Invariants of contact structures and Reeb dynamicsnelson/2015_handout.pdf · dy @ @y = 1: Jo Nelson...

Invariants of contact structures and Reebdynamics

Jo Nelson

the IAS and Columbia University

March 26, 2015

Jo Nelson Invariants of contact structures and Reeb dynamics

2-plane fields

A 2-plane field ξ on M3 can be written as the kernel of a 1-form.

It is a smooth choice of an R2 subspace in TpM at each point p.

Definition

ξ is integrable if at each point p there is a small open chunk of asurface S in M containing p for which TpS = ξp.

Nice and integrable WTF!?!?Jo Nelson Invariants of contact structures and Reeb dynamics

First Contact with Contact StructuresA 2-plane field ξ on M3 is a contact structure if it is maximally non-integrable.

Rotate a line of planes from +∞ to -∞. Sweep left-right and up-down.

(images from sketches of topology blog)


Describing contact structures on manifolds

The kernel of a 1-form α on M2n−1 is a contact structure whenever

α ∧ (dα)n−1 is a volume form ⇔ dα|ξ is nondegenerate.

Let α = dz − ydx and ξ = kerα.

ξ = Span{∂∂y , y

∂∂z + ∂

∂x

}α ∧ dα = dx ∧ dy ∧ dz .

Alternatively,

dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.


Considerations of the Contactomorphic

Definition

(M, ξ1) and (N, ξ2) are contactomorphic whenever there exists adiffeomorphism f : M → N such that df (ξ1) = ξ2.

Equivalently, ξ1 = kerα1 and ξ2 = kerα2 are contactomorphicwhenever there is a g ∈ C∞(M,R+) such that f ∗α2 = gα1.

Example

(R3, ker(dz + r2dθ)) and (R3, ker(dz − ydx)) are contactomorphic.

images/video: Patrick Massot


All things are locally equal!

Theorem (Darboux’s theorem)

Let α be a contact form on the M2n+1 and p ∈ M. Then there arecoordinates (x1, y1, ..., xn, yn, z) on a neighbourhood U ⊂ M of p suchthat p = (0, ..., 0) and

α|U = dz −n∑

i=1

yidxi .

Thus locally all contact structures (and contact forms) look like the onewe saw on the last slide!

Contact structures are also “local in time,” e.g. compact deformations ofthe contact structure do not produce new contact structures.

Theorem (Gray’s Stability Theorem)

Let ξt , t ∈ [0, 1] be a family of contact structures on M2n+1 that agreeoff some compact subset of M. Then there is a family of diffeomorphismsφt : M → M such that dφt(ξt) = ξ0.


Overtwisted

Take cylindrical coordinates (r , θ, z) on R3; define

αOT = cos rdz + r sin rdθ.

αOT is smooth since r2dθ and the following function are smooth

r 7→

sin rr for r 6= 0,

1 for r = 0,

We see that αOT is a contact form by computing

αOT ∧ dαOT =

(1 +

sin r

rcos r

)rdr ∧ dθ ∧ dz

ξOT = kerαOT is called the standard overtwisted contactstructure on R3.


Overtwisted versus Tight

αOT = cos rdz + r sin rdθ

Overtwisted (Patrick Massot)

αstd = dz + r2dθ

Standard (Otto van Koert)

ξstd and ξOT are horizontal along the z-axis. Along any ray both turncounterclockwise () as one moves outwards from the z-axis.

The rotation angle of ξstd approaches (but never reaches) π/2, but thecontact planes of ξOT make infinitely many complete turns.

Bennequin proved (R3, ξ) and (R3, ξOT ) are NOT contactomorphic (’83).This result can be regarded as the birth of contact topology!

Generalized by Eliashberg (‘89), Giroux (‘99), Borman-Eliashberg-Murphy (‘14).


Reeb vector fields and flow

Definition

The Reeb vector field Rα on (M, α) is uniquely determined by

α(Rα) = 1,

dα(Rα, ·) = 0.

The Reeb flow, ϕt : M → M is defined by ddt (ϕt)(x) = Rα(ϕt(x)).

For α = dz − ydx and ξ = kerα

dα = dx ∧ dy ⇒ Rα = ∂∂z

ϕt(x , y , z) = (x , y , z + t)

Rα(p) does not necessarily agreewith the normal direction to ξp!


Reeb orbits on S3

(S3 := {|u|2 + |v |2 = 1}, α = i

2(udu − udu + vdv − vdv)).

The Reeb vector field is

Rα = iu∂

∂u− i u

∂

∂u+ iv

∂

∂v− i v

∂

∂v= (iu, iv),

with flow ϕt(u, v) = (e itu, e itv). Hence the orbits are the Hopffibers of S3.

Patrick Massot http://www.nilesjohnson.net/hopf.html


A video!

Hopf fibration video.

https://www.youtube.com/watch?v=AKotMPGFJYk


The irrational ellipsoid

The 3-ellipsoid is E (a, b) := f −1(1), f := |u|2a + |v |2

b ; a, b ∈ R>0.The standard contact form is

αE = −1

4df ◦ J0 =

i

2

(1

a(udu − udu) +

1

b(vdv − vdv)

).

The Reeb vector field

RE =1

a

(u∂

∂u− u

∂

∂u

)+

1

b

(v∂

∂v− v

∂

∂v

),

rotates the u-plane at angular speed 1a and the v -plane at speed 1

b .

If a/b is irrational, there are only two nondegenerate simple Reeborbits living in the u = 0 and v = 0 planes.

But (S3, α) and (E , αE ) are contactomorphic!


Global Invariants

The Weinstein Conjecture

Let (M, ξ) be a closed oriented contact manifold. Then for anycontact form α for ξ, the Reeb vector field Rα has a closed orbit.

Proven in dimension 3 by Taubes in 2007.

Conjecture

The tight contact 3-sphere admits either 2 or infinitely manysimple Reeb orbits.

Cristofaro-Gardiner – Hutchings proved at least 2 simple orbits.

Hofer-Wysocki-Zehnder proved for dynamically convex 3-spheres

Conjecture (Hutchings-Taubes)

The only contact 3-manifolds that admit exactly two simple Reeborbits must be either a sphere or a lens space...Otherwise there arealways infinitely many simple periodic Reeb orbits!


A dream for a chain complex

Assume: M closed and α nondegenerate

“Do” Morse theory on

A : C∞(S1,M) → R,

γ 7→∫γα.

Proposition

γ ∈ Crit(A)⇔ γ is a closed Reeb orbit.

Grading on orbits given by Conley-Zehnder index,

C∗(α) = {closed Reeb orbits} \ {bad Reeb orbits}


A dream...

Gradient flow lines no go; use finite energy pseudoholomorphiccylinders u ∈MJ (γ+; γ−), where γ± are T±-periodic Reeb orbits.

u := (a, f ) : (R× S1, j)→ (R×M, J )

∂j ,J u := du + J ◦ du ◦ j ≡ 0

lims→±∞

a(s, t) = ±∞

lims→±∞

f (s, t) = γ±(T±t)

up to reparametrization.

∂ : C∗ → C∗−1 is a weighted count of pseudoholomorphiccylinders up to reparametrization

Hope this is independent of our choices.

Conjeorem (Eliashberg-Givental-Hofer ’00)

Assume a minimal amount of things. Then (C∗(α), ∂)) forms achain complex and H(C∗(α), ∂) is independent of α and J .


The nightmare of contact homology

Transversality for multiply covered curves...good luck

Is MJ (γ+; γ−) more than a letter?

MJ (γ+; γ−) can have nonpositive virtual dimension!?!

Compactness issues are severe

1

ind= 2

1

Desired compactificationfor CZ (x)− CZ (z) = 2.

x

−3

y1 y2

2

y3

0

y4

2

y5

2

−1

0

y6

z

Adding to 2 becomes hard


Hope: the big reveal

Automatic transversality results of Wendl,Hutchings, and Taubes in dimension 3.

Understand basic arithmetic and theConley-Zehnder index

Realize your original thesis projectcontained a useful geometric perturbation

Definition

Assume c1(ξ) = 0. For today restrict to when Rα has only contractibleorbits. We say a contact form is dynamically separated whenever

(i) All closed simple contractible Reeb orbits γ satisfy 3 ≤ µCZ (γ) ≤ 5.

(ii) µCZ (γk) = µCZ (γk−1) + 4, γk is the k-th iterate of a simple orbit γ.

Theorem (N.)

∂2 = 0, invariance under choice of J and dynamically separated α.


A better reveal

Do more index calculations

Learn some intersection theory

Team up with Hutchings

Remaining obstruction to ∂2 = 0can be excluded!

γd+1

ind= 0

γ

2

γd

0γd

Definition

A nondegenerate (M3, ξ = kerα) is dynamically convex wheneverc1(ξ)|π2(M) = 0 and every contractible γ satisfies µCZ (γ) ≥ 3.

Any convex hypersurface transverse to the radial vector field Y in(R4, ω0) admits a dynamically convex contact form α := ω0(Y , ·).

Theorem (Hutchings-N.)

If (M3, α) is dynamically convex and every contractible Reeb orbit γ hasµCZ (γ) = 3 only if γ is simple then ∂2 = 0.


Too legit to quit

Still stuck on Invariance....

Throw in the entire kitchen sink

Non-equivariant formulations, domain dependent almostcomplex structures, obstruction bundle gluing

Family Floer homology constructions to get an S1-equivariant

theory which should be SHS1,+∗ over Z.

Tensor with Q to get back CH∗

Theorem (Hutchings-N; in progress)

INVARIANCE! Obtained for dynamically convex (M3, α) wherein acontractible γ has µCZ (γ) = 3 only if γ is simple.


Onwards

Computations for Seifert fiber spaces

Connections to Chen-Ruan orbifold homology and stringtopology

Look at dimensions > 3??

Other dynamical questions involving contact structures


The end!

Thanks!


Invariants of contact structures and Reeb dynamicsnelson/2015_handout.pdf · dy @ @y = 1: Jo Nelson...

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