an introduction to symplectic and contact topology and the ... · 1 introduction to symplectic and...

An Introduction toSymplectic and Contact Topology

and theTechnique of Generating Families

Lisa Traynor

Bryn Mawr College

March 2014

Lisa Traynor (Bryn Mawr) Symplectic and Contact Topology Banff 2013 1 / 40

Outline

1 Introduction to Symplectic and Contact Topology

2 Introduction to Generating Families

3 Invariants from Generating FamiliesGenerating Family Homology for a Legendrian SubmanifoldWrapped Generating Family Cohomology for a LagrangianCobordism

Outline

Where Are We?

Important Symplectic and Contact Objects

Symplectic Manifold (X 2n,!)

Symplectomorphism ⇤! = !

Lagrangian SubmanifoldLn : !|TL ⌘ 0

Contact Manifold (Y 2n+1, ⇠)

Contactomorphism⇤⇠ = ⇠

Legendrian Submanifold⇤n : T⇤ ⇢ ⇠

Symplectic Manifolds:�

X 2n,!�

! is a closed, non-degenerate 2-form:

d! = 0, !n is a volume form.

Example

Any orientable surface ⌃ with area form !;�

R2n,!0 = dx1 ^ dy1 + · · ·+ dxn ^ dyn�

;Cotangent Bundle, T ⇤M, has a canonical 1-form �0,!0 = d�0 is a symplectic form.

! S4 is not a symplectic manifold: H2(S4) = 0.

Theorem (Darboux’s Theorem)All symplectic manifolds are locally equivalent to

R2n,!0�

X 2n,!�

ExampleAny orientable surface ⌃ with area form !;

R2n,!0 = dx1 ^ dy1 + · · ·+ dxn ^ dyn�

R2n,!0�

X 2n,!�

ExampleAny orientable surface ⌃ with area form !;�

R2n,!0 = dx1 ^ dy1 + · · ·+ dxn ^ dyn�

Cotangent Bundle, T ⇤M, has a canonical 1-form �0,!0 = d�0 is a symplectic form.

R2n,!0�

X 2n,!�

R2n,!0 = dx1 ^ dy1 + · · ·+ dxn ^ dyn�

R2n,!0�

X 2n,!�

R2n,!0 = dx1 ^ dy1 + · · ·+ dxn ^ dyn�

R2n,!0�

X 2n,!�

R2n,!0 = dx1 ^ dy1 + · · ·+ dxn ^ dyn�

R2n,!0�

Symplectic Diffeomorphisms: !(v ,w) = !( ⇤v , ⇤w).

Example

(⌃2,!): Any area-preserving transformation;

(R2n,!0): Products of area-preserving transformations of thexiyi -planes;

(X 2n,!): Hamiltonian Functions and Vector FieldsFor any smooth Ht : X ! R, define vt by

!(vt , ·) = dHt ;

The flow of vt defines a symplectic isotopy t : (X ,!) ! (X ,!).

Example(⌃2,!): Any area-preserving transformation;

!(vt , ·) = dHt ;

Lagrangian Submanifolds: Ln : !|TL ⌘ 0

Example

(⌃2,!): Any embedded curve is Lagrangian;

(R2n,!0): The x1x2x3 . . . xn-plane is Lagrangianthe x1y1x3 . . . xn-plane is not Lagrangian;

(T ⇤M, d�0): For any function f : M ! R,the section �df ⇢ T ⇤M is a Lagrangian submanifold.

! There is no embedded Lagrangian S2 in R4.

Example(⌃2,!): Any embedded curve is Lagrangian;

Contact Manifolds: (Y 2n+1, ⇠)

⇠ is a field of maximally non-integrable tangent hyperplanes:

⇠ = ker↵ =) ↵ ^ (d↵)n 6= 0.

Example

R3, ⇠0 = ker(dz � ydx)�

R2n+1, ⇠0 = ker(dz �P

yidxi)�

;(1-Jet Bundles) J1M = T ⇤M ⇥ R has a canonical ⇠0;[Lutz-Martinet] On any closed, orientable 3-manifold, every2-plane field is homotopic to a contact structure.

⇠ = ker↵ =) ↵ ^ (d↵)n 6= 0.

Example�

R3, ⇠0 = ker(dz � ydx)�

R2n+1, ⇠0 = ker(dz �P

yidxi)�

⇠ = ker↵ =) ↵ ^ (d↵)n 6= 0.

Example�

R3, ⇠0 = ker(dz � ydx)�

R2n+1, ⇠0 = ker(dz �P

yidxi)�

(1-Jet Bundles) J1M = T ⇤M ⇥ R has a canonical ⇠0;[Lutz-Martinet] On any closed, orientable 3-manifold, every2-plane field is homotopic to a contact structure.

⇠ = ker↵ =) ↵ ^ (d↵)n 6= 0.

Example�

R3, ⇠0 = ker(dz � ydx)�

R2n+1, ⇠0 = ker(dz �P

yidxi)�

;(1-Jet Bundles) J1M = T ⇤M ⇥ R has a canonical ⇠0;

[Lutz-Martinet] On any closed, orientable 3-manifold, every2-plane field is homotopic to a contact structure.

⇠ = ker↵ =) ↵ ^ (d↵)n 6= 0.

Example�

R3, ⇠0 = ker(dz � ydx)�

R2n+1, ⇠0 = ker(dz �P

yidxi)�

Local and Global Equivalence of Contact Structures

Theorem (Darboux’s Theorem)All contact manifolds are locally equivalent to

R2n+1, ⇠0�

Important Problem: Understand the different contact structureson a fixed smooth manifold.

⇠0, ⇠1 : ⇠0 and ⇠1 are homotopic as plane fields,6 9' : (Y , ⇠0) ! (Y , ⇠1).

Theorem (Gray Stability)If ⇠t is a smooth family of contact structures on a closed manifold Y ,then there is an isotopy t : (Y , ⇠0) ! (Y , ⇠t) with (t)⇤⇠0 = ⇠t .

R2n+1, ⇠0�

Tight vs. Overtwisted (Y 3, ⇠)

A contact 3-manifold is overtwisted if it contains an overtwisted disk,D: interior of D is transversal to ⇠ except at 1 point; @D is tangent to ⇠.

Theorem (Eliashberg)Two overtwisted structures on Y are isotopic if they are homotopic asplane fields.

Tight vs. Overtwisted (Y 3, ⇠)

A contact 3-manifold is overtwisted if it contains an overtwisted disk,D: interior of D is transversal to ⇠ except at 1 point; @D is tangent to ⇠.

Theorem (Eliashberg)Two overtwisted structures on Y are isotopic if they are homotopic asplane fields.

Contact Diffeomorphisms: ⇤⇠ = ⇠

Contact Vector Field: flow preserves the contact structure

contact vector field �! contact isotopy

Example

Reeb Vector Field: If ⇠ = ker↵, the Reeb vector field R↵ definedby:

R↵ 2 ker d↵, ↵(R↵) = 1,

is a contact vector field.Contact Hamiltonian Vector Field: For any smooth H : Y ! R,there is a unique vector field KH ⇢ ⇠ = ker↵ so that

VH = HR↵ + KH

is a contact vector field.

Also: time-dependent Hamiltonian functions, Ht , and vector fields VHt

ExampleReeb Vector Field: If ⇠ = ker↵, the Reeb vector field R↵ definedby:

R↵ 2 ker d↵, ↵(R↵) = 1,

Contact Hamiltonian Vector Field: For any smooth H : Y ! R,there is a unique vector field KH ⇢ ⇠ = ker↵ so that

VH = HR↵ + KH

R↵ 2 ker d↵, ↵(R↵) = 1,

VH = HR↵ + KH

R↵ 2 ker d↵, ↵(R↵) = 1,

VH = HR↵ + KH

Legendrian Submanifolds: T⇤n ⇢ ⇠

Legendrians are abundant!

Any knot/link in R3 has a Legendrian representation;

In higher dimensions, any closed sphere, torus, or n-manifold thatsatisfies certain homotopy-theoretic conditions can beC0-approximated by a Legendrian.

Interested in the classification of Legendrians up to Legendrian isotopy(equivalently, up to ambient contact isotopy).

Basic Examples of Legendrian Curves

R3, ker(dz � ydx)�

, ⇤f = Legendrian lift of the graph of f : R ! R.!"#"$%&'

("#")$

)&%&'"

&("#"+

("#",-("#".

⇤f =

⇢✓

x ,dfdx

(x), f (x)◆�

⇢ R3.

Basic Examples of Legendrian Curves

("#")$

)&%&'"

&("#"+

("#",-("#". (

⇤f =

⇢✓

x ,dfdx

(x), f (x)◆�

⇢ R3.

More Complicated Legendrian Shapes

⇤ = lift of the graph of this “multi-valued" function to R3

No vertical tangents allowed;

Lift is smooth (semi-cubical cusps);

At crossing, branch with lesser slope is on top.

Legendrian Knots

Every topological knot/link has a Legendrian representative.

!"#"$#"%&'()*+

,*-.&'"/ 01

2"%&'()*+

Legendrian Surfaces

Legendrian sphere in R5:

More possibilities for singularities:

Legendrian Surfaces

Legendrian sphere in R5:

More possibilities for singularities:

Invariants of Legendrian Submanifolds

Important Problem: Understand the equivalence ofLegendrian submanifolds

Classical Invariants: for ⇤n ⇢ R2n+1,The diffeomorphism type of ⇤;

The rotation class r(⇤) 2 [⇤,U(n)];

The Thurston-Bennequin Invariant tb(⇤) 2 Z;

( n � 4 and even) Relative invariant tb⇤(⇤1,⇤2) 2 Z/2Z.

When ⇤ = S2n, r(⇤) = 0, tb(⇤) determined from �(⇤).

Classical Invariants: for ⇤n ⇢ R2n+1,

The diffeomorphism type of ⇤;

Reeb Chords

Question: Is it possible to construct non-classical invariantsof Legendrian submanifolds from the Reeb chords?

Reeb Chords

Question: Is it possible to construct non-classical invariantsof Legendrian submanifolds from the Reeb chords?

Techniques to Construct Invariants from Reeb Chords

J-Holomorphic CurvesInitiated by Gromov, ’85

Applies to more general manifolds.

Generating Families of FunctionsClassic technique;

Modernized by Sikorav, Laudenbach, Chaperon, Viterbo,80’s and 90’s

Analysis is finite dimensional.Arguments apply to all dimensions.

Interesting parallels between results!Combinatorics as a bridge.

Interesting parallels between results!

Combinatorics as a bridge.

Where Are We?

Idea of Generating Families

Manifolds of Focus:

ContactR2n+1

J1(M) = T ⇤M ⇥ R

SymplecticR2n

T ⇤(M)

In J1(M), describe Legendrian submanifolds as the “1-jet" offunctions F : M ⇥ RN ! R.

In T ⇤(M), describe Lagrangian submanifolds as the “derivatives"of functions F : M ⇥ RN ! R.

Strategy: Apply analysis/Morse theoretic arguments to these functionsto obtain invariants of the Lagrangian and Legendrian submanifolds.

Manifolds of Focus:

ContactR2n+1

J1(M) = T ⇤M ⇥ R

SymplecticR2n

T ⇤(M)

Manifolds of Focus:

ContactR2n+1

J1(M) = T ⇤M ⇥ R

SymplecticR2n

T ⇤(M)

Manifolds of Focus:

ContactR2n+1

J1(M) = T ⇤M ⇥ R

SymplecticR2n

T ⇤(M)

Basic Examples

("#")$

)&%&'"

&("#"+

("#",-("#".

!"#"$%&'

("#")$

)&%&'"

&("#"+

("#",-("#". (

⇤f =

⇢✓

x ,dfdx

(x), f (x)◆�

⇢ R3.

f : R ! R “generates" ⇤f .

Basic Examples

("#")$

)&%&'"

&("#"+

("#",-("#". (

⇤f =

⇢✓

x ,dfdx

(x), f (x)◆�

⇢ R3.

f : R ! R “generates" ⇤f .

⇤ = lift of the graph of this “multi-valued" function to R3 = J1R

⇤ is not the 1-jet of f : R ! R.

BUT, ⇤ can be viewed as the “1-jet" of a 1-parameter family of functions

F : R⇥ RN = {(x , e)} ! R.

⇤ = lift of the graph of this “multi-valued" function to R3 = J1R

⇤ is not the 1-jet of f : R ! R.

BUT, ⇤ can be viewed as the “1-jet" of a 1-parameter family of functions

F : R⇥ RN = {(x , e)} ! R.

Explicit Construction

Idea: Construct a family of functions Fx : RN = {e} ! R, for x 2 R.

This family of functions “generates" ⇤.9F : R⇥ RN ! R so that

⇢✓

x ,@F@x

(x , e),F (x , e)◆

(x , e) = 0�

9 choices: Canchange N by stabilizing with a quadratic: eF (x , e, e) = F (x , e) +Q(e);apply a fiber-preserving diffeomorphism: eF (x , e) = F (x ,�x(e)).

⇢✓

x ,@F@x

(x , e),F (x , e)◆

(x , e) = 0�

⇢✓

x ,@F@x

(x , e),F (x , e)◆

(x , e) = 0�

GF Domain has Finite Dimension

6 9F : R⇥ R1 ! R

that generates ⇤ ⇢ R3 = J1R.

9F : R⇥ R2 ! R.

In general, for ⇤ ⇢ J1(M), consider F : M ⇥ RN ! R for large N.

Usually want F “nice" (quadratic or linear)outside a compact set.

6 9F : R⇥ R1 ! R

that generates ⇤ ⇢ R3 = J1R. But,

9F : R⇥ R2 ! R.

6 9F : R⇥ R1 ! R

that generates ⇤ ⇢ R3 = J1R. But,

9F : R⇥ R2 ! R.

Generating Families for Lagrangian Submanifolds

Similar set up for Lagrangians:

9F : Rn ! R so that L =��

x , @F@x (x)

: x 2 Rn .

9F : Rn ⇥ RN ! R so that L =��

x , @F@x (x , e)

: @F@e (x , e) = 0

Generating Families for Lagrangian Submanifolds

Similar set up for Lagrangians:

9F : Rn ! R so that L =��

x , @F@x (x)

: x 2 Rn .

9F : Rn ⇥ RN ! R so that L =��

x , @F@x (x , e)

: @F@e (x , e) = 0

Important Points

Not all Legendrian (Lagrangian) submanifolds have generatingfamilies.

Sabloff, Fuchs, Pushkar, and others show that for ⇤1 ⇢ R3:

9 generating family () 9 augmentation of the holomorphic-DGA.

If a Legendrian (Lagrangian) submanifold has a generating familyand t is a contact (symplectic) isotopy, then 1(⇤) has agenerating family. “Persistence / Serre Fibration"

Important Points

Not all Legendrian (Lagrangian) submanifolds have generatingfamilies.

Sabloff, Fuchs, Pushkar, and others show that for ⇤1 ⇢ R3:

9 generating family () 9 augmentation of the holomorphic-DGA.

If a Legendrian (Lagrangian) submanifold has a generating familyand t is a contact (symplectic) isotopy, then 1(⇤) has agenerating family. “Persistence / Serre Fibration"

Where Are We?

Difference Functions of Generating Families

Given a generating family f : M ⇥ RN ! R for ⇤ ⇢ J1M,define the difference function � : M ⇥ RN ⇥ RN ! R by:

�(x , e, e) = f (x , e)� f (x , e).

Critical points of � are of two types:For each Reeb chord of ⇤, there are two critical points of � withcritical values ± (Reeb chord length);There is a non-degenerate critical submanifold diffeomorphic to ⇤with critical value 0.

As Legendrian ⇤ is isotoped, critical values born/die/change.

Apply Morse-theoretic constructions to get an invariant of ⇤ from �.

�(x , e, e) = f (x , e)� f (x , e).

Critical points of � are of two types:

For each Reeb chord of ⇤, there are two critical points of � withcritical values ± (Reeb chord length);There is a non-degenerate critical submanifold diffeomorphic to ⇤with critical value 0.

�(x , e, e) = f (x , e)� f (x , e).

Critical points of � are of two types:For each Reeb chord of ⇤, there are two critical points of � withcritical values ± (Reeb chord length);

There is a non-degenerate critical submanifold diffeomorphic to ⇤with critical value 0.

�(x , e, e) = f (x , e)� f (x , e).

Generating Family Homology and Cohomology

For ! � 0, H⇤(�!, ��!) = 0, for all ⇤.

Choose ✏ sufficiently small:

Definition (LT, Fuchs-Rutherford, Sabloff-LT)

GHk (f ) = Hk+N+1(�!, �✏), gGHk (f ) = Hk+N+1(�

!, ��✏),

GHk (f ) = Hk+N+1(�!, �✏), ]GHk (f ) = Hk+N+1(�!, ��✏).

Theorem (LT, Fuchs-Rutherford){GH⇤(f ) : f generates ⇤} is an invariant of ⇤.

{GH⇤(f )},n

gGH⇤(f )o

]GH⇤(f )o

are also invariants of ⇤.

For ! � 0, H⇤(�!, ��!) = 0, for all ⇤.

!, ��✏),

GHk (f ) = Hk+N+1(�!, �✏), ]GHk (f ) = Hk+N+1(�!, ��✏).

{GH⇤(f )},n

gGH⇤(f )o

]GH⇤(f )o

For ! � 0, H⇤(�!, ��!) = 0, for all ⇤.

!, ��✏),

GHk (f ) = Hk+N+1(�!, �✏), ]GHk (f ) = Hk+N+1(�!, ��✏).

{GH⇤(f )},n

gGH⇤(f )o

]GH⇤(f )o

For ! � 0, H⇤(�!, ��!) = 0, for all ⇤.

!, ��✏),

GHk (f ) = Hk+N+1(�!, �✏), ]GHk (f ) = Hk+N+1(�!, ��✏).

{GH⇤(f )},n

gGH⇤(f )o

]GH⇤(f )o

Generating Family Polynomials

Work over a field F and encode GH⇤(f ) by its Poincaré polynomial:

�f (t) =X

dim GHi(f )t i

ExampleThere are two different Legendrian m(52) knots with the same tb and rvalues:

Generating Family Polynomials

Work over a field F and encode GH⇤(f ) by its Poincaré polynomial:

�f (t) =X

dim GHi(f )t i

ExampleThere are two different Legendrian m(52) knots with the same tb and rvalues:

P = t + t + t-2 2}{

P = {2+ t}

Duality for Generating Family Homology

There is a structure to the gf-polynomials:

t + 4t2, 4t�2 + 4t2

cannot occur as gf-polynomials for ⇤1 ⇢ R3.

Duality Theorem:

Theorem (Sabloff-LT)There is a long exact sequence:

· · · ! GHk�1(f ) ! GHn�k (f ) ! Hk (⇤) ! · · ·

Duality for Generating Family Homology

There is a structure to the gf-polynomials:

t + 4t2, 4t�2 + 4t2

cannot occur as gf-polynomials for ⇤1 ⇢ R3.

Duality Theorem:

Theorem (Sabloff-LT)There is a long exact sequence:

· · · ! GHk�1(f ) ! GHn�k (f ) ! Hk (⇤) ! · · ·

Open Problems for GF-Homology

Important Directions:Combinatorially calculate GF-Homology;

Construct GF-DGA.

[Henry and Rutherford] A combinatorial-DGA for Legendrian knotsmotivated by generating families

Open Problems for GF-Homology

Important Directions:Combinatorially calculate GF-Homology;

Construct GF-DGA.

[Henry and Rutherford] A combinatorial-DGA for Legendrian knotsmotivated by generating families

Lagrangian Cobordisms between Legendrians

⇤� �L ⇤+

denotes an exact Lagrangian cobordismthat is cylindrical over ⇤± at ±1.

Wrapped Cohomology for Lagrangian Cobordisms

Lagrangian Floer Cohomology: For closed Lagrangians, define acohomology with underlying cochain complex the intersection points ofthe Lagrangians.

Wrapped Floer Cohomology:[Abbondodolo-Schwarz, Fukaya-Seidel-Smith, Abouzaid-Smith]For Lagrangian cobordisms, cochain complex generated byintersection between the compact pieces of two Lagrangiancobordisms and the Reeb chords at the Legendrian ends.

How to study with generating families?

GF-Compatible Lagrangian Cobordisms

Assume Legendrians and Lagrangian can be described by compatiblegenerating families:

(⇤�, f�) �(L,F ) (⇤+, f+).

: R⇥ R2n+1 ! T ⇤(R+ ⇥ Rn)

L 7! (L)

9 generating families

f± : Rn ⇥ RN ! R for ⇤± (linear-at-1); andF : R+ ⇥ Rn ⇥ RN ! R for (L) ⇢ T ⇤(R+ ⇥ Rn) that correlates tof± outside a compact set of R+.

(⇤�, f�) �(L,F ) (⇤+, f+).

: R⇥ R2n+1 ! T ⇤(R+ ⇥ Rn)

L 7! (L)

9 generating familiesf± : Rn ⇥ RN ! R for ⇤± (linear-at-1); and

F : R+ ⇥ Rn ⇥ RN ! R for (L) ⇢ T ⇤(R+ ⇥ Rn) that correlates tof± outside a compact set of R+.

(⇤�, f�) �(L,F ) (⇤+, f+).

: R⇥ R2n+1 ! T ⇤(R+ ⇥ Rn)

L 7! (L)

9 generating familiesf± : Rn ⇥ RN ! R for ⇤± (linear-at-1); andF : R+ ⇥ Rn ⇥ RN ! R for (L) ⇢ T ⇤(R+ ⇥ Rn) that correlates tof± outside a compact set of R+.

Wrapped Generating Family Cohomology Groups

Given a generating family F : R+ ⇥ M ⇥ RN ! R for (L) ⇢ T ⇤(R+ ⇥ M), and a function H : R+ ! R,

define the sheared difference function � : M ⇥ RN ⇥ RN ! R by:

�(t , x , e, e) = F (t , x , e)� F (t , x , e) + H(t).

KEY: There is a non-degenerate critical submanifold with value 0diffeomorphic to the compact L, and non-degenerate critical pointscorresponding to the Reeb chords of ⇤±.

WGH⇤(F ) := H⇤(�1,�✏).

In fact, WGH⇤(F ) ' H⇤(L,⇤+).

�(t , x , e, e) = F (t , x , e)� F (t , x , e) + H(t).

WGH⇤(F ) := H⇤(�1,�✏).

�(t , x , e, e) = F (t , x , e)� F (t , x , e) + H(t).

WGH⇤(F ) := H⇤(�1,�✏).

�(t , x , e, e) = F (t , x , e)� F (t , x , e) + H(t).

WGH⇤(F ) := H⇤(�1,�✏).

(�1,�✏) as a Relative Mapping Cone

Classical Mapping Cone:

Given f : X ! Y , C(f ) := C(X ) [ Y/ ⇠. Induced long exact sequence:

· · · ! Hk (Y )f⇤! Hk (X ) ! Hk+1(C(f )) ! Hk+1(Y )

f⇤! . . . .

Relative Mapping Cone: Given a map g : (X ,A) ! (Y ,B), there is arelative mapping cone C(g) and a long exact sequence:

· · · ! Hk (Y ,B)g⇤! Hk (X ,A) ! Hk+1(C(g)) ! Hk+1(Y ,B)

g⇤! · · · .

(�1,�✏) can be viewed as C( ) where :⇣

�1f+ , �✏f+

�1f� , �✏f�

So, there is a Cobordism Exact Sequence!

· · · ! Hk (Y )f⇤! Hk (X ) ! Hk+1(C(f )) ! Hk+1(Y )

f⇤! . . . .

g⇤! · · · .

�1f+ , �✏f+

�1f� , �✏f�

So, there is a Cobordism Exact Sequence!

· · · ! Hk (Y )f⇤! Hk (X ) ! Hk+1(C(f )) ! Hk+1(Y )

f⇤! . . . .

g⇤! · · · .

�1f+ , �✏f+

�1f� , �✏f�

So, there is a Cobordism Exact Sequence!Lisa Traynor (Bryn Mawr) Symplectic and Contact Topology Banff 2013 38 / 40

Cobordism Long Exact Sequence

Theorem ([Sabloff-LT])Given (⇤�, f�) �(L,F ) (⇤+, f+), there is a long exact sequence:

· · · ! GHk (f�) F! GHk (f+) ! Hk+1(L,⇤+) ! · · · .

F is non-trivial, natural, and behaves nicely under gluings ofcobordisms.

If ⇤� = ; (filling), then GHk (f+) ' Hk+1(L,⇤+).

If L is a concordance, then GHk (f�) ' GHk (f+).

There is a similar story for linearized Legendrian contact homology;[Ekholm, Ekholm-Honda-Kálmán, Golovko]

· · · ! GHk (f�) F! GHk (f+) ! Hk+1(L,⇤+) ! · · · .

Application: Obstructions to Lagrangian Cobordisms

!"# $% P = t + t + t

-2 2}{&'()

!"# $% P = + t2{

Application: Obstructions to Lagrangian Cobordisms

!"# $% P = t + t + t

-2 2}{&'()

!"# $% P = + t2{

!"# $%

!"# $& !"# $%

!"# $&

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