Tight contact structures on the solid torus Jessica Zhang • October 18, 2020 Mentor: Zhenkun Li • MIT PRIMES Conference Informally, contact geometry is concerned with contact structures, which are geometric structures defined on odd-dimensional spaces. It helps us better understand and prove results in low-dimensional topology, but many fundamental questions still remain unanswered. Open question Can we classify the contact structures on a given 3-manifold? Introduction: Why do we care? Jessica Zhang Tight contact structures on the solid torus Page 1 of 13 It helps us better understand and prove results in low-dimensional topology, but many fundamental questions still remain unanswered. Open question Can we classify the contact structures on a given 3-manifold? Introduction: Why do we care? Informally, contact geometry is concerned with contact structures, which are geometric structures defined on odd-dimensional spaces. Jessica Zhang Tight contact structures on the solid torus Page 1 of 13 Open question Can we classify the contact structures on a given 3-manifold? Introduction: Why do we care? Informally, contact geometry is concerned with contact structures, which are geometric structures defined on odd-dimensional spaces. It helps us better understand and prove results in low-dimensional topology, but many fundamental questions still remain unanswered. Jessica Zhang Tight contact structures on the solid torus Page 1 of 13 Introduction: Why do we care? Informally, contact geometry is concerned with contact structures, which are geometric structures defined on odd-dimensional spaces. It helps us better understand and prove results in low-dimensional topology, but many fundamental questions still remain unanswered. Open question Can we classify the contact structures on a given 3-manifold? Jessica Zhang Tight contact structures on the solid torus Page 1 of 13 Section 1 Contact geometry background A smooth n-manifold is a (Hausdorff) topological space M without n edges or corners which locally looks like an open set of R . If M contains its boundary ∂M, we call it an n-manifold with boundary. The surface of a ball is a A solid torus is a 3-manifold 2-manifold without boundary with boundary Smooth manifolds Jessica Zhang Tight contact structures on the solid torus Page 2 of 13 If M contains its boundary ∂M, we call it an n-manifold with boundary. A solid torus is a 3-manifold with boundary Smooth manifolds A smooth n-manifold is a (Hausdorff) topological space M without n edges or corners which locally looks like an open set of R . The surface of a ball is a 2-manifold without boundary Jessica Zhang Tight contact structures on the solid torus Page 2 of 13 Smooth manifolds A smooth n-manifold is a (Hausdorff) topological space M without n edges or corners which locally looks like an open set of R . If M contains its boundary ∂M, we call it an n-manifold with boundary. The surface of a ball is a A solid torus is a 3-manifold 2-manifold without boundary with boundary Jessica Zhang Tight contact structures on the solid torus Page 2 of 13 At each point x ∈ M, we can define the tangent space TxM as the n-dimensional vector space consisting of all vectors tangent to M at x. M x TxM For the rest of this talk, M will be a smooth 3-manifold with boundary (n = 3). Tangent spaces Jessica Zhang Tight contact structures on the solid torus Page 3 of 13 M x TxM For the rest of this talk, M will be a smooth 3-manifold with boundary (n = 3). Tangent spaces At each point x ∈ M, we can define the tangent space TxM as the n-dimensional vector space consisting of all vectors tangent to M at x. Jessica Zhang Tight contact structures on the solid torus Page 3 of 13 M x TxM Tangent spaces At each point x ∈ M, we can define the tangent space TxM as the n-dimensional vector space consisting of all vectors tangent to M at x. For the rest of this talk, M will be a smooth 3-manifold with boundary (n = 3). Jessica Zhang Tight contact structures on the solid torus Page 3 of 13 A contact structure ξ on M is a way to smoothly assign a plane ξx ⊂ TxM to every x ∈ M such that no surface S ⊂ M is tangent to ξx for every x ∈ S. 3 The standard contact structure ξst on R . A contact structure is just a very “twisty” way to assign a plane to each point. Contact structures Jessica Zhang Tight contact structures on the solid torus Page 4 of 13 A contact structure is just a very “twisty” way to assign a plane to each point. Contact structures A contact structure ξ on M is a way to smoothly assign a plane ξx ⊂ TxM to every x ∈ M such that no surface S ⊂ M is tangent to ξx for every x ∈ S. 3 The standard contact structure ξst on R . Jessica Zhang Tight contact structures on the solid torus Page 4 of 13 Contact structures A contact structure ξ on M is a way to smoothly assign a plane ξx ⊂ TxM to every x ∈ M such that no surface S ⊂ M is tangent to ξx for every x ∈ S. 3 The standard contact structure ξst on R . A contact structure is just a very “twisty” way to assign a plane to each point. Jessica Zhang Tight contact structures on the solid torus Page 4 of 13 A vector field X on a 3-manifold M is a way to smoothly assign a tangent vector Xx ∈ TxM to every x ∈ M. We call X a contact vector field if pushing ξ from x to a nearby point y along X takes ξx to ξy. Vector fields Jessica Zhang Tight contact structures on the solid torus Page 5 of 13 We call X a contact vector field if pushing ξ from x to a nearby point y along X takes ξx to ξy. Vector fields A vector field X on a 3-manifold M is a way to smoothly assign a tangent vector Xx ∈ TxM to every x ∈ M. Jessica Zhang Tight contact structures on the solid torus Page 5 of 13 Vector fields A vector field X on a 3-manifold M is a way to smoothly assign a tangent vector Xx ∈ TxM to every x ∈ M. We call X a contact vector field if pushing ξ from x to a nearby point y along X takes ξx to ξy. Jessica Zhang Tight contact structures on the solid torus Page 5 of 13 The dividing set on a convex surface Σ is ΓΣ = {x ∈ Σ: Xx ∈ ξx}. ΓΣ Σ Call Σ ⊂ M be a convex surface if there exists a contact vector field X such that Xx 6∈ TxΣ for any x ∈ Σ. The dividing set Jessica Zhang Tight contact structures on the solid torus Page 6 of 13 ΓΣ Σ The dividing set on a convex surface Σ is ΓΣ = {x ∈ Σ: Xx ∈ ξx}. The dividing set Call Σ ⊂ M be a convex surface if there exists a contact vector field X such that Xx 6∈ TxΣ for any x ∈ Σ. Jessica Zhang Tight contact structures on the solid torus Page 6 of 13 ΓΣ Σ The dividing set Call Σ ⊂ M be a convex surface if there exists a contact vector field X such that Xx 6∈ TxΣ for any x ∈ Σ. The dividing set on a convex surface Σ is ΓΣ = {x ∈ Σ: Xx ∈ ξx}. Jessica Zhang Tight contact structures on the solid torus Page 6 of 13 Giroux Flexibility Theorem Let M be a fixed 3-manifold with boundary. If Σ ⊂ M is convex, then ΓΣ encodes all relevant contact topological information about (M, ξ) on the neighborhood of Σ. Main question For every possible dividing set ΓΣ on the boundary Σ = ∂M, how many contact structures have the dividing set ΓΣ on Σ? Giroux flexibility Jessica Zhang Tight contact structures on the solid torus Page 7 of 13 Main question For every possible dividing set ΓΣ on the boundary Σ = ∂M, how many contact structures have the dividing set ΓΣ on Σ? Giroux flexibility Giroux Flexibility Theorem Let M be a fixed 3-manifold with boundary. If Σ ⊂ M is convex, then ΓΣ encodes all relevant contact topological information about (M, ξ) on the neighborhood of Σ. Jessica Zhang Tight contact structures on the solid torus Page 7 of 13 Giroux flexibility Giroux Flexibility Theorem Let M be a fixed 3-manifold with boundary. If Σ ⊂ M is convex, then ΓΣ encodes all relevant contact topological information about (M, ξ) on the neighborhood of Σ. Main question For every possible dividing set ΓΣ on the boundary Σ = ∂M, how many contact structures have the dividing set ΓΣ on Σ? Jessica Zhang Tight contact structures on the solid torus Page 7 of 13 Call ξ tight if it is not overtwisted, i.e, if it does not contain an overtwisted disk: Overtwisted manifolds are fully understood, so we focus on tight contact structures, where the only complete result is on the 3-ball. Tight contact structures Jessica Zhang Tight contact structures on the solid torus Page 8 of 13 Overtwisted manifolds are fully understood, so we focus on tight contact structures, where the only complete result is on the 3-ball. Tight contact structures Call ξ tight if it is not overtwisted, i.e, if it does not contain an overtwisted disk: Jessica Zhang Tight contact structures on the solid torus Page 8 of 13 Tight contact structures Call ξ tight if it is not overtwisted, i.e, if it does not contain an overtwisted disk: Overtwisted manifolds are fully understood, so we focus on tight contact structures, where the only complete result is on the 3-ball.