### 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

### 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

### 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?

### 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

### 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?*

### 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

### 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.

### 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

### 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.

### 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

### 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

.

### 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

### 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?)

### 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

### 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?)

### 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

### 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.

### 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

### 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.

### 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

### 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

### 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

### 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 ^{′}

### 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

### 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

+

### 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

### 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

+

### 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

### 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

+

### 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 _{ℓ} , ξ _{ℓ} ) ⇒ ℓ ≥ k.

**!**

### S ^{1} ×

### S ^{1} ×

### (M _{1} , ξ _{1} )

### (M _{2} , ξ _{2} )

### 7

### 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 _{ℓ} , ξ _{ℓ} ) ⇒ ℓ ≥ k .

**!**

### S ^{1} ×

### S ^{1} ×

### (M _{1} , ξ _{1} )

### (M _{2} , ξ _{2} )

### 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

### 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℄℄.

### 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

### 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 [[ ~ ]].

### 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

### 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

+

### 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

### 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

+

### 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

### 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

### 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

### 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.

### 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

### 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.

### 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

### 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.

### 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

### 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.

### 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

### 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}.

### 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

### 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

+

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Definition of H Γ ^{±} := ( γ _{1} ^{±} , . . . , γ _{k} ^{±}

### ± ) lists of Reeb orbits

### M _{g} (Γ ^{+} , 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.

### Definition of H Γ ^{±} := ( γ _{1} ^{±} , . . . , γ _{k} ^{±}

### ± ) lists of Reeb orbits

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

### parametrization H := ^{X}

### g, Γ ^{+} , Γ ^{−}

### # ^{} M _{g} (Γ ^{+} , 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

### Definition of H Γ ^{±} := ( γ _{1} ^{±} , . . . , γ _{k} ^{±}

### ± ) lists of Reeb orbits

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

### parametrization H := ^{X}

### g, Γ ^{+} , Γ ^{−}

### # ^{} M _{g} (Γ ^{+} , Γ ^{−} )/ R ^{} ~ ^{g−1} q ^{Γ} ^{−} p ^{Γ} ^{+} [0

### R × M Γ ^{+}

### Γ ^{−} *SFT* *compactness* *theorem:*

### M _{g} (Γ ^{+} , Γ ^{−} ) = {J -holomorphic buildings}

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

### Definition of H Γ ^{±} := ( γ _{1} ^{±} , . . . , γ _{k} ^{±}

### ± ) lists of Reeb orbits

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

### parametrization H := ^{X}

### g, Γ ^{+} , Γ ^{−}

### # ^{} M _{g} (Γ ^{+} , Γ ^{−} )/ R ^{} ~ ^{g−1} q ^{Γ} ^{−} p ^{Γ} ^{+} [0

### R × M Γ ^{+}

### Γ ^{−} *SFT* *compactness* *theorem:*

### M _{g} (Γ ^{+} , Γ ^{−} ) = {J -holomorphic buildings}

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

### 13

### Definition of H Γ ^{±} := ( γ _{1} ^{±} , . . . , γ _{k} ^{±}

### ± ) lists of Reeb orbits

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

### parametrization H := ^{X}

### g, Γ ^{+} , Γ ^{−}

### # ^{} M _{g} (Γ ^{+} , Γ ^{−} )/ R ^{} ~ ^{g−1} q ^{Γ} ^{−} p ^{Γ} ^{+} [0

### R × M Γ ^{+}

### Γ ^{−} *SFT* *compactness* *theorem:*

### M _{g} (Γ ^{+} , Γ ^{−} ) = {J -holomorphic buildings}

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

### Definition of H Γ ^{±} := ( γ _{1} ^{±} , . . . , γ _{k} ^{±}

### ± ) lists of Reeb orbits

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

### parametrization H := ^{X}

### g, Γ ^{+} , Γ ^{−}

### # ^{} M _{g} (Γ ^{+} , Γ ^{−} )/ R ^{} ~ ^{g−1} q ^{Γ} ^{−} p ^{Γ} ^{+} [0

### R × M

### R × M Γ ^{+}

### Γ ^{−} Γ ^{0}

*SFT* *compactness* *theorem:*

### M _{g} (Γ ^{+} , Γ ^{−} ) = {J -holomorphic buildings}

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

### 13

### Definition of H Γ ^{±} := ( γ _{1} ^{±} , . . . , γ _{k} ^{±}

### ± ) lists of Reeb orbits

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

### parametrization H := ^{X}

### g, Γ ^{+} , Γ ^{−}

### # ^{} M _{g} (Γ ^{+} , Γ ^{−} )/ R ^{} ~ ^{g−1} q ^{Γ} ^{−} p ^{Γ} ^{+} [0

### R × M

### R × M Γ ^{+}

### Γ ^{−} Γ ^{0}

*SFT* *compactness* *theorem:*

### M _{g} (Γ ^{+} , Γ ^{−} ) = {J -holomorphic buildings}

### H ^{2} counts the boundary of a 1-dimensional

### space ⇒ H ^{2} = 0.

### 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

### 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.

### 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.