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.
