A 21st Century Geometry: Contact Geometry

Bahar Acu

University of Southern California

California State University Channel IslandsSeptember 23, 2015

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 1 / 12

Manifolds

DefinitionA smooth n-manifold is a topological space that looks locally like Rn and admitsa global differentiable structure.

Examples:

Trivial example; Rn; Euclidean space

S1; compact 1-manifold

More generally, Sn; compact n-manifold

Torus (doughnut!); closed n-manifold

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 2 / 12

Manifolds

DefinitionA smooth n-manifold is a topological space that looks locally like Rn and admitsa global differentiable structure.

Examples:

Trivial example; Rn; Euclidean space

S1; compact 1-manifold

More generally, Sn; compact n-manifold

Torus (doughnut!); closed n-manifold

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 2 / 12

Tangent Spaces and Differential Forms

DefinitionThe tangent space of Mn is a vector space at a point p ∈ M diffeomorphic to Rn.It is denoted by TpM,

DefinitionA 1-form is a linear function: TpM → R

Differential forms are a coordinate independent approach to calculus.

They’re great for defining integrals over curves, surfaces, and manifolds!

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 3 / 12

Tangent Spaces and Differential Forms

DefinitionThe tangent space of Mn is a vector space at a point p ∈ M diffeomorphic to Rn.It is denoted by TpM,

DefinitionA 1-form is a linear function: TpM → R

Differential forms are a coordinate independent approach to calculus.

They’re great for defining integrals over curves, surfaces, and manifolds!

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 3 / 12

Tangent Spaces and Differential Forms

DefinitionThe tangent space of Mn is a vector space at a point p ∈ M diffeomorphic to Rn.It is denoted by TpM,

DefinitionA 1-form is a linear function: TpM → R

Differential forms are a coordinate independent approach to calculus.

They’re great for defining integrals over curves, surfaces, and manifolds!

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 3 / 12

2-Plane Fields

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

Definitionξ is integrable if at each point p ∈ M there is a small open chunk of a surface Sin M containing p for which TpS = ξp.

Nice and integrable

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 4 / 12

2-Plane Fields

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

Definitionξ is integrable if at each point p ∈ M there is a small open chunk of a surface Sin M containing p for which TpS = ξp.

Nice and integrable

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 4 / 12

2-Plane Fields

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

Definitionξ is integrable if at each point p ∈ M there is a small open chunk of a surface Sin M containing p for which TpS = ξp.

Nice and integrable Nonintegrable!!

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 4 / 12

First Contact with Contact Manifolds

A 2-plane field ξ is a contact structure if it is nowhere integrable. This isequivalent to saying hyperplanes “twist too much” to be tangent to hypersurfaces.

Rotate a line of planes from +∞ to -∞.

Sweep left-right and up-down.

(images from sketches oftopology blog)

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 5 / 12

First Contact with Contact Manifolds

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 5 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

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.

Here α = dz − ydx and ξ = kerα

Span(ξ) ={∂∂y , y

∂∂z + ∂

∂x

}dα = −dy ∧ dx = dx ∧ dy

⇒ α ∧ dα = dz ∧ dx ∧ dy= dx ∧ dy ∧ dz .

Also, dα(∂∂y , y

∂∂z + ∂

∂x

)= dx ∧ dy

(∂∂y ,

∂∂x

)= dx

(∂∂y

)dy(∂∂x

)− dx

(∂∂x

)dy(∂∂y

)= −1.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 6 / 12

Standard Contact Structure on R3

x

y

z

λ = dz − ydx .These planes appear to twist along the y-axis.

Another example: All odd dimensional spheres are contact manifolds!

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 7 / 12

Standard Contact Structure on R3

x

y

z

λ = dz − ydx .These planes appear to twist along the y-axis.

Another example: All odd dimensional spheres are contact manifolds!

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 7 / 12

Hopf Fibration Video

Fun Fact: Reeb orbits of S3 are the Hopf fibers of S3!!

The Hopf fibration shows how the three-sphere can be built by a collection ofcircles arranged like points on a two-sphere. We see specific points on thetwo-sphere synchronized with the circles (fibers) over them.

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

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 8 / 12

Open Book Decomposition and Giroux Correspondence

Definition (Informal)

An open book decomposition (or simply an open book) is a decomposition of aclosed 3-manifold M into a union of surfaces (necessarily with boundary) and solidtori.

Surfaces = Pages, F , of the open book of MSolid Tori = Binding, B, of the open book of M

Theorem (Giroux, 2000)

Let M be a compact oriented 3-manifold. Then there is a bijection between theset of oriented contact structures on M and the set of open book decompositionsof M.

That is to say, contact geometry can be studied from an entirely topologicalviewpoint.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 9 / 12

Open Book Decomposition and Giroux Correspondence

Definition (Informal)

An open book decomposition (or simply an open book) is a decomposition of aclosed 3-manifold M into a union of surfaces (necessarily with boundary) and solidtori.

Surfaces = Pages, F , of the open book of MSolid Tori = Binding, B, of the open book of M

Theorem (Giroux, 2000)

Let M be a compact oriented 3-manifold. Then there is a bijection between theset of oriented contact structures on M and the set of open book decompositionsof M.

That is to say, contact geometry can be studied from an entirely topologicalviewpoint.

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 9 / 12

Dehn Twists as monodromy

(Photo courtesy: Jonny Evans)

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 10 / 12

Factorization of Monodromy is Possible!

Theorem (Acu-Avdek’14)

Given

a homogeneous polynomial f ∈ C[z0, . . . , zn] of degree k with an isolatedsingularity at 0,

a 2n-dimensional Weinstein domain (W , dβ) whereW = {f (z0, . . . , zn) = 0} ∩ B2n+2

Then the contact manifold ∂W has an open book OB(F ,Φ∂) such that a fiberedDehn twist Φ∂ along ∂W can be expressed as a product of k(k − 1)n

right-handed Dehn twists Φ1 . . .Φk(k−1)n(up to symplectic isotopy).

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 11 / 12

Thanks for listening!

Special thanks to Jo Nelson for sharing contact plane distribution images with me!

Bahar Acu (University of Southern California) A 21st Century Geometry CSUCI Undergraduate Seminar 12 / 12