On Contact Topology, Symplectic Field Theory and the …wendl/pub/Durham_20121203.pdf · On Contact Topology, Symplectic Field Theory and the ... (mathematical physics): w Ho trivial

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(q1, p1, . . . , qn, pn) is a time-independent

Hamiltonian and H−1(c) is convex, does

qj =∂H

∂pj, pj = −

∂H

∂qj

have a periodic orbit in H−1(c)?Problem 2 (topology):Is a given losed manifold M the boundary ofany ompa t manifold W?How unique is W?Problem 3 ( omplex geometry / PDE):Given a Riemann surfa e � and omplex man-ifold W , what is the spa e of holomorphi maps �!W?(Finite dimensional? Smooth? Compa t?)Problem 4 (mathemati al physi s):How trivial is my TQFT?1





qj =∂H

∂pj, pj = −

∂H

∂qj

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 surfa e � and omplex man-ifold W , what is the spa e of holomorphi maps �!W?(Finite dimensional? Smooth? Compa t?)Problem 4 (mathemati al physi s):How trivial is my TQFT?1





qj =∂H

∂pj, pj = −

∂H

∂qj





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 (mathemati al physi s):How trivial is my TQFT?1





qj =∂H

∂pj, pj = −

∂H

∂qj





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?

1

Theorem (Rabinowitz-Weinstein ’78).

Every star-shaped hypersurface in R2n ad-

mits a periodic orbit.

De�nition.A symple ti stru ture on a 2n-dimensional manifold W is a system of lo- al oordinate systems (q1; p1; : : : ; qn; pn) inwhi h Hamilton's equations are invariant.It arries a natural volume form:dp1 dq1 : : : dpn dqn:�W is onvex if it is transverse to a ve tor�eld Y that dilates the symple ti stru ture.2












Definition.A symplectic structure on a 2n-

dimensional manifold W is a system of lo-

cal coordinate systems (q1, p1, . . . , qn, pn) in

which Hamilton’s equations are invariant.

It carries a natural volume form:

dp1 dq1 . . . dpn dqn.�W is onvex if it is transverse to a ve tor�eld Y that dilates the symple ti stru ture.2




Definition.A symplectic structure on a 2n-

dimensional manifold W is a system of lo-

cal coordinate systems (q1, p1, . . . , qn, pn) in

which Hamilton’s equations are invariant.

It carries a natural volume form:

dp1 dq1 . . . dpn dqn.

∂W is convex if it is transverse to a vector

field Y that dilates the symplectic structure.

2

M := ∂W convex ; contact structure

ξ ⊂ TM,

a field of tangent hyperplanes that are

“locally twisted” (maximally nonintegrable),

and transverse to the Reeb (i.e. Hamilto-

nian) vector field.Example: T3 := S1 � S1 � S1

PSfrag repla ementsS1��WW(M+; �+)(M�; ��)S1�(M1; �1)(M2; �2)[ ℄ = 0 2 HSFT� (M; �)[ ℄ 6= 0 2 HSFT� (;)AT =1AT � 0AT � 1AT � 24-dimensional2-handle�[0;1℄�MMM 0index 0index 1index 2�M� � � 1 2 k�+��0[0;1)�M+(�1;0℄�M�W�M��M+= boundary of T2 �D = D�T2 � T �T2.3

M := ∂W convex ; contact structure

ξ ⊂ TM,

a field of tangent hyperplanes that are

“locally twisted” (maximally nonintegrable),

and transverse to the Reeb (i.e. Hamilto-

nian) vector field.

Example: T3 := S1 × S1 × S1

= boundary of T2 × D = D∗T2 ⊂ T ∗T2.

3

Some hard problems in contact topology

1. Classification of contact structures:

given ξ1, ξ2 on M , is there a diffeomor-

phism ϕ : M → M mapping ξ1 to ξ2?2. Weinstein onje ture:Every Reeb ve tor �eld on every losed onta t manifold has a periodi orbit?3. Partial orders: say (M�; ��) � (M+; �+)if there is a (symple ti , exa t or Stein) obordism between them.

PSfrag repla ementsS1��WW

(M+; �+)

(M�; ��)

S1�(M1; �1)(M2; �2)[ ℄ = 0 2 HSFT� (M; �)[ ℄ 6= 0 2 HSFT� (;)AT =1AT � 0AT � 1AT � 24-dimensional2-handle�[0;1℄�MMM 0index 0index 1index 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




phism ϕ : M → M mapping ξ1 to ξ2?

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 (symple ti , exa t or Stein) obordism between them.

PSfrag repla ementsS1��WW

(M+; �+)

(M�; ��)

S1�(M1; �1)(M2; �2)[ ℄ = 0 2 HSFT� (M; �)[ ℄ 6= 0 2 HSFT� (;)AT =1AT � 0AT � 1AT � 24-dimensional2-handle�[0;1℄�MMM 0index 0index 1index 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








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?)4








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 ξ1 and ξ2 are both overtwisted, then(M, ξ1)

∼= (M, ξ2) ⇔ ξ1 and ξ2 are homotopic.\Overtwisted onta t stru tures are exible."

1]×

Theorem (Gromov '85 and Eliashberg '89).� overtwisted ) (M; �) not �llable.Non-overtwisted onta t stru tures are alled\tight".They are not fully understood.5



∼= (M, ξ2) ⇔ ξ1 and ξ2 are homotopic.

“Overtwisted contact structures are flexible.”

1]×

Theorem (Gromov '85 and Eliashberg '89).� overtwisted ) (M; �) not �llable.Non-overtwisted onta t stru tures are alled\tight".They are not fully understood.5





1]×

Theorem (Gromov ’85 and Eliashberg ’89).ξ overtwisted ⇒ (M, ξ) not fillable.Non-overtwisted onta t stru tures are alled\tight".They are not fully understood.

5





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 atta hing obordism:D y D

zylinder

PSfrag repla ementsS1��WW(M+; �+)(M�; ��)S1�(M1; �1)(M2; �2)[ ℄ = 0 2 HSFT� (M; �)[ ℄ 6= 0 2 HSFT� (;)AT =1AT � 0AT � 1AT � 2

4-dimensional2-handle

�[0;1℄�M

MM

M 0index 0index 1index 2�M� � � 1 2 k�+��0[0;1)�M+(�1;0℄�M�W�M��M+�([0;1℄�M) = �M tM

6

Conjecture.


Then (M, ξ) tight ⇒ (M ′, ξ′) tight.

Surgery ; handle attaching cobordism:


D× D

[0,1]×M

M

M

∂([0,1]×M) = −M ⊔M

6

Conjecture.


Then (M, ξ) tight ⇒ (M ′, ξ′) tight.

Surgery ; handle attaching cobordism:


D× D

[0,1]×M

M

M ′

∂(([0,1]×M) ∪ (D× D)) = −M ⊔M ′

6

Recent results: ∃ “degrees of tightness”.

Theorem (Latschev-W. 2010).

overtwisted

fillable

tight

Corollary:(Mk; �k) onta t surgery��! (M`; �`)) ` � k.!

PSfrag repla ementsS1��WW(M+; �+)(M�; ��)

S1�S1� (M1; �1)

(M2; �2)

[ ℄ = 0 2 HSFT� (M; �)[ ℄ 6= 0 2 HSFT� (;)AT =1AT � 0AT � 1AT � 24-dimensional2-handle�[0;1℄�MMM 0index 0index 1index 2�M� � � 1 2 k�+��0[0;1)�M+(�1;0℄�M�W�M��M+7



algebraically overtwisted

fillable

AT = ∞

AT ≥ 0

AT ≥ 1AT ≥ 2



S1�S1� (M1; �1)

(M2; �2)




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 algebrai ally overtwisted� (M; �) �llable ) AT(M; �) =1� 8k, 9(Mk; �k) with AT(Mk; �k) = k.Corollary:(Mk; �k) onta t surgery��! (M`; �`)) ` � k.

!


S1�S1� (M1; �1)

(M2; �2)






• (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(Mk; �k) with AT(Mk; �k) = k.Corollary:(Mk; �k) onta t surgery��! (M`; �`)) ` � k.

!


S1�S1� (M1; �1)

(M2; �2)






• (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(Mk; �k) with AT(Mk; �k) = k.Corollary:(Mk; �k) onta t surgery��! (M`; �`)) ` � k.

!


S1�S1� (M1; �1)

(M2; �2)






• (M−, ξ−) ≺ (M+, ξ+) ⇒

AT(M−, ξ−) ≤ AT(M+, ξ+)

• AT(M, ξ) = 0 ⇔


• (M, ξ) fillable ⇒ AT(M, ξ) = ∞

• ∀k, ∃(Mk, ξk) with AT(Mk, ξk) = k.



S1�S1� (M1; �1)

(M2; �2)






• (M−, ξ−) ≺ (M+, ξ+) ⇒

AT(M−, ξ−) ≤ AT(M+, ξ+)

• AT(M, ξ) = 0 ⇔


• (M, ξ) fillable ⇒ AT(M, ξ) = ∞

• ∀k, ∃(Mk, ξk) with AT(Mk, ξk) = k.

Corollary:

(Mk, ξk)contact surgery−−−−−−−−−−−→ (Mℓ, ξℓ) ⇒ ℓ ≥ k.

!

S1×

S1×

(M1, ξ1)

(M2, ξ2)

7



algebraically overtwisted

fillable

AT = ∞

AT ≥ 0

AT ≥ 1AT ≥ 2

Corollary:

(Mk, ξk)contact surgery−−−−−−−−−−−→ (Mℓ, ξℓ) ⇒ ℓ ≥ k.

!

S1×

S1×

(M1, ξ1)

(M2, ξ2)

7

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 {qγ}γ∈P.W := fformal power series F(q ; p ; h)g with,[p ; q 0℄ = Æ ; 0h:F 2 W, substitute p := h� ��q � ! operatorDF : A[[h℄℄! A[[h℄℄\Theorem": There exists H 2 W withH2 = 0 su h that DH(1) = 0 andHSFT� (M; �) := H�(A[[h℄℄; DH) := kerDHimDHis a onta t invariant.Symple ti obordism (M�; ��) � (M+; �+)! natural mapHSFT� (M+; �+)! HSFT� (M�; ��)preserving elements of R[[h℄℄.8






and generators {qγ}γ∈P.

W := {formal power series F(qγ, pγ, ~)} with,

[pγ, qγ′] = δγ,γ′~.

F ∈ W, substitute pγ := ~∂

∂qγ; operator

DF : A[[~]] → A[[~]]\Theorem": There exists H 2 W withH2 = 0 su h that DH(1) = 0 andHSFT� (M; �) := H�(A[[h℄℄; DH) := kerDHimDHis a onta t invariant.Symple ti obordism (M�; ��) � (M+; �+)) natural mapHSFT� (M+; �+)! HSFT� (M�; ��)preserving elements of R[[h℄℄.8








[pγ, qγ′] = δγ,γ′~.


∂qγ; operator

DF : A[[~]] → A[[~]]

“Theorem”: There exists H ∈ W with

H2 = 0 such that DH(1) = 0 and

HSFT∗ (M, ξ) := H∗(A[[~]], DH) :=

kerDH

imDH

is a contact invariant.Symple ti obordism (M�; ��) � (M+; �+)) natural mapHSFT� (M+; �+)! HSFT� (M�; ��)preserving elements of R[[h℄℄.8








[pγ, qγ′] = δγ,γ′~.


∂qγ; operator

DF : A[[~]] → A[[~]]

“Theorem”: There exists H ∈ W with

H2 = 0 such that DH(1) = 0 and

HSFT∗ (M, ξ) := H∗(A[[~]], DH) :=

kerDH

imDH

is a contact invariant.

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

; natural map

HSFT∗ (M+, ξ+) → HSFT

∗ (M−, ξ−)

preserving elements of R[[~]].

8

Example

If no periodic orbits, then HSFT∗ (M, ξ) = R[[~]].De�nition (Lats hev-W.).(M; �) has algebrai k-torsion if[hk℄ = 0 2 HSFT� (M; �).AT(M; �) := sup nk �� [hk�1℄ 6= 0 2 HSFT� (M; �)o

ExampleOvertwisted )all \interesting" onta t invariants vanish:HSFT� (M; �) = f0g ) [1℄ = 0) AT(M; �) = 0:Theorem.Algebrai k-torsion ) not �llable.notemptr

emptr

!


S1�

(M1; �1)(M2; �2)[ ℄ = 0 2 HSFT� (M; �)[ ℄ 6= 0 2 HSFT� (;)AT =1AT � 0AT � 1AT � 24-dimensional2-handle�[0;1℄�MMM 0index 0index 1index 2�M� � � 1 2 k�+��0[0;1)�M+(�1;0℄�M�W�M��M+9

Example

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

Definition (Latschev-W.).

We say (M, ξ) has algebraic k-torsion if

[~k] = 0 ∈ HSFT∗ (M, ξ).AT(M; �) := sup nk �� [hk�1℄ 6= 0 2 HSFT� (M; �)o


emptr

!


S1�


Example




[~k] = 0 ∈ HSFT∗ (M, ξ).

AT(M, ξ) := sup{

k∣

∣

∣ [~k−1] 6= 0 ∈ HSFT∗ (M, ξ)

}


emptr

!


S1�


Example




[~k] = 0 ∈ HSFT∗ (M, ξ).

AT(M, ξ) := sup{

k∣

∣

∣ [~k−1] 6= 0 ∈ HSFT∗ (M, ξ)

}

Example

Overtwisted ⇒

all “interesting” contact invariants vanish:

HSFT∗ (M, ξ) = {0} ⇒ [1] = 0 ⇒ AT(M, ξ) = 0.Theorem.Algebrai k-torsion ) not �llable.

notemptr

emptr

!


S1�


Example




[~k] = 0 ∈ HSFT∗ (M, ξ).

AT(M, ξ) := sup{

k∣

∣

∣ [~k−1] 6= 0 ∈ HSFT∗ (M, ξ)

}

Example

Overtwisted ⇒


HSFT∗ (M, ξ) = {0} ⇒ [1] = 0 ⇒ AT(M, ξ) = 0.

Theorem.Algebraic k-torsion ⇒ not fillable.

notemptr

emptr

!


S1�


Example




[~k] = 0 ∈ HSFT∗ (M, ξ).

AT(M, ξ) := sup{

k∣

∣

∣ [~k−1] 6= 0 ∈ HSFT∗ (M, ξ)

}

Example

Overtwisted ⇒


HSFT∗ (M, ξ) = {0} ⇒ [1] = 0 ⇒ AT(M, ξ) = 0.

Theorem.Algebraic k-torsion ⇒ not fillable.

!

S1× [~] = 0 ∈ HSFT∗ (M, ξ)

[~] 6= 0 ∈ HSFT∗ (∅)

1

9

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), df

)

,

where df counts rigid gradient flow lines,

x(t) +∇f(x(t)) = 0.

PSfrag repla ementsS1��WW(M+; �+)(M�; ��)S1�(M1; �1)(M2; �2)[ ℄ = 0 2 HSFT� (M; �)[ ℄ 6= 0 2 HSFT� (;)AT =1AT � 0AT � 1AT � 24-dimensional2-handle�[0;1℄�MMM 0

index 0index 1index 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 onta t a tion fun tional� : C1(S1;M)! R : x 7! ZS1 x��;with Crit(�) = fperiodi Reeb orbitsg.10




H∗(X;Z) ∼= H∗

(

Z#Crit(f), df

)

,


x(t) +∇f(x(t)) = 0.

index 0

index 1

index 2

SFT of (M; � = ker�):\1-dimensional Morse theory" for the onta t a tion fun tional� : C1(S1;M)! R : x 7! ZS1 x��;with Crit(�) = fperiodi Reeb orbitsg.10




H∗(X;Z) ∼= H∗

(

Z#Crit(f), df

)

,


x(t) +∇f(x(t)) = 0.

index 0

index 1

index 2





H∗(X;Z) ∼= H∗

(

Z#Crit(f), df

)

,


x(t) +∇f(x(t)) = 0.

index 0

index 1

index 2





H∗(X;Z) ∼= H∗

(

Z#Crit(f), df

)

,


x(t) +∇f(x(t)) = 0.

index 0

index 1

index 2





H∗(X;Z) ∼= H∗

(

Z#Crit(f), df

)

,


x(t) +∇f(x(t)) = 0.

index 0

index 1

index 2





H∗(X;Z) ∼= H∗

(

Z#Crit(f), df

)

,


x(t) +∇f(x(t)) = 0.

index 0

index 1

index 2





H∗(X;Z) ∼= H∗

(

Z#Crit(f), df

)

,


x(t) +∇f(x(t)) = 0.

index 0

index 1

index 2





H∗(X;Z) ∼= H∗

(

Z#Crit(f), df

)

,


x(t) +∇f(x(t)) = 0.

index 0

index 1index 1

index 2





H∗(X;Z) ∼= H∗

(

Z#Crit(f), df

)

,


x(t) +∇f(x(t)) = 0.

index 0

index 1index 1

index 2

SFT of (M, ξ = kerα):

“∞-dimensional Morse theory” for the

contact action functional

Φ : C∞(S1,M) → R : x 7→∫

S1x∗α,

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

10

Gradient flow:

Consider 1-parameter families of loops

{us ∈ C∞(S1,M)}s∈R with

∂sus +∇Φ(us) = 0.! ylinders u : R�S1! R�M satisfying thenonlinear Cau hy-Riemann equation�su+ J(u) �tu = 0for an almost omplex stru ture J on R�M .For a symple ti obordism W and Riemannsurfa e �, onsider J-holomorphi urvesu : � n fz1; : : : ; zng !Wapproa hing Reeb orbits at the pun tures.

PSfrag repla ementsS1��WW(M+; �+)(M�; ��)S1�(M1; �1)(M2; �2)[ ℄ = 0 2 HSFT� (M; �)[ ℄ 6= 0 2 HSFT� (;)AT =1AT � 0AT � 1AT � 24-dimensional2-handle�[0;1℄�MMM 0index 0index 1index 2�M� � � 1 2 k�+��0[0;1)�M+(�1;0℄�M�W�M��M+11

Gradient flow:



∂sus +∇Φ(us) = 0.

; cylinders u : R×S1 → R×M satisfying the

nonlinear Cauchy-Riemann equation

∂su+ J(u) ∂tu = 0

for an almost complex structure J on R×M .For a symple ti obordism W and Riemannsurfa e �, onsider J-holomorphi urvesu : � n fz1; : : : ; zng !Wapproa hing Reeb orbits at the pun tures.

PSfrag repla ementsS1��WW(M+; �+)(M�; ��)S1�(M1; �1)(M2; �2)[ ℄ = 0 2 HSFT� (M; �)[ ℄ 6= 0 2 HSFT� (;)AT =1AT � 0AT � 1AT � 24-dimensional2-handle�[0;1℄�MMM 0index 0index 1index 2�M� � � 1 2 k�+��0[0;1)�M+(�1;0℄�M�W�M��M+11

Gradient flow:



∂sus +∇Φ(us) = 0.

; cylinders u : R×S1 → R×M satisfying the

nonlinear Cauchy-Riemann equation

∂su+ J(u) ∂tu = 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 \ {z1, . . . , zn} → W

approaching Reeb orbits at the punctures.

AT

11

The Cauchy-Riemann equation is elliptic:

‖u‖W1,p ≤ ‖u‖Lp + ‖∂su+ i ∂tu‖Lp

⇒ Spaces of holomorphic curves are (often)

• smooth finite-dimensional manifolds,

• compact up to bubbling / breaking.

AT

12






AT

[0,∞)×M+

(−∞,0]×M−

W

12






AT

[0,∞)×M+

(−∞,0]×M−

W

12






AT

[0,∞)×M+

(−∞,0]×M−

W

12






AT

[0,∞)×M+

(−∞,0]×M−

W

12






AT

W

R×M−

R×M−

R×M−

R×M+

12

Definition of H

Γ± := (γ±1 , . . . , γ±k±) lists of Reeb orbits

Mg(Γ+,Γ−) := { rigid J-holomorphic curves

in R×M with genus g, ends at Γ±}/

parametrization

H := Xg;�+;��# �Mg(�+;��)=R�hg�1q��p�+

[0

R×M

Γ+

Γ−

SFT ompa tness theorem:Mg(�+;��) = fJ-holomorphi buildingsgH2 ounts the boundary of a 1-dimensionalspa e )H2 = 0.13

Definition of H




parametrization

H :=∑

g,Γ+,Γ−

#(

Mg(Γ+,Γ−)/R

)

~g−1qΓ

−pΓ

+

[0

R×M

Γ+

Γ−

SFT ompa tness theorem:Mg(�+;��) = fJ-holomorphi buildingsgH2 ounts the boundary of a 1-dimensionalspa e )H2 = 0.13

Definition of H




parametrization

H :=∑

g,Γ+,Γ−

#(

Mg(Γ+,Γ−)/R

)

~g−1qΓ

−pΓ

+

[0

R×M

Γ+

Γ−

SFT compactness theorem:

Mg(Γ+,Γ−) = {J-holomorphic buildings}H2 ounts the boundary of a 1-dimensionalspa e )H2 = 0.13

Definition of H




parametrization

H :=∑

g,Γ+,Γ−

#(

Mg(Γ+,Γ−)/R

)

~g−1qΓ

−pΓ

+

[0

R×M

Γ+

Γ−



Definition of H




parametrization

H :=∑

g,Γ+,Γ−

#(

Mg(Γ+,Γ−)/R

)

~g−1qΓ

−pΓ

+

[0

R×M

Γ+

Γ−



Definition of H




parametrization

H :=∑

g,Γ+,Γ−

#(

Mg(Γ+,Γ−)/R

)

~g−1qΓ

−pΓ

+

[0

R×M

R×M

Γ+

Γ−

Γ0



Definition of H




parametrization

H :=∑

g,Γ+,Γ−

#(

Mg(Γ+,Γ−)/R

)

~g−1qΓ

−pΓ

+

[0

R×M

R×M

Γ+

Γ−

Γ0


Mg(Γ+,Γ−) = {J-holomorphic buildings}

H2 counts the boundary of a 1-dimensional

space ⇒ H2 = 0.

13

Example

Suppose R×M has exactly one rigid J-holomorphic

curve, with genus 0, no negative ends, and

positive ends at orbits γ1, . . . , γk.

R×M

· · ·

· · ·

γ1 γ2 γk

Then H = h�1p 1 : : : p k:Substituting p i = h� ��q i� givesDH �q 1 : : : q k�= hk�1) [hk�1℄ = 0 2 HSFT� (M; �)) AT(M; �) � k � 1.14

Example




R×M

· · ·

· · ·

γ1 γ2 γk

Then

H = ~−1pγ1 . . . pγk.Substituting p i = h� ��q i� givesDH �q 1 : : : q k�= hk�1) [hk�1℄ = 0 2 HSFT� (M; �)) AT(M; �) � k � 1.

14

Example




R×M

· · ·

· · ·

γ1 γ2 γk

Then

H = ~−1pγ1 . . . pγk.

Substituting pγi = ~∂

∂qγigives

DH(

qγ1 . . . qγk)

= ~k−1) [hk�1℄ = 0 2 HSFT� (M; �)) AT(M; �) � k � 1.

14

Example




R×M

· · ·

· · ·

γ1 γ2 γk

Then

H = ~−1pγ1 . . . pγk.

Substituting pγi = ~∂

∂qγigives

DH(

qγ1 . . . qγk)

= ~k−1

⇒ [~k−1] = 0 ∈ HSFT∗ (M, ξ)

⇒ AT(M, ξ) ≤ k − 1.

14

Why (M2, ξ2) ≺ (M1, ξ1) is not true:

S1×

S1×

(M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


!

S1×

S1×

(M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


S1×

S1× (M1, ξ1)

(M2, ξ2)

D D

15


!

S1×

S1×

(M1, ξ1)

(M2, ξ2)

D D

15

Some open questions and partial answers

1. What geometric conditions correspond to

AT(M, ξ) = k?� Overtwistedness, Giroux torsion, planar torsion:C. Wendl, A hierar hy of lo al �lling obstru tions for onta t 3-manifolds, Preprint 2010, arXiv:1009.2746.2. Interesting examples beyond dimension 3?� Higher-dimensional overtwisted disks:K. Niederkr�uger, The plastikstufe|a generalization ofthe overtwisted disk to higher dimensions, Algebr. Geom.Topol. 6 (2006), 2473-2508.F. Bourgeois, K. Niederkr�uger, PS-overtwisted onta tmanifolds are algebrai ally overtwisted, in preparation.� Higher-dimensional Giroux torsion:P. Massot, K. Niederkr�uger and C. Wendl, Weak andstrong �llability of higher dimensional onta t mani-folds, to appear in Invent. Math., Preprint 2011,arXiv:1111.6008.3. Can onta t stru tures with AT(M; �) �k be lassi�ed???� Overtwisted onta t stru tures are exible:Y. Eliashberg, Classi� ation of overtwisted onta t stru -tures on 3-manifolds, Invent. Math. 98 (1989), 623-637.� Coarse lassi� ation|�nitely many have AT(M; �) � 2:V. Colin, E. Giroux and K. Honda, Finitude homo-topique et isotopique des stru tures de onta t ten-dues, Publ. Math. Inst. Hautes �Etudes S i. 109 (2009),no. 1, 245-293.16



AT(M, ξ) = k?

• Overtwistedness, Giroux torsion, planar torsion:C. Wendl, A hierarchy of local filling obstructions forcontact 3-manifolds, Preprint 2010, arXiv:1009.2746.2. Interesting examples beyond dimension 3?� Higher-dimensional overtwisted disks:K. Niederkr�uger, The plastikstufe|a generalization ofthe overtwisted disk to higher dimensions, Algebr. Geom.Topol. 6 (2006), 2473-2508.F. Bourgeois, K. Niederkr�uger, PS-overtwisted onta tmanifolds are algebrai ally overtwisted, in preparation.� Higher-dimensional Giroux torsion:P. Massot, K. Niederkr�uger and C. Wendl, Weak andstrong �llability of higher dimensional onta t mani-folds, to appear in Invent. Math., Preprint 2011,arXiv:1111.6008.3. Can onta t stru tures with AT(M; �) �k be lassi�ed???� Overtwisted onta t stru tures are exible:Y. Eliashberg, Classi� ation of overtwisted onta t stru -tures on 3-manifolds, Invent. Math. 98 (1989), 623-637.� Coarse lassi� ation|�nitely many have AT(M; �) � 2:V. Colin, E. Giroux and K. Honda, Finitude homo-topique et isotopique des stru tures de onta t ten-dues, Publ. Math. Inst. Hautes �Etudes S i. 109 (2009),no. 1, 245-293.

16



AT(M, ξ) = k?

• Overtwistedness, Giroux torsion, planar torsion:C. Wendl, A hierarchy of local filling obstructions forcontact 3-manifolds, Preprint 2010, arXiv:1009.2746.

2. Interesting examples beyond dimension 3?� Higher-dimensional overtwisted disks:K. Niederkr�uger, The plastikstufe|a generalization ofthe overtwisted disk to higher dimensions, Algebr. Geom.Topol. 6 (2006), 2473-2508.F. Bourgeois, K. Niederkr�uger, PS-overtwisted onta tmanifolds are algebrai ally overtwisted, in preparation.� Higher-dimensional Giroux torsion:P. Massot, K. Niederkr�uger and C. Wendl, Weak andstrong �llability of higher dimensional onta t mani-folds, to appear in Invent. Math., Preprint 2011,arXiv:1111.6008.3. Can onta t stru tures with AT(M; �) �k be lassi�ed???� Overtwisted onta t stru tures are exible:Y. Eliashberg, Classi� ation of overtwisted onta t stru -tures on 3-manifolds, Invent. Math. 98 (1989), 623-637.� Coarse lassi� ation|�nitely many have AT(M; �) � 2:V. Colin, E. Giroux and K. Honda, Finitude homo-topique et isotopique des stru tures de onta t ten-dues, Publ. Math. Inst. Hautes �Etudes S i. 109 (2009),no. 1, 245-293.16



AT(M, ξ) = k?


2. Interesting examples beyond dimension 3?

• Higher-dimensional overtwisted disks:K. Niederkruger, The plastikstufe—a generalization ofthe overtwisted disk to higher dimensions, Algebr. Geom.Topol. 6 (2006), 2473-2508.F. Bourgeois, K. Niederkr�uger, PS-overtwisted onta tmanifolds are algebrai ally overtwisted, in preparation.� Higher-dimensional Giroux torsion:P. Massot, K. Niederkr�uger and C. Wendl, Weak andstrong �llability of higher dimensional onta t mani-folds, to appear in Invent. Math., Preprint 2011,arXiv:1111.6008.3. Can onta t stru tures with AT(M; �) �k be lassi�ed???� Overtwisted onta t stru tures are exible:Y. Eliashberg, Classi� ation of overtwisted onta t stru -tures on 3-manifolds, Invent. Math. 98 (1989), 623-637.� Coarse lassi� ation|�nitely many have AT(M; �) � 2:V. Colin, E. Giroux and K. Honda, Finitude homo-topique et isotopique des stru tures de onta t ten-dues, Publ. Math. Inst. Hautes �Etudes S i. 109 (2009),no. 1, 245-293.

16



AT(M, ξ) = k?



• Higher-dimensional overtwisted disks:K. Niederkruger, The plastikstufe—a generalization ofthe overtwisted disk to higher dimensions, Algebr. Geom.Topol. 6 (2006), 2473-2508.F. Bourgeois, K. Niederkruger, PS-overtwisted contactmanifolds are algebraically overtwisted, in preparation.� Higher-dimensional Giroux torsion:P. Massot, K. Niederkr�uger and C. Wendl, Weak andstrong �llability of higher dimensional onta t mani-folds, to appear in Invent. Math., Preprint 2011,arXiv:1111.6008.3. Can onta t stru tures with AT(M; �) �k be lassi�ed???� Overtwisted onta t stru tures are exible:Y. Eliashberg, Classi� ation of overtwisted onta t stru -tures on 3-manifolds, Invent. Math. 98 (1989), 623-637.� Coarse lassi� ation|�nitely many have AT(M; �) � 2:V. Colin, E. Giroux and K. Honda, Finitude homo-topique et isotopique des stru tures de onta t ten-dues, Publ. Math. Inst. Hautes �Etudes S i. 109 (2009),no. 1, 245-293.

16



AT(M, ξ) = k?



• Higher-dimensional overtwisted disks:K. Niederkruger, The plastikstufe—a generalization ofthe overtwisted disk to higher dimensions, Algebr. Geom.Topol. 6 (2006), 2473-2508.F. Bourgeois, K. Niederkruger, PS-overtwisted contactmanifolds are algebraically overtwisted, in preparation.

• Higher-dimensional Giroux torsion:P. Massot, K. Niederkruger and C. Wendl, Weak andstrong fillability of higher dimensional contact mani-folds, to appear in Invent. Math., Preprint 2011,arXiv:1111.6008.3. Can onta t stru tures with AT(M; �) �k be lassi�ed???� Overtwisted onta t stru tures are exible:Y. Eliashberg, Classi� ation of overtwisted onta t stru -tures on 3-manifolds, Invent. Math. 98 (1989), 623-637.� Coarse lassi� ation|�nitely many have AT(M; �) � 2:V. Colin, E. Giroux and K. Honda, Finitude homo-topique et isotopique des stru tures de onta t ten-dues, Publ. Math. Inst. Hautes �Etudes S i. 109 (2009),no. 1, 245-293.

16



AT(M, ξ) = k?




• Higher-dimensional Giroux torsion:P. Massot, K. Niederkruger and C. Wendl, Weak andstrong fillability of higher dimensional contact mani-folds, to appear in Invent. Math., Preprint 2011,arXiv:1111.6008.

3. Can contact structures with AT(M, ξ) ≥k be classified???� Overtwisted onta t stru tures are exible:Y. Eliashberg, Classi� ation of overtwisted onta t stru -tures on 3-manifolds, Invent. Math. 98 (1989), 623-637.� Coarse lassi� ation|�nitely many have AT(M; �) � 2:V. Colin, E. Giroux and K. Honda, Finitude homo-topique et isotopique des stru tures de onta t ten-dues, Publ. Math. Inst. Hautes �Etudes S i. 109 (2009),no. 1, 245-293.

16



AT(M, ξ) = k?





3. Can contact structures with AT(M, ξ) ≥k be classified???

• Overtwisted contact structures are flexible:Y. Eliashberg, Classification of overtwisted contact struc-tures on 3-manifolds, Invent. Math. 98 (1989), 623-637.� Coarse lassi� ation|�nitely many have AT(M; �) � 2:V. Colin, E. Giroux and K. Honda, Finitude homo-topique et isotopique des stru tures de onta t ten-dues, Publ. Math. Inst. Hautes �Etudes S i. 109 (2009),no. 1, 245-293.

16



AT(M, ξ) = k?





3. Can contact structures with AT(M, ξ) ≥k be classified???

• Overtwisted contact structures are flexible:Y. Eliashberg, Classification of overtwisted contact struc-tures on 3-manifolds, Invent. Math. 98 (1989), 623-637.

• Coarse classification—finitely many have AT(M, ξ) ≥ 2:V. Colin, E. Giroux and K. Honda, Finitude homo-topique et isotopique des structures de contact ten-dues, Publ. Math. Inst. Hautes Etudes Sci. 109 (2009),no. 1, 245-293.

16

Main reference

• Janko Latschev and Chris Wendl, Algebraic tor-sion in contact manifolds, Geom. Funct. Anal.21 (2011), no. 5, 1144-1195, with an appendixby Michael Hutchings.

Acknowledgment

Contact structure illustrations by Patrick Massot:

http://www.math.u-psud.fr/~pmassot/

-dimensional

These slides are available at:

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

17

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