# On Contact Topology, Symplectic Field Theory and the PDE That Unites Them

## Full text

(1)

(2)

### ∂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?

(3)

### 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)

### (Finite dimensional? Smooth? Compact?)

Problem 4 (mathematial physis):

How trivial is my TQFT?

(5)

(6)

### Every star-shaped hypersurface in R 2nad-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.

(7)

### Every star-shaped hypersurface in R 2nad-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)

### Every star-shaped hypersurface in R 2nad-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.

(9)

### 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)

(11)

### 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)

(13)

### 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)

### 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?)

(15)

### (M − , ξ − ) -dimensional

When is (M ; ) (M

+

;

+ )?

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

(16)

(17)

### (M, ξ 1 ) = (M, ξ ∼ 2 ) ⇔ ξ 1and ξ 2arehomotopic.

\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)

### 1] ×

Theorem (Gromov '85 and Eliashberg '89).

overtwisted ) (M;) not llable.

Non-overtwisted ontat strutures are alled

\tight".

They are not fully understood.

(19)

### ξ overtwisted ⇒ ( M, ξ ) not fillable.

Non-overtwisted ontat strutures are alled

\tight".

They are not fully understood.

(20)

(21)

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

Surgery ! handle attahing obordism:

### 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)

(23)

(24)

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

+

(25)

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)

### • (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

+

(27)

### ( M, ξ ) isalgebraically 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)

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

+

(29)

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)

(31)

(32)

### 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℄℄.

(33)

### ∂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)

### 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℄℄.

(35)

(36)

### 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.

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

+

(37)

### 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.

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)

### 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.

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

+

(39)

### H ∗SFT ( M, ξ ) = {0} ⇒  = 0 ⇒ AT( M, ξ ) = 0 .

Theorem. Algebrai k-torsion ) not llable.

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)

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

+

(41)

(42)

### 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.

(43)

### 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)

### 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.

(45)

### 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)

### 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.

(47)

### 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)

### 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.

(49)

### 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)

### 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.

(51)

(52)

### ∂ 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

+

(53)

### 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)

(55)

(56)

(57)

(58)

(59)

(60)

(61)

H :=

X

g;

+

;

#

M

g (

+

; )=R

h g 1

q p +

### Γ −

SFT ompatness theorem:

M

g (

+

; ) = fJ-holomorphi buildingsg

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

(62)

### Γ −

SFT ompatness theorem:

M

g (

+

; ) = fJ-holomorphi buildingsg

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

(63)

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

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

(64)

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

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

(65)

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

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

(66)

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

H 2

ounts the boundary of a 1-dimensional

spae ) H 2

= 0.

(67)

(68)

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.

(69)

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)

) [h k 1

℄ = 0 2 H

SFT

(M;)

) AT(M;) k 1.

(71)

(72)

(73)

(74)

(75)

(76)

(77)

(78)

(79)

(80)

(81)

(82)

(83)

(84)

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## References

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