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On Contact Topology,

Symplectic Field Theory and the PDE That Unites Them

index 0

Chris Wendl

University College London

Slides available at:

http://www.homepages.ucl.ac.uk/~ucahcwe/publications.html#talks

(2)

How are the following related?

Problem 1 (dynamics):

If H ( q 1 , p 1 , . . . , q n , p n ) is a time-independent Hamiltonian and H −1 (c) is convex, does

q ˙ j = ∂H

∂p j , p ˙ j = − ∂H

∂q j have a periodic orbit in H −1 (c)?

Problem 2 (topology):

Is a given losed manifold M the boundary of

any ompat manifold W?

How unique is W?

Problem 3 (omplex geometry / PDE):

Given a Riemann surfae and omplex man-

ifold W, what is the spae of holomorphi

maps ! W?

(Finite dimensional? Smooth? Compat?)

Problem 4 (mathematial physis):

How trivial is my TQFT?

1

(3)

How are the following related?

Problem 1 (dynamics):

If H ( q 1 , p 1 , . . . , q n , p n ) is a time-independent Hamiltonian and H −1 (c) is convex, does

q ˙ j = ∂H

∂p j , p ˙ j = − ∂H

∂q j have a periodic orbit in H −1 (c)?

Problem 2 (topology):

Is a given closed manifold M the boundary of any compact manifold W ?

How unique is W ?

Problem 3 (omplex geometry / PDE):

Given a Riemann surfae and omplex man-

ifold W, what is the spae of holomorphi

maps ! W?

(Finite dimensional? Smooth? Compat?)

Problem 4 (mathematial physis):

How trivial is my TQFT?

(4)

How are the following related?

Problem 1 (dynamics):

If H ( q 1 , p 1 , . . . , q n , p n ) is a time-independent Hamiltonian and H −1 (c) is convex, does

q ˙ j = ∂H

∂p j , p ˙ j = − ∂H

∂q j have a periodic orbit in H −1 (c)?

Problem 2 (topology):

Is a given closed manifold M the boundary of any compact manifold W ?

How unique is W ?

Problem 3 (complex geometry / PDE):

Given a Riemann surface Σ and complex man- ifold W , what is the space of holomorphic maps Σ → W ?

(Finite dimensional? Smooth? Compact?)

Problem 4 (mathematial physis):

How trivial is my TQFT?

1

(5)

How are the following related?

Problem 1 (dynamics):

If H ( q 1 , p 1 , . . . , q n , p n ) is a time-independent Hamiltonian and H −1 (c) is convex, does

q ˙ j = ∂H

∂p j , p ˙ j = − ∂H

∂q j have a periodic orbit in H −1 (c)?

Problem 2 (topology):

Is a given closed manifold M the boundary of any compact manifold W ?

How unique is W ?

Problem 3 (complex geometry / PDE):

Given a Riemann surface Σ and complex man- ifold W , what is the space of holomorphic maps Σ → W ?

(Finite dimensional? Smooth? Compact?) Problem 4 (mathematical physics):

How trivial is my TQFT?

(6)

Theorem (Rabinowitz-Weinstein ’78).

Every star-shaped hypersurface in R 2n ad- mits a periodic orbit.

Denition. A sympleti struture on a 2n-

dimensional manifold W is a system of lo-

al oordinate systems (q

1

;p

1

;:::;q

n

;p

n

) in

whih Hamilton's equations are invariant.

It arries a natural volume form:

dp

1 dq

1

:::dp

n dq

n :

W is onvex if it is transverse to a vetor

eld Y that dilates the sympleti struture.

2

(7)

Theorem (Rabinowitz-Weinstein ’78).

Every star-shaped hypersurface in R 2n ad- mits a periodic orbit.

Denition. A sympleti struture on a 2n-

dimensional manifold W is a system of lo-

al oordinate systems (q

1

;p

1

;:::;q

n

;p

n

) in

whih Hamilton's equations are invariant.

It arries a natural volume form:

dp

1 dq

1

:::dp

n dq

n :

W is onvex if it is transverse to a vetor

eld Y that dilates the sympleti struture.

(8)

Theorem (Rabinowitz-Weinstein ’78).

Every star-shaped hypersurface in R 2n ad- mits a periodic orbit.

Denition. A sympleti struture on a 2n-

dimensional manifold W is a system of lo-

al oordinate systems (q

1

;p

1

;:::;q

n

;p

n

) in

whih Hamilton's equations are invariant.

It arries a natural volume form:

dp

1 dq

1

:::dp

n dq

n :

W is onvex if it is transverse to a vetor

eld Y that dilates the sympleti struture.

2

(9)

Theorem (Rabinowitz-Weinstein ’78).

Every star-shaped hypersurface in R 2n ad- mits a periodic orbit.

Definition. A symplectic structure on a 2n- dimensional manifold W is a system of lo- cal coordinate systems ( q 1 , p 1 , . . . , q n , p n ) in which Hamilton’s equations are invariant.

It carries a natural volume form:

dp 1 dq 1 . . . dp n dq n .

W is onvex if it is transverse to a vetor

eld Y that dilates the sympleti struture.

(10)

Theorem (Rabinowitz-Weinstein ’78).

Every star-shaped hypersurface in R 2n ad- mits a periodic orbit.

Definition. A symplectic structure on a 2n- dimensional manifold W is a system of lo- cal coordinate systems ( q 1 , p 1 , . . . , q n , p n ) in which Hamilton’s equations are invariant.

It carries a natural volume form:

dp 1 dq 1 . . . dp n dq n .

∂W is convex if it is transverse to a vector field Y that dilates the symplectic structure.

2

(11)

M := ∂W convex ; contact structure ξ ⊂ T M,

a field of tangent hyperplanes that are

“locally twisted” (maximally nonintegrable ),

and transverse to the Reeb (i.e. Hamilto- nian) vector field.

Example: T 3

:= S 1

S 1

S 1 S

1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

= boundary of T 2

D = D

T 2

T

T 2

.

(12)

M := ∂W convex ; contact structure ξ ⊂ T M,

a field of tangent hyperplanes that are

“locally twisted” (maximally nonintegrable ),

and transverse to the Reeb (i.e. Hamilto- nian) vector field.

Example: T 3 := S 1 × S 1 × S 1

= boundary of T 2 × D = D T 2 ⊂ T T 2 .

3

(13)

Some hard problems in contact topology

1. Classification of contact structures:

given ξ 1 , ξ 2 on M , is there a diffeomor- phism ϕ : M → M mapping ξ 1 to ξ 2 ?

2. Weinstein onjeture:

Every Reeb vetor eld on every losed

ontat manifold has a periodi orbit?

3. Partial orders: say (M ; ) (M

+

;

+ )

if there is a (sympleti, exat or Stein)

obordism between them.

S 1

W

W

(M

+

;

+ )

(M ; ) S

1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

When is (M ; ) (M

+

;

+ )?

When is ; (M;)? (Is it llable?)

(14)

Some hard problems in contact topology

1. Classification of contact structures:

given ξ 1 , ξ 2 on M , is there a diffeomor- phism ϕ : M → M mapping ξ 1 to ξ 2 ?

2. Weinstein conjecture:

Every Reeb vector field on every closed contact manifold has a periodic orbit?

3. Partial orders: say (M ; ) (M

+

;

+ )

if there is a (sympleti, exat or Stein)

obordism between them.

S 1

W

W

(M

+

;

+ )

(M ; ) S

1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

When is (M ; ) (M

+

;

+ )?

When is ; (M;)? (Is it llable?)

4

(15)

Some hard problems in contact topology

1. Classification of contact structures:

given ξ 1 , ξ 2 on M , is there a diffeomor- phism ϕ : M → M mapping ξ 1 to ξ 2 ?

2. Weinstein conjecture:

Every Reeb vector field on every closed contact manifold has a periodic orbit?

3. Partial orders: say ( M , ξ ) ≺ ( M + , ξ + ) if there is a (symplectic, exact or Stein) cobordism between them.

(M + , ξ + )

(M , ξ ) -dimensional

When is (M ; ) (M

+

;

+ )?

When is ; (M;)? (Is it llable?)

(16)

Some hard problems in contact topology

1. Classification of contact structures:

given ξ 1 , ξ 2 on M , is there a diffeomor- phism ϕ : M → M mapping ξ 1 to ξ 2 ?

2. Weinstein conjecture:

Every Reeb vector field on every closed contact manifold has a periodic orbit?

3. Partial orders: say ( M , ξ ) ≺ ( M + , ξ + ) if there is a (symplectic, exact or Stein) cobordism between them.

(M + , ξ + )

(M , ξ ) -dimensional

When is ( M , ξ ) ≺ ( M + , ξ + )?

When is ∅ ≺ ( M, ξ )? (Is it fillable ?)

4

(17)

Overtwisted vs. tight

Theorem (Eliashberg ’89).

If ξ 1 and ξ 2 are both overtwisted, then

(M, ξ 1 ) = (M, ξ ∼ 2 ) ⇔ ξ 1 and ξ 2 are homotopic.

\Overtwisted ontat strutures are exible."

1] ×

Theorem (Gromov '85 and Eliashberg '89).

overtwisted ) (M;) not llable.

Non-overtwisted ontat strutures are alled

\tight".

They are not fully understood.

(18)

Overtwisted vs. tight

Theorem (Eliashberg ’89).

If ξ 1 and ξ 2 are both overtwisted, then

(M, ξ 1 ) = (M, ξ ∼ 2 ) ⇔ ξ 1 and ξ 2 are homotopic.

“Overtwisted contact structures are flexible.”

1] ×

Theorem (Gromov '85 and Eliashberg '89).

overtwisted ) (M;) not llable.

Non-overtwisted ontat strutures are alled

\tight".

They are not fully understood.

5

(19)

Overtwisted vs. tight

Theorem (Eliashberg ’89).

If ξ 1 and ξ 2 are both overtwisted, then

(M, ξ 1 ) = (M, ξ ∼ 2 ) ⇔ ξ 1 and ξ 2 are homotopic.

“Overtwisted contact structures are flexible.”

1] ×

Theorem (Gromov ’85 and Eliashberg ’89).

ξ overtwisted ⇒ ( M, ξ ) not fillable.

Non-overtwisted ontat strutures are alled

\tight".

They are not fully understood.

(20)

Overtwisted vs. tight

Theorem (Eliashberg ’89).

If ξ 1 and ξ 2 are both overtwisted, then

(M, ξ 1 ) = (M, ξ ∼ 2 ) ⇔ ξ 1 and ξ 2 are homotopic.

“Overtwisted contact structures are flexible.”

1] ×

Theorem (Gromov ’85 and Eliashberg ’89).

ξ overtwisted ⇒ ( M, ξ ) not fillable.

Non-overtwisted contact structures are called

“tight”.

They are not fully understood.

5

(21)

Conjecture.

Suppose ( M, ξ ) contact surgery

−−−−−−−−−−−→ ( M , ξ ).

Then ( M, ξ ) tight ⇒ ( M , ξ ) tight.

Surgery ! handle attahing obordism:

D y D

zylinder

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M M M

0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

([0;1℄ M) = M t M

(22)

Conjecture.

Suppose ( M, ξ ) contact surgery

−−−−−−−−−−−→ ( M , ξ ).

Then ( M, ξ ) tight ⇒ ( M , ξ ) tight.

Surgery ; handle attaching cobordism:

4-dimensional 2-handle D × D

[0 , 1] × M

M

M

∂ ([0, 1] × M ) = −M ⊔ M

6

(23)

Conjecture.

Suppose ( M, ξ ) contact surgery

−−−−−−−−−−−→ ( M , ξ ).

Then ( M, ξ ) tight ⇒ ( M , ξ ) tight.

Surgery ; handle attaching cobordism:

4-dimensional 2-handle D × D

[0 , 1] × M

M M

∂(([0, 1] × M ) ∪ ( D × D )) = −M ⊔ M

(24)

Recent results: ∃ “degrees of tightness”.

Theorem (Latschev-W. 2010) .

overtwisted

fillable tight

Corollary:

(M

k

;

k )

ontat surgery

! (M

`

;

`

) ) ` k.

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

S

1

(M

1

;

1 )

(M

2

;

2 ) [ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0; 1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

7

(25)

Recent results: ∃ “degrees of tightness”.

Theorem (Latschev-W. 2010) .

algebraically overtwisted

fillable

AT = ∞

AT ≥ 0 AT ≥ 1

AT ≥ 2

Corollary:

(M

k

;

k )

ontat surgery

! (M

`

;

`

) ) ` k.

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

S

1

(M

1

;

1 )

(M

2

;

2 ) [ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0; 1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

(26)

Recent results: ∃ “degrees of tightness”.

Theorem (Latschev-W. 2010).

There exists a numerical contact invariant AT( M, ξ ) ∈ N ∪ {0 , ∞} such that:

• (M , ξ ) ≺ (M + , ξ + ) ⇒ AT(M , ξ ) ≤ AT(M + , ξ + )

AT(M;) = 0 ,

(M;) is algebraially overtwisted

(M;) llable ) AT(M;) = 1

8k, 9(M

k

;

k

) with AT(M

k

;

k

) = k.

Corollary:

(M

k

;

k )

ontat surgery

! (M

`

;

`

) ) ` k.

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

S

1

(M

1

;

1 )

(M

2

;

2 ) [ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0; 1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

7

(27)

Recent results: ∃ “degrees of tightness”.

Theorem (Latschev-W. 2010).

There exists a numerical contact invariant AT( M, ξ ) ∈ N ∪ {0 , ∞} such that:

• (M , ξ ) ≺ (M + , ξ + ) ⇒ AT(M , ξ ) ≤ AT(M + , ξ + )

• AT( M, ξ ) = 0 ⇔

( M, ξ ) is algebraically overtwisted

(M;) llable ) AT(M;) = 1

8k, 9(M

k

;

k

) with AT(M

k

;

k

) = k.

Corollary:

(M

k

;

k )

ontat surgery

! (M

`

;

`

) ) ` k.

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

S

1

(M

1

;

1 )

(M

2

;

2 ) [ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0; 1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

(28)

Recent results: ∃ “degrees of tightness”.

Theorem (Latschev-W. 2010).

There exists a numerical contact invariant AT( M, ξ ) ∈ N ∪ {0 , ∞} such that:

• (M , ξ ) ≺ (M + , ξ + ) ⇒ AT(M , ξ ) ≤ AT(M + , ξ + )

• AT( M, ξ ) = 0 ⇔

( M, ξ ) is algebraically overtwisted

• (M, ξ) fillable ⇒ AT(M, ξ) = ∞

8k, 9(M

k

;

k

) with AT(M

k

;

k

) = k.

Corollary:

(M

k

;

k )

ontat surgery

! (M

`

;

`

) ) ` k.

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

S

1

(M

1

;

1 )

(M

2

;

2 ) [ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0; 1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

7

(29)

Recent results: ∃ “degrees of tightness”.

Theorem (Latschev-W. 2010).

There exists a numerical contact invariant AT( M, ξ ) ∈ N ∪ {0 , ∞} such that:

• (M , ξ ) ≺ (M + , ξ + ) ⇒ AT(M , ξ ) ≤ AT(M + , ξ + )

• AT( M, ξ ) = 0 ⇔

( M, ξ ) is algebraically overtwisted

• (M, ξ) fillable ⇒ AT(M, ξ) = ∞

• ∀k , ∃( M k , ξ k ) with AT( M k , ξ k ) = k .

Corollary:

(M

k

;

k )

ontat surgery

! (M

`

;

`

) ) ` k.

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

S

1

(M

1

;

1 )

(M

2

;

2 ) [ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0; 1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

(30)

Recent results: ∃ “degrees of tightness”.

Theorem (Latschev-W. 2010).

There exists a numerical contact invariant AT( M, ξ ) ∈ N ∪ {0 , ∞} such that:

• (M , ξ ) ≺ (M + , ξ + ) ⇒ AT(M , ξ ) ≤ AT(M + , ξ + )

• AT( M, ξ ) = 0 ⇔

( M, ξ ) is algebraically overtwisted

• (M, ξ) fillable ⇒ AT(M, ξ) = ∞

• ∀k , ∃( M k , ξ k ) with AT( M k , ξ k ) = k . Corollary :

(M k , ξ k ) contact surgery

−−−−−−−−−−−→ (M , ξ ) ⇒ ℓ ≥ k.

!

S 1 ×

S 1 ×

(M 1 , ξ 1 )

(M 2 , ξ 2 )

7

(31)

Recent results: ∃ “degrees of tightness”.

Theorem (Latschev-W. 2010) .

algebraically overtwisted

fillable

AT = ∞

AT ≥ 0 AT ≥ 1

AT ≥ 2

Corollary :

( M k , ξ k ) contact surgery

−−−−−−−−−−−→ ( M , ξ ) ⇒ ℓ ≥ k .

!

S 1 ×

S 1 ×

(M 1 , ξ 1 )

(M 2 , ξ 2 )

(32)

Symplectic Field Theory

(Eliashberg-Givental-Hofer ’00 + Cieliebak-Latschev ’09)

(M, ξ) with Reeb vector field ; P := {periodic Reeb orbits on M }.

A := graded commutative algebra with unit and generators {q γ } γ∈P .

W := fformal power series F(q

;p

;h)g with,

[p

;q

0

℄ = Æ

; 0

h:

F 2 W, substitute p

:= h

q

! operator

D

F

: A[[h℄℄ ! A[[h℄℄

\Theorem": There exists H 2 W with

H 2

= 0 suh that D

H

(1) = 0 and

H

SFT

(M;) := H

(A[[h℄℄; D

H

) :=

kerD

H

im D

H

is a ontat invariant.

Sympleti obordism (M ; ) (M

+

;

+ )

! natural map

H

SFT

(M

+

;

+

) ! H

SFT

(M ; )

preserving elements of R [[h℄℄.

8

(33)

Symplectic Field Theory

(Eliashberg-Givental-Hofer ’00 + Cieliebak-Latschev ’09)

(M, ξ) with Reeb vector field ; P := {periodic Reeb orbits on M }.

A := graded commutative algebra with unit and generators {q γ } γ∈P .

W := {formal power series F ( q γ , p γ , ~ )} with, [p γ , q γ ′ ] = δ γ,γ ′ ~ .

F ∈ W , substitute p γ := ~

∂q γ ; operator D F : A[[ ~ ]] → A[[ ~ ]]

\Theorem": There exists H 2 W with

H 2

= 0 suh that D

H

(1) = 0 and

H

SFT

(M;) := H

(A[[h℄℄; D

H

) :=

kerD

H

im D

H

is a ontat invariant.

Sympleti obordism (M ; ) (M

+

;

+ )

) natural map

H

SFT

(M

+

;

+

) ! H

SFT

(M ; )

preserving elements of R [[h℄℄.

(34)

Symplectic Field Theory

(Eliashberg-Givental-Hofer ’00 + Cieliebak-Latschev ’09)

(M, ξ) with Reeb vector field ; P := {periodic Reeb orbits on M }.

A := graded commutative algebra with unit and generators {q γ } γ∈P .

W := {formal power series F ( q γ , p γ , ~ )} with, [p γ , q γ ′ ] = δ γ,γ ′ ~ .

F ∈ W , substitute p γ := ~

∂q γ ; operator D F : A[[ ~ ]] → A[[ ~ ]]

“Theorem”: There exists H ∈ W with H 2 = 0 such that D H (1) = 0 and

H SFT ( M, ξ ) := H (A[[ ~ ]] , D H ) := ker D H im D H is a contact invariant.

Sympleti obordism (M ; ) (M

+

;

+ )

) natural map

H

SFT

(M

+

;

+

) ! H

SFT

(M ; )

preserving elements of R [[h℄℄.

8

(35)

Symplectic Field Theory

(Eliashberg-Givental-Hofer ’00 + Cieliebak-Latschev ’09)

(M, ξ) with Reeb vector field ; P := {periodic Reeb orbits on M }.

A := graded commutative algebra with unit and generators {q γ } γ∈P .

W := {formal power series F ( q γ , p γ , ~ )} with, [p γ , q γ ′ ] = δ γ,γ ′ ~ .

F ∈ W , substitute p γ := ~

∂q γ ; operator D F : A[[ ~ ]] → A[[ ~ ]]

“Theorem”: There exists H ∈ W with H 2 = 0 such that D H (1) = 0 and

H SFT ( M, ξ ) := H (A[[ ~ ]] , D H ) := ker D H im D H is a contact invariant.

Symplectic cobordism (M , ξ ) ≺ (M + , ξ + )

; natural map

H SFT (M + , ξ + ) → H SFT (M , ξ )

preserving elements of R [[ ~ ]].

(36)

Example

If no periodic orbits, then H SFT ( M, ξ ) = R [[ ~ ]].

Denition (Latshev-W.).

(M;) has algebrai k-torsion if

[h k

℄ = 0 2 H

SFT

(M;).

AT(M;) := sup n

k

[h k 1

℄ 6= 0 2 H

SFT

(M;) o

Example

Overtwisted )

all \interesting" ontat invariants vanish:

H

SFT

(M;) = f0g ) [1℄ = 0 ) AT(M;) = 0:

Theorem. Algebrai k-torsion ) not llable.

notemptr

emptr

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

9

(37)

Example

If no periodic orbits, then H SFT ( M, ξ ) = R [[ ~ ]].

Definition (Latschev-W.) .

We say (M, ξ) has algebraic k -torsion if [ ~ k ] = 0 ∈ H SFT (M, ξ).

AT(M;) := sup n

k

[h k 1

℄ 6= 0 2 H

SFT

(M;) o

Example

Overtwisted )

all \interesting" ontat invariants vanish:

H

SFT

(M;) = f0g ) [1℄ = 0 ) AT(M;) = 0:

Theorem. Algebrai k-torsion ) not llable.

notemptr

emptr

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

(38)

Example

If no periodic orbits, then H SFT ( M, ξ ) = R [[ ~ ]].

Definition (Latschev-W.) .

We say (M, ξ) has algebraic k -torsion if [ ~ k ] = 0 ∈ H SFT (M, ξ).

AT( M, ξ ) := sup n k [ ~ k−1 ] 6= 0 ∈ H SFT ( M, ξ ) o

Example

Overtwisted )

all \interesting" ontat invariants vanish:

H

SFT

(M;) = f0g ) [1℄ = 0 ) AT(M;) = 0:

Theorem. Algebrai k-torsion ) not llable.

notemptr

emptr

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

9

(39)

Example

If no periodic orbits, then H SFT ( M, ξ ) = R [[ ~ ]].

Definition (Latschev-W.) .

We say (M, ξ) has algebraic k -torsion if [ ~ k ] = 0 ∈ H SFT (M, ξ).

AT( M, ξ ) := sup n k [ ~ k−1 ] 6= 0 ∈ H SFT ( M, ξ ) o

Example

Overtwisted ⇒

all “interesting” contact invariants vanish:

H SFT ( M, ξ ) = {0} ⇒ [1] = 0 ⇒ AT( M, ξ ) = 0 .

Theorem. Algebrai k-torsion ) not llable.

notemptr

emptr

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

(40)

Example

If no periodic orbits, then H SFT ( M, ξ ) = R [[ ~ ]].

Definition (Latschev-W.) .

We say (M, ξ) has algebraic k -torsion if [ ~ k ] = 0 ∈ H SFT (M, ξ).

AT( M, ξ ) := sup n k [ ~ k−1 ] 6= 0 ∈ H SFT ( M, ξ ) o

Example

Overtwisted ⇒

all “interesting” contact invariants vanish:

H SFT ( M, ξ ) = {0} ⇒ [1] = 0 ⇒ AT( M, ξ ) = 0 . Theorem. Algebraic k-torsion ⇒ not fillable.

notemptr

emptr

!

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

9

(41)

Example

If no periodic orbits, then H SFT ( M, ξ ) = R [[ ~ ]].

Definition (Latschev-W.) .

We say (M, ξ) has algebraic k -torsion if [ ~ k ] = 0 ∈ H SFT (M, ξ).

AT( M, ξ ) := sup n k [ ~ k−1 ] 6= 0 ∈ H SFT ( M, ξ ) o

Example

Overtwisted ⇒

all “interesting” contact invariants vanish:

H SFT ( M, ξ ) = {0} ⇒ [1] = 0 ⇒ AT( M, ξ ) = 0 . Theorem. Algebraic k-torsion ⇒ not fillable.

!

S 1 × [ ~ ] = 0 ∈ H SFT ( M, ξ )

[ ~ ] 6= 0 ∈ H SFT (∅)

1

(42)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0 index 1

index 2 M

1

2

k

+

0

[0;1) M

+

( 1;0℄ M

W

M

M

+

SFT of (M; = ker ):

\1-dimensional Morse theory" for the

ontat ation funtional

: C 1

(S 1

;M) ! R : x 7!

Z

S 1

x

;

with Crit() = fperiodi Reeb orbitsg.

10

(43)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

index 0 index 1

index 2

SFT of (M; = ker ):

\1-dimensional Morse theory" for the

ontat ation funtional

: C 1

(S 1

;M) ! R : x 7!

Z

S 1

x

;

with Crit() = fperiodi Reeb orbitsg.

(44)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

index 0 index 1

index 2

SFT of (M; = ker ):

\1-dimensional Morse theory" for the

ontat ation funtional

: C 1

(S 1

;M) ! R : x 7!

Z

S 1

x

;

with Crit() = fperiodi Reeb orbitsg.

10

(45)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

index 0 index 1

index 2

SFT of (M; = ker ):

\1-dimensional Morse theory" for the

ontat ation funtional

: C 1

(S 1

;M) ! R : x 7!

Z

S 1

x

;

with Crit() = fperiodi Reeb orbitsg.

(46)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

index 0 index 1

index 2

SFT of (M; = ker ):

\1-dimensional Morse theory" for the

ontat ation funtional

: C 1

(S 1

;M) ! R : x 7!

Z

S 1

x

;

with Crit() = fperiodi Reeb orbitsg.

10

(47)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

index 0 index 1

index 2

SFT of (M; = ker ):

\1-dimensional Morse theory" for the

ontat ation funtional

: C 1

(S 1

;M) ! R : x 7!

Z

S 1

x

;

with Crit() = fperiodi Reeb orbitsg.

(48)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

index 0 index 1

index 2

SFT of (M; = ker ):

\1-dimensional Morse theory" for the

ontat ation funtional

: C 1

(S 1

;M) ! R : x 7!

Z

S 1

x

;

with Crit() = fperiodi Reeb orbitsg.

10

(49)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

index 0 index 1

index 2

SFT of (M; = ker ):

\1-dimensional Morse theory" for the

ontat ation funtional

: C 1

(S 1

;M) ! R : x 7!

Z

S 1

x

;

with Crit() = fperiodi Reeb orbitsg.

(50)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

index 0

index 1 index 1

index 2

SFT of (M; = ker ):

\1-dimensional Morse theory" for the

ontat ation funtional

: C 1

(S 1

;M) ! R : x 7!

Z

S 1

x

;

with Crit() = fperiodi Reeb orbitsg.

10

(51)

A beautiful idea (Witten ’82 + Floer ’88):

( X, g ) Riemannian manifold, f : X → R generic Morse function. Then singular homology

H (X ; Z ) = ∼ H Z # Crit( f ) , d f ,

where d f counts rigid gradient flow lines, x(t) + ˙ ∇f (x(t)) = 0.

index 0

index 1 index 1

index 2

SFT of (M, ξ = ker α):

“∞-dimensional Morse theory” for the contact action functional

Φ : C ( S 1 , M ) → R : x 7→

Z

S 1 x α,

with Crit(Φ) = {periodic Reeb orbits}.

(52)

Gradient flow:

Consider 1-parameter families of loops {u s ∈ C (S 1 , M )} s∈ R with

s u s + ∇Φ( u s ) = 0 .

! ylinders u : RS 1

! RM satisfying the

nonlinear Cauhy-Riemann equation

s

u + J(u)

t

u = 0

for an almost omplex struture J on RM.

For a sympleti obordism W and Riemann

surfae , onsider J-holomorphi urves

u : n fz

1

;:::;z

n

g ! W

approahing Reeb orbits at the puntures.

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0; 1) M

+

( 1;0℄ M

W

M

M

+

11

(53)

Gradient flow:

Consider 1-parameter families of loops {u s ∈ C (S 1 , M )} s∈ R with

s u s + ∇Φ( u s ) = 0 .

; cylinders u : R × S 1 → R × M satisfying the nonlinear Cauchy-Riemann equation

s u + J (u) ∂ t u = 0

for an almost complex structure J on R × M .

For a sympleti obordism W and Riemann

surfae , onsider J-holomorphi urves

u : n fz

1

;:::;z

n

g ! W

approahing Reeb orbits at the puntures.

S 1

W

W

(M

+

;

+ )

(M ; )

S 1

(M

1

;

1 )

(M

2

;

2 )

[ ℄ = 0 2 H

SFT

(M;)

[ ℄ 6= 0 2 H

SFT

(;)

AT = 1

AT 0

AT 1

AT 2

4-dimensional

2-handle

[0;1℄ M

M

M 0

index 0

index 1

index 2

M

1

2

k

+

0

[0; 1) M

+

( 1;0℄ M

W

M

M

+

(54)

Gradient flow:

Consider 1-parameter families of loops {u s ∈ C (S 1 , M )} s∈ R with

s u s + ∇Φ( u s ) = 0 .

; cylinders u : R × S 1 → R × M satisfying the nonlinear Cauchy-Riemann equation

s u + J (u) ∂ t u = 0

for an almost complex structure J on R × M . For a symplectic cobordism W and Riemann surface Σ, consider J -holomorphic curves

u : Σ \ {z 1 , . . . , z n } → W

approaching Reeb orbits at the punctures.

AT

11

(55)

The Cauchy-Riemann equation is elliptic:

kuk W 1,p ≤ kuk L p + k∂ s u + i ∂ t uk L p

⇒ Spaces of holomorphic curves are (often)

• smooth finite-dimensional manifolds,

• compact up to bubbling / breaking.

AT

(56)

The Cauchy-Riemann equation is elliptic:

kuk W 1,p ≤ kuk L p + k∂ s u + i ∂ t uk L p

⇒ Spaces of holomorphic curves are (often)

• smooth finite-dimensional manifolds,

• compact up to bubbling / breaking.

AT

[0, ∞) × M +

(−∞, 0] × M W

12

(57)

The Cauchy-Riemann equation is elliptic:

kuk W 1,p ≤ kuk L p + k∂ s u + i ∂ t uk L p

⇒ Spaces of holomorphic curves are (often)

• smooth finite-dimensional manifolds,

• compact up to bubbling / breaking.

AT

[0, ∞) × M +

(−∞, 0] × M

W

(58)

The Cauchy-Riemann equation is elliptic:

kuk W 1,p ≤ kuk L p + k∂ s u + i ∂ t uk L p

⇒ Spaces of holomorphic curves are (often)

• smooth finite-dimensional manifolds,

• compact up to bubbling / breaking.

AT

[0, ∞) × M +

(−∞, 0] × M W

12

(59)

The Cauchy-Riemann equation is elliptic:

kuk W 1,p ≤ kuk L p + k∂ s u + i ∂ t uk L p

⇒ Spaces of holomorphic curves are (often)

• smooth finite-dimensional manifolds,

• compact up to bubbling / breaking.

AT

[0, ∞) × M +

(−∞, 0] × M

W

(60)

The Cauchy-Riemann equation is elliptic:

kuk W 1,p ≤ kuk L p + k∂ s u + i ∂ t uk L p

⇒ Spaces of holomorphic curves are (often)

• smooth finite-dimensional manifolds,

• compact up to bubbling / breaking.

AT

W

R × M R × M

R × M R × M +

12

(61)

Definition of H Γ ± := ( γ 1 ± , . . . , γ k ±

± ) lists of Reeb orbits

M g+ , Γ ) := { rigid J -holomorphic curves in R ×M with genus g, ends at Γ ± }

parametrization

H :=

X

g;

+

;

#

M

g (

+

; )=R

h g 1

q p +

[0

R × M Γ +

Γ

SFT ompatness theorem:

M

g (

+

; ) = fJ-holomorphi buildingsg

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

(62)

Definition of H Γ ± := ( γ 1 ± , . . . , γ k ±

± ) lists of Reeb orbits

M g+ , Γ ) := { rigid J -holomorphic curves in R ×M with genus g, ends at Γ ± }

parametrization H := X

g, Γ + , Γ

# M g+ , Γ )/ R ~ g−1 q Γ p Γ + [0

R × M Γ +

Γ

SFT ompatness theorem:

M

g (

+

; ) = fJ-holomorphi buildingsg

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

13

(63)

Definition of H Γ ± := ( γ 1 ± , . . . , γ k ±

± ) lists of Reeb orbits

M g+ , Γ ) := { rigid J -holomorphic curves in R ×M with genus g, ends at Γ ± }

parametrization H := X

g, Γ + , Γ

# M g+ , Γ )/ R ~ g−1 q Γ p Γ + [0

R × M Γ +

Γ SFT compactness theorem:

M g+ , Γ ) = {J -holomorphic buildings}

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

(64)

Definition of H Γ ± := ( γ 1 ± , . . . , γ k ±

± ) lists of Reeb orbits

M g+ , Γ ) := { rigid J -holomorphic curves in R ×M with genus g, ends at Γ ± }

parametrization H := X

g, Γ + , Γ

# M g+ , Γ )/ R ~ g−1 q Γ p Γ + [0

R × M Γ +

Γ SFT compactness theorem:

M g+ , Γ ) = {J -holomorphic buildings}

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

13

(65)

Definition of H Γ ± := ( γ 1 ± , . . . , γ k ±

± ) lists of Reeb orbits

M g+ , Γ ) := { rigid J -holomorphic curves in R ×M with genus g, ends at Γ ± }

parametrization H := X

g, Γ + , Γ

# M g+ , Γ )/ R ~ g−1 q Γ p Γ + [0

R × M Γ +

Γ SFT compactness theorem:

M g+ , Γ ) = {J -holomorphic buildings}

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

(66)

Definition of H Γ ± := ( γ 1 ± , . . . , γ k ±

± ) lists of Reeb orbits

M g+ , Γ ) := { rigid J -holomorphic curves in R ×M with genus g, ends at Γ ± }

parametrization H := X

g, Γ + , Γ

# M g+ , Γ )/ R ~ g−1 q Γ p Γ + [0

R × M

R × M Γ +

Γ Γ 0

SFT compactness theorem:

M g+ , Γ ) = {J -holomorphic buildings}

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

13

(67)

Definition of H Γ ± := ( γ 1 ± , . . . , γ k ±

± ) lists of Reeb orbits

M g+ , Γ ) := { rigid J -holomorphic curves in R ×M with genus g, ends at Γ ± }

parametrization H := X

g, Γ + , Γ

# M g+ , Γ )/ R ~ g−1 q Γ p Γ + [0

R × M

R × M Γ +

Γ Γ 0

SFT compactness theorem:

M g+ , Γ ) = {J -holomorphic buildings}

H 2 counts the boundary of a 1-dimensional

space ⇒ H 2 = 0.

(68)

Example

Suppose R ×M has exactly one rigid J -holomorphic curve, with genus 0, no negative ends, and

positive ends at orbits γ 1 , . . . , γ k .

R × M

· · ·

· · ·

γ 1 γ 2 γ k

Then

H = h 1

p

1

:::p

k :

Substituting p

i

= h

q

i

gives

D

H q

1

:::q

k

= h

k 1

) [h k 1

℄ = 0 2 H

SFT

(M;)

) AT(M;) k 1.

14

(69)

Example

Suppose R ×M has exactly one rigid J -holomorphic curve, with genus 0, no negative ends, and

positive ends at orbits γ 1 , . . . , γ k .

R × M

· · ·

· · ·

γ 1 γ 2 γ k

Then

H = ~ −1 p γ 1 . . . p γ k .

Substituting p

i

= h

q

i

gives

D

H q

1

:::q

k

= h

k 1

) [h k 1

℄ = 0 2 H

SFT

(M;)

) AT(M;) k 1.

(70)

Example

Suppose R ×M has exactly one rigid J -holomorphic curve, with genus 0, no negative ends, and

positive ends at orbits γ 1 , . . . , γ k .

R × M

· · ·

· · ·

γ 1 γ 2 γ k

Then

H = ~ −1 p γ 1 . . . p γ k . Substituting p γ i = ~ ∂

∂q γ i gives

D H q γ 1 . . . q γ k = ~ k−1

) [h k 1

℄ = 0 2 H

SFT

(M;)

) AT(M;) k 1.

14

(71)

Example

Suppose R ×M has exactly one rigid J -holomorphic curve, with genus 0, no negative ends, and

positive ends at orbits γ 1 , . . . , γ k .

R × M

· · ·

· · ·

γ 1 γ 2 γ k

Then

H = ~ −1 p γ 1 . . . p γ k . Substituting p γ i = ~ ∂

∂q γ i gives

D H q γ 1 . . . q γ k = ~ k−1

⇒ [ ~ k−1 ] = 0 ∈ H SFT ( M, ξ )

⇒ AT(M, ξ) ≤ k − 1.

(72)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 ×

(M 1 , ξ 1 )

(M 2 , ξ 2 ) D D

15

(73)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 )

D D

(74)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 ) D D

15

(75)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 )

D D

(76)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 ) D D

15

(77)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 )

D D

(78)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

!

S 1 ×

S 1 ×

(M 1 , ξ 1 )

(M 2 , ξ 2 ) D D

15

(79)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 )

D D

(80)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 ) D D

15

(81)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 )

D D

(82)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 ) D D

15

(83)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

S 1 ×

S 1 × (M 1 , ξ 1 )

(M 2 , ξ 2 )

D D

(84)

Why (M 2 , ξ 2 ) ≺ (M 1 , ξ 1 ) is not true:

!

S 1 ×

S 1 ×

(M 1 , ξ 1 )

(M 2 , ξ 2 ) D D

15

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