Lagrangian tori $L ≅ T^{n}$ with $H^{1} (T^{n}; Z) = Z^{n}$ , the Maslov class has $n$ independent integer components, and the cobordism group is genuinely higher-rank.

Full proof set Master

Proposition (cylindrical-ends define an equivalence relation). Lagrangian cobordism between compact oriented Lagrangians in $(M, ω)$ is an equivalence relation: reflexive, symmetric, and transitive.

Reflexivity. The product cylinder $Λ = L \times R \times {r_{0}}$ is Lagrangian in $M \times R^{2}$ (the $R^{2}$ -part contributes only the $\partial_{s}$ -direction to the tangent space, and $d s \land d r$ vanishes on it), and outside the compact region ${∣ s ∣ \leq 0}$ both ends coincide with $L$ . Symmetry. If $Λ$ cobords $L_{0}$ to $L_{1}$ , then the reflection $σ (x, s, r) = (x, - s, r)$ produces a Lagrangian $σ (Λ)$ cobording $L_{1}$ to $L_{0}$ — the reflection preserves $ω \oplus d s \land d r$ up to a sign on $d s$ , which can be absorbed by reorienting. Transitivity. If $Λ$ cobords $L_{0}$ to $L_{1}$ and $Λ^{'}$ cobords $L_{1}$ to $L_{2}$ , truncate each at its cylindrical ends, translate so the $L_{1}$ -cylindrical regions overlap, and glue along the common Lagrangian $L_{1} \times [T, T + ϵ] \times {r_{1}}$ . The glued submanifold $Λ \cup_{L_{1}} Λ^{'}$ is smooth (the cylindrical-ends condition gives matching collar neighbourhoods), Lagrangian (the form is unchanged on each piece), and has cylindrical ends $L_{0}$ on the far left and $L_{2}$ on the far right. The smoothing of the gluing region uses the standard collar-neighbourhood smoothing of the boundary-from-cobordism construction. $□$

Proposition ( $Ω^{L} (R^{2})$ is generated by the standard circle and standard figure eight). Every immersed Legendrian curve in $J^{1} (R)$ is Legendrian-cobordant to a disjoint union of $r$ standard circles and $μ$ standard figure eights, where $r$ is the rotation index of the front projection and $μ$ is the signed cusp count.

Let $γ : S^{1} \to R^{2}$ be the front projection of a Legendrian curve $Λ \subset J^{1} (R)$ . Smooth $γ$ near its cusps to obtain a smooth immersion $\tilde{γ}$ (the smoothing introduces additional intersections, but each intersection is paired with a cusp in such a way that the original Maslov index is preserved). The rotation index of $\tilde{γ}$ equals $r$ by construction, and the cusp count of the original curve equals $μ$ . Apply the Smale-Whitney classification of immersed plane curves: $\tilde{γ}$ is regularly homotopic to a connected sum of $r$ standard circles (each rotation index $1$ , no cusps) with itself, and the cusp data $μ$ is added by a sequence of cusp-creation moves. Each move (regular homotopy step, cusp creation, cusp cancellation) is realised by a Legendrian cobordism in $J^{1} (R) \times R$ , by an h-principle argument ^{[Arnold 1980]}. The conclusion follows from concatenating the resulting cobordisms via the transitivity proposition. $□$

Proposition (Polterovich-surgery trace is a Lagrangian cobordism). The trace of the Polterovich Lagrangian-surgery move at a transverse double point $p \in L_{0} \cap L_{1}$ is a Lagrangian cobordism in $M \times R^{2}$ between $L_{0} ⊔ L_{1}$ and $L_{0} #_{p} L_{1}$ .

By the Darboux-Weinstein theorem, a neighbourhood of $p$ in $(M, ω)$ is symplectomorphic to a neighbourhood of $0$ in $(R^{2 n}, ω_{0})$ such that $L_{0}$ becomes $R^{n} \times {0}$ and $L_{1}$ becomes ${0} \times R^{n}$ . In this model, the Polterovich handle $H_{ϵ} = {(q, p) : q \cdot p = ϵ, ∣ q ∣^{2} = ∣ p ∣^{2}}$ for $ϵ > 0$ is a Lagrangian submanifold of $R^{2 n}$ diffeomorphic to $T^{*} S^{n - 1}$ . The verification is direct: $H_{ϵ}$ is half-dimensional, and $ω_{0} = d q \land d p$ restricts to zero on the tangent space by the constraint equations.

The trace $$ V = \bigl{ (q, p, s, r) \in \mathbb{R}^{2n} \times \mathbb{R}^2 : (q, p) \in H_{\phi(s)},\ r \text{ free}\bigr} $$ for a smooth cutoff $ϕ : R \to [0, ϵ_{0}]$ with $ϕ (s) = 0$ for $s \leq - 1$ and $ϕ (s) = ϵ_{0}$ for $s \geq 1$ is a Lagrangian in $R^{2 n} \times R^{2}$ of dimension $n + 1$ : the constraint $q \cdot p = ϕ (s)$ has differential $p d q + q d p - ϕ^{'} (s) d s = 0$ on the tangent space, which combined with $ω_{0} + d s \land d r$ gives a Lagrangian condition. Outside the strip $- 1 \leq s \leq 1$ the trace is cylindrical: on the left ( $s \leq - 1$ , $ϕ (s) = 0$ ) it is the singular union $L_{0} \cup L_{1}$ scaled to the limiting two transverse planes; on the right ( $s \geq 1$ , $ϕ (s) = ϵ_{0}$ ) it is the smoothed $L_{0} #_{p} L_{1}$ . The trace satisfies the cylindrical-ends condition modulo the singular point at $ϵ = 0$ , which is resolved by a further smoothing of the cutoff near $s = - 1$ ^{[Polterovich 1991]}. $□$

Proposition (Maslov-index homomorphism). The Maslov index of a Lagrangian loop is a homomorphism from $Ω^{L} (R^{2})$ to $Z$ , factoring through the cobordism quotient.

The Maslov class is the pull-back of the universal class $μ \in H^{1} (Λ (n); Z)$ along the Gauss map $L \to Λ (n)$ , by 05.12.01. For a cobordism $Λ$ between $L_{0}$ and $L_{1}$ , the Gauss map extends to $Λ \to Λ (n)$ , and the pull-back $μ_{Λ} \in H^{1} (Λ; Z)$ restricts to $μ_{L_{0}} - μ_{L_{1}}$ on the boundary in the relative-cohomology sense. Since $H^{1} (Λ; Z)$ surjects onto $H^{1} (\partial Λ; Z) = H^{1} (L_{0} ⊔ L_{1}; Z)$ , the equality $μ_{L_{0}} = μ_{L_{1}}$ holds modulo coboundaries. For the cobordism group $Ω^{L} (R^{2})$ , where the underlying Lagrangians are curves $S^{1} \to R^{2}$ and $H^{1} (S^{1}; Z) = Z$ , the boundary obstruction vanishes: the Maslov index on each end is a single integer, and the cobordism relation forces these integers to coincide. The Maslov index thus descends to a homomorphism $Ω^{L} (R^{2}) \to Z$ . Additivity under disjoint union follows from the corresponding additivity of $H^{1}$ pull-backs ^{[Arnold 1980]}. $□$

Connections Master

The universal Maslov class of 05.12.01 is the source of the integer cobordism invariant $μ : Ω^{L} (R^{2}) \to Z$ via pull-back along the Gauss map. The intersection-theoretic Maslov-cycle picture from that unit specialises directly to the signed cusp-count formula on a Legendrian-cobordism representative.
The Legendrian singularity theory of 05.12.03 supplies the front-projection and generating-family machinery: a Lagrangian cobordism between Lagrangians in $T^{*} Q$ is locally a discriminant of a generating family on $Q \times R$ , and the Picard-Lefschetz monodromy of the family along the cobordism direction is the Lagrangian-vanishing-cycle calculus underlying Biran-Cornea's categorical lift.
The unoriented bordism foundation of 03.06.12 is the topological-bordism antecedent: Audin's identification of $Ω_{*}^{L, imm}$ with the homotopy of a Thom-spectrum-style object specialises the Pontryagin-Thom collapse to the Lagrangian setting. The Audin-Eliashberg theory is the symplectic refinement of the Thom 1954 result.
The monotone Lagrangian Floer homology of 05.08.02 is the home of the Biran-Cornea categorical lift: cobordism descends to an exact triangle in the derived Fukaya category, with the connecting morphism constructed from the Floer complex of the cobordism Lagrangian. Monotonicity is required to control disc bubbling, exactly the same condition used to define Floer homology in the first place.
The Maslov-index grading of 05.08.03 for Lagrangian Floer homology is the cobordism-invariant interpretation of $μ$ : the grading of the Floer complex of a Lagrangian is determined by the Maslov class, and the cobordism invariance of $μ$ implies the Floer-homology grading is a cobordism invariant.

Historical & philosophical context Master

Arnold introduced Lagrangian and Legendrian cobordism in the 1980 paper Lagrange and Legendre cobordisms I, II in Funct. Anal. Appl. 14, completing a programme begun in his 1967 paper on the universal Maslov class ^{[Arnold 1967]} and extended through the 1972 Wave fronts and the topology of caustics in Funct. Anal. Appl. 6. The 1980 paper gave the definition of cobordism via cylindrical ends, computed the plane-curve case $Ω^{L} (R^{2}) = Z \oplus Z$ , and posed the problem of computing the cobordism group in higher dimensions ^{[Arnold 1980]}. Eliashberg 1984 Funct. Anal. Appl. 18 ^{[Eliashberg 1984]} settled the higher-dimensional case for a point using the contact h-principle, his foundational work that grew into the joint Introduction to the h-Principle (Eliashberg-Mishachev, AMS Graduate Studies in Mathematics 48, 2002).

Audin 1985 Comm. Math. Helv. 60 ^{[Audin 1985]} translated Eliashberg's result into the Pontryagin-Thom framework, identifying the Lagrangian-immersion cobordism group with the homotopy of a Thom-spectrum-style object built from $U / O$ . Polterovich 1991 Internat. J. Math. 2 ^{[Polterovich 1991]} introduced the Lagrangian-surgery construction that produces explicit cobordisms by smoothing transverse double points; the surgery move became the basic building block of the Lagrangian-cobordism category. Biran and Cornea 2013 Geom. Topol. 17 ^{[Biran-Cornea 2013]} lifted the theory to a categorical statement: a monotone Lagrangian cobordism induces an exact triangle in the derived Fukaya category, recasting cobordism as a structural feature of $D Fuk (M, ω)$ . The Biran-Cornea result fits into the broader programme of homological mirror symmetry, where Lagrangian cobordisms on the symplectic side correspond to exact triangles of coherent sheaves on the mirror algebraic variety.

Bibliography Master

@article{Arnold1980Cobordism,
  author = {Arnold, V. I.},
  title = {Lagrange and Legendre cobordisms. {I}, {II}},
  journal = {Functional Analysis and its Applications},
  volume = {14},
  number = {3, 4},
  year = {1980},
  pages = {1--13, 8--17}
}

@article{Eliashberg1984Cobordism,
  author = {Eliashberg, Ya. M.},
  title = {Cobordisme des solutions de relations différentielles},
  journal = {Séminaire Sud-Rhodanien de Géométrie, Travaux en Cours},
  publisher = {Hermann, Paris},
  year = {1984},
  note = {Originally Funct. Anal. Appl. 18; see also Eliashberg-Mishachev 2002 Ch. 16-17}
}

@article{Audin1985Cobordismes,
  author = {Audin, M.},
  title = {Cobordismes d'immersions lagrangiennes et legendriennes},
  journal = {Commentarii Mathematici Helvetici},
  volume = {60},
  year = {1985},
  pages = {622--645}
}

@incollection{ArnoldGiventalEMSCh6,
  author = {Arnold, V. I. and Givental, A. B.},
  title = {Symplectic Geometry, Ch. 6: Lagrangian and Legendrian Cobordism},
  booktitle = {Dynamical Systems IV: Symplectic Geometry and its Applications},
  editor = {Arnold, V. I. and Novikov, S. P.},
  series = {Encyclopaedia of Mathematical Sciences},
  volume = {4},
  publisher = {Springer},
  edition = {2nd},
  year = {2001}
}

@book{EliashbergMishachev2002,
  author = {Eliashberg, Y. and Mishachev, N.},
  title = {Introduction to the h-Principle},
  series = {Graduate Studies in Mathematics},
  volume = {48},
  publisher = {American Mathematical Society},
  year = {2002}
}

@article{Polterovich1991Surgery,
  author = {Polterovich, L.},
  title = {The surgery of Lagrange submanifolds},
  journal = {Geometric and Functional Analysis (GAFA)},
  volume = {1},
  number = {2},
  year = {1991},
  pages = {198--210}
}

@article{BiranCornea2013,
  author = {Biran, P. and Cornea, O.},
  title = {Lagrangian cobordism. {I}},
  journal = {Geometry \& Topology},
  volume = {17},
  number = {5},
  year = {2013},
  pages = {2881--2989}
}

@article{Arnold1967Maslov,
  author = {Arnold, V. I.},
  title = {On a characteristic class entering into conditions of quantization},
  journal = {Functional Analysis and its Applications},
  volume = {1},
  number = {1},
  year = {1967},
  pages = {1--14}
}

Prerequisites

05.12.01
05.12.03
03.06.12

Tier anchors

beginner: Arnold *Catastrophe Theory* (Springer 1992) — strip-cobordism intuition; Polterovich-Rosen *Function Theory on Symplectic Manifolds* (CRM Monograph 34, 2014) introductory chapter on cobordism
intermediate: Arnold-Givental EMS Vol 4 Ch 6; Audin *Cobordismes d'immersions lagrangiennes* Comm. Math. Helv. 60 (1985)
master: Arnold 1980 *Funct. Anal. Pril.* 14; Eliashberg 1984 *Funct. Anal. Pril.* 18; Audin 1985 *Comm. Math. Helv.* 60; Eliashberg-Mishachev *Introduction to the h-Principle* AMS 2002; Biran-Cornea 2013 *Geom. Topol.* 17

References

Arnold 1980 — Lagrange and Legendre cobordisms I, II · Funct. Anal. Appl. 14, no. 3, 1-13 and no. 4, 8-17
Eliashberg 1984 — Cobordisme des solutions de relations différentielles · Funct. Anal. Appl. 18 (Russian Math. Surveys translation), see also Séminaire Sud-Rhodanien 1984
Audin 1985 — Cobordismes d'immersions lagrangiennes et legendriennes · Comment. Math. Helv. 60, 622-645
Arnold-Givental — Symplectic Geometry (EMS Vol. 4, Ch. 6) · EMS Vol 4 Ch 6 §6.1-6.5
Eliashberg-Mishachev — Introduction to the h-Principle · Graduate Studies in Mathematics 48, AMS 2002, Ch. 16-17
Polterovich 1991 — Lagrangian surgery · Internat. J. Math. 2, no. 2, 195-204
Biran-Cornea 2013 — Lagrangian cobordism I · Geom. Topol. 17, no. 5, 2881-2989
Arnold 1967 — One-dimensional cohomology of a Lagrange manifold · Funct. Anal. Appl. 1, no. 1, 1-14 (Maslov class as cobordism invariant)

Estimated time

beginner: 18m
intermediate: 45m
master: 80m